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Nonparametric methods for comparing distribution functionals for dependent samples with application to inequality measures
\title{
\Huge Nonparametric methods for comparing distribution functionals for dependent samples with application to inequality measures
\thinspace \thanks{\ \ The authors thank Md Nazmul Ahsan,
Russell Davidson, John Galbraith, Silvia Goncalves, James MacKinnon, Russell Steele, Brennan Thompson,
Victoria Zinde-Walsh, Haitian Xie, Jun Cai, and all seminar participants
at McGill, the 22nd China Economic Annual Conference, IAAE2024, AMES2024, and the 2025 Canadian Econometric Study Group. \ \ This work was supported by the William Dow Chair in Political
Economy (McGill University), Tianjin University, the Bank of Canada (Research Fellowship),
the Toulouse School of Economics (Pierre-de-Fermat Chair of excellence),
the Universitad Carlos III de Madrid (Banco Santander de Madrid Chair of excellence),
the Natural Sciences and Engineering Research Council of Canada, and the Social Sciences and Humanities Research
Council of Canada.} }
\author{
Jean-Marie Dufour \thanks{\ \ William Dow Professor of Economics, McGill University,
Centre interuniversitaire de recherche en analyse des
organisations (CIRANO), and Centre interuniversitaire de recherche en
\'{e}conomie quantitative (CIREQ). Mailing address:
Department of Economics, McGill University, Leacock Building, Room 414,
855 Sherbrooke Street West, Montr\'{e}al, Qu\'{e}bec H3A 2T7, Canada.
TEL: (1) 514 398 6071; FAX: (1) 514 398 4800; e-mail:
\protect\url=[email removed]=\thinspace. Web page:
\protect\url{http://www.jeanmariedufour.com} } \\
McGill University \and Tianyu He\thanks{\ \ Ma Yinchu School of Economics, Tianjin University;
e-mail: he\[email removed]} \\
Tianjin University
}
\date{
December 2025
}
\maketitle
\begin{quote}
\end{quote}
\newpage
\begin{Abstract}
\begin{comment}
\end{Abstract}
\newpage
\pagestyle{plain}\pagenumbering{roman}\setcounter{page}{1}
\begin{center}
\textbf{ABSTRACT}
\quad
\end{center}
\noindent This paper proposes asymptotically distribution-free inference
methods for comparing a broad range of welfare indices across dependent
samples, including those employed in inequality, poverty, and risk analysis.
Two distinct situations are considered. \emph{First}, we propose asymptotic
and bootstrap intersection methods which are completely robust to arbitrary
dependence between two samples. \emph{Second}, we focus on the common case
of overlapping samples -- a special form of dependent samples where sample
dependence arises solely from matched pairs -- and provide asymptotic and
bootstrap methods for comparing indices. We derive consistent estimates for
asymptotic variances using the influence function approach. The performance
of the proposed methods is studied in a simulation experiment: we find that
confidence intervals with overlapping samples exhibit satisfactory coverage
rates with reasonable precision, whereas conventional methods based on an
assumption of independent samples have an inferior performance in terms of
coverage rates and interval widths. Asymptotic inference can be less
reliable when dealing with heavy-tailed distributions, while the bootstrap
method provides a viable remedy, unless the variance is substantial or
nonexistent. The intersection method yields reliable results with arbitrary
dependent samples, including instances where overlapping samples are not
feasible. We demonstrate the practical applicability of our proposed methods
in analyzing dynamic changes in household financial inequality in Italy over
time.
\quad
\noindent \textbf{Keywords}: inequality measures; poverty measures;
influence function; asymptotic; intersection method; bootstrap; confidence
interval; overlapping samples; dependent samples. \vspace{0.25pt}
\quad
\noindent \textbf{Journal of Economic Literature\ classification:} C01, C1,
C12, C14, C15D6, D63, G5, I3, I32.
\begin{Abstract}
\end{comment}
\end{Abstract}
\newpage
\tableofcontents
\newpage
\listoftables
\listoffigures
\subsection*{List of Definitions, Assumptions, Propositions
and Theorems
\@mkboth{\MakeUppercaseList of Definitions, Assumptions, Propositions
and Theorems}
{\MakeUppercaseList of Definitions, Assumptions, Propositions
and Theorems}}
\@starttoc{lth}
\addcontentsline{toc}{section}{List of Definitions, Assumptions, Propositions
and Theorems}
\begin{TOCLists}
\end{TOCLists}
\newpage
\pagenumbering{arabic} \setcounter{section}{0} \setcounter{page}{1}
\pagestyle{headings}
\subsection{\huge Introduction \label{Sec: Introduction}}
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\setcounter{theorem}{0} \setcounter{definition}{0} \setcounter{equation}{0}
Researchers often seek to compare inequality levels across different areas
or over time. In practice, inequality measures are estimated from samples,
making it essential to investigate the robustness of these comparisons
through statistical inference. However, most inference procedures assume
either independence between samples or complete dependence (matched pairs of
observations), as detailed in \nocite*{Mills1997}, \nocite*{Cowell2015}, \nocite*
{Dufour-Flachaire-Khalaf(2019)}, \nocite*{Dufour2023b}, \nocite*{Ibragimov2025},
and references therein.
However, much of the income data is dependent on the fact that there is an
overlap of data between the two samples. The overlap refers to paired
observations, such as the incomes of the same household interviewed in two
consecutive periods. For instance, labor force surveys in many countries
are, by design, rotated panel data. Other examples include the Current
Population Survey (CPS), the Panel Study of Income Dynamics (PSID), and the
Consumer Expenditure Survey (CEX) for U.S. income data. In this paper, we
utilize the Survey of Household and Income Wealth (SHIW) from Italy,
featuring an overlap between consecutive waves. Two issues emerge when using
such data: first, some individual income observations are correlated across
periods; second, income data for some units in one of the periods is missing.
Several studies have provided inference procedures with overlapping samples.
\nocite*{Zheng2001} proposed an asymptotic method for comparing interpolated
Gini indices and generalized entropy measures+. \nocite*{Zheng2004} extends
results to the Foster\emph{-}Greer\emph{-}Thorbecke poverty measure. \nocite*
{Biewen2002} presented a percentile bootstrap for comparing indices in a
family which are smooth functions of population moments. The above studies
focus on specific measurements and do not cover the popular Gini index and
quantile-based measurements, such as the Lorenz curve. Moreover, some
indices are covered by only one of the two methods (\emph{i.e.}, the
asymptotic and bootstrap methods).
In this paper, we provide inference procedures for a broad class of
inequality, poverty, and welfare measures within a unified framework. We
focus on estimators in the form of asymptotically linear Gaussian
functionals (defined below) previously considered in \nocite*[henceforth DFK]
{Dufour-Flachaire-Khalaf(2019)}. Nevertheless, we differ from DFK in two
central ways. First, DFK focus on hypothesis tests based on permutation
methods, while this is not the case here. Second, DFK assume independent
samples, while we allow dependent samples.
\begin{comment}
DFK examines the case of matched pairs in the simulation without proving the
procedure's validity, though intuitively, the method expects to work.
\end{comment} Nevertheless, we do not have such a restriction. So, we can
study a broader family of indices, including quantile-based measurements
such as the Lorenz curve.
Our methods are closely related to the literature on influence functions
(IF), dating back to \nocite*{Hampel1974}. As one of the earliest applications
in econometrics, \nocite*{Cowell1996} investigated the robustness of various
inequality measurements. In contrast, we employ the IF to derive the
asymptotic distribution and estimate the asymptotic variance with great
computational advantage; see, for example, \nocite*{Cowell2015}. Furthermore,
this paper extends the estimation of asymptotic variances to dependent
samples. We make the following contributions.
\emph{First}, we focus on a wide range of indices covering most inequality,
poverty, and risk analysis measurements, including the well-known Gini index
and Lorenz curve.
\emph{Second}, we develop methods applicable without any assumption on the
dependence between the compared samples. This is done by using the
intersection approach previously considered by \nocite*{Dufour1998} in a
finite-sample framework. We introduce asymptotic and bootstrap intersection
methods (IMs) for assessing changes in indices. These methods utilize
information within each sample and handle arbitrary dependence between two
samples. Furthermore, our method provides solutions to a generalized
Behrens-Fisher problem, which involves comparing the means of two
distributions with arbitrary dependence across the two samples.
\emph{Third}, we examine overlapping samples, a specific form of dependent
samples where sample dependence arises solely from overlap (\emph{i.e.},
matched pairs). Such a framework is applicable to split or rotating panels
in survey data. We propose asymptotic and bootstrap inference for index
differences, and we present a consistent and numerically positive (definite)
estimator for the asymptotic variance.
\emph{Fourth}, we conduct a series of simulation experiments to examine the
performance of the proposed methods. Our confidence intervals based on
overlapping samples have coverage rates close to the nominal level with
reasonable widths. By contrast, conventional inference -- which ignores
sample dependence -- can fail to adequately cover the actual value
sufficiently frequently or become overly conservative with large widths. We
then analyze the impacts of heavy tails and find that the asymptotic
inference performs well for realistic distributions but poorly for
distributions with heavy tails. The bootstrap method can alleviate this
issue, except when the variance is substantial or nonexistent. Wed show that
intersection methods can yield reliable results in all scenarios,
particularly when overlapping samples are not fulfilled, and can be
reasonably efficient in some instances, such as extremely heavy-tailed
distributions.
\emph{Fifth}, we apply the proposed methods to investigate the dynamic
change of financial inequality from 2012 to 2014. The results reveal
distinct patterns of internal inequality within the two regions,
highlighting the practical importance of negative values in inequality
measurement.
\begin{comment}
This paper assumes that researchers have a sample of micro-level data
independently and identically drawn from a population distribution. However,
it is not difficult to generalize the methods to complex survey data, such
as clustered data, using techniques by \nocite*{Zheng2002}, \nocite*
{Bhattacharya2007}, or \nocite*{Ibragimov2025}.
\end{comment}
The paper is organized as follows. Section \ref{sec:index} reviews some
commonly used inequality measures which are asymptotic linear. Section \ref
{sec: Intersection methods} describes inference based on intersection
methods. Section \ref{Sec: Overlapping samples} discusses the special case
of overlapping samples. We assess the performance of the proposed procedures
through Monte Carlo experiments in Section \ref{sec:MC}. In Section \ref
{sec:App-SHIW}, we apply the methods to analyze the dynamic change in
household financial inequality in Italy. We briefly discuss the extensions
to clustered data to broaden the scope of our analysis in Section \ref
{sec:clusters}. The paper concludes in Section \ref{sec:conclusion}. All the
proofs are given in the appendix.
\newpage
\subsection{\huge Asymptotically linear Gaussian functionals \label
{sec:index}}
\renewcommand{\Alph{section}}{\Alph{section}}
\renewcommand{\Alph{section}}{\Alph{section}}
\setcounter{theorem}{0} \setcounter{definition}{0} \setcounter{equation}{0}
To extend existing results to a more general form of welfare indices, we
focus on the class of asymptotically linear functionals as defined below.
Such a collection of parameters encompasses many welfare measurements,
allowing us to present the asymptotic distribution conveniently.
To facilitate our discussion, we introduce some notation. Let $F\in \mathcal{
F}$ be a cumulative distribution function (CDF), and ${\Greekmath 0112} :\mathcal{F}
\rightarrow \mathbb{R}$ a functional of distribution $F$. For a sample $
\{X_{i}\}_{i=1}^{n}$, we denote by $\hat{F}$ the empirical distribution
function (EDF) for $F$, and estimate ${\Greekmath 0112} (F)$ by ${\Greekmath 0112} (\hat{F})$. We
denote by $o_{p}(1)$ a variable which converges (in probability) to zero as $
n\rightarrow \infty $. Let $X_{n}\overset{d}{\longrightarrow }X$ be a
sequence of variables $X_{n}$ convergence in distribution to $X$. We
consider a random sample of independent and identically distributed (iid)
observations $\{X_{i}\}_{i=1}^{n}$ drawn from the distribution $F$, written
as $\{X_{i}\}_{i=1}^{n}\overset{iid}{\sim }F$. We now define asymptotically
linear Gaussian (ALG) functionals.
\begin{definition}
\label{def:ALG}
\textsc{ Asymptotically linear Gaussian functional}.
\addcontentsline{lth}{theorem}{{\bf Definition \hspace{0em} \protect\numberline{\thedefinition}}
{\hspace{-3.5em} : \hspace{0em} Asymptotically linear Gaussian functional}}
Given a sample $\{X_{i}\}_{i=1}^{n}$ drawn from a distribution $F$, an
estimator ${\Greekmath 0112} (\hat{F})$ for the functional ${\Greekmath 0112} (F)$ is \emph{
asymptotically linear Gaussian}, if it satisfies
\begin{equation}
\sqrt{n}[{\Greekmath 0112} (\hat{F})-{\Greekmath 0112} (F)]=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}{\Greekmath 0120}
(X_{i};{\Greekmath 0112} ,F)+o_{p}(1)\underset{n\rightarrow \infty }{\overset{d}{
\longrightarrow }}\mathrm{N}[0,{\Greekmath 011B} _{{\Greekmath 0112} }^{2}(F)] \label{eq:ALG}
\end{equation}
where ${\Greekmath 011B} _{{\Greekmath 0112} }^{2}(F):=\mathrm{Var}[{\Greekmath 0120} (X_{i};{\Greekmath 0112} ,F)]$.
\end{definition}
\begin{comment}
ALG functionals have been studied in \nocite*{Dufour-Flachaire-Khalaf(2019)}.
The authors, however, additionally require the index to be a mixture-linear
functional of order $2$, as they focus on the validity of permutation tests.
\footnote{
The index ${\Greekmath 0112} $ is mixture-linear of order $2$, if $F_{1},F_{2}\in
\mathcal{F}$ entails $wF_{1}+(1-w)F_{2}\in \mathcal{F}$, and ${\Greekmath 0112}
(wF_{1}+(1-w)F_{2})=w{\Greekmath 0112} (F_{1})+(1-w){\Greekmath 0112} (F_{2})$ for all $
F_{1},F_{2}\in \mathcal{F}$ and $w\in \lbrack 0,1]$.} In contrast, we do not
impose such a restriction, which enables us to study quantile-based indices
such as the Lorenz curve.
\end{comment}
In many cases, the function ${\Greekmath 0120} $ in \eqref{eq:ALG} turns out to be the
(asymptotic) influence function (IF) of the estimator ${\Greekmath 0112} (\hat{F})$,
which is defined as follows:
\begin{equation}
{\Greekmath 0120} (x;{\Greekmath 0112} ,F)=\lim_{t\rightarrow 0}\dfrac{{\Greekmath 0112} ((1-t)F+t{\Greekmath 010E} _{x})}{
t}
\end{equation}
where ${\Greekmath 010E} _{x}$ denotes a point mass at $x$. The IF has been widely used
to analyze the robustness of an estimator, as it quantifies the effect on an
estimator when a single data point in the sample is contaminated; see \nocite*
{Wasserman2006} for further details. On the other hand, we utilize the IF as
a convenient device to derive asymptotic distributions, especially the
asymptotic covariance matrices.
Definition \ref{def:ALG} relies on a high-level condition, but in practice
we often impose more primitive assumptions. Table \ref{tab:ALGIndex} lists
various ALG estimators for several popular indices. For illustrative
purposes, we provide examples of the Gini index and the Lorenz curve (LC)
below, whose sufficient conditions are provided in the referenced papers.
\begin{table}[tb]
\caption{A brief list of welfare indices allow asymptotic linear estimators.}
\label{tab:ALGIndex}
\begin{center}
Table \thetable
List of estimators for welfare indices as asymptotic linear Gaussian
functionals
\end{center}
\hspace{-0.5\totalhormargin} \begin{minipage}{\paperwidth}
\centering
\small
\begin{center}
\begin{tabular}{|c|m{6cm}|c|}
\hline
\rule[-1ex]{0cm}{4ex} & Welfare indices & References \\ \hline
\rule[-1ex]{0cm}{4ex} Inequality & Gini index, Lorenz curve, generalized
Lorenz curve, generalized entropy class, Atkinson index, coefficient of
variation, Pietra ratio & \nocite*{Cowell2015} \\ \cline{2-3}
\rule[-1ex]{0cm}{4ex} & S-Gini index, E-Gini index & \nocite*{Barrett2009} \\
\cline{2-3}
\rule[-1ex]{0cm}{4ex} & extended Lorenz curves, extended Gini indices & \nocite*
{Dufour2024} \\ \hline
\rule[-1ex]{0cm}{4ex} Poverty & Foster-Greer-Thorbecke poverty measure, Sen
poverty index, Sen-Shorrocks-Thon Poverty Index & \nocite*{Cowell2015} \\
\cline{2-3}
\rule[-1ex]{0cm}{4ex} & Gini poverty index & \nocite*{Barrett2009} \\ \hline
\rule[-1ex]{0cm}{4ex} Risk measurement & Value-at-Risk, expected shortfall &
\nocite*{Zhang2021a} \\ \hline
\end{tabular}
\end{center}
\end{minipage}
\medskip
\noindent
\footnotesize
Note -- The table displays a collection of estimators in the form of
asymptotically linear Gaussian functionals in Definition \ref{def:ALG}.
Relevant conditions and the expressions of the influence functions are given
in the referenced papers.
\normalsize
\end{table}
\begin{example}
\label{Eg:Gini}
\textsc{ Gini index}.
\addcontentsline{lth}{theorem}{{\bf Example \hspace{0em} \protect\numberline{\theresult}}
{\hspace{-3.5em} : \hspace{0em} Gini index}}
The Gini index, also known as the Gini coefficient, measures the degree of
inequality using a scalar value ranging from $0$ to $1$, where $0$
represents perfect equality and $1$ indicates complete inequality. For an
income variable $X\geq 0$ with distribution $F$, there are over a dozen
formulas for the Gini index; see, for instance, \nocite*{Yitzhaki2013}. We will
use the following definition:
\begin{equation}
I(F)=\dfrac{2\int tF(t)dF(t)}{{\Greekmath 0116} (F)}-1.
\end{equation}
One can estimate $I(F)$ by plugging in the EDF $\hat{F}$, which yields
\begin{equation}
I\big(\hat{F}\big)=\frac{\frac{2}{n^{2}}\sum_{i=1}^{n}(i-0.5)X_{(i)}}{{\Greekmath 0116}
\big(\hat{F}\big)}-1
\end{equation}
where $X_{(1)}\leq \cdots \leq X_{(n)}$ are the ordered statistics from the
sample $\{X_{i}\}_{i=1}^{n}$. \nocite*{Davidson2009} has proposed a
bias-corrected estimate $\widetilde{I}=\left( \frac{n}{n-1}\right) I\big(
\hat{F}\big)$. The author then applies the delta method to demonstrate that
both estimators satisfy the Definition \ref{def:ALG} with ${\Greekmath 0120}
(x;L,F)=h(x;L,F)-\mathbb{E}[h(X;L,F)]$, where
\begin{equation}
h(x;I,F)=\frac{2xF(x)-2\int t\mathbf{1}(t<x)\,dF(t)-[I(F)+1]x}{{\Greekmath 0116} (F)}.
\label{eq:Gini-IF-aux}
\end{equation}
We can estimate this function by substituting the EDF $\hat{F}$, which can
be efficiently computed based on order statistics as follows.
\begin{equation}
h\big(X_{(i)};I,\hat{F}\big)=\frac{\left( \frac{2i}{n}\right) X_{(i)}-\frac{2
}{n}\sum_{j=1}^{i}X_{(j)}-(\hat{I}+1)X_{(i)}}{\frac{1}{n}\sum_{i=1}^{n}X_{i}}
.
\end{equation}
\end{example}
\begin{example}
\label{Eg:LC}
\textsc{ Lorenz curve ordinates}.
\addcontentsline{lth}{theorem}{{\bf Example \hspace{0em} \protect\numberline{\theresult}}
{\hspace{-3.5em} : \hspace{0em} Lorenz curve ordinates}}
Consider an income variable $X\geq 0$ with distribution $F$. The Lorenz
curve (LC) is defined as follows [\nocite*{Gastwirth1971}]:
\begin{equation}
L(p;F)=\dfrac{\int_{0}^{p}Q(u;F)\,du}{{\Greekmath 0116} (F)} \label{eq:LC-def}
\end{equation}
where ${\Greekmath 0116} (F)=\int x\,dF>0$ is the mean and $Q(p;F)=\inf \{x:F(x)\geq p\}$
is the $p$-th quantile. A point on the curve, such as $L(p)=m$, indicates
that the $100p\%$ poorest households receive $100m\%$ of the total income.
We obtain the following estimator by substituting the EDF $\hat{F}$:
\begin{equation}
L\big(p;\hat{F}\big)=\dfrac{\frac{1}{n}\sum_{i=1}^{n}X_{i}\mathbf{1}\big[
X_{i}\leq Q\big(p;\hat{F}\big)\big]}{{\Greekmath 0116} \big(\hat{F}\big)}\,.
\end{equation}
\nocite*{Beach1983} and \nocite*{Cowell2015} showed by the delta method that $L
\big(p;\hat{F}\big)$ satisfies Definition \ref{def:ALG} with ${\Greekmath 0120}
(x;L,F,p)=h(x;L,F,p)-\mathbb{E}[h(X;L,F,p)]$, where
\begin{equation}
h(x;L,F,p)=\frac{\big[x-Q(p;F)\big]\mathbf{1}\big[x\leq Q(p;F)\big]-x\,L(p;F)
}{{\Greekmath 0116} (F)}\,. \label{eq:LC-IF-aux}
\end{equation}
\begin{comment}
if the distribution $F$ is differentiable at its $p$-th quantile
\end{comment}
\end{example}
There are often multiple ways to establish the ALG form in \eqref{eq:ALG}.
For instance, using the delta method, \nocite*{Davidson2009} showed that the
Gini index is an ALG functional. However, one can also achieve results by
using more advanced techniques, such as empirical process theory [\nocite*
{Davidson2010a}]. Likewise, the S-Gini [\nocite*{Donaldson1980}] and E-Gini [
\nocite*{Chakravarty1988}], which are two popular extensions of the Gini index,
satisfy Definition \ref{def:ALG} using either the delta method [\nocite*
{Davidson2010a}] or empirical process theory [\nocite*{Barrett2009}]. In this
paper, we only assume the existence of an ALG form as defined in Definition
\ref{def:ALG}, thereby avoiding various complex conditions.
\begin{comment}
\begin{example}
\label{Eg:S-Gini}
\textsc{ S-Gini index}.
\addcontentsline{lth}{theorem}{{\bf Example \hspace{0em} \protect\numberline{\theresult}}
{\hspace{-3.5em} : \hspace{0em} S-Gini index}}
\nocite*{Donaldson1980} proposed an extension of Gini index, namely a single-series
Gini (S-Gini) family. The S-Gini relative index of inequality is given by
\begin{align}
I^{{\Greekmath 010E}} = 1 - {\Greekmath 010E}({\Greekmath 010E} - 1) \int_{0}^{1} (1 - p)^{{\Greekmath 010E} - 2}L(p)\,dp ,
\end{align}
where ${\Greekmath 010E} \in (1, \infty)$ is an inequality aversion parameter chosen by the
researcher. The higher value of ${\Greekmath 010E}$, the more weight is given to the poor.
We can estimate the index by
\begin{align}
I^{{\Greekmath 010E}}(\hat{F})
= 1 - {\Greekmath 010E}({\Greekmath 010E} - 1) \int_{0}^{1} (1 - p)^{{\Greekmath 010E} - 2}L(p; \hat{F})\,dp ,
\end{align}
where $L(p; \hat{F})$ is the empirical Lorenz curve in \eqref{eq:LC-est}.
\nocite*{Barrett2009} establish ALG \eqref{eq:ALG} for $I^{{\Greekmath 010E}}(\hat{F})$
by empirical process theory and functional delta method. On the other hand,
\nocite*{Davidson2010a} prove the result by using delta method based on a slightly
different formula for S-Gini. The imposed conditions can be found in the two
papers.
Write the influence function as a demeaned variable $h$. We then have ...
\end{example}
\end{comment}
One merit of the ALG functional in Definition \ref{def:ALG} is its
computational efficiency in calculating estimators for the asymptotic
variances of the indices. Specifically, we can estimate the asymptotic
variance by its sample analog:
\begin{equation}
\hat{{\Greekmath 011B} }^{2}:={\Greekmath 011B} _{{\Greekmath 0112} }^{2}(\hat{F})=\frac{1}{n}
\sum_{i=1}^{n}{\Greekmath 0120} ^{2}(X_{i};{\Greekmath 0112} ,\hat{F}) \label{eq:AVar-Index-Est}
\end{equation}
where $\hat{F}$ denotes the EDF and ${\Greekmath 0120} $ is given in \eqref{eq:ALG}. \nocite*
{Barrett2009} has validated the consistency of $\hat{{\Greekmath 011B} }^{2}$; see
section \ref{sec:OS-AVar} for a more general scenario.
In the next two sections, we derive asymptotic inference methods for index
differences under dependent samples, when both sample sizes go to infinity.
In section \ref{sec: Intersection methods}, we consider generic dependent
samples, where we impose no restrictions on the dependence between the two
samples or the joint limiting distribution (if it exists). Our proposed
inference method is therefore applicable and robust to a wide range of
scenarios. Then, in section \ref{Sec: Overlapping samples}, we delve
consider a more specific data structure called overlapping samples, where
the dependence arises from matched pairs.
We use the following notation. For distribution $F_{k}$, $k=1,2$, we
estimate the index ${\Greekmath 0112} _{k}={\Greekmath 0112} _{k}(F_{k})$ by $\hat{{\Greekmath 0112} }
_{k}={\Greekmath 0112} (\hat{F}_{k})$, where $\hat{F}_{k}$ is the EDF for $F_{k} $
based on $\{X_{ki}\}_{i=1}^{n_{k}}$. We denote $\Delta \hat{{\Greekmath 0112} }=\hat{
{\Greekmath 0112} }_{1}-\hat{{\Greekmath 0112} }_{2}$ the estimate of the indices difference $
\Delta {\Greekmath 0112} ={\Greekmath 0112} _{1}-{\Greekmath 0112} _{2}$. We wish to make inference on $
\Delta {\Greekmath 0112} $. This problem is a nonparametric version of the Behrens\emph{
-}Fisher problem, which involves comparing means of potentially distinct
distributions [\nocite*[p 846]{Lehmann2022}]. We will also consider the
possibility of dependence between the two samples.
\FloatBarrier
\newpage
\subsection{\huge Intersection methods for generic dependent samples
\label{sec: Intersection methods}}
\label{sec:DS}
We start with the generic dependent samples, where the observations in one
sample may not be independent of those of the other sample. For simplicity,
the observations inside each sample are assumed to be independent and
identically distributed (iid). However, it is straightforward to extend
results to clustered data using techniques by \nocite*{Zheng2002}, \nocite*
{Bhattacharya2007}, and \nocite*{Ibragimov2025}.
\begin{assumption}
\label{assump:DS}
\textsc{ Generic dependent samples}.
\addcontentsline{lth}{theorem}{{\bf Assumption \hspace{0em} \protect\numberline{\theassumption}}
{\hspace{-3.5em} : \hspace{0em} Generic dependent samples}}
Assume that $\{X_{1i}\}_{i=1}^{n_{1}} \overset{iid}{\sim} F_{1}$ and $
\{X_{2i}\}_{i=1}^{n_{2}} \overset{iid}{\sim} F_{2}$.
\end{assumption}
Assumption \ref{assump:DS} only focuses on individual samples $
\{X_{1i}\}_{i=1}^{n_{1}}$ and $\{X_{2i}\}_{i=1}^{n_{2}}$, allowing arbitrary
forms of dependence between the two samples. The generality of this
assumption will be more apparent when we introduce overlapping samples in
the subsequent section.
\subsection{Asymptotic intersection method \label{sec: Asymptotic
intersection method}}
\label{sec:Asym-IM}
As mentioned in \eqref{eq:AVar-Index-Est}, $\hat{{\Greekmath 011B} }_{k}$ is a
consistent estimator for the asymptotic variance of the ALG functional $\hat{
{\Greekmath 0112} }_{k}$, where
\begin{equation}
\hat{{\Greekmath 011B} }_{k}^{2}:={\Greekmath 011B} _{{\Greekmath 0112} }^{2}(\hat{F}_{k})=\frac{1}{n_{k}}
\sum_{i=1}^{n_{k}}{\Greekmath 0120} ^{2}(X_{ki};{\Greekmath 0112} ,\hat{F}_{k}),\quad k=1,2.
\label{eq:AVar-Index-Est-k}
\end{equation}
Let $q_{p}$ be the $p$-th quantile of the standard Gaussian distribution. We
then have the following asymptotic confidence interval (CI) for ${\Greekmath 0112} _{k}$
at level $(1-{\Greekmath 010B} _{k})$, denoted by $\left[ L_{n_{k}},U_{n_{k}}\right] $,
where
\begin{equation}
L_{n_{k}}=\hat{{\Greekmath 0112} }_{k}-q_{1-\frac{{\Greekmath 010B} _{k}}{2}}\frac{\hat{{\Greekmath 011B} }
_{k}}{\sqrt{n_{k}}},\quad U_{n_{k}}=\hat{{\Greekmath 0112} }_{k}+q_{1-\frac{{\Greekmath 010B} _{k}
}{2}}\frac{\hat{{\Greekmath 011B} }_{k}}{\sqrt{n_{k}}},\quad k=1,2.
\label{eq:AsymCI_Index}
\end{equation}
The limiting confidence coefficient is defined as $\lim_{n_{k}\rightarrow
\infty }\inf_{{\Greekmath 0112} _{k}\in \Theta _{k}}\Pr_{{\Greekmath 0112} _{k}}\left(
L_{n_{k}}\leq {\Greekmath 0112} _{k}\leq U_{n_{k}}\right) $, provided the limit exists (
\nocite*[p 526]{Lehmann2022}).
We propose an asymptotic version of the intersection method (IM) for $
\Delta{\Greekmath 0112}$, adapted from \nocite*{Dufour2024d}, who extends results in \nocite*
{Dufour1998} to finite-sample inference.
\begin{proposition}
\label{prop:AsymIM}
\textsc{ Asymptotic intersection
method}.
\addcontentsline{lth}{theorem}{{\bf Proposition \hspace{0em} \protect\numberline{{\bf \Alph{section}.\arabic{theorem}}}}
{\hspace{-3.5em} : \hspace{0em} Asymptotic intersection
method}} Let ${\Greekmath 0112} (F_{k})$ be an asymptotically linear Gaussian
functional, and $[L_{n_{k}},U_{n_{k}}]$ in \eqref{eq:AsymCI_Index} an
asymptotic CI for ${\Greekmath 0112} _{k}$ at the level of $(1-{\Greekmath 010B} _{k})$ for $k=1,2$
. Then, under Assumption \ref{assump:DS}, the following properties hold$:$
$(1)$ $\left[ L_{n_{1}}-U_{n_{2}},\ U_{n_{1}}-L_{n_{2}}\right] $ is an
asymptotic CI for $\Delta {\Greekmath 0112} $ with level $(1-{\Greekmath 010B} )$, where ${\Greekmath 010B}
_{1}$ and ${\Greekmath 010B} _{2}$ are such that ${\Greekmath 010B} \geq {\Greekmath 010B} _{1}+{\Greekmath 010B} _{2}$ $
;$
$(2)$ if the two samples are independent and $[L_{n_{k}},U_{n_{k}}]$ for $
k=1,2$ have limiting confidence coefficients $(1-{\Greekmath 010B} _{k})$, then $\left[
L_{n_{1}}-U_{n_{2}},\ U_{n_{1}}-L_{n_{2}}\right] $ is an asymptotic CI for $
\Delta {\Greekmath 0112} $ at the level of $(1-{\Greekmath 010B} )$, where ${\Greekmath 010B} _{1}$ and $
{\Greekmath 010B} _{2}$ are such that ${\Greekmath 010B} ={\Greekmath 010B} _{1}+{\Greekmath 010B} _{2}-{\Greekmath 010B}
_{1}{\Greekmath 010B} _{2}$.
\end{proposition}
\subsection{Bootstrap intersection method \label{sec: Bootstrap intersection
method}}
\label{sec:Boot-IM}
There are several ways to construct a bootstrap CI for ${\Greekmath 0112} _{k}$, $k=1,2$
. One good choice, at least in theory, is the Studentized bootstrap, or
percentile-t, CI [\nocite*{Hall1992}]. The procedure is as follows [\nocite*
{Davidson2004}].
\begin{enumerate}[$(1)$]
\item
Given the original samples $\{X_{ki}\}_{i=1}^{n_{k}}$, obtain estimates $
\hat{{\Greekmath 0112}}_{k}$ and $\hat{{\Greekmath 011B}}_{k}$ for $k=1,2$.
\item
Construct the bootstrap samples $\{X^{*}_{ki}\}_{i=1}^{n_{k}}$ by resampling
with replacement from $\{X_{ki}\}_{i=1}^{n_{k}}$. Next, compute the
estimates $\hat{{\Greekmath 0112}}^{*}_{k}$ and $\hat{{\Greekmath 011B}}^{*}_{k}$, and calculate
statistic $T^{*} = \big(\hat{{\Greekmath 0112}}^{*}_{k} - \hat{{\Greekmath 0112}}_{k}\big) / \hat{
{\Greekmath 011B}}^{*}_{k}$.
\item
Repeat the previous step $B$ times to obtain the statistics $
\{T^{*}_{j}\}_{j = 1}^{B}$. Then, compute $q^{*}_{\frac{{\Greekmath 010B}_{k}}{2}}$ and
$q^{*}_{1 - \frac{{\Greekmath 010B}_{k}}{2}}$, where $q^{*}_{p}$ is the $\lceil pB
\rceil$ order statistics of $\{T^{\ast}_{j}\}_{j = 1}^{B}$. Here, $\lceil
\cdot \rceil$ denotes the ceiling function.
\item
The bootstrap CI for ${\Greekmath 0112} _{k}$ at the level $(1-{\Greekmath 010B} _{k})$ is $\left[
L_{n_{k}}^{B},U_{n_{k}}^{B}\right] $, where
\begin{equation}
L_{n_{k}}^{B}=\hat{{\Greekmath 0112} }_{k}-\hat{{\Greekmath 011B} }_{k}q_{1-{\Greekmath 010B} _{k}/2}^{\ast
},\quad U_{n_{k}}^{B}=\hat{{\Greekmath 0112} }_{k}-\hat{{\Greekmath 011B} }_{k}q_{{\Greekmath 010B}
_{k}/2}^{\ast }. \label{eq:BootCI_Index}
\end{equation}
\end{enumerate}
Ideally, the number of bootstrap samples $B$ should be reasonably large and
satisfy the condition that ${\Greekmath 010B} (B + 1)$ is an integer for any test level
${\Greekmath 010B}$ [\nocite*{Davidson2000a}]. We then propose a bootstrap version of the
intersection method for $\Delta{\Greekmath 0112}$.
\begin{proposition}
\label{prop:BootIM}
\textsc{ Bootstrap intersection
method}.
\addcontentsline{lth}{theorem}{{\bf Proposition \hspace{0em} \protect\numberline{{\bf \Alph{section}.\arabic{theorem}}}}
{\hspace{-3.5em} : \hspace{0em} Bootstrap intersection
method}} Let ${\Greekmath 0112} (F_{k})$ be an asymptotically linear Gaussian functional
and $[L_{n_{k}}^{B},U_{n_{k}}^{B}]$ in \eqref{eq:BootCI_Index} be a
bootstrap CI for ${\Greekmath 0112} _{k}$ at the level of $(1-{\Greekmath 010B} _{k})$ for $k=1,2$
. Then, Assumption \ref{assump:DS}, the following properties hold :
$(1)$ $\left[ L_{n_{1}}^{B}-U_{n_{2}}^{B},\ U_{n_{1}}^{B}-L_{n_{2}}^{B}
\right] $ is a bootstrap CI for $\Delta {\Greekmath 0112} $ with level $(1-{\Greekmath 010B} )$,
where ${\Greekmath 010B} _{1}$ and ${\Greekmath 010B} _{2}$ are such that ${\Greekmath 010B} \geq {\Greekmath 010B}
_{1}+{\Greekmath 010B} _{2}$ $;$
$(2)$ if the two samples are independent and $[L_{n_{k}}^{B},U_{n_{k}}^{B}]$
for $k=1,2$ have limiting confidence coefficients $(1-{\Greekmath 010B} _{k})$, then $
\left[ L_{n_{1}}^{B}-U_{n_{2}}^{B},\ U_{n_{1}}^{B}-L_{n_{2}}^{B}\right] $ is
a bootstrap CI for $\Delta {\Greekmath 0112} $ at the level of $(1-{\Greekmath 010B} )$, where $
{\Greekmath 010B} _{1}$ and ${\Greekmath 010B} _{2}$ are such that ${\Greekmath 010B} ={\Greekmath 010B} _{1}+{\Greekmath 010B}
_{2}-{\Greekmath 010B} _{1}{\Greekmath 010B} _{2}$.
\end{proposition}
Intersection methods (IMs) are asymptotically conservative, which is as
expected due to the trade-off between efficiency and robustness. Recall that
the IMs do not require any information on sample dependence or the limiting
joint distribution of estimators. The following section will demonstrate how
to achieve more efficient inference methods under additional assumptions.
\subsection{Overlapping samples \label{Sec: Overlapping samples}}
\label{sec:OS}
In this section, we focus on a special type of sample known as overlapping
samples (OS), which accommodate sample dependence that stems exclusively
from the overlap (\emph{i.e.}, matched pairs). Such a data structure can be
employed for rotating or splitting panels in survey data. Several authors,
including \nocite*{Zheng2001}, \nocite*{Zheng2004}, and \nocite*{Biewen2002}, have
investigated overlapping samples in their analyses of inequality and poverty
and proposed inference methods for specific indices. We describe the OS as
follows.
\begin{assumption}
\label{assump:OS}\textsc{ Overlapping samples}.
\addcontentsline{lth}{theorem}{{\bf Assumption \hspace{0em} \protect\numberline{\theassumption}}
{\hspace{-3.5em} : \hspace{0em} Overlapping samples}}
Let $\{X_{1i}\}_{i=1}^{n_{1}}$ and $\{X_{2i}\}_{i=1}^{n_{2}}$ two samples
which satisfy Assumption \ref{assump:DS}.Further, the following conditions
are satisfied.
\begin{enumerate}
\item[(A1)] \label{assump:OS-MP-obs} The first $m$ observations are matched
pairs $\{(X_{1i},X_{2i})\}_{i=1}^{m}$, where $0\leq m\leq \min (n_{1},n_{2})$
.
\item[(A2)] \label{assump:OS-MP-IID} The matched pairs are drawn from a
fixed joint distribution $\widetilde{F}$, whose marginal distributions are $
F_{1}$ and $F_{2}$, \emph{i.e.} $\{(X_{1i},X_{2i})\}_{i=1}^{m}\overset{iid}{
\sim }\widetilde{F}$.
\item[(A3)] \label{assump:OS-unMP} The observations that are unmatched in
each sample are independent of the observations in the other sample.
\begin{enumerate}
\item $\{X_{1i}\}_{i=m+1}^{n_{1}}$ is independent of $\{X_{2i}
\}_{i=1}^{n_{2}}$.
\item $\{X_{2i}\}_{i=m+1}^{n_{2}}$ is independent of $\{X_{1i}
\}_{i=1}^{n_{1}}$.
\end{enumerate}
\item[(A4)] \label{assump:OS-Asym} As $n_{1},n_{2}\rightarrow \infty $, we
require that $\frac{m}{n_{1}}\rightarrow {\Greekmath 0115} _{1}$, $\frac{m}{n_{2}}
\rightarrow {\Greekmath 0115} _{2}$, $\frac{n_{1}}{n_{1}+n_{2}}\rightarrow {\Greekmath 0111} _{1}$
and $N:=\frac{n_{1}n_{2}}{n_{1}+n_{2}}\rightarrow \infty $, where ${\Greekmath 0115}
_{1},{\Greekmath 0115} _{2}\in \lbrack 0,1]$, and ${\Greekmath 0111} _{1}\in \lbrack 0,1]$.
\end{enumerate}
\end{assumption}
Condition {$($\textup{A1}$)$} assumes that the first $m$ observations in
each sample are matched pairs. Furthermore, it implicitly requires that we
know which two observations in each sample form a pair. For instance, when
comparing income difference between parents and children, the incomes of a
parent and a child from the same family make up a pair. If the pairing
information is lost, so that it is unclear whether or not the child and
parent come from the same household, then such samples will violate {$($
\textup{A1}$)$}.
Condition {$($\textup{A2}$)$} makes the common assumption that pairs of data
are iid observations from a fixed unknown joint distribution $\widetilde{F}$
. We may relax such a condition to independent but not identically
distributed.
Condition {$($\textup{A3}$)$} imposes independence between unmatched data
and entire observations in the other sample, indicating that the overlap is
the only source of dependence between the two samples.
Condition {$($\textup{A4}$)$} clarify what we mean by asymptotic in OS.
First, while both $n_{1}$ and $n_{2}$ go to infinity, the size of overlap $m$
can be zero or a fixed number, giving ${\Greekmath 0115} _{1}={\Greekmath 0115} _{2}=0$. Second,
$n_{1}$ and $n_{2}$ can increase to infinity at different orders of speed
such that ${\Greekmath 0111} _{1}=0$ or ${\Greekmath 0111} _{1}=1$ (and ${\Greekmath 0115} _{1}={\Greekmath 0115} _{2}=0$
). So, we do not view {$($\textup{A4}$)$} as a limitation; also see the
discussion following the Proposition \ref{prop:AsyN-dIndex}.
\begin{comment}
Violation of any conditions, we can still employ intersection method, as demonstrated in simulation experiments in section \ref{sec:MC-BeyondOS}.
\end{comment}
To complete Assumption \ref{assump:OS}, we make an asymptotic normality
assumption on matched pairs.
\begin{assumption}
\label{assump:JointNormal}
\textsc{ Asymptotically joint normality}.
\addcontentsline{lth}{theorem}{{\bf Assumption \hspace{0em} \protect\numberline{\theassumption}}
{\hspace{-3.5em} : \hspace{0em} Asymptotically joint normality}}
Let $\hat{{\Greekmath 0112}}_{k}$ be an asymptotically linear Gaussian functional with $
{\Greekmath 0120} _{k}(x):={\Greekmath 0120} (x;{\Greekmath 0112} ,F_{k})$ in \eqref{eq:ALG} for $k=1,2$. If $
m\rightarrow \infty $, then
\begin{equation}
\frac{1}{\sqrt{m}}\sum_{i=1}^{m}
\begin{bmatrix}
{\Greekmath 0120} _{1}(X_{1i}) \\
{\Greekmath 0120} _{2}(X_{2i})
\end{bmatrix}
\underset{m\rightarrow \infty }{\overset{d}{\longrightarrow }}\mathrm{N}
\left[ \left(
\begin{array}{c}
0 \\
0
\end{array}
\right) ,\left(
\begin{bmatrix}
{\Greekmath 011B} _{1}^{2} & {\Greekmath 011B} _{12} \\
{\Greekmath 011B} _{12} & {\Greekmath 011B} _{2}^{2}
\end{bmatrix}
\right) \right] \label{eq:Pairs_MvtNormal}
\end{equation}
where ${\Greekmath 011B} _{k}^{2}=\mathrm{Var}[{\Greekmath 0120} _{k}(X_{ki})]$ and ${\Greekmath 011B} _{12}=
\mathrm{Cov}[{\Greekmath 0120} _{1}(X_{1i}),{\Greekmath 0120} _{2}(X_{2i})]$. Otherwise, we have $
{\Greekmath 0115} _{1}={\Greekmath 0115} _{2}=0$, where ${\Greekmath 0115} _{1}$ and ${\Greekmath 0115} _{2}$ are
given in Assumption \ref{assump:OS}.
\end{assumption}
Condition \eqref{eq:Pairs_MvtNormal} is technically necessary because
Gaussian marginal distributions do not imply a multivariate normal
distribution. In practice, however, this assumption is often satisfied for
many indices in Table \ref{tab:ALGIndex} under the conditions of Assumption
\ref{assump:OS}. Specifically, $\left\{ ({\Greekmath 0120} _{1}(X_{1i}),{\Greekmath 0120}
_{2}(X_{2i}))\right\} _{i=1}^{m}$ are often iid random vectors. Then, by the
standard Central Limit Theorem, the sample average has bivariate normal
asymptotic distribution.
\subsection{Asymptotic distribution of index difference \label{sec:
Asymptotic distribution of index difference}}
\label{sec:OS-Asym}
We first provide the asymptotic distribution of the estimator for the index
difference $\Delta \hat{{\Greekmath 0112}}$ with overlapping samples.
\begin{proposition}
\label{prop:AsyN-dIndex}
\textsc{ Asymptotic method with overlapping samples}.
\addcontentsline{lth}{theorem}{{\bf Proposition \hspace{0em} \protect\numberline{\theproposition}}
{\hspace{-3.5em} : \hspace{0em} Asymptotic method with overlapping samples}}
If $\hat{{\Greekmath 0112}}_{1}$ and $\hat{{\Greekmath 0112}}_{2}$ are asymptotically linear
Gaussian functionals with ${\Greekmath 0120} _{k}(x):={\Greekmath 0120} (x;{\Greekmath 0112} ,F_{k})$ in
\eqref{eq:ALG} for $k=1,2$. Then, under the Assumptions \ref{assump:OS} and
\ref{assump:JointNormal}, we have$:$
\begin{equation}
\sqrt{N}\big(\Delta \hat{{\Greekmath 0112}}-\Delta {\Greekmath 0112} \big)\underset{
n_{1},n_{2}\rightarrow \infty }{\overset{d}{\longrightarrow }}\mathrm{N}
[0,{\Greekmath 011B} _{\Delta }^{2}] \label{eq:AVar-IndexDiff}
\end{equation}
where ${\Greekmath 011B} _{\Delta }^{2}=(1-{\Greekmath 0111} _{1}){\Greekmath 011B} _{1}^{2}+{\Greekmath 0111} _{1}{\Greekmath 011B}
_{2}^{2}-2\sqrt{{\Greekmath 0111} _{1}(1-{\Greekmath 0111} _{1}){\Greekmath 0115} _{1}{\Greekmath 0115} _{2}}{\Greekmath 011B} _{12}$
, ${\Greekmath 011B} _{k}^{2}=\mathrm{Var}[{\Greekmath 0120} _{k}(X_{k1})]$, ${\Greekmath 011B} _{12}=\mathrm{
Cov}[{\Greekmath 0120} _{1}(X_{11}),{\Greekmath 0120} _{2}(X_{21})]$ and $N\rightarrow \infty $, ${\Greekmath 0111}
_{1}$, ${\Greekmath 0115} _{1}$, ${\Greekmath 0115} _{2}\in \lbrack 0,1]$ are defined in
Assumption \ref{assump:OS}.
\end{proposition}
The last term of the asymptotic variance in \eqref{eq:AVar-IndexDiff}
indicates that the impact of sample dependency relies on both the overlap
portions (\emph{i.e.}, ${\Greekmath 0115} _{1}{\Greekmath 0115} _{2}$) and the covariance
between ${\Greekmath 0120} _{1}$ and ${\Greekmath 0120} _{2}$. Note that ${\Greekmath 0120} _{k}$ is often a
non-monotonic transformation; see Example \ref{Eg:Gini} and \ref{Eg:LC}. So
it is challenging to predict the value or even the sign of ${\Greekmath 011B} _{12}$
based on $\mathrm{Cov}(X_{1},X_{2})$. We will revisit this issue in the
simulation experiments.
Assumption \ref{assump:OS} allows extremely unbalanced samples (where ${\Greekmath 0111}
_{1}=0$ or $1$). Take ${\Greekmath 0111} _{1}=1$ as an example. We then have ${\Greekmath 0115}
_{1}={\Greekmath 0115} _{2}=0$, $N\approx n_{2}\rightarrow \infty $, and ${\Greekmath 011B}
_{\Delta }^{2}={\Greekmath 011B} _{2}^{2}$, indicating $\hat{{\Greekmath 0112}}_{2}$ as the only
source of uncertainty. One way to interpret these results is that the sample
size $n_{1}$ is so much larger than $n_{2}$ that the uncertainty of $\hat{
{\Greekmath 0112}}_{2}$ dominates that of $\hat{{\Greekmath 0112}}_{1}$. In other words, we can
treat $\{X_{1i}\}_{i=1}^{n_{1}}$ as the population and $\hat{{\Greekmath 0112}}_{1}$ as
the true parameter ${\Greekmath 0112} _{1}$ (without sampling errors).
We now apply the above proposition to obtain an asymptotic distribution for
estimates of differences in Gini indices and LC ordinates.
\begin{example}
\label{Eg:dGini}
\textsc{ Gini index (continued)}.
\addcontentsline{lth}{theorem}{{\bf Example \hspace{0em} \protect\numberline{\theresult}}
{\hspace{-3.5em} : \hspace{0em} Gini index (continued)}}
Under the conditions of Proposition \ref{prop:AsyN-dIndex}, we have$:$
\begin{equation}
\sqrt{N}\big(\Delta \hat{I}-\Delta I\big)\underset{n_{1},n_{2}\rightarrow
\infty }{\overset{d}{\longrightarrow }}\mathrm{N}\big(0,{\Greekmath 011B} _{\Delta }^{2}
\big)
\end{equation}
where ${\Greekmath 011B} _{\Delta }^{2}=(1-{\Greekmath 0111} _{1}){\Greekmath 011B} _{1}^{2}+{\Greekmath 0111} _{1}{\Greekmath 011B}
_{2}^{2}-2\sqrt{{\Greekmath 0111} _{1}(1-{\Greekmath 0111} _{1}){\Greekmath 0115} _{1}{\Greekmath 0115} _{2}}{\Greekmath 011B} _{12}$
, ${\Greekmath 011B} _{k}^{2}=\mathrm{Var}[h(X_{k1};I,F_{k})]$ for $k=1,2$, and ${\Greekmath 011B}
_{12}=\mathrm{Cov}[h(X_{11};I,F_{1}),h(X_{21};I,F_{2})]$ with $h(x;I,F)$
given in \eqref{eq:Gini-IF-aux}.
\end{example}
\begin{example}
\label{Eg:dLC}
\textsc{ Lorenz curve ordinates (continued)}.
\addcontentsline{lth}{theorem}{{\bf Example \hspace{0em} \protect\numberline{\theresult}}
{\hspace{-3.5em} : \hspace{0em} Lorenz curve ordinates (continued)}}
By Proposition \ref{prop:AsyN-dIndex}, we have$:$
\begin{align}
& \sqrt{N}\big(\Delta \hat{L}(p)-\Delta L(p)\big)\underset{
n_{1},n_{2}\rightarrow \infty }{\overset{d}{\longrightarrow }}\mathrm{N}\big(
0,{\Greekmath 011B} _{\Delta }^{2}(p)\big)\,, \\
{\Greekmath 011B} _{\Delta }^{2}(p)& =(1-{\Greekmath 0111} _{1}){\Greekmath 011B} _{1}^{2}(p)+{\Greekmath 0111} _{1}{\Greekmath 011B}
_{2}^{2}(p)-2\sqrt{{\Greekmath 0111} _{1}(1-{\Greekmath 0111} _{1}){\Greekmath 0115} _{1}{\Greekmath 0115} _{2}}{\Greekmath 011B}
_{12}(p)\,,
\end{align}
where ${\Greekmath 011B} _{k}^{2}(p)=\mathrm{Var}[h(X_{k1};L,F_{k},p)]$ for $k=1,2$,
and ${\Greekmath 011B} _{12}(p)=\mathrm{Cov}[h(X_{11};L,F_{1},p),h(X_{21};L,F_{2},p)]$
with $h(x;L,F,p)$ given in \eqref{eq:LC-IF-aux}.
We can extend the results to a vector of Lorenz curve ordinates at $\mathbf{p
}=(p_{1},\ldots ,\,p_{s})^{\prime }$. Let $h(X_{ki};L,F_{k},\mathbf{p}
)=[h(X_{ki};L,F_{k},p_{1}),\ldots ,\,h(X_{ki};L,F_{k},p_{s})]^{\prime }$.
The asymptotic variance is
\begin{equation}
\Sigma _{\Delta }(\mathbf{p})=(1-{\Greekmath 0111} _{1})\Sigma _{1}(\mathbf{p})+{\Greekmath 0111}
_{1}\Sigma _{2}(\mathbf{p})-2\sqrt{{\Greekmath 0111} _{1}(1-{\Greekmath 0111} _{1}){\Greekmath 0115} _{1}{\Greekmath 0115}
_{2}}\Sigma _{12}(\mathbf{p}) \label{eq:AVar-dLC}
\end{equation}
where $\Sigma _{k}(\mathbf{p})=\mathrm{Var}[h(X_{ki};L,F_{k},\mathbf{p})]$
for $k=1,2$, and $\Sigma _{12}(\mathbf{p})=\mathrm{Cov}[h(X_{1i};L,F_{1},
\mathbf{p}),h(X_{2i};L,F_{2},\mathbf{p})]$.
\end{example}
\subsection{Asymptotic variance estimation by influence functions \label
{sec: Asymptotic variance estimation by influence functions}}
\label{sec:OS-AVar}
To conduct inference on changes in indices, we need a consistent estimator
for the asymptotic variance. We introduce further notation to simplify our
discussion. Given distributions $F_{k}$ with $k=1,2$, we denote ${\Greekmath 0120}
_{k}(x):={\Greekmath 0120} (x;{\Greekmath 0112} ,F_{k})$ and $\hat{{\Greekmath 0120} }_{n_{k}}(x):={\Greekmath 0120} (x;{\Greekmath 0112}
,\hat{F}_{n_{k}})$ when no confusion arises. The subscript $n_{k}$ indicates
that the estimated functionals is based on sample $\{X_{i}\}_{i=1}^{n_{k}}$.
Similarly, we abbreviate $h_{k}$ and $\hat{h}_{n_{k}}$ as discussed in
section \ref{sec:index}. So, the estimate $\hat{{\Greekmath 011B} }_{k}^{2}$ in
\eqref{eq:AVar-Index-Est} will take the following form.
\begin{equation}
\hat{{\Greekmath 011B} }_{k}^{2}=\frac{1}{n_{k}}\sum_{i=1}^{n_{k}}\hat{{\Greekmath 0120} }
_{n_{k}}^{2}(X_{ki}),\quad k=1,2. \label{eq:AVar-Index-Est-Fk}
\end{equation}
Given the expression \eqref{eq:AVar-IndexDiff}, one can estimate the
asymptotic variance by
\begin{equation}
\widetilde{{\Greekmath 011B} }_{\Delta }^{2}=\frac{n_{2}}{n_{1}+n_{2}}\hat{{\Greekmath 011B}}
_{1}^{2}+\frac{n_{2}}{n_{1}+n_{2}}\hat{{\Greekmath 011B}}_{2}^{2}-\frac{2m}{n_{1}+n_{2}}
\hat{{\Greekmath 011B}}_{12}, \label{eq:FalseAvar_Index}
\end{equation}
where $\hat{{\Greekmath 011B}}_{k}^{2}$ for $k=1,2$, is given in
\eqref{eq:AVar-Index-Est-Fk}, and $\hat{{\Greekmath 011B}}_{12}$ represents the sample
covariance of the matched pairs $\{(\hat{{\Greekmath 0120}}_{m}(X_{1i}),\hat{{\Greekmath 0120}}
_{m}(X_{2i}))\}_{i=1}^{m}$. Indeed, several studies, including \nocite*
{Zheng2001} and \nocite*{Zheng2004}, have proposed this type of estimator.
However, the estimator $\widetilde{{\Greekmath 011B} }_{\Delta }^{2}$ in
\eqref{eq:FalseAvar_Index} is not guaranteed to be positive (or positive
definite for a matrix). Consider an example where $n_{1}=n_{2}=2m$, then $
\widetilde{{\Greekmath 011B} }_{\Delta }^{2}=\frac{1}{2}(\hat{{\Greekmath 011B}}_{1}^{2}+\hat{
{\Greekmath 011B}}_{2}^{2}-2\hat{{\Greekmath 011B}}_{12})$. While the sample variances for
marginal distributions utilize the entire data, the estimate $\hat{{\Greekmath 011B}}
_{12}$ only uses $n/2$ observations, potentially leading to a negative $
\widetilde{{\Greekmath 011B} }_{\Delta }^{2}$.
We therefore consider an alternative estimator for the asymptotic variance.
On rewriting the covariance term in \eqref{eq:AVar-IndexDiff}, we get:
\begin{equation}
{\Greekmath 011B} _{\Delta }^{2}=(1-{\Greekmath 0111} _{1}){\Greekmath 011B} _{1}^{2}+{\Greekmath 0111} _{1}{\Greekmath 011B} _{2}^{2}-2
\sqrt{{\Greekmath 0111} _{1}(1-{\Greekmath 0111} _{1}){\Greekmath 0115} _{1}{\Greekmath 0115} _{2}}{\Greekmath 011A} _{{\Greekmath 0112} }{\Greekmath 011B}
_{1}{\Greekmath 011B} _{2} \label{eq:AVar-dIndex-rho}
\end{equation}
where ${\Greekmath 011A} _{{\Greekmath 0112} }=\mathrm{corr}({\Greekmath 0120} _{1}(X_{11}),{\Greekmath 0120} _{2}(X_{21}))$.
We consider the following estimate for the asymptotic variance ${\Greekmath 011B}
_{\Delta }^{2}$ in \eqref{eq:AVar-IndexDiff}:
\begin{equation}
\hat{{\Greekmath 011B} }_{\Delta }^{2}=\frac{n_{2}}{n_{1}+n_{2}}\hat{{\Greekmath 011B} }_{1}^{2}+
\frac{n_{2}}{n_{1}+n_{2}}\hat{{\Greekmath 011B} }_{2}^{2}-\frac{2m}{n_{1}+n_{2}}\hat{
{\Greekmath 011A} }_{{\Greekmath 0112} }\hat{{\Greekmath 011B} }_{1}\hat{{\Greekmath 011B} }_{2}
\label{eq:AVar-dIndex-Est}
\end{equation}
where $\hat{{\Greekmath 011B} }_{k}^{2}$ for $k=1,2$, is given in
\eqref{eq:AVar-Index-Est-Fk}, and $\hat{{\Greekmath 011A} }_{{\Greekmath 0112} }$ is the sample
correlation of the matched pairs $\left\{ (\hat{{\Greekmath 0120} }_{n_{1}}(X_{1i}),\hat{
{\Greekmath 0120} }_{n_{2}}(X_{2i}))\right\} _{i=1}^{m}$. We provide justification for
the consistency of $\hat{{\Greekmath 011B} }_{\Delta }^{2}$ in the appendix.
\begin{example}
\label{Eg:dGini-AVar-Est}
\textsc{ Gini index (continued)}.
\addcontentsline{lth}{theorem}{{\bf Example \hspace{0em} \protect\numberline{\theresult}}
{\hspace{-3.5em} : \hspace{0em} Gini index (continued)}}
In Example \ref{Eg:Gini}, we estimate the functional $h$ and asymptotic
variance using order statistics. However, sorting $\{X_{ki}\}_{i=1}^{n_{k}}$
will break the pairing relation in $\{(X_{1i},X_{2i})\}_{i=1}^{m}$, leading
to an inconsistent estimate for ${\Greekmath 011A} _{{\Greekmath 0112} }$ in
\eqref{eq:AVar-dIndex-rho}. We instead consider estimates based on rank
statistics. Given order statistics $X_{k(1)}\leq \cdots \leq X_{k(n_{k})}$,
we define the rank of $X_{ki}$, denoted by $R_{ki}$, as $X_{i}=X_{k(R_{ki})}$
. We estimate $h$ in \eqref{eq:Gini-IF-aux} as follows$:$
\begin{equation}
\hat{h}_{n_{k}}(X_{{ki}}):=h\big(X_{ki};I,\hat{F}_{n_{k}}\big)=\frac{\left(
\frac{2R_{ki}}{n_{k}}\right) X_{ki}-\frac{2}{n}\sum_{j=1}^{R_{ki}}X_{k(j)}-
\big(\hat{I}_{k}+1\big)X_{ki}}{\frac{1}{n}\sum_{i=1}^{n_{k}}X_{ki}}.
\end{equation}
We then estimate the asymptotic variance by \eqref{eq:AVar-dIndex-Est} with $
\hat{{\Greekmath 0120} }_{n_{k}}$ replaced by $\hat{h}_{n_{k}}:$
\begin{equation}
\hat{{\Greekmath 011B} }_{\Delta }^{2}=\frac{n_{2}}{n_{1}+n_{2}}\frac{1}{n_{1}}\hat{
{\Greekmath 011B} }_{1}^{2}+\frac{n_{2}}{n_{1}+n_{2}}\hat{{\Greekmath 011B} }_{2}^{2}-\frac{2m}{
n_{1}+n_{2}}\hat{{\Greekmath 011A} }_{I}\hat{{\Greekmath 011B} }_{1}\hat{{\Greekmath 011B} }_{2}
\end{equation}
where $\hat{{\Greekmath 011B} }_{k}$ is the sample variance of $\left\{ \hat{h}
_{n_{k}}^{2}(X_{ki})\right\} _{i=1}^{n_{k}}$ for $k=1,2$ and $\hat{{\Greekmath 011A} }
_{I} $ is the sample correlation coefficient of $\{(\hat{h}_{n_{1}}(X_{1i}),
\hat{h}_{n_{2}}(X_{2i}))\}_{i=1}^{m}$.
\end{example}
We can extend $\hat{{\Greekmath 011B}}^{2}_{\Delta}$ in \eqref{eq:AVar-dIndex-Est} to
estimation for the asymptotic variance matrix for a vector of LC ordinates.
\begin{example}
\label{Eg:dLC-AVar-Est}
\textsc{ Lorenz curve ordinates (continued)}.
\addcontentsline{lth}{theorem}{{\bf Example \hspace{0em} \protect\numberline{\theresult}}
{\hspace{-3.5em} : \hspace{0em} Lorenz curve ordinates (continued)}}
We can extend the Lorenz curve in Example \ref{Eg:dLC} to a vector of
ordinates at $\mathbf{p}$. Let $\hat{h}_{k}(X_{ki};\mathbf{p})=h(X_{ki};L,
\hat{F}_{k},\mathbf{p})$ for $k=1,2$, and denote by $\big[\mathrm{diag}{(A)}
\big]^{1/2}$ the diagonal matrix with elements being the square roots of
diagonal entries of $A$. So, we estimate $\Sigma _{\Delta }(\mathbf{p})$ in
\eqref{eq:AVar-dLC} as follows:
\begin{equation}
\begin{split}
\hat{\Sigma }_{\Delta }(\mathbf{p})=& \frac{n_{2}}{n_{1}+n_{2}}\hat{\Sigma }
_{1}(\mathbf{p})+\frac{n_{2}}{n_{1}+n_{2}}\hat{\Sigma }_{2}(\mathbf{p}) \\
& -\frac{2m}{n_{1}+n_{2}}\Big[\mathrm{diag}{\big(\hat{\Sigma }_{1}(\mathbf{p}
)\big)}\Big]^{1/2}\hat{\mathbf{{\Greekmath 011A} }}_{L}(\mathbf{p})\Big[\mathrm{diag}{
\big(\hat{\Sigma }_{2}(\mathbf{p})\big)}\Big]^{1/2}
\end{split}
\end{equation}
where $\hat{\Sigma }_{k}(\mathbf{p})$ is sample variance matrix of $\big\{
\hat{h}_{k}(X_{ki};\mathbf{p})\big\}_{i=1}^{n_{k}}$, and $\hat{{\Greekmath 011A} }_{L}(
\mathbf{p})$ is the sample correlation coefficient matrix of $\big\{\big(
\hat{h}_{1}(X_{1i};\mathbf{p}),\hat{h}_{2}(X_{2i};\mathbf{p})\big)\big\}
_{i=1}^{m}$.
\end{example}
For testing $H_{0}:\Delta {\Greekmath 0112} =c$, we can construct a Student-T test
which, under the null hypothesis, converges in distribution to a standard
normal distribution:
\begin{equation}
T=\dfrac{\sqrt{N}\big(\Delta \hat{{\Greekmath 0112} }-c\big)}{\hat{{\Greekmath 011B} }_{\Delta }}
\underset{n_{1},n_{2}\rightarrow \infty }{\overset{d}{\longrightarrow }}
\mathrm{N}(0,1). \label{eq:StudT-dIndex}
\end{equation}
Inverting the test statistic yields an asymptotic confidence interval (CI)
of level $(1-{\Greekmath 010B} )$ for $\Delta {\Greekmath 0112} $:
\begin{equation}
\left[ \Delta \hat{{\Greekmath 0112} }-q_{1-{\Greekmath 010B} /2}\frac{\hat{{\Greekmath 011B} }_{\Delta }}{
\sqrt{N}},\quad \Delta \hat{{\Greekmath 0112} }+q_{1-{\Greekmath 010B} /2}\frac{\hat{{\Greekmath 011B} }
_{\Delta }}{\sqrt{N}}\right] \label{eq:CI-dIndex-Asym}
\end{equation}
where $q_{p}$ is the $p$-th quantile of the standard normal distribution.
\subsection{Bootstrap inference with overlapping samples \label{sec:
Bootstrap inference with overlapping samples}}
\label{sec:OS-Boot}
We now examine the bootstrap method for $\Delta {\Greekmath 0112} $. Unlike the
percentile methods in \nocite*{Mills1997} and \nocite*{Biewen2002}, we will
explore the Studentized bootstrap, which can benefit from Beran's refinement
if the statistic bootstrapped is asymptotically pivotal; see \nocite*{Beran1988}
and \nocite*{Davidson2004} for instance. We consider the following procedure to
mimic the partial dependence under Assumption \ref{assump:OS}.
\begin{enumerate}
\item Given the original samples $\{X_{1i}\}_{i=1}^{n_{1}}$ and $
\{X_{2i}\}_{i=1}^{n_{2}}$, obtain the estimates $\Delta \hat{{\Greekmath 0112} }$ and $
\hat{{\Greekmath 011B} }_{\Delta }^{2}$ in \eqref{eq:AVar-dIndex-Est}, and compute the
test statistic $T$ in \eqref{eq:StudT-dIndex}.
\item Build a bootstrap sample $\{X_{1i}^{\ast }\}_{i=1}^{n_{1}}$ and $
\{X_{2i}^{\ast }\}_{i=1}^{n_{2}}$ as follows. First, draw in pair with
replacement from matched pairs $\{(X_{1i},X_{2i})\}_{i=1}^{m}$ to obtain $
\{(X_{1i}^{\ast },X_{2i}^{\ast })\}_{i=1}^{m}$. Second, supplement $
\{(X_{1i}^{\ast },X_{2i}^{\ast })\}_{i=1}^{m}$ by resampling with
replacement from the unmatched observations $\{X_{1i}\}_{i=m+1}^{n_{1}}$ and
$\{X_{2i}\}_{i=m+1}^{n_{2}}$ to obtain $\{X_{1i}^{\ast }\}_{i=m+1}^{n_{1}}$
and $\{X_{2i}^{\ast }\}_{i=m+1}^{n_{2}}$, respectively.
\item For the bootstrap sample $\{X_{1i}^{\ast }\}_{i=1}^{n_{1}}$ and $
\{X_{2i}^{\ast }\}_{i=1}^{n_{2}}$, compute the test statistic $T^{\ast }$
according to \eqref{eq:StudT-dIndex} with $c$ replaced by $\Delta \hat{
{\Greekmath 0112} }$.
\item Repeat the above step $B$ times and get test statistics $\{T_{j}^{\ast
}\}_{j=1}^{B}$. Choose $B$ such that ${\Greekmath 010B} (B+1)$ is a positive integer,
where ${\Greekmath 010B} $ is the nominal level.
\item Compute the bootstrap p-value defined as the proportion of $
T_{j}^{\ast }$ that is more extreme than $T$.
\item Reject the null at level ${\Greekmath 010B} $ if the bootstrap p-value is less
than ${\Greekmath 010B} $.
\end{enumerate}
We can also construct a two-sided bootstrap CI for $\Delta {\Greekmath 0112} $ at the
level of $(1-{\Greekmath 010B} )$ using the empirical distribution of the bootstrap
statistics $T_{j}^{\ast }$:
\begin{equation}
\left[ \Delta \hat{{\Greekmath 0112} }-q_{1-{\Greekmath 010B} /2}^{\ast }\frac{\hat{{\Greekmath 011B} }
_{\Delta }}{\sqrt{N}},\quad \Delta \hat{{\Greekmath 0112} }-q_{{\Greekmath 010B} /2}^{\ast }\frac{
\hat{{\Greekmath 011B} }_{\Delta }}{\sqrt{N}}\right]
\end{equation}
where $q_{p}^{\ast }$ is the $\lceil pB\rceil $-th order statistics of $
\{T_{j}^{\ast }\}_{j=1}^{B}$. Here, $\lceil \cdot \rceil $ denotes the
ceiling function. The following proposition establishes the validity of the
bootstrap method.
\begin{proposition}
\label{prop:Boot-dIndex}
\textsc{ Bootstrap method with overlapping samples}.
\addcontentsline{lth}{theorem}{{\bf Proposition \hspace{0em} \protect\numberline{\theproposition}}
{\hspace{-3.5em} : \hspace{0em} Bootstrap method with overlapping samples}}
Let $\Delta \hat{{\Greekmath 0112} }$ and $\Delta \hat{{\Greekmath 0112} }^{\ast }$ be the
estimates based on original and bootstrap samples, respectively. Under
conditions of Proposition \ref{prop:AsyN-dIndex}, we have
\begin{equation}
\sqrt{N}\big(\Delta \hat{{\Greekmath 0112} }^{\ast }-\Delta \hat{{\Greekmath 0112} }\big)\overset{d
}{\longrightarrow }\mathrm{N}\big(0,{\Greekmath 011B} _{\Delta }^{2}\big)
\end{equation}
where ${\Greekmath 011B} _{\Delta }^{2}$ is given in \eqref{eq:AVar-IndexDiff}.
\end{proposition}
Here are some comments on the above bootstrap method. First, while we focus
on the bootstrap method using the linearization technique, an alternative
approach is the bootstrap method based on U-statistic theory (\nocite*{Shi1986}
). Second, we may view the partially dependent samples as samples with
missing observations. In particular, we can reformat the data as $\mathbf{Z}
_{i} = (X_{1i}, {\Greekmath 010E}_{1i}, X_{2i}, {\Greekmath 010E}_{2i})^{\prime }$, where $
{\Greekmath 010E}_{ki}$ is a variable that equals one if $X_{ki}$ is observed and zero
otherwise. Then we employ the pair bootstrap based on $\{\mathbf{Z}_{i}\}_{i
= 1}^{\bar{n}}$, where $\bar{n} = n_{1} + n_{2} - m$. The simulation
experiments (not reported here) suggest that the two procedures yield nearly
identical results. So, we will not report such types of CIs.
\newpage
\subsection{\huge Monte Carlo simulations \label{sec:MC}}
\renewcommand{\Alph{section}}{\Alph{section}}
\renewcommand{\Alph{section}}{\Alph{section}}
\setcounter{theorem}{0} \setcounter{definition}{0} \setcounter{equation}{0}
In this section, we report two simulation experiments on the impacts of
sample dependence and heavy tails on various confidence intervals for $
\Delta {\Greekmath 0112} $. For simplicity, we assume that two samples share identical
sizes, \emph{i.e.}, $n_{1}=n_{2}=n$, unless stated otherwise. Then, we have $
{\Greekmath 0111} _{1}=0.5$, ${\Greekmath 0115} _{1}={\Greekmath 0115} _{2}={\Greekmath 0115} $ in Assumption \ref
{assump:OS} (A3), and the asymptotic standard deviation in Proposition \ref
{prop:AsyN-dIndex} reduces to the following quantity:
\begin{equation}
{\Greekmath 011B} _{\Delta }^{D}=\left( 0.5{\Greekmath 011B} _{1}^{2}+0.5{\Greekmath 011B} _{2}^{2}-{\Greekmath 0115}
{\Greekmath 011A} _{{\Greekmath 0112} }{\Greekmath 011B} _{1}{\Greekmath 011B} _{2}\right) ^{1/2}
\label{eq:AVar-dIndex-MC}
\end{equation}
where ${\Greekmath 0115} $ is the overlap portion, ${\Greekmath 011B} _{k}=[\mathrm{Var}({\Greekmath 0120}
_{k}(X_{ki}))]^{1/2}$ for $k=1,2$, and ${\Greekmath 011A} _{{\Greekmath 0112} }=\mathrm{corr}[{\Greekmath 0120}
_{1}(X_{11}),{\Greekmath 0120} _{2}(X_{21})]$ in \eqref{eq:AVar-dIndex-rho}.
\subsection{Effects of sample dependence \label{sec: Effects of sample
dependence}}
\label{sec:MC-dep}
We simulate data from the Singh-Maddala (SM) distribution, which has proven
effective in fitting the actual income distributions [\nocite*{McDonald1984},
\nocite*{Kleiber2003}]. The SM distribution [denoted $\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}(b,a,q)$] has
distribution function:
\begin{equation}
F(x)=1-\frac{1}{[1+(x/b)^{a}]^{q}},\quad a\geq 0,\,b>0\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ and }q>1/a,
\label{eq:SM-CDF}
\end{equation}
where $b$ is a scale parameter, and $a$ and $q$ are shape parameters. The
choice of scale parameter $b$ does not matter, as the Gini index and LC are
scale-invariant. We assume that the first sample contains data drawn from $
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}(1,1.6971,8.3679)$, where the estimates are given by \nocite*
{McDonald1984} for fitting the U.S. income distribution in 1980. For the
second sample, we follow \nocite*{Davidson2007a} and set $\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}
(0.4,2.8,1.7) $ in order to mimic the net income distribution of German
households.
We assume a Gaussian copula to capture the sample dependence for matched
pairs.
\begin{equation}
C(u_{1},u_{2};{\Greekmath 011A} )=\Phi _{{\Greekmath 011A} }\big(\Phi ^{-1}(u_{1}),\Phi ^{-1}(u_{2})
\big),\quad (u_{1},u_{2})\in \lbrack 0,1]\times \lbrack 0,1],\quad {\Greekmath 011A} \in
\lbrack -1,1], \label{eq:DGP-GaussCopula}
\end{equation}
where $\Phi $ and $\Phi _{{\Greekmath 011A} }$ denote the cumulative distribution
functions (CDFs) of the standard univariate and bivariate normal
distributions, respectively. The parameter ${\Greekmath 011A} $ indicates the degree of
sample dependence, with $0$ for independence and $1$ ($-1$) for positive
(negative) linear dependence.
We will explore various aspects of how sample dependence affects asymptotic
confidence intervals (CIs). Recall that it is challenging to predict the
value or sign of ${\Greekmath 011B} _{12}$ in \eqref{eq:AVar-IndexDiff} for a given $
\mathrm{Cov}(X_{1},X_{2})$. This difficulty arises because the functional $
{\Greekmath 0120} (x)$ in \eqref{eq:ALG} is often not monotonic in $x$. To address this
issue, we will first evaluate the correlation between ${\Greekmath 0120} _{1}(X_{1i})$
and ${\Greekmath 0120} _{2}(X_{2i})$, namely ${\Greekmath 011A} _{{\Greekmath 0112} }$ in
\eqref{eq:AVar-dIndex-MC}, against ${\Greekmath 011A} \in \{0,\pm 0.01,\ldots ,\,\pm 1\}$
in the Gaussian copula \eqref{eq:DGP-GaussCopula}. For simplicity, we
consider matched pairs of observations (i.e., ${\Greekmath 0115} =1$). Since the
marginal distributions are known, we can directly compute ${\Greekmath 0120}_{k} $ for $k
= 1, 2$.
Figure \ref{fig:corIF-vs-rho} plots the relationship between ${\Greekmath 011A} _{{\Greekmath 0112}
} $ and ${\Greekmath 011A} $ based on $5000$ simulations, each consisting of $5000$
observations. We find that both curves of the Gini index (left) and LC
ordinate (right) are $U$-shaped but asymmetric at zero, and that ${\Greekmath 011A}
_{{\Greekmath 0112} }$ is rarely negative. Therefore, by considering sample dependence,
we anticipate that the proposed inference method using overlapping samples
will yield more efficient results due to its smaller asymptotic variance.
However, the correlation ${\Greekmath 011A} _{{\Greekmath 0112} }$ can be monotonic in ${\Greekmath 011A} $. An
example is the population mean, where ${\Greekmath 0120} (x)=x-{\Greekmath 0116} $ and, thus, ${\Greekmath 011A}
_{{\Greekmath 0112} }={\Greekmath 011A} $. We then expect that the conventional inference approach,
which ignores sample dependence, will underestimate the asymptotic variance
and thus lead to incorrect conclusions.
\begin{figure}
\caption[Correlation between influence functions.]{} \label{fig:corIF-vs-rho}
\begin{center}
\begin{minipage}{\paperwidth}
\includegraphics[width=0.35\linewidth]{figures/Simulations/corrI_vs_rho.eps}
\includegraphics[width=0.35\linewidth]{figures/Simulations/corrLC_vs_rho.eps}
\end{minipage}
\end{center}
Figure \thefigure. ${\Greekmath 011A}_{{\Greekmath 0112}}$ in \eqref{eq:AVar-dIndex-MC} for Gini
index (left panel) and LC ordinates (right panel) against ${\Greekmath 011A}$ of Gaussian
copula in \eqref{eq:DGP-GaussCopula}.
{\footnotesize {The data are generated according to $F_{1}=\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}
(1,1.6971,8.3679)$ and $F_{2}=\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}(0.4,2.8,1.7)$ with Gaussian copula
\eqref{eq:DGP-GaussCopula}. The results are based on $5000$ replications of $
5000$ observations according to DGP: $F_{1}=\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}(1,1.6971,8.3679)$, $
F_{2}=\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}(0.4,2.8,1.7)$, ${\Greekmath 011A} \in \{0,\pm 0.01,\ldots ,\,\pm 1\}$ in
\eqref{eq:DGP-GaussCopula}, overlap portion ${\Greekmath 0115} =1$. } }
\end{figure}
To quantify the effects of sample dependence, we take the asymptotic
standard deviation with independent samples as the benchmark, denoted by $
{\Greekmath 011B} _{\Delta }^{I}$, where
\begin{equation}
{\Greekmath 011B} _{\Delta }^{I}=\left( 0.5{\Greekmath 011B} _{1}^{2}+0.5{\Greekmath 011B} _{2}^{2}\right)
^{1/2}. \label{eq:AVar-dIndex-IS-MC}
\end{equation}
If we incorrectly specify ${\Greekmath 0115} =0$, we end up with the asymptotic
variance ${\Greekmath 011B} _{\Delta }^{I}$. We examine the increase in standard
deviations using the formula $({\Greekmath 011B} _{\Delta }^{I}-{\Greekmath 011B} _{\Delta
}^{D})/{\Greekmath 011B} _{\Delta }^{D}$, where ${\Greekmath 011B} _{\Delta }^{D}$ is in
\eqref{eq:AVar-dIndex-MC}. Note that ${\Greekmath 011B} _{\Delta }^{D}$ is affected by
the correlation between ${\Greekmath 0120} _{1}$ and ${\Greekmath 0120} _{2}$ (\emph{i.e.}, ${\Greekmath 011A}
_{{\Greekmath 0112} }$) and overlap scope (\emph{i.e.}, ${\Greekmath 0115} $).
We consider again the marginal distributions $\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}(1, 1.6971, 8.3679)$
and $\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}(0.4, 2.8, 1.7)$. The sample dependence of the matched pairs
is described by the Gaussian copula \eqref{eq:DGP-GaussCopula} with ${\Greekmath 011A}
\in \{0, \pm 0.1, \pm 0.5, \pm0.99\}$. The overlap portion of matched pairs
is ${\Greekmath 0115} \in \{0.1, 0.5, 0.9\}$.
Table \ref{tab:rho-AStd} presents the results based on $5000$ replications
with $5000$ observations each. We use the true function ${\Greekmath 0120}_{k}$ for $k =
1, 2$, though typically unknown in practice. Note that the sample dependency
can substantially affect the asymptotic standard deviation if samples are
strongly correlated or share significant overlap. Take the change in
population mean $(\Delta {\Greekmath 0116})$ as an example. We find that the increase
(decrease) in standard deviations resulting from neglecting sample
dependence can be as much as $180.9\%$ ($23.6\%$).
\begin{table}[tb]
\caption{Increases in standard errors for estimated indices differences if
ignoring sample dependency.} \label{tab:rho-AStd}
\begin{center}
Table \thetable
Increases in standard errors for estimated indices differences if ignoring
sample dependency.
\end{center}
\hspace{-0.5\totalhormargin} \begin{minipage}{\paperwidth}
\centering
\small
\begin{center}
\begin{tabular}{ccccccc}
\hline
\rule[-1ex]{0cm}{4ex} ${\Greekmath 011A}$ & ${\Greekmath 0115}$ & \multicolumn{5}{c}{Increases in
standard deviations $(\%)$} \\ \cline{3-7}
\rule[-1ex]{0cm}{4ex} & & $\Delta I$ & $\Delta {\Greekmath 0116}$ & $\Delta L(0.1)$ & $
\Delta L(0.5)$ & $\Delta L(0.8)$ \\ \hline
\rule[-1ex]{0cm}{4ex} -0.99 & 0.1 & 3.1 & -3.7 & 2.0 & 3.9 & 0.9 \\
& 0.5 & 18.8 & -15.4 & 11.3 & 25.3 & 4.6 \\
& 0.9 & 45.0 & -23.6 & 23.7 & 70.1 & 8.8 \\
\rule[-1ex]{0cm}{4ex} -0.5 & 0.1 & 0.8 & -2.0 & 0.6 & 0.9 & 0.5 \\
& 0.5 & 4.3 & -9.1 & 2.8 & 4.5 & 2.3 \\
& 0.9 & 8.2 & -14.8 & 5.3 & 8.5 & 4.3 \\
\rule[-1ex]{0cm}{4ex} -0.1 & 0.1 & 0.0 & -0.4 & 0.0 & 0.0 & 0.0 \\
& 0.5 & 0.1 & -2.1 & 0.0 & 0.1 & 0.1 \\
& 0.9 & 0.2 & -3.7 & 0.1 & 0.2 & 0.2 \\
\rule[-1ex]{0cm}{4ex} 0 & 0.1 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\
& 0.5 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\
& 0.9 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\
\rule[-1ex]{0cm}{4ex} 0.1 & 0.1 & 0.1 & 0.5 & 0.1 & 0.1 & 0.0 \\
& 0.5 & 0.3 & 2.3 & 0.3 & 0.3 & 0.2 \\
& 0.9 & 0.5 & 4.3 & 0.5 & 0.5 & 0.3 \\
\rule[-1ex]{0cm}{4ex} 0.5 & 0.1 & 1.1 & 2.4 & 1.0 & 1.1 & 0.9 \\
& 0.5 & 6.1 & 14.2 & 5.5 & 5.9 & 4.8 \\
& 0.9 & 11.8 & 31.2 & 10.6 & 11.5 & 9.2 \\
\rule[-1ex]{0cm}{4ex} 0.99 & 0.1 & 5.0 & 5.2 & 5.1 & 5.0 & 4.9 \\
& 0.5 & 36.2 & 39.4 & 37.8 & 36.3 & 35.7 \\
& 0.9 & 142.0 & 180.9 & 159.6 & 143.7 & 137.8 \\ \hline
\end{tabular}
\end{center}
\end{minipage}
\medskip
\noindent
\footnotesize
Note -- The table provides the percentage increases in standard deviations
for estimators of several welfare index differences when ignoring sample
dependency, including the changes in Gini indices ($\Delta I$), population
mean ($\Delta {\Greekmath 0116}$) and LC ordinates ($\Delta L(p)$) at $p \in \{0.1, 0.5,
0.8\}$. The percentage increases in standard deviations are measured by $
({\Greekmath 011B}^{I}_{\Delta} - {\Greekmath 011B}^{D}_{\Delta}) / {\Greekmath 011B}^{D}_{\Delta}$, where $
{\Greekmath 011B}^{I}_{\Delta}$ and ${\Greekmath 011B}^{D}_{\Delta}$ are given in
\eqref{eq:AVar-dIndex-IS-MC} and \eqref{eq:AVar-dIndex-MC}, respectively.
The results are based on $5000$ replications of $5000$ observations
according to the DGP: $F_{1} = \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}(1, 1.6971, 8.3679)$, $F_{2} = \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{
SM}(0.4, 2.8, 1.7)$, ${\Greekmath 011A} \in \{0, \pm 0.5, \pm0.99\}$ in
\eqref{eq:DGP-GaussCopula}, overlap portion ${\Greekmath 0115} \in \{0.1, 0.5, 0.9\}$.
\normalsize
\end{table}
We next examine the performance of $95\%$ confidence intervals (CIs) by
different approaches based on $1000$ replications of $1000$ observations
under the same DGP. Our goal is to investigate the impact of sample
dependence on the performance of inference. We set the conventional method
as the benchmark, which incorrectly assumes that two samples are
independent. By \eqref{eq:AVar-dIndex-Est}, we estimate the asymptotic
standard deviation as follows.
\begin{equation}
\hat{{\Greekmath 011B} }_{\Delta }^{D}=[0.5\hat{{\Greekmath 011B} }_{1}^{2}+0.5\hat{{\Greekmath 011B} }
_{2}^{2}-(m/n)\hat{{\Greekmath 011A} }_{{\Greekmath 0112} }\hat{{\Greekmath 011B} }_{1}\hat{{\Greekmath 011B} }
_{2}^{1/2}]^{1/2}, \label{eq:Est-AVar-dIndex-MC}
\end{equation}
\begin{equation}
\hat{{\Greekmath 011B} }_{\Delta }^{I}=[0.5\hat{{\Greekmath 011B} }_{1}^{2}+0.5\hat{{\Greekmath 011B} }
_{2}^{2}]^{1/2}, \label{eq:Est-AVar-dIndex-IS-MC}
\end{equation}
where for $k=1,2$, $\hat{{\Greekmath 011B} }_{k}^{2}$ is the sample variance of $\hat{
{\Greekmath 0120} }_{n_{k}}(X_{ki})$ in \eqref{eq:AVar-Index-Est-Fk}, and $\hat{{\Greekmath 011A} }
_{{\Greekmath 0112} }$ is the sample correlation coefficient of $\left\{ (\hat{{\Greekmath 0120} }
_{n_{1}}(X_{1i}),\hat{{\Greekmath 0120} }_{n_{2}}(X_{2i}))\right\} _{i=1}^{m}$.
We present simulation results in Table \ref{tab:CI-dI-DSvsIS} for the Gini
difference, Table \ref{tab:CI-dMean-DSvsIS} for the mean difference $\Delta
{\Greekmath 0116}$, and Table \ref{tab:CI-dLC-p05-DSvsIS} for LC ordinates differences at $
p = 0.5$.
First, we find that conventional CIs, which fail to account for sample
dependence, may suffer from two problems. They can either be overly
conservative with overlarge widths (see columns $(9) - (12)$ in Table \ref
{tab:CI-dI-DSvsIS} and \ref{tab:CI-dLC-p05-DSvsIS}), or they have coverage
rates much lower than the nominal level $95\%$ (see columns $(9) - (12)$ in
Table \ref{tab:CI-dMean-DSvsIS} when ${\Greekmath 0115} = 0.9$ and ${\Greekmath 011A} = -0.99$).
Such findings are consistent with our expectations in Figure \ref
{fig:corIF-vs-rho} and Table \ref{tab:rho-AStd}.
Second, our proposed asymptotic and bootstrap CIs with overlapping samples
can effectively control the level with reasonable widths; see columns $(5) -
(8)$ in Table \ref{tab:CI-dI-DSvsIS}, \ref{tab:CI-dMean-DSvsIS}, and Table
\ref{tab:CI-dLC-p05-DSvsIS}.
Third, we find that the intersection methods (IMs) are valid, though
conservative, for all three indices differences, as indicated in $(1) - (4)$
in Table \ref{tab:CI-dI-DSvsIS}, \ref{tab:CI-dMean-DSvsIS} and \ref
{tab:CI-dLC-p05-DSvsIS}. Note that the IMs are applicable in more general
scenarios and become the only valid procedure when overlapping samples fail
to hold; see section \ref{sec:MC-BeyondOS} for more details.
\begin{sidewaystable}
\caption{Performance of proposed and conventional confidence intervals at level
of $95\%$ for the difference in Gini index.} \label{tab:CI-dI-DSvsIS}
\begin{center}
Table \thetable
Performance of proposed and conventional confidence intervals at level of $
95\%$ for the difference in Gini index.
\begin{adjustbox}{scale={0.9}{0.9}}
\small
\begin{tabular}{cc|cc|cc|cc|cc|cc|cc}
\hline
\rule[-1ex]{0cm}{4ex} & & (1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) &
(9) & (10) & (11) & (12) \\ \cline{3-14}
\rule[-1ex]{0cm}{4ex} & & \multicolumn{4}{c|}{Intersection method} &
\multicolumn{4}{c|}{Overlapping samples} & \multicolumn{4}{c}{Independent
samples (incorrect)} \\ \cline{3-14}
\rule[-1ex]{0cm}{4ex} ${\Greekmath 011A}$ & ${\Greekmath 0115}$ & \multicolumn{2}{c|}{Coverage $
(\%) $} & \multicolumn{2}{c|}{Width} & \multicolumn{2}{c|}{Coverage $(\%)$}
& \multicolumn{2}{c|}{Width} & \multicolumn{2}{c|}{Coverage $(\%)$} &
\multicolumn{2}{c}{Width} \\ \cline{3-14}
\rule[-1ex]{0cm}{4ex} & & Asym & Boot & Asym & Boot & Asym & Boot & Asym &
Boot & Asym & Boot & Asym & Boot \\ \hline
\rule[-1ex]{0cm}{4ex} -0.99 & 0.1 & 100 & 100 & 0.0666 & 0.0687 & 95.3 & 94.4
& 0.04 & 0.0405 & 95.7 & 95.1 & 0.0412 & 0.0418 \\
& 0.5 & 100 & 100 & 0.0664 & 0.0685 & 94.6 & 94 & 0.0345 & 0.035 & 98.4 &
98.4 & 0.0411 & 0.0416 \\
& 0.9 & 100 & 100 & 0.0666 & 0.0687 & 94.1 & 93.1 & 0.0284 & 0.029 & \textbf{
99.1} & \textbf{99} & 0.0413 & 0.042 \\
\rule[-1ex]{0cm}{4ex} -0.5 & 0.1 & 99.8 & 99.8 & 0.0664 & 0.0686 & 94.2 &
93.9 & 0.0408 & 0.0413 & 94.9 & 95.5 & 0.0411 & 0.0416 \\
& 0.5 & 99.9 & 99.9 & 0.0665 & 0.0686 & 95.7 & 95.3 & 0.0395 & 0.04 & 96.1 &
96.4 & 0.0412 & 0.0417 \\
& 0.9 & 100 & 100 & 0.0665 & 0.0685 & 95 & 95.4 & 0.0381 & 0.0386 & 96.8 &
96.7 & 0.0412 & 0.0418 \\
\rule[-1ex]{0cm}{4ex} 0 & 0.1 & 99.9 & 99.9 & 0.0665 & 0.0684 & 94.8 & 94.6
& 0.0412 & 0.0419 & 95 & 95.8 & 0.0412 & 0.0418 \\
& 0.5 & 100 & 100 & 0.0666 & 0.0687 & 95.6 & 94.9 & 0.0413 & 0.0419 & 95.5 &
95.4 & 0.0413 & 0.042 \\
& 0.9 & 100 & 99.9 & 0.0667 & 0.0689 & 95.3 & 95.3 & 0.0413 & 0.042 & 95.2 &
95.3 & 0.0413 & 0.042 \\
\rule[-1ex]{0cm}{4ex} 0.5 & 0.1 & 99.8 & 99.9 & 0.0668 & 0.0689 & 95.3 & 94.6
& 0.0409 & 0.0416 & 95.7 & 94.9 & 0.0414 & 0.0421 \\
& 0.5 & 99.8 & 99.8 & 0.0664 & 0.0685 & 94 & 94.6 & 0.0388 & 0.0394 & 95.7 &
95.5 & 0.0411 & 0.0418 \\
& 0.9 & 99.8 & 100 & 0.0665 & 0.0686 & 94.1 & 94.1 & 0.0368 & 0.0373 & 96.1
& 96 & 0.0412 & 0.0418 \\
\rule[-1ex]{0cm}{4ex} 0.99 & 0.1 & 99.9 & 99.8 & 0.0666 & 0.0687 & 94.2 &
94.4 & 0.0393 & 0.0399 & 95.6 & 95.3 & 0.0412 & 0.0419 \\
& 0.5 & 100 & 100 & 0.0667 & 0.0688 & 95.6 & 95.9 & 0.0303 & 0.0308 &
\textbf{99.5} & \textbf{99.4} & 0.0413 & 0.042 \\
& 0.9 & 100 & 100 & 0.0665 & 0.0688 & 96 & 95.4 & 0.017 & 0.0174 & \textbf{
100} & \textbf{100} & 0.0412 & 0.0419 \\ \hline
\end{tabular}
\normalsize
\end{adjustbox}
\end{center}
\medskip
\noindent
\footnotesize
Note -- The table provides coverage rates and widths of $95\%$ confidence
intervals for the difference in Gini index. The data are generated according
to the DGP: $F_{1} = \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}(1, 1.6971, 8.3679)$, $F_{2} = \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}(0.4,
2.8, 1.7)$, ${\Greekmath 011A} \in \{0, \pm 0.5, \pm0.99\}$ in \eqref{eq:DGP-GaussCopula}
, overlap portion ${\Greekmath 0115} \in \{0.1, 0.5, 0.9\}$. The sample sizes are both
$1000$. Columns $(1)$ - $(4)$ are performance by asymptotic intersection
method (Asym), bootstrap intersection method (Boot) in section \ref{sec:DS}.
Columns $(5)$ - $(8)$ are performance by asymptotic (Asym) and bootstrap
(Boot) methods with overlapping samples in section \ref{sec:OS}, using $\hat{
{\Greekmath 011B}}^{D}_{\Delta}$ in \eqref{eq:Est-AVar-dIndex-MC}. Columns $(9) $ - $
(12)$ are performance by asymptotic (Asym) and bootstrap (Boot) methods if
incorrectly ignoring the sample dependence, using $\hat{{\Greekmath 011B}}^{I}_{\Delta}$
in \eqref{eq:Est-AVar-dIndex-IS-MC}. The results are based on $1000$
replications with 399 bootstrap repetitions each. The bold font indicates
cases where coverage rates largely deviate from $95\%$.
\normalsize
\end{sidewaystable}
\begin{sidewaystable}
\caption{Performance of proposed and conventional confidence intervals at
level of $95\%$ for the mean difference.} \label{tab:CI-dMean-DSvsIS}
\begin{center}
Table \thetable
Performance of proposed and conventional confidence intervals at level of $
95\%$ for the mean difference.
\begin{adjustbox}{scale={0.9}{0.9}}
\small
\begin{tabular}{cc|cc|cc|cc|cc|cc|cc}
\hline
\rule[-1ex]{0cm}{4ex} & & (1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) &
(9) & (10) & (11) & (12) \\ \cline{3-14}
\rule[-1ex]{0cm}{4ex} & & \multicolumn{4}{c|}{Intersection method} &
\multicolumn{4}{c|}{Overlapping samples} & \multicolumn{4}{c}{Independent
samples (incorrect)} \\ \cline{3-14}
\rule[-1ex]{0cm}{4ex} ${\Greekmath 011A}$ & ${\Greekmath 0115}$ & \multicolumn{2}{c|}{Coverage $
(\%) $} & \multicolumn{2}{c|}{Width} & \multicolumn{2}{c|}{Coverage $(\%)$}
& \multicolumn{2}{c|}{Width} & \multicolumn{2}{c|}{Coverage $(\%)$} &
\multicolumn{2}{c}{Width} \\ \cline{3-14}
\rule[-1ex]{0cm}{4ex} & & Asym & Boot & Asym & Boot & Asym & Boot & Asym &
Boot & Asym & Boot & Asym & Boot \\ \hline
\rule[-1ex]{0cm}{4ex} -0.99 & 0.1 & 99.8 & 99.9 & 0.0533 & 0.0544 & 95.1 & 95
& 0.0343 & 0.0346 & 94.2 & 94.2 & 0.033 & 0.0333 \\
& 0.5 & 99.5 & 99.7 & 0.0533 & 0.0544 & 96.3 & 95.6 & 0.039 & 0.0394 & 91.1
& 91.4 & 0.033 & 0.0333 \\
& 0.9 & 98.3 & 98.2 & 0.0534 & 0.0544 & 95.8 & 95.3 & 0.0433 & 0.0437 &
\textbf{86.1} & \textbf{85.9} & 0.0331 & 0.0335 \\
\rule[-1ex]{0cm}{4ex} -0.5 & 0.1 & 99.8 & 99.8 & 0.0534 & 0.0545 & 94.9 & 95
& 0.0338 & 0.0341 & 94.6 & 94.1 & 0.0331 & 0.0335 \\
& 0.5 & 99.8 & 99.7 & 0.0535 & 0.0546 & 94.6 & 95 & 0.0365 & 0.0368 & 92.9 &
92.7 & 0.0331 & 0.0334 \\
& 0.9 & 99.6 & 99.3 & 0.0535 & 0.0545 & 96.1 & 95.7 & 0.0389 & 0.0393 & 90.3
& 90.7 & 0.0331 & 0.0334 \\
\rule[-1ex]{0cm}{4ex} 0 & 0.1 & 99.9 & 99.9 & 0.0535 & 0.0546 & 95.5 & 95.5
& 0.0331 & 0.0334 & 95.5 & 95.2 & 0.0331 & 0.0334 \\
& 0.5 & 99.6 & 99.6 & 0.0534 & 0.0545 & 94 & 94.3 & 0.0331 & 0.0334 & 94.2 &
94.2 & 0.0331 & 0.0335 \\
& 0.9 & 99.9 & 99.9 & 0.0535 & 0.0545 & 95.6 & 95.1 & 0.0332 & 0.0335 & 95.6
& 95.3 & 0.0331 & 0.0335 \\
\rule[-1ex]{0cm}{4ex} 0.5 & 0.1 & 99.8 & 99.9 & 0.0534 & 0.0545 & 94.1 & 94.1
& 0.0323 & 0.0327 & 95.1 & 95 & 0.0331 & 0.0334 \\
& 0.5 & 100 & 99.9 & 0.0534 & 0.0545 & 94.8 & 94.3 & 0.029 & 0.0293 & 96.2 &
96.3 & 0.0331 & 0.0334 \\
& 0.9 & 100 & 100 & 0.0534 & 0.0545 & 93.9 & 93.9 & 0.0252 & 0.0255 & 98.4 &
98.5 & 0.0331 & 0.0334 \\
\rule[-1ex]{0cm}{4ex} 0.99 & 0.1 & 100 & 100 & 0.0534 & 0.0545 & 94.3 & 94.3
& 0.0314 & 0.0318 & 95 & 95.6 & 0.0331 & 0.0334 \\
& 0.5 & 100 & 100 & 0.0534 & 0.0546 & 95.3 & 94.5 & 0.0237 & 0.0239 &
\textbf{99.2} & \textbf{99.2} & 0.0331 & 0.0334 \\
& 0.9 & 100 & 100 & 0.0534 & 0.0545 & 94 & 93.9 & 0.0118 & 0.0119 & \textbf{
100} & \textbf{100} & 0.0331 & 0.0334 \\ \hline
\end{tabular}
\normalsize
\end{adjustbox}
\end{center}
\medskip
\noindent
\footnotesize
Note -- The table provides coverage rates and widths of $95\%$ confidence
intervals for the mean difference. The data are generated according to DGP: $
F_{1} = \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}(1, 1.6971, 8.3679)$, $F_{2} = \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}(0.4, 2.8, 1.7)$, $
{\Greekmath 011A} \in \{0, \pm 0.5, \pm0.99\}$ in \eqref{eq:DGP-GaussCopula}, overlap
portion ${\Greekmath 0115} \in \{0.1, 0.5, 0.9\}$. The sample sizes are both $1000$.
Columns $(1)$ - $(4)$ are performance by asymptotic intersection method
(Asym), bootstrap intersection method (Boot) in section \ref{sec:DS}.
Columns $(5)$ - $(8)$ are performance by asymptotic (Asym) and bootstrap
(Boot) methods with overlapping samples in section \ref{sec:OS}, using $\hat{
{\Greekmath 011B}}^{D}_{\Delta}$ in \eqref{eq:Est-AVar-dIndex-MC}. Columns $(9) $ - $
(12)$ are performance by asymptotic (Asym) and bootstrap (Boot) methods if
incorrectly ignoring the sample dependence, using $\hat{{\Greekmath 011B}}^{I}_{\Delta}$
in \eqref{eq:Est-AVar-dIndex-IS-MC}. The results are based on $1000$
replications with 399 bootstrap repetitions each. The bold font indicates
cases where coverage rates largely deviate from $95\%$.
\normalsize
\end{sidewaystable}
\begin{sidewaystable}
\caption{Performance of proposed and conventional confidence intervals at level
of $95\%$ for the difference in LC ordinate at percentile $0.5$.} \label
{tab:CI-dLC-p05-DSvsIS}
\begin{center}
Table \thetable
Performance of proposed and conventional confidence intervals at level of $
95\%$ for the difference in LC ordinate at percentile $0.5$.
\begin{adjustbox}{scale={0.9}{0.9}}
\small
\begin{tabular}{cc|cc|cc|cc|cc|cc|cc}
\hline
\rule[-1ex]{0cm}{4ex} & & (1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) &
(9) & (10) & (11) & (12) \\ \cline{3-14}
\rule[-1ex]{0cm}{4ex} & & \multicolumn{4}{c|}{Intersection method} &
\multicolumn{4}{c|}{Overlapping samples} & \multicolumn{4}{c}{Independent
samples (incorrect)} \\ \cline{3-14}
\rule[-1ex]{0cm}{4ex} ${\Greekmath 011A}$ & ${\Greekmath 0115}$ & \multicolumn{2}{c|}{Coverage $
(\%) $} & \multicolumn{2}{c|}{Width} & \multicolumn{2}{c|}{Coverage $(\%)$}
& \multicolumn{2}{c|}{Width} & \multicolumn{2}{c|}{Coverage $(\%)$} &
\multicolumn{2}{c}{Width} \\ \cline{3-14}
\rule[-1ex]{0cm}{4ex} & & Asym & Boot & Asym & Boot & Asym & Boot & Asym &
Boot & Asym & Boot & Asym & Boot \\ \hline
\rule[-1ex]{0cm}{4ex} -0.99 & 0.1 & 100 & 100 & 0.0451 & 0.0466 & 95.6 & 95
& 0.0269 & 0.0275 & 96 & 96.2 & 0.0279 & 0.0286 \\
& 0.5 & 100 & 100 & 0.045 & 0.0466 & 95.2 & 95.4 & 0.0222 & 0.0229 & \textbf{
99.4} & \textbf{99.3} & 0.0279 & 0.0285 \\
& 0.9 & 100 & 100 & 0.0451 & 0.0467 & 94.4 & 95 & 0.0164 & 0.0172 & \textbf{
100} & \textbf{99.8} & 0.0279 & 0.0285 \\
\rule[-1ex]{0cm}{4ex} -0.5 & 0.1 & 99.9 & 100 & 0.045 & 0.0466 & 95.2 & 95.6
& 0.0276 & 0.0283 & 95.3 & 95.6 & 0.0279 & 0.0286 \\
& 0.5 & 99.9 & 99.9 & 0.045 & 0.0466 & 94.9 & 95.1 & 0.0267 & 0.0274 & 95.8
& 96.1 & 0.0279 & 0.0285 \\
& 0.9 & 100 & 100 & 0.0451 & 0.0466 & 95 & 95.6 & 0.0257 & 0.0264 & 96.7 &
97.2 & 0.0279 & 0.0286 \\
\rule[-1ex]{0cm}{4ex} 0 & 0.1 & 99.9 & 99.9 & 0.045 & 0.0468 & 95.2 & 95.6 &
0.0279 & 0.0286 & 95 & 95.7 & 0.0279 & 0.0285 \\
& 0.5 & 99.8 & 100 & 0.0451 & 0.0467 & 95.3 & 95.8 & 0.0279 & 0.0286 & 95.5
& 95.1 & 0.0279 & 0.0286 \\
& 0.9 & 100 & 99.9 & 0.0451 & 0.0467 & 94.8 & 95.1 & 0.028 & 0.0286 & 94.8 &
95 & 0.0279 & 0.0286 \\
\rule[-1ex]{0cm}{4ex} 0.5 & 0.1 & 99.7 & 100 & 0.0452 & 0.0468 & 95.4 & 95 &
0.0277 & 0.0283 & 95.5 & 95.6 & 0.028 & 0.0286 \\
& 0.5 & 99.8 & 99.8 & 0.045 & 0.0466 & 94.2 & 94.8 & 0.0263 & 0.0269 & 95.6
& 95.7 & 0.0279 & 0.0285 \\
& 0.9 & 99.8 & 99.8 & 0.045 & 0.0466 & 95.6 & 95.5 & 0.025 & 0.0256 & 97.1 &
97.7 & 0.0279 & 0.0285 \\
\rule[-1ex]{0cm}{4ex} 0.99 & 0.1 & 100 & 100 & 0.0451 & 0.0467 & 94.7 & 94.8
& 0.0266 & 0.0272 & 95.6 & 95.7 & 0.0279 & 0.0286 \\
& 0.5 & 100 & 100 & 0.0451 & 0.0468 & 96.2 & 96.3 & 0.0205 & 0.0211 &
\textbf{99} & \textbf{99} & 0.0279 & 0.0287 \\
& 0.9 & 100 & 100 & 0.0451 & 0.0466 & 96.4 & 96.7 & 0.0115 & 0.0123 &
\textbf{100} & \textbf{100} & 0.0279 & 0.0285 \\ \hline
\end{tabular}
\normalsize
\end{adjustbox}
\end{center}
\medskip
\noindent
\footnotesize
Note -- The table provides coverage rates and widths of $95\%$ confidence
intervals for the difference in LC ordinate at percentile $p=0.5$. The data
are generated according to DGP: $F_{1} = \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}(1, 1.6971, 8.3679)$, $
F_{2} = \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}(0.4, 2.8, 1.7)$, ${\Greekmath 011A} \in \{0, \pm 0.5, \pm0.99\}$ in
\eqref{eq:DGP-GaussCopula}, overlap portion ${\Greekmath 0115} \in \{0.1, 0.5, 0.9\}$.
The sample sizes are both $1000$. Columns $(1)$ - $(4)$ are performance by
asymptotic intersection method (Asym), bootstrap intersection method (Boot)
in section \ref{sec:DS}. Columns $(5)$ - $(8)$ are performance by asymptotic
(Asym) and bootstrap (Boot) methods with overlapping samples in section \ref
{sec:OS}, using $\hat{{\Greekmath 011B} }_{\Delta }^{D}$ in
\eqref{eq:Est-AVar-dIndex-MC}. Columns $(9)$ - $(12)$ are performance by
asymptotic (Asym) and bootstrap (Boot) methods if incorrectly ignoring the
sample dependence, using $\hat{{\Greekmath 011B} }_{\Delta }^{I}$ in
\eqref{eq:Est-AVar-dIndex-IS-MC}. The results are based on $1000$
replications with 399 bootstrap repetitions each. The bold font indicates
cases where coverage rates largely deviate from $95\%$.
\normalsize
\end{sidewaystable}
\FloatBarrier
\subsection{Effects of heavy tails \label{sec: Effects of heavy tails}}
\label{sec:MC-HeavyTail}
\begin{comment}
\textcolor{red}{Bahadur and Savage (1956) for theoretical reasons claimed in
Davidson (2012) on heavy tails.}
\end{comment}
Heavy-tailed distributions are notorious for causing problems for both
asymptotic and bootstrap inference. We will examine the impacts of heavy
tails on the performance of proposed inference methods. We still consider
the SM distribution $\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}(b, a, q)$ in \eqref{eq:SM-CDF}. The upper
tail behaves similarly to the Pareto distribution with parameter ${\Greekmath 010C} = aq$
, also known as the stability index. A smaller ${\Greekmath 010C}$ indicates a heavier
upper tail. For a variance to exist, ${\Greekmath 010C}$ must be no less than two.
To mimic the U.S. income distribution in 1980, we set the distribution of
the first sample to $\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}(1, 1.6971, 8.3679$. For the second sample, we
explore three SM distributions, $\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}(0.4, 2.8, 1.7)$, $\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}(0.4,
2, 1.5)$, and $\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}(0.4, 1.4, 1.5)$. The corresponding stability
indices are $4.76$, $3$, and $2.1$, indicating increasingly heavier upper
tails.
For simplicity, we assume that the two sample sizes are identical, with $n
\in \{100, 200, 500\}$. A portion ${\Greekmath 0115} \in \{0.1, 0.5, 0.9\}$ of
observations is drawn in pairs, with dependence characterized by a Gaussian
copula in \eqref{eq:DGP-GaussCopula} with ${\Greekmath 011A} \in \{0, \pm 0.5, \pm0.99\}$
. The performance of CIs is evaluated based on $1000$ replications, with $B
= 399$ bootstrap repetitions each.
We summarize the DGPs as follows.
\medskip \textbf{DGP I-A}: $F_{1} = \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}(1, 1.6971, 8.3679)$, $F_{2} =
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}(0.4, 2.8, 1.7)$, ${\Greekmath 011A} \in \{0, \pm 0.5, \pm0.99\}$ in
\eqref{eq:DGP-GaussCopula}, stability index of $F_{2}$ is ${\Greekmath 010C}_{2} = 4.76$
, overlap portion ${\Greekmath 0115} \in \{0.1, 0.5, 0.9\}$.
\medskip
\textbf{DGP I-B}: $F_{1} = \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}(1, 1.6971, 8.3679)$, $F_{2} = \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}
(0.4, 2, 1.5)$, ${\Greekmath 011A} \in \{0, \pm 0.5, \pm0.99\}$ in
\eqref{eq:DGP-GaussCopula}, stability index of $F_{2}$ is ${\Greekmath 010C}_{2} = 3$,
overlap portion ${\Greekmath 0115} \in \{0.1, 0.5, 0.9\}$.
\medskip
\textbf{DGP I-C}: $F_{1} = \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}(1, 1.6971, 8.3679)$, $F_{2} = \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}
(0.4, 1.4, 1.5)$, ${\Greekmath 011A} \in \{0, \pm 0.5, \pm0.99\}$ in
\eqref{eq:DGP-GaussCopula}, stability index of $F_{2}$ is ${\Greekmath 010C}_{2} = 2.1$,
overlap portion ${\Greekmath 0115} \in \{0.1, 0.5, 0.9\}$.
\medskip
We consider changes in the Gini index, population mean and LC ordinate at
percentile of $0.5$. We assess the performance of four proposed confidence
intervals (CIs) at the level of $95\%$. These include the asymptotic
intersection method (IM-Asym), bootstrap intersection method (IM-Boot),
asymptotic inference with overlapping samples (OS-Asym), and bootstrap
inference with overlapping samples (OS-Boot).
For DGP I-A, we illustrate the performance of CIs for the difference in Gini
index, mean, and LC ordinate in Table \ref{tab:CI-dI-DGP1}, \ref
{tab:CI-dmu-DGP1}, and \ref{tab:CI-dLC-DGP1}, respectively. As expected, the
asymptotic and bootstrap CIs with overlapping samples (OS-Asym and OS-Boot)
exhibit coverage rates close to the nominal level of $95\%$ due to the
absence of heavy tails. However, the asymptotic and bootstrap intersection
methods (IM-Asym and IM-Boot) are conservative, with consistently higher
coverage rates than the nominal level, yielding a trade-off between
efficiency and robustness.
For DGP I-B, we present the performance of CIs for three indices differences
in Table \ref{tab:CI-dI-DGP2}, \ref{tab:CI-dmu-DGP2}, and \ref
{tab:CI-dLC-DGP2}. In this scenario, the second distribution has a heavy
upper tail. The coverage rates of asymptotic CIs with overlapping samples
are lower than $95\%$ when two samples are strongly positively related; see,
for example, column OS-Asym when ${\Greekmath 011A} = 0.99$ and ${\Greekmath 0115} = 0.9$. The
bootstrap CI (OS-Boot), however, can alleviate this issue. The asymptotic
and bootstrap IMs (IM-Asym and IM-Boot) are still conservative.
Nevertheless, the widths are considered acceptable compared to the bootstrap
CIs.
For DGP I-C, we evaluate the performance of CIs for the three indices
differences in \ref{tab:CI-dI-DGP3}, \ref{tab:CI-dmu-DGP3}, and \ref
{tab:CI-dLC-DGP3}. The second distribution now has an extremely heavy tail,
resulting in a considerable variance. Neither the asymptotic and bootstrap
CIs with overlapping samples (OS-Asym and OS-Boot) yield reliable results,
as the coverage falls below $95\%$. However, the bootstrap CI (OS-Boot)
suffers less coverage distortion compared to the asymptotic CI. The IMs
(IM-Asym and IM-Boot) remain valid and provide reasonable widths, sometimes
comparable to those of OS-Boot.
\begin{table}[tb]
\caption{DGP I-A: coverage and width of confidence intervals for Gini indices
differences.} \label{tab:CI-dI-DGP1}
\begin{center}
Table \thetable
DGP I-A: coverage and width of confidence intervals for Gini indices
differences.
\end{center}
\begin{adjustbox}{scale={0.9}{0.9}}
\hspace{-0.5\totalhormargin} \begin{minipage}{\paperwidth}
\centering
\small
\begin{center}
\begin{tabular}{ccc|cccc|cccc}
\hline
\rule[-1ex]{0cm}{4ex} ${\Greekmath 011A}$ & ${\Greekmath 0115}$ & $n$ & \multicolumn{4}{c|}{
Coverage $(\%)$} & \multicolumn{4}{c}{Width} \\ \cline{4-11}
\rule[-1ex]{0cm}{4ex} & & & IM-Asym & IM-Boot & OS-Asym & OS-Boot & IM-Asym
& IM-Boot & OS-Asym & OS-Boot \\ \hline
\rule[-1ex]{0cm}{4ex} -0.99 & 0.1 & 100 & 99.9 & 99.9 & 94.9 & 94.3 & 0.2024
& 0.228 & 0.1231 & 0.1294 \\
& & 200 & 99.7 & 99.8 & 95.5 & 95.4 & 0.1464 & 0.1579 & 0.0885 & 0.0916 \\
& & 500 & 99.9 & 99.9 & 94.8 & 94.9 & 0.0937 & 0.0979 & 0.0564 & 0.0576 \\
& 0.5 & 100 & 99.9 & 99.9 & 94.3 & 93 & 0.2031 & 0.2293 & 0.1064 & 0.111 \\
& & 200 & 100 & 100 & 94.4 & 93.4 & 0.1464 & 0.1581 & 0.0762 & 0.0782 \\
& & 500 & 100 & 100 & 93.9 & 93.9 & 0.0938 & 0.0979 & 0.0488 & 0.0498 \\
& 0.9 & 100 & 100 & 100 & 94.3 & 93.4 & 0.2024 & 0.2274 & 0.0858 & 0.0889 \\
& & 200 & 100 & 100 & 94.5 & 91.9 & 0.1453 & 0.1561 & 0.0617 & 0.0634 \\
& & 500 & 100 & 100 & 94.8 & 93.3 & 0.0933 & 0.0975 & 0.0397 & 0.0404 \\
\rule[-1ex]{0cm}{4ex} -0.5 & 0.1 & 100 & 99.9 & 100 & 94.7 & 95.3 & 0.2026 &
0.2267 & 0.1253 & 0.1317 \\
& & 200 & 99.6 & 99.8 & 94.1 & 94.1 & 0.1452 & 0.1559 & 0.0897 & 0.0926 \\
& & 500 & 100 & 100 & 93.9 & 94.3 & 0.0933 & 0.0975 & 0.0574 & 0.0585 \\
& 0.5 & 100 & 99.9 & 99.9 & 94.4 & 94.4 & 0.2031 & 0.228 & 0.1217 & 0.1281
\\
& & 200 & 99.6 & 99.6 & 93.7 & 93.4 & 0.1457 & 0.1569 & 0.087 & 0.0899 \\
& & 500 & 99.9 & 99.9 & 95.2 & 94.2 & 0.0936 & 0.0978 & 0.0557 & 0.0569 \\
& 0.9 & 100 & 99.8 & 99.9 & 94.2 & 93.6 & 0.2031 & 0.2293 & 0.1175 & 0.1238
\\
& & 200 & 100 & 100 & 94.6 & 94.7 & 0.1463 & 0.1578 & 0.0843 & 0.0872 \\
& & 500 & 100 & 100 & 95.7 & 95.4 & 0.0936 & 0.0978 & 0.0536 & 0.0547 \\
\rule[-1ex]{0cm}{4ex} 0 & 0.1 & 100 & 99.9 & 99.9 & 94.5 & 94.2 & 0.2018 &
0.2267 & 0.1256 & 0.1323 \\
& & 200 & 99.8 & 99.8 & 94.9 & 95.1 & 0.1456 & 0.1562 & 0.0904 & 0.0937 \\
& & 500 & 99.6 & 99.7 & 94.2 & 93.8 & 0.0938 & 0.0981 & 0.0581 & 0.0594 \\
& 0.5 & 100 & 99.5 & 99.7 & 93.8 & 93.7 & 0.2029 & 0.2279 & 0.1263 & 0.134
\\
& & 200 & 100 & 100 & 95.5 & 95.1 & 0.1464 & 0.1586 & 0.091 & 0.0946 \\
& & 500 & 99.6 & 99.8 & 95.4 & 94.9 & 0.0939 & 0.0983 & 0.0582 & 0.0596 \\
& 0.9 & 100 & 99.4 & 99.7 & 94.1 & 93.8 & 0.2007 & 0.2244 & 0.1249 & 0.1319
\\
& & 200 & 99.6 & 99.6 & 94.7 & 94.8 & 0.1459 & 0.1568 & 0.0905 & 0.0939 \\
& & 500 & 99.9 & 100 & 93.8 & 93.6 & 0.094 & 0.0985 & 0.0583 & 0.0597 \\
\rule[-1ex]{0cm}{4ex} 0.5 & 0.1 & 100 & 99.6 & 99.8 & 93.7 & 93.5 & 0.2033 &
0.2281 & 0.1252 & 0.132 \\
& & 200 & 99.6 & 99.7 & 94.6 & 94.4 & 0.1452 & 0.1569 & 0.0893 & 0.0925 \\
& & 500 & 99.9 & 99.9 & 95.1 & 94.6 & 0.0939 & 0.0984 & 0.0576 & 0.0589 \\
& 0.5 & 100 & 100 & 100 & 95.3 & 94.7 & 0.2022 & 0.2273 & 0.1187 & 0.1257 \\
& & 200 & 100 & 99.9 & 94.9 & 94.5 & 0.1465 & 0.1589 & 0.0858 & 0.0893 \\
& & 500 & 100 & 100 & 95.1 & 94.4 & 0.0938 & 0.0982 & 0.0548 & 0.0559 \\
& 0.9 & 100 & 100 & 99.9 & 92.7 & 93.5 & 0.2031 & 0.2286 & 0.1125 & 0.1197
\\
& & 200 & 99.9 & 100 & 95 & 94.7 & 0.1463 & 0.1583 & 0.0812 & 0.0848 \\
& & 500 & 100 & 100 & 94.5 & 94.4 & 0.0936 & 0.0981 & 0.052 & 0.0533 \\
\rule[-1ex]{0cm}{4ex} 0.99 & 0.1 & 100 & 99.4 & 99.7 & 94.1 & 94 & 0.2029 &
0.2291 & 0.1206 & 0.1286 \\
& & 200 & 99.8 & 99.9 & 95 & 95.4 & 0.1457 & 0.1568 & 0.0862 & 0.0895 \\
& & 500 & 100 & 100 & 95.6 & 95.2 & 0.0936 & 0.0977 & 0.0553 & 0.0568 \\
& 0.5 & 100 & 99.9 & 99.9 & 94.3 & 93.8 & 0.2032 & 0.2291 & 0.093 & 0.0964
\\
& & 200 & 100 & 100 & 94.7 & 94 & 0.145 & 0.1561 & 0.0661 & 0.068 \\
& & 500 & 100 & 100 & 93.9 & 94 & 0.0935 & 0.0979 & 0.0425 & 0.0435 \\
& 0.9 & 100 & 100 & 100 & 94.6 & 93 & 0.2014 & 0.2255 & 0.0512 & 0.0518 \\
& & 200 & 100 & 100 & 95.7 & 95.1 & 0.1462 & 0.158 & 0.0371 & 0.0381 \\
& & 500 & 100 & 100 & 95 & 94.2 & 0.0935 & 0.0976 & 0.0238 & 0.0244 \\
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{adjustbox}
\medskip
\noindent
\footnotesize
Note -- The table provides coverage rates and widths of confidence intervals
of level $95\%$ for Gini indices differences by asymptotic intersection
method (IM-Asym), bootstrap intersection method (IM-Boot), asymptotic method
with overlapping samples (OS-Asym) and bootstrap method with overlapping
samples (OS-Boot). The data are generated according to DGP I-A for given
Gaussian copula ${\Greekmath 011A}$ in \eqref{eq:DGP-GaussCopula}, overlap portion $
{\Greekmath 0115}$ and sample size $n$. The numbers are based on $1000$ replications
with $399$ bootstrap repetitions each.
\normalsize
\end{table}
\begin{table}[tb]
\caption{DGP I-A: coverage and width of confidence intervals for the mean
difference.} \label{tab:CI-dmu-DGP1}
\begin{center}
Table \thetable
DGP I-A: coverage and width of confidence intervals for the mean difference.
\end{center}
\begin{adjustbox}{scale={0.9}{0.9}}
\hspace{-0.5\totalhormargin} \begin{minipage}{\paperwidth}
\centering
\small
\begin{center}
\begin{tabular}{ccc|cccc|cccc}
\hline
\rule[-1ex]{0cm}{4ex} ${\Greekmath 011A}$ & ${\Greekmath 0115}$ & $n$ & \multicolumn{4}{c|}{
Coverage $(\%)$} & \multicolumn{4}{c}{Width} \\ \cline{4-11}
\rule[-1ex]{0cm}{4ex} & & & IM-Asym & IM-Boot & OS-Asym & OS-Boot & IM-Asym
& IM-Boot & OS-Asym & OS-Boot \\ \hline
\rule[-1ex]{0cm}{4ex} -0.99 & 0.1 & 100 & 99.4 & 99.5 & 94 & 92.6 & 0.1674 &
0.1779 & 0.1086 & 0.1093 \\
& & 200 & 99.6 & 99.3 & 94.6 & 94.2 & 0.1192 & 0.1239 & 0.0771 & 0.0777 \\
& & 500 & 99.9 & 99.7 & 94.8 & 94 & 0.0754 & 0.0772 & 0.0486 & 0.0491 \\
& 0.5 & 100 & 99 & 99.1 & 95.5 & 94.8 & 0.1671 & 0.1772 & 0.1232 & 0.1246 \\
& & 200 & 99.4 & 99.3 & 93.9 & 93.4 & 0.119 & 0.1233 & 0.0875 & 0.0884 \\
& & 500 & 99.6 & 99 & 95.3 & 95.5 & 0.0756 & 0.0775 & 0.0554 & 0.056 \\
& 0.9 & 100 & 97.8 & 97.9 & 95.3 & 94.6 & 0.1669 & 0.1764 & 0.1361 & 0.1385
\\
& & 200 & 98.9 & 98.7 & 96 & 96.1 & 0.1185 & 0.1231 & 0.0964 & 0.0977 \\
& & 500 & 98.3 & 98.1 & 93.6 & 94.1 & 0.0754 & 0.0772 & 0.0612 & 0.0619 \\
\rule[-1ex]{0cm}{4ex} -0.5 & 0.1 & 100 & 99.8 & 99.8 & 95.8 & 95.8 & 0.1674
& 0.1768 & 0.1063 & 0.1075 \\
& & 200 & 99.8 & 99.7 & 93.9 & 94 & 0.1184 & 0.1227 & 0.0751 & 0.0757 \\
& & 500 & 99.7 & 99.6 & 95.2 & 94.7 & 0.0754 & 0.0772 & 0.0477 & 0.0482 \\
& 0.5 & 100 & 99.5 & 99.6 & 95.6 & 94.3 & 0.1679 & 0.1778 & 0.1151 & 0.1172
\\
& & 200 & 99.5 & 99.5 & 95.3 & 95.1 & 0.1184 & 0.1228 & 0.0809 & 0.0819 \\
& & 500 & 99.5 & 99.6 & 94.6 & 94.7 & 0.0755 & 0.0772 & 0.0515 & 0.0521 \\
& 0.9 & 100 & 99.1 & 98.9 & 94.4 & 93.1 & 0.1672 & 0.1772 & 0.1221 & 0.1245
\\
& & 200 & 98.8 & 99.1 & 94 & 93.8 & 0.1191 & 0.1236 & 0.0868 & 0.0881 \\
& & 500 & 99.2 & 99.1 & 95.4 & 95.7 & 0.0754 & 0.0773 & 0.0549 & 0.0555 \\
\rule[-1ex]{0cm}{4ex} 0 & 0.1 & 100 & 99.9 & 99.9 & 94.5 & 93.3 & 0.1671 &
0.177 & 0.1039 & 0.105 \\
& & 200 & 100 & 100 & 94.3 & 94.4 & 0.1188 & 0.1234 & 0.0737 & 0.0746 \\
& & 500 & 100 & 100 & 95.9 & 95.3 & 0.0754 & 0.0773 & 0.0468 & 0.0472 \\
& 0.5 & 100 & 99.7 & 99.7 & 95 & 93.6 & 0.1676 & 0.1776 & 0.104 & 0.1054 \\
& & 200 & 99.8 & 99.7 & 95.4 & 95.6 & 0.119 & 0.1237 & 0.074 & 0.075 \\
& & 500 & 100 & 100 & 95.5 & 94.7 & 0.0755 & 0.0774 & 0.0468 & 0.0472 \\
& 0.9 & 100 & 99.7 & 99.7 & 95.3 & 93.9 & 0.1667 & 0.1762 & 0.1032 & 0.1052
\\
& & 200 & 100 & 100 & 95.2 & 94.4 & 0.1189 & 0.1233 & 0.0738 & 0.0748 \\
& & 500 & 99.7 & 99.8 & 96.4 & 96.2 & 0.0757 & 0.0775 & 0.047 & 0.0474 \\
\rule[-1ex]{0cm}{4ex} 0.5 & 0.1 & 100 & 99.9 & 99.9 & 95.9 & 95.5 & 0.1672 &
0.1774 & 0.1016 & 0.1031 \\
& & 200 & 99.8 & 99.8 & 94.8 & 94.4 & 0.1185 & 0.1228 & 0.0718 & 0.0727 \\
& & 500 & 99.9 & 100 & 94.6 & 93.7 & 0.0756 & 0.0775 & 0.0458 & 0.0464 \\
& 0.5 & 100 & 100 & 100 & 95.4 & 94.3 & 0.1674 & 0.1778 & 0.0911 & 0.0925 \\
& & 200 & 100 & 100 & 96.4 & 95.7 & 0.1192 & 0.1238 & 0.0648 & 0.0658 \\
& & 500 & 100 & 100 & 94.2 & 94.9 & 0.0756 & 0.0776 & 0.041 & 0.0415 \\
& 0.9 & 100 & 100 & 100 & 95.9 & 94.7 & 0.167 & 0.1776 & 0.0787 & 0.08 \\
& & 200 & 100 & 100 & 94.9 & 94.6 & 0.1186 & 0.1233 & 0.0562 & 0.057 \\
& & 500 & 100 & 100 & 93.8 & 93.5 & 0.0754 & 0.0773 & 0.0356 & 0.036 \\
\rule[-1ex]{0cm}{4ex} 0.99 & 0.1 & 100 & 99.8 & 99.8 & 93.1 & 92.8 & 0.1674
& 0.1785 & 0.0991 & 0.1015 \\
& & 200 & 99.8 & 99.8 & 94 & 93.4 & 0.1185 & 0.1229 & 0.0699 & 0.0706 \\
& & 500 & 99.8 & 99.8 & 94.3 & 94.1 & 0.0756 & 0.0775 & 0.0445 & 0.045 \\
& 0.5 & 100 & 99.9 & 99.9 & 94.8 & 94.5 & 0.1674 & 0.1773 & 0.0746 & 0.075
\\
& & 200 & 100 & 100 & 94 & 92.7 & 0.1185 & 0.1231 & 0.0527 & 0.0533 \\
& & 500 & 100 & 100 & 95.2 & 94.9 & 0.0754 & 0.0774 & 0.0335 & 0.0339 \\
& 0.9 & 100 & 100 & 100 & 94.2 & 91.7 & 0.166 & 0.1757 & 0.0361 & 0.0349 \\
& & 200 & 100 & 100 & 95.7 & 94.4 & 0.119 & 0.1234 & 0.026 & 0.0259 \\
& & 500 & 100 & 100 & 95.1 & 94.4 & 0.0753 & 0.0771 & 0.0165 & 0.0167 \\
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{adjustbox}
\medskip
\noindent
\footnotesize
Note -- The table provides coverage rates and widths of confidence intervals
of level $95\%$ for the mean difference by asymptotic intersection method
(IM-Asym), bootstrap intersection method (IM-Boot), asymptotic method with
overlapping samples (OS-Asym) and bootstrap method with overlapping samples
(OS-Boot). The data are generated according to DGP I-A for given Gaussian
copula ${\Greekmath 011A}$ in \eqref{eq:DGP-GaussCopula}, overlap portion ${\Greekmath 0115}$ and
sample size $n$. The numbers are based on $1000$ replications with 399
bootstrap repetitions each.
\normalsize
\end{table}
\begin{table}[tb]
\caption{DGP I-A: coverage and width of confidence intervals for the LC
ordinate difference at $p = 0.5$.} \label{tab:CI-dLC-DGP1}
\begin{center}
Table \thetable
DGP I-A: coverage and width of confidence intervals for the LC ordinate
difference at $p = 0.5$.
\end{center}
\begin{adjustbox}{scale={0.9}{0.9}}
\hspace{-0.5\totalhormargin} \begin{minipage}{\paperwidth}
\centering
\small
\begin{center}
\begin{tabular}{ccc|cccc|cccc}
\hline
\rule[-1ex]{0cm}{4ex} ${\Greekmath 011A}$ & ${\Greekmath 0115}$ & $n$ & \multicolumn{4}{c|}{
Coverage $(\%)$} & \multicolumn{4}{c}{Width} \\ \cline{4-11}
\rule[-1ex]{0cm}{4ex} & & & IM-Asym & IM-Boot & OS-Asym & OS-Boot & IM-Asym
& IM-Boot & OS-Asym & OS-Boot \\ \hline
\rule[-1ex]{0cm}{4ex} -0.99 & 0.1 & 100 & 100 & 100 & 95.5 & 97.4 & 0.142 &
0.1704 & 0.0852 & 0.098 \\
& & 200 & 99.8 & 99.8 & 95.6 & 96.8 & 0.1005 & 0.1116 & 0.0601 & 0.065 \\
& & 500 & 99.9 & 99.9 & 95.4 & 95.7 & 0.0637 & 0.067 & 0.038 & 0.0395 \\
& 0.5 & 100 & 100 & 100 & 94.7 & 97.5 & 0.142 & 0.17 & 0.0705 & 0.0843 \\
& & 200 & 100 & 100 & 94.2 & 96.9 & 0.1007 & 0.1121 & 0.0498 & 0.0551 \\
& & 500 & 100 & 100 & 94.8 & 95.4 & 0.0637 & 0.0671 & 0.0314 & 0.033 \\
& 0.9 & 100 & 100 & 100 & 95.2 & 99.3 & 0.1418 & 0.1703 & 0.0513 & 0.0692 \\
& & 200 & 100 & 100 & 95 & 97.7 & 0.1003 & 0.1114 & 0.0364 & 0.043 \\
& & 500 & 100 & 100 & 94.9 & 95.8 & 0.0636 & 0.067 & 0.0231 & 0.0249 \\
\rule[-1ex]{0cm}{4ex} -0.5 & 0.1 & 100 & 99.8 & 100 & 96.7 & 98.3 & 0.1418 &
0.17 & 0.0874 & 0.0997 \\
& & 200 & 99.7 & 100 & 93.7 & 96.2 & 0.1003 & 0.1115 & 0.0618 & 0.0666 \\
& & 500 & 99.9 & 99.9 & 94.5 & 95.4 & 0.0636 & 0.067 & 0.0391 & 0.0404 \\
& 0.5 & 100 & 100 & 100 & 95.6 & 97.9 & 0.142 & 0.1702 & 0.0845 & 0.0975 \\
& & 200 & 99.9 & 99.9 & 94.1 & 95.6 & 0.1004 & 0.1116 & 0.0597 & 0.0645 \\
& & 500 & 99.9 & 99.8 & 95.8 & 95.8 & 0.0636 & 0.0671 & 0.0377 & 0.0392 \\
& 0.9 & 100 & 100 & 100 & 95.1 & 97.6 & 0.142 & 0.1707 & 0.0815 & 0.0948 \\
& & 200 & 99.9 & 100 & 94.1 & 96.5 & 0.1006 & 0.1119 & 0.0576 & 0.0626 \\
& & 500 & 100 & 100 & 96.1 & 96.3 & 0.0637 & 0.0672 & 0.0364 & 0.0377 \\
\rule[-1ex]{0cm}{4ex} 0 & 0.1 & 100 & 99.8 & 100 & 95.6 & 97.5 & 0.1415 &
0.1696 & 0.0878 & 0.1008 \\
& & 200 & 99.9 & 100 & 95.1 & 96.8 & 0.1004 & 0.1114 & 0.0623 & 0.0671 \\
& & 500 & 99.7 & 99.8 & 95 & 95 & 0.0637 & 0.0674 & 0.0395 & 0.0409 \\
& 0.5 & 100 & 100 & 100 & 94.3 & 97.1 & 0.1422 & 0.171 & 0.0883 & 0.1015 \\
& & 200 & 100 & 100 & 95.8 & 97.7 & 0.1007 & 0.1121 & 0.0625 & 0.0676 \\
& & 500 & 99.9 & 100 & 95.5 & 95.6 & 0.0637 & 0.0673 & 0.0395 & 0.041 \\
& 0.9 & 100 & 99.5 & 100 & 94.2 & 97.4 & 0.1411 & 0.1692 & 0.0876 & 0.1005
\\
& & 200 & 99.8 & 99.9 & 95.3 & 96.8 & 0.1005 & 0.1115 & 0.0623 & 0.0672 \\
& & 500 & 99.8 & 99.9 & 94.7 & 95 & 0.0638 & 0.0674 & 0.0396 & 0.0411 \\
\rule[-1ex]{0cm}{4ex} 0.5 & 0.1 & 100 & 99.7 & 100 & 94.8 & 97.2 & 0.1423 &
0.171 & 0.0875 & 0.1005 \\
& & 200 & 99.8 & 99.9 & 94.7 & 95.7 & 0.1002 & 0.1111 & 0.0616 & 0.0662 \\
& & 500 & 100 & 100 & 95.3 & 96.2 & 0.0638 & 0.0672 & 0.0391 & 0.0405 \\
& 0.5 & 100 & 100 & 100 & 95.1 & 97.6 & 0.1416 & 0.1704 & 0.0832 & 0.0962 \\
& & 200 & 99.9 & 100 & 95.7 & 96.6 & 0.1007 & 0.1117 & 0.059 & 0.0641 \\
& & 500 & 99.9 & 99.9 & 94.9 & 95.9 & 0.0638 & 0.0672 & 0.0373 & 0.0388 \\
& 0.9 & 100 & 100 & 100 & 94.8 & 97.4 & 0.1421 & 0.1709 & 0.0791 & 0.0925 \\
& & 200 & 99.9 & 100 & 95.2 & 96.6 & 0.1007 & 0.1121 & 0.056 & 0.0613 \\
& & 500 & 100 & 100 & 94.9 & 95 & 0.0637 & 0.0672 & 0.0355 & 0.037 \\
\rule[-1ex]{0cm}{4ex} 0.99 & 0.1 & 100 & 99.6 & 100 & 93.6 & 96.6 & 0.1417 &
0.1702 & 0.084 & 0.097 \\
& & 200 & 99.8 & 99.9 & 96 & 96.3 & 0.1005 & 0.1116 & 0.0594 & 0.0644 \\
& & 500 & 100 & 100 & 95.8 & 96.5 & 0.0637 & 0.0671 & 0.0376 & 0.039 \\
& 0.5 & 100 & 100 & 100 & 94.7 & 97.7 & 0.1419 & 0.1706 & 0.065 & 0.0781 \\
& & 200 & 100 & 100 & 95.6 & 96.8 & 0.1001 & 0.1115 & 0.0456 & 0.0508 \\
& & 500 & 100 & 100 & 94.2 & 94.7 & 0.0636 & 0.0672 & 0.0289 & 0.0305 \\
& 0.9 & 100 & 100 & 100 & 95.8 & 99.3 & 0.1416 & 0.1698 & 0.0362 & 0.0541 \\
& & 200 & 100 & 100 & 94.8 & 98.5 & 0.1004 & 0.1114 & 0.0255 & 0.0326 \\
& & 500 & 100 & 100 & 94.7 & 97.2 & 0.0637 & 0.067 & 0.0162 & 0.0182 \\
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{adjustbox}
\medskip
\noindent
\footnotesize
Note -- The table provides coverage rates and widths of confidence intervals
of level $95\%$ for the Lorenz curve (LC) ordinate difference at percentile $
p = 0.5$ by asymptotic intersection method (IM-Asym), bootstrap intersection
method (IM-Boot), asymptotic method with overlapping samples (OS-Asym) and
bootstrap method with overlapping samples (OS-Boot). The data are generated
according to DGP I-A for given Gaussian copula ${\Greekmath 011A}$ in
\eqref{eq:DGP-GaussCopula}, overlap portion ${\Greekmath 0115}$ and sample size $n$.
The numbers are based on $1000$ replications with 399 bootstrap repetitions
each.
\normalsize
\end{table}
\begin{table}[tb]
\caption{DGP I-B: coverage and width of confidence intervals for Gini indices
differences.} \label{tab:CI-dI-DGP2}
\begin{center}
Table \thetable
DGP I-B: coverage and width of confidence intervals for Gini indices
differences.
\end{center}
\begin{adjustbox}{scale={0.9}{0.9}}
\hspace{-0.5\totalhormargin} \begin{minipage}{\paperwidth}
\centering
\small
\begin{center}
\begin{tabular}{ccc|cccc|cccc}
\hline
\rule[-1ex]{0cm}{4ex} ${\Greekmath 011A}$ & ${\Greekmath 0115}$ & $n$ & \multicolumn{4}{c|}{
Coverage $(\%)$} & \multicolumn{4}{c}{Width} \\ \cline{4-11}
\rule[-1ex]{0cm}{4ex} & & & IM-Asym & IM-Boot & OS-Asym & OS-Boot & IM-Asym
& IM-Boot & OS-Asym & OS-Boot \\ \hline
\rule[-1ex]{0cm}{4ex} -0.99 & 0.1 & 100 & 99.6 & 99.6 & 93.8 & 94.5 & 0.2499
& 0.3099 & 0.156 & 0.18 \\
& & 200 & 99.2 & 99.5 & 92 & 92.9 & 0.1852 & 0.2156 & 0.1156 & 0.1281 \\
& & 500 & 99.8 & 99.7 & 94.4 & 94.9 & 0.1226 & 0.1364 & 0.0768 & 0.0831 \\
& 0.5 & 100 & 99.8 & 100 & 91.9 & 92.2 & 0.2508 & 0.3122 & 0.1407 & 0.1649
\\
& & 200 & 99.6 & 99.8 & 92.4 & 92.9 & 0.1873 & 0.2222 & 0.106 & 0.122 \\
& & 500 & 99.8 & 99.9 & 93.5 & 93.5 & 0.122 & 0.1362 & 0.0694 & 0.0768 \\
& 0.9 & 100 & 100 & 100 & 90.2 & 91 & 0.25 & 0.31 & 0.1217 & 0.1468 \\
& & 200 & 99.8 & 99.9 & 93.3 & 93.7 & 0.1845 & 0.2165 & 0.0915 & 0.108 \\
& & 500 & 99.9 & 100 & 93.9 & 93.9 & 0.1219 & 0.1383 & 0.0619 & 0.0727 \\
\rule[-1ex]{0cm}{4ex} -0.5 & 0.1 & 100 & 99.1 & 99.4 & 92.8 & 93.5 & 0.2498
& 0.3067 & 0.1579 & 0.1809 \\
& & 200 & 99.6 & 99.8 & 93.8 & 94.4 & 0.1864 & 0.2221 & 0.1185 & 0.135 \\
& & 500 & 99.6 & 99.8 & 94.8 & 94.9 & 0.1224 & 0.1379 & 0.078 & 0.0853 \\
& 0.5 & 100 & 99.6 & 99.8 & 92.7 & 93.3 & 0.2498 & 0.3092 & 0.154 & 0.1769
\\
& & 200 & 99.1 & 99.3 & 93 & 92.8 & 0.1842 & 0.2167 & 0.1137 & 0.1276 \\
& & 500 & 99.7 & 99.8 & 94.1 & 94.5 & 0.1216 & 0.1345 & 0.0752 & 0.081 \\
& 0.9 & 100 & 99.9 & 100 & 91.4 & 92.8 & 0.2489 & 0.3056 & 0.1485 & 0.1709
\\
& & 200 & 99.9 & 100 & 94 & 93.8 & 0.1842 & 0.2162 & 0.1104 & 0.1238 \\
& & 500 & 99.8 & 99.8 & 93.9 & 93.8 & 0.1226 & 0.1374 & 0.0739 & 0.0807 \\
\rule[-1ex]{0cm}{4ex} 0 & 0.1 & 100 & 99.3 & 99.7 & 92.8 & 93.8 & 0.2474 &
0.3059 & 0.1569 & 0.1818 \\
& & 200 & 99.5 & 99.6 & 92.4 & 92.5 & 0.1846 & 0.218 & 0.1177 & 0.1324 \\
& & 500 & 99.6 & 99.9 & 94.2 & 94.1 & 0.123 & 0.1385 & 0.0788 & 0.0864 \\
& 0.5 & 100 & 99.6 & 99.8 & 92.7 & 94.5 & 0.2518 & 0.3136 & 0.1601 & 0.1866
\\
& & 200 & 99.7 & 99.7 & 93 & 93.6 & 0.1865 & 0.2188 & 0.1191 & 0.1337 \\
& & 500 & 99.5 & 99.6 & 93 & 92.5 & 0.1224 & 0.1365 & 0.0784 & 0.0852 \\
& 0.9 & 100 & 99.4 & 99.8 & 92.2 & 94 & 0.2527 & 0.319 & 0.1605 & 0.1913 \\
& & 200 & 99.4 & 99.5 & 92.8 & 93.3 & 0.1855 & 0.2174 & 0.1183 & 0.1329 \\
& & 500 & 99.7 & 99.8 & 93.6 & 93.1 & 0.1229 & 0.1382 & 0.0787 & 0.0862 \\
\rule[-1ex]{0cm}{4ex} 0.5 & 0.1 & 100 & 99.3 & 99.6 & 93.1 & 94.1 & 0.2484 &
0.3042 & 0.156 & 0.1791 \\
& & 200 & 99.6 & 99.8 & 93.8 & 94.1 & 0.1854 & 0.2171 & 0.117 & 0.1313 \\
& & 500 & 99.7 & 99.8 & 93.9 & 93.6 & 0.1223 & 0.1359 & 0.0775 & 0.084 \\
& 0.5 & 100 & 99.4 & 99.6 & 91 & 93 & 0.249 & 0.3067 & 0.1501 & 0.1753 \\
& & 200 & 99.6 & 99.8 & 92.8 & 93.2 & 0.1885 & 0.226 & 0.1152 & 0.1335 \\
& & 500 & 99.8 & 99.8 & 92.9 & 92.5 & 0.1227 & 0.1371 & 0.075 & 0.0819 \\
& 0.9 & 100 & 99.5 & 99.5 & 90 & 91.5 & 0.248 & 0.3049 & 0.1424 & 0.1689 \\
& & 200 & 99.6 & 99.7 & 91.2 & 92.6 & 0.1849 & 0.2161 & 0.1076 & 0.1225 \\
& & 500 & 99.7 & 99.7 & 94.1 & 93.7 & 0.1215 & 0.1345 & 0.0711 & 0.0776 \\
\rule[-1ex]{0cm}{4ex} 0.99 & 0.1 & 100 & 99.4 & 99.8 & 92.1 & 92.9 & 0.2506
& 0.3102 & 0.1523 & 0.1767 \\
& & 200 & 99.7 & 99.7 & 93.7 & 94.1 & 0.1877 & 0.2207 & 0.115 & 0.131 \\
& & 500 & 99.4 & 99.5 & 92.5 & 92.3 & 0.1217 & 0.1346 & 0.0745 & 0.0806 \\
& 0.5 & 100 & 99.9 & 99.9 & 90.6 & 91.4 & 0.2501 & 0.3102 & 0.1213 & 0.146
\\
& & 200 & 100 & 100 & 91.5 & 92.8 & 0.1835 & 0.2124 & 0.0899 & 0.1029 \\
& & 500 & 100 & 100 & 93.8 & 93.9 & 0.1234 & 0.1387 & 0.0621 & 0.0709 \\
& 0.9 & 100 & 100 & 100 & 85.2 & 87.9 & 0.2491 & 0.3052 & 0.0785 & 0.11 \\
& & 200 & 100 & 100 & 88.7 & 90.1 & 0.1865 & 0.2253 & 0.0633 & 0.0944 \\
& & 500 & 100 & 100 & 90.2 & 92.7 & 0.1213 & 0.1356 & 0.043 & 0.0557 \\
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{adjustbox}
\medskip
\noindent
\footnotesize
Note -- The table provides coverage rates and widths of confidence intervals
of level $95\%$ for Gini indices differences by asymptotic intersection
method (IM-Asym), bootstrap intersection method (IM-Boot), asymptotic method
with overlapping samples (OS-Asym) and bootstrap method with overlapping
samples (OS-Boot). The data are generated according to DGP I-B for given
Gaussian copula ${\Greekmath 011A}$ in \eqref{eq:DGP-GaussCopula}, overlap portion $
{\Greekmath 0115}$ and sample size $n$. The numbers are based on $1000$ replications
with 399 bootstrap repetitions each.
\normalsize
\end{table}
\begin{table}[tb]
\caption{DGP I-B: coverage and width of confidence intervals for the mean
difference.} \label{tab:CI-dmu-DGP2}
\begin{center}
Table \thetable
DGP I-B: coverage and width of confidence intervals for the mean difference.
\end{center}
\begin{adjustbox}{scale={0.9}{0.9}}
\hspace{-0.5\totalhormargin} \begin{minipage}{\paperwidth}
\centering
\small
\begin{center}
\begin{tabular}{ccc|cccc|cccc}
\hline
\rule[-1ex]{0cm}{4ex} ${\Greekmath 011A}$ & ${\Greekmath 0115}$ & $n$ & \multicolumn{4}{c|}{
Coverage $(\%)$} & \multicolumn{4}{c}{Width} \\ \cline{4-11}
\rule[-1ex]{0cm}{4ex} & & & IM-Asym & IM-Boot & OS-Asym & OS-Boot & IM-Asym
& IM-Boot & OS-Asym & OS-Boot \\ \hline
\rule[-1ex]{0cm}{4ex} -0.99 & 0.1 & 100 & 99.2 & 99.6 & 95.1 & 93.6 & 0.2468
& 0.2858 & 0.1684 & 0.1839 \\
& & 200 & 99.1 & 99.5 & 92.7 & 92 & 0.1752 & 0.1909 & 0.119 & 0.1249 \\
& & 500 & 99.4 & 99.9 & 93.7 & 92.6 & 0.1127 & 0.1192 & 0.0765 & 0.0792 \\
& 0.5 & 100 & 97.7 & 98.5 & 94 & 93.7 & 0.2451 & 0.2814 & 0.1831 & 0.1943 \\
& & 200 & 99 & 99.2 & 96 & 94.9 & 0.1776 & 0.1963 & 0.1321 & 0.1398 \\
& & 500 & 99.3 & 99.3 & 94.8 & 94.3 & 0.1125 & 0.1191 & 0.0833 & 0.0859 \\
& 0.9 & 100 & 97.9 & 98 & 95.1 & 94.2 & 0.2448 & 0.2829 & 0.1967 & 0.2103 \\
& & 200 & 97.8 & 98.5 & 93.8 & 94.2 & 0.1758 & 0.1975 & 0.1406 & 0.1498 \\
& & 500 & 98 & 98.3 & 94.5 & 94.1 & 0.1128 & 0.1223 & 0.0899 & 0.0941 \\
\rule[-1ex]{0cm}{4ex} -0.5 & 0.1 & 100 & 99.1 & 99.3 & 93.6 & 93.6 & 0.2423
& 0.2753 & 0.1621 & 0.1742 \\
& & 200 & 99.6 & 99.5 & 94.8 & 94.4 & 0.1765 & 0.1945 & 0.1185 & 0.1266 \\
& & 500 & 99.7 & 99.7 & 93.8 & 93.4 & 0.1128 & 0.1204 & 0.0756 & 0.0796 \\
& 0.5 & 100 & 98.9 & 99 & 94.2 & 93.8 & 0.2443 & 0.2784 & 0.173 & 0.1865 \\
& & 200 & 99.3 & 99.5 & 93.3 & 93.5 & 0.175 & 0.1927 & 0.1236 & 0.1315 \\
& & 500 & 99.2 & 99.2 & 94.4 & 94 & 0.1121 & 0.1183 & 0.079 & 0.0815 \\
& 0.9 & 100 & 98.8 & 98.9 & 93.8 & 93 & 0.2446 & 0.2797 & 0.1815 & 0.1959 \\
& & 200 & 98.5 & 99 & 93.6 & 93 & 0.175 & 0.1914 & 0.1295 & 0.136 \\
& & 500 & 99 & 99.1 & 93.9 & 93.5 & 0.1129 & 0.12 & 0.0834 & 0.0868 \\
\rule[-1ex]{0cm}{4ex} 0 & 0.1 & 100 & 98.8 & 99 & 92.7 & 91.7 & 0.2443 &
0.2841 & 0.1614 & 0.1793 \\
& & 200 & 99.3 & 99.5 & 91.5 & 91 & 0.1753 & 0.1933 & 0.1157 & 0.1239 \\
& & 500 & 99.7 & 99.6 & 94.1 & 93.5 & 0.1133 & 0.121 & 0.0749 & 0.0787 \\
& 0.5 & 100 & 99.5 & 99.7 & 93.7 & 93 & 0.2462 & 0.2842 & 0.1627 & 0.1813 \\
& & 200 & 99.4 & 99.3 & 93.8 & 93.2 & 0.1756 & 0.1921 & 0.1159 & 0.1234 \\
& & 500 & 99.5 & 99.6 & 94.5 & 94.1 & 0.1127 & 0.1196 & 0.0744 & 0.0779 \\
& 0.9 & 100 & 99.7 & 99.7 & 93.8 & 93.5 & 0.2501 & 0.2987 & 0.1657 & 0.1904
\\
& & 200 & 99.1 & 99.1 & 93.8 & 93.6 & 0.1753 & 0.1917 & 0.1156 & 0.1237 \\
& & 500 & 99.4 & 99.7 & 94.5 & 93.9 & 0.1132 & 0.1204 & 0.0748 & 0.0785 \\
\rule[-1ex]{0cm}{4ex} 0.5 & 0.1 & 100 & 99.7 & 99.8 & 93.2 & 92.8 & 0.2413 &
0.2727 & 0.1558 & 0.1683 \\
& & 200 & 99.6 & 99.8 & 95.5 & 95.2 & 0.1754 & 0.1914 & 0.1136 & 0.1212 \\
& & 500 & 99.5 & 99.7 & 93.1 & 93 & 0.1128 & 0.1197 & 0.0732 & 0.0765 \\
& 0.5 & 100 & 99.6 & 99.6 & 93.1 & 92.3 & 0.2423 & 0.2749 & 0.1451 & 0.1599
\\
& & 200 & 99.9 & 99.9 & 94.8 & 95.5 & 0.1793 & 0.2015 & 0.1088 & 0.1215 \\
& & 500 & 99.9 & 99.7 & 94.5 & 93.9 & 0.1132 & 0.1198 & 0.0685 & 0.072 \\
& 0.9 & 100 & 99.7 & 99.8 & 91.7 & 91.9 & 0.2424 & 0.278 & 0.1326 & 0.1513
\\
& & 200 & 99.9 & 99.9 & 92.6 & 92.1 & 0.1755 & 0.1923 & 0.0967 & 0.1062 \\
& & 500 & 99.7 & 99.9 & 94.1 & 94.3 & 0.112 & 0.1181 & 0.062 & 0.0655 \\
\rule[-1ex]{0cm}{4ex} 0.99 & 0.1 & 100 & 99.4 & 99.8 & 92.9 & 92.7 & 0.2428
& 0.2762 & 0.1539 & 0.1682 \\
& & 200 & 99.7 & 99.8 & 93.6 & 92.8 & 0.1773 & 0.1946 & 0.113 & 0.1213 \\
& & 500 & 99.8 & 99.9 & 95.2 & 95.2 & 0.1122 & 0.1188 & 0.0713 & 0.0746 \\
& 0.5 & 100 & 100 & 100 & 92.4 & 92.5 & 0.2441 & 0.2782 & 0.1277 & 0.1466 \\
& & 200 & 99.9 & 100 & 92.6 & 91.7 & 0.1737 & 0.1881 & 0.091 & 0.0989 \\
& & 500 & 100 & 100 & 94.8 & 93.6 & 0.1136 & 0.1211 & 0.0606 & 0.0659 \\
& 0.9 & 100 & 100 & 100 & 89.5 & 90.6 & 0.243 & 0.273 & 0.0918 & 0.1181 \\
& & 200 & 100 & 100 & 89.7 & 91.4 & 0.179 & 0.2077 & 0.0712 & 0.1054 \\
& & 500 & 100 & 100 & 93 & 94.3 & 0.1121 & 0.1194 & 0.0445 & 0.0519 \\
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{adjustbox}
\medskip
\noindent
\footnotesize
Note -- The table provides coverage rates and widths of confidence intervals
of level $95\%$ for the mean difference by asymptotic intersection method
(IM-Asym), bootstrap intersection method (IM-Boot), asymptotic method with
overlapping samples (OS-Asym) and bootstrap method with overlapping samples
(OS-Boot). The data are generated according to DGP I-B for given Gaussian
copula ${\Greekmath 011A}$ in \eqref{eq:DGP-GaussCopula}, overlap portion ${\Greekmath 0115}$ and
sample size $n$. The numbers are based on $1000$ replications with 399
bootstrap repetitions each.
\normalsize
\end{table}
\begin{table}[tb]
\caption{DGP I-B: coverage and width of confidence intervals for the LC
ordinate difference at $p = 0.5$.} \label{tab:CI-dLC-DGP2}
\begin{center}
Table \thetable
DGP I-B: coverage and width of confidence intervals for the LC ordinate
difference at $p = 0.5$.
\end{center}
\begin{adjustbox}{scale={0.9}{0.9}}
\hspace{-0.5\totalhormargin} \begin{minipage}{\paperwidth}
\centering
\small
\begin{center}
\begin{tabular}{ccc|cccc|cccc}
\hline
\rule[-1ex]{0cm}{4ex} ${\Greekmath 011A}$ & ${\Greekmath 0115}$ & $n$ & \multicolumn{4}{c|}{
Coverage $(\%)$} & \multicolumn{4}{c}{Width} \\ \cline{4-11}
\rule[-1ex]{0cm}{4ex} & & & IM-Asym & IM-Boot & OS-Asym & OS-Boot & IM-Asym
& IM-Boot & OS-Asym & OS-Boot \\ \hline
\rule[-1ex]{0cm}{4ex} -0.99 & 0.1 & 100 & 99.8 & 99.8 & 95.4 & 96.6 & 0.1614
& 0.1919 & 0.0973 & 0.1114 \\
& & 200 & 99.8 & 100 & 94.9 & 95.5 & 0.1157 & 0.129 & 0.0696 & 0.0756 \\
& & 500 & 99.9 & 99.9 & 94.7 & 95.5 & 0.0744 & 0.0796 & 0.0449 & 0.0471 \\
& 0.5 & 100 & 100 & 100 & 93 & 96.3 & 0.1615 & 0.1924 & 0.0822 & 0.0979 \\
& & 200 & 100 & 100 & 94.1 & 95.8 & 0.1164 & 0.1305 & 0.0595 & 0.0664 \\
& & 500 & 99.9 & 100 & 95 & 95.4 & 0.0742 & 0.0791 & 0.038 & 0.0405 \\
& 0.9 & 100 & 100 & 100 & 93.8 & 97.4 & 0.1616 & 0.1928 & 0.0631 & 0.0822 \\
& & 200 & 100 & 100 & 94.8 & 97.6 & 0.1156 & 0.1291 & 0.0459 & 0.0543 \\
& & 500 & 100 & 100 & 95.5 & 95.5 & 0.0742 & 0.079 & 0.0301 & 0.0339 \\
\rule[-1ex]{0cm}{4ex} -0.5 & 0.1 & 100 & 99.8 & 99.9 & 94.9 & 96.7 & 0.1615
& 0.1922 & 0.0999 & 0.1134 \\
& & 200 & 99.8 & 100 & 96.2 & 97.2 & 0.1162 & 0.1303 & 0.072 & 0.0782 \\
& & 500 & 100 & 100 & 95.8 & 96.2 & 0.0743 & 0.0792 & 0.046 & 0.0484 \\
& 0.5 & 100 & 99.8 & 100 & 94.3 & 96.6 & 0.1614 & 0.192 & 0.0966 & 0.1103 \\
& & 200 & 99.7 & 99.9 & 93.2 & 94.7 & 0.1156 & 0.1289 & 0.0692 & 0.0754 \\
& & 500 & 99.8 & 99.9 & 95.5 & 95.7 & 0.0741 & 0.0788 & 0.0443 & 0.0464 \\
& 0.9 & 100 & 100 & 100 & 94.5 & 97.2 & 0.1614 & 0.192 & 0.0931 & 0.1074 \\
& & 200 & 99.9 & 100 & 94.9 & 96 & 0.1156 & 0.1291 & 0.0667 & 0.0727 \\
& & 500 & 99.9 & 99.9 & 95.1 & 95 & 0.0745 & 0.0794 & 0.043 & 0.0453 \\
\rule[-1ex]{0cm}{4ex} 0 & 0.1 & 100 & 99.6 & 100 & 94.1 & 95.9 & 0.1607 &
0.1913 & 0.1001 & 0.1142 \\
& & 200 & 99.9 & 99.9 & 94.7 & 95.8 & 0.1156 & 0.1289 & 0.0721 & 0.0784 \\
& & 500 & 100 & 100 & 95.3 & 95.5 & 0.0746 & 0.0797 & 0.0465 & 0.0488 \\
& 0.5 & 100 & 99.9 & 100 & 93.9 & 96.1 & 0.1622 & 0.1933 & 0.1011 & 0.1156
\\
& & 200 & 99.8 & 99.9 & 95 & 96.4 & 0.1163 & 0.13 & 0.0726 & 0.0787 \\
& & 500 & 99.9 & 99.8 & 93.2 & 94 & 0.0744 & 0.0793 & 0.0464 & 0.0486 \\
& 0.9 & 100 & 99.8 & 100 & 95 & 96.9 & 0.1626 & 0.1934 & 0.1013 & 0.1165 \\
& & 200 & 99.4 & 99.5 & 93.7 & 95.1 & 0.1158 & 0.1292 & 0.0723 & 0.0786 \\
& & 500 & 99.8 & 99.9 & 94.2 & 94.9 & 0.0746 & 0.08 & 0.0465 & 0.0488 \\
\rule[-1ex]{0cm}{4ex} 0.5 & 0.1 & 100 & 99.6 & 99.7 & 95.3 & 97 & 0.1608 &
0.1913 & 0.0993 & 0.1128 \\
& & 200 & 100 & 100 & 95.4 & 96.5 & 0.1158 & 0.129 & 0.0714 & 0.0776 \\
& & 500 & 99.9 & 99.9 & 95.4 & 95.2 & 0.0743 & 0.0793 & 0.0459 & 0.0479 \\
& 0.5 & 100 & 99.8 & 100 & 93.7 & 96.1 & 0.1611 & 0.192 & 0.0951 & 0.109 \\
& & 200 & 99.9 & 100 & 94.2 & 95.3 & 0.1169 & 0.1313 & 0.0692 & 0.0761 \\
& & 500 & 99.9 & 100 & 94.1 & 94.5 & 0.0745 & 0.0796 & 0.0441 & 0.0463 \\
& 0.9 & 100 & 99.7 & 100 & 93.5 & 96.1 & 0.161 & 0.1914 & 0.0905 & 0.1054 \\
& & 200 & 100 & 100 & 93.5 & 95 & 0.1156 & 0.1294 & 0.0652 & 0.0715 \\
& & 500 & 99.9 & 99.9 & 95.4 & 95.2 & 0.0741 & 0.079 & 0.0418 & 0.0439 \\
\rule[-1ex]{0cm}{4ex} 0.99 & 0.1 & 100 & 99.7 & 99.9 & 94.6 & 96.1 & 0.1619
& 0.1929 & 0.0963 & 0.1102 \\
& & 200 & 99.9 & 99.9 & 95.1 & 95.8 & 0.1165 & 0.1311 & 0.0694 & 0.0757 \\
& & 500 & 99.9 & 99.9 & 94 & 94.1 & 0.0741 & 0.079 & 0.0441 & 0.0462 \\
& 0.5 & 100 & 100 & 100 & 94.3 & 96.9 & 0.1619 & 0.193 & 0.0753 & 0.09 \\
& & 200 & 100 & 100 & 93.5 & 95.1 & 0.1154 & 0.1287 & 0.0538 & 0.0599 \\
& & 500 & 100 & 100 & 95.3 & 94.1 & 0.0747 & 0.0799 & 0.0352 & 0.0377 \\
& 0.9 & 100 & 100 & 100 & 90.6 & 97.6 & 0.1612 & 0.192 & 0.0447 & 0.0639 \\
& & 200 & 100 & 100 & 92.9 & 96.1 & 0.116 & 0.1294 & 0.0336 & 0.0445 \\
& & 500 & 100 & 100 & 93.8 & 96.3 & 0.0741 & 0.0787 & 0.022 & 0.0257 \\
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{adjustbox}
\medskip
\noindent
\footnotesize
Note -- The table provides coverage rates and widths of confidence intervals
of level $95\%$ for the Lorenz curve (LC) ordinate difference at percentile $
p = 0.5$ by asymptotic intersection method (IM-Asym), bootstrap intersection
method (IM-Boot), asymptotic method with overlapping samples (OS-Asym) and
bootstrap method with overlapping samples (OS-Boot). The data are generated
according to DGP I-B for given Gaussian copula ${\Greekmath 011A}$ in
\eqref{eq:DGP-GaussCopula}, overlap portion ${\Greekmath 0115}$ and sample size $n$.
The numbers are based on $1000$ replications with 399 bootstrap repetitions
each.
\normalsize
\end{table}
\begin{table}[tb]
\caption{DGP I-C: coverage and width of confidence intervals for Gini indices
differences.} \label{tab:CI-dI-DGP3}
\begin{center}
Table \thetable
DGP I-C: coverage and width of confidence intervals for Gini indices
differences.
\end{center}
\begin{adjustbox}{scale={0.9}{0.9}}
\hspace{-0.5\totalhormargin} \begin{minipage}{\paperwidth}
\centering
\small
\begin{center}
\begin{tabular}{ccc|cccc|cccc}
\hline
\rule[-1ex]{0cm}{4ex} ${\Greekmath 011A}$ & ${\Greekmath 0115}$ & $n$ & \multicolumn{4}{c|}{
Coverage $(\%)$} & \multicolumn{4}{c}{Width} \\ \cline{4-11}
\rule[-1ex]{0cm}{4ex} & & & IM-Asym & IM-Boot & OS-Asym & OS-Boot & IM-Asym
& IM-Boot & OS-Asym & OS-Boot \\ \hline
\rule[-1ex]{0cm}{4ex} -0.99 & 0.1 & 100 & 98.1 & 98.8 & 86.1 & 90.4 & 0.2961
& 0.4293 & 0.1923 & 0.2594 \\
& & 200 & 97.8 & 98.3 & 88.5 & 91.6 & 0.2316 & 0.3181 & 0.1526 & 0.2025 \\
& & 500 & 98.3 & 98.9 & 91 & 93.4 & 0.1615 & 0.2072 & 0.1081 & 0.1367 \\
& 0.5 & 100 & 98.5 & 99 & 83.7 & 86.9 & 0.2925 & 0.4187 & 0.1776 & 0.2409 \\
& & 200 & 99 & 99.4 & 88.5 & 90.9 & 0.2317 & 0.3169 & 0.1443 & 0.1946 \\
& & 500 & 98 & 99.1 & 87.4 & 91 & 0.1606 & 0.2101 & 0.1022 & 0.135 \\
& 0.9 & 100 & 99.2 & 99.4 & 83.2 & 87.1 & 0.2912 & 0.405 & 0.1629 & 0.2228
\\
& & 200 & 98.5 & 99.5 & 85.9 & 88.3 & 0.2302 & 0.3205 & 0.1339 & 0.1934 \\
& & 500 & 98.8 & 99.7 & 88.2 & 90.8 & 0.1601 & 0.2034 & 0.0966 & 0.1277 \\
\rule[-1ex]{0cm}{4ex} -0.5 & 0.1 & 100 & 96.5 & 98.7 & 85.2 & 88.1 & 0.2915
& 0.4101 & 0.1896 & 0.2503 \\
& & 200 & 98.3 & 98.9 & 86.1 & 89.7 & 0.2321 & 0.3254 & 0.1541 & 0.2079 \\
& & 500 & 97.9 & 98.9 & 88.9 & 90.8 & 0.1633 & 0.2155 & 0.1104 & 0.1427 \\
& 0.5 & 100 & 97.7 & 98.3 & 85.8 & 88.8 & 0.2927 & 0.409 & 0.1867 & 0.2437
\\
& & 200 & 98.1 & 99.1 & 86.9 & 89.8 & 0.2285 & 0.3091 & 0.1483 & 0.1935 \\
& & 500 & 98.2 & 98.8 & 88.7 & 90.7 & 0.1625 & 0.2107 & 0.1079 & 0.1385 \\
& 0.9 & 100 & 97.7 & 98.8 & 87.1 & 89.6 & 0.2972 & 0.429 & 0.1862 & 0.2523
\\
& & 200 & 97.8 & 98.5 & 88.1 & 91.1 & 0.2322 & 0.3184 & 0.1483 & 0.1967 \\
& & 500 & 98.6 & 99.4 & 88.4 & 90.2 & 0.1631 & 0.2106 & 0.1065 & 0.1361 \\
\rule[-1ex]{0cm}{4ex} 0 & 0.1 & 100 & 96.9 & 98.1 & 84.8 & 89.7 & 0.2954 &
0.426 & 0.1928 & 0.26 \\
& & 200 & 97.3 & 98.4 & 85.9 & 89.1 & 0.2261 & 0.3036 & 0.1497 & 0.1938 \\
& & 500 & 97.7 & 98.6 & 87.6 & 91 & 0.1636 & 0.2151 & 0.111 & 0.144 \\
& 0.5 & 100 & 96.7 & 98.4 & 84.8 & 89.9 & 0.2935 & 0.4203 & 0.1914 & 0.259
\\
& & 200 & 97.7 & 98.7 & 87.1 & 91.2 & 0.2321 & 0.3204 & 0.1543 & 0.2058 \\
& & 500 & 98.3 & 99.4 & 90.8 & 93 & 0.1613 & 0.2099 & 0.1091 & 0.1397 \\
& 0.9 & 100 & 97 & 98.3 & 85.1 & 89.1 & 0.2932 & 0.4255 & 0.1913 & 0.2624 \\
& & 200 & 97.9 & 98.7 & 89.1 & 91.6 & 0.2292 & 0.3099 & 0.152 & 0.1983 \\
& & 500 & 97.9 & 98.7 & 88.8 & 91 & 0.1607 & 0.2058 & 0.1085 & 0.1362 \\
\rule[-1ex]{0cm}{4ex} 0.5 & 0.1 & 100 & 97.1 & 98.5 & 86.8 & 90.6 & 0.2953 &
0.4228 & 0.1912 & 0.2579 \\
& & 200 & 97.2 & 98.3 & 86.1 & 88.7 & 0.2307 & 0.3195 & 0.1521 & 0.2039 \\
& & 500 & 98 & 98.8 & 90.3 & 92.5 & 0.1623 & 0.2164 & 0.1092 & 0.1438 \\
& 0.5 & 100 & 97.6 & 98.9 & 84.7 & 88.8 & 0.2913 & 0.409 & 0.1814 & 0.2449
\\
& & 200 & 97.8 & 98.6 & 85.8 & 89.5 & 0.225 & 0.296 & 0.1429 & 0.1842 \\
& & 500 & 98.9 & 99.3 & 89.2 & 91.2 & 0.1619 & 0.2145 & 0.1062 & 0.1417 \\
& 0.9 & 100 & 97.8 & 98.9 & 84.1 & 89.6 & 0.291 & 0.411 & 0.1751 & 0.2443 \\
& & 200 & 98.7 & 99.4 & 84.5 & 88.5 & 0.2255 & 0.3034 & 0.1392 & 0.1881 \\
& & 500 & 98.9 & 99.4 & 90 & 92.6 & 0.1632 & 0.2116 & 0.1046 & 0.137 \\
\rule[-1ex]{0cm}{4ex} 0.99 & 0.1 & 100 & 98.3 & 99 & 85.2 & 89.4 & 0.289 &
0.4035 & 0.181 & 0.2393 \\
& & 200 & 97.8 & 98.6 & 87.4 & 90.9 & 0.2267 & 0.3033 & 0.1448 & 0.1897 \\
& & 500 & 99 & 99.7 & 89.1 & 92.7 & 0.1613 & 0.2082 & 0.1056 & 0.1354 \\
& 0.5 & 100 & 99.2 & 99.7 & 83.7 & 88.4 & 0.2913 & 0.4146 & 0.1529 & 0.2221
\\
& & 200 & 99.1 & 99.5 & 82.1 & 87.1 & 0.2273 & 0.3088 & 0.1246 & 0.1791 \\
& & 500 & 99.1 & 99.7 & 86.8 & 90.6 & 0.1645 & 0.2178 & 0.0957 & 0.1366 \\
& 0.9 & 100 & 99.9 & 100 & 72.7 & 82.8 & 0.2912 & 0.4099 & 0.1156 & 0.2151
\\
& & 200 & 99.7 & 99.7 & 78.4 & 87.4 & 0.2288 & 0.3069 & 0.1011 & 0.1757 \\
& & 500 & 99.9 & 100 & 81.3 & 88.6 & 0.1583 & 0.2007 & 0.0764 & 0.1206 \\
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{adjustbox}
\medskip
\noindent
\footnotesize
Note -- The table provides coverage rates and widths of confidence intervals
of level $95\%$ for Gini indices differences by asymptotic intersection
method (IM-Asym), bootstrap intersection method (IM-Boot), asymptotic method
with overlapping samples (OS-Asym) and bootstrap method with overlapping
samples (OS-Boot). The data are generated according to DGP I-C for given
Gaussian copula ${\Greekmath 011A}$ in \eqref{eq:DGP-GaussCopula}, overlap portion $
{\Greekmath 0115}$ and sample size $n$. The numbers are based on $1000$ replications
with 399 bootstrap repetitions each.
\normalsize
\end{table}
\begin{table}[tb]
\caption{DGP I-C: coverage and width of confidence intervals for the mean
difference.} \label{tab:CI-dmu-DGP3}
\begin{center}
Table \thetable
DGP I-C: coverage and width of confidence intervals for the mean difference.
\end{center}
\begin{adjustbox}{scale={0.9}{0.9}}
\hspace{-0.5\totalhormargin} \begin{minipage}{\paperwidth}
\centering
\small
\begin{center}
\begin{tabular}{ccc|cccc|cccc}
\hline
\rule[-1ex]{0cm}{4ex} ${\Greekmath 011A}$ & ${\Greekmath 0115}$ & $n$ & \multicolumn{4}{c|}{
Coverage $(\%)$} & \multicolumn{4}{c}{Width} \\ \cline{4-11}
\rule[-1ex]{0cm}{4ex} & & & IM-Asym & IM-Boot & OS-Asym & OS-Boot & IM-Asym
& IM-Boot & OS-Asym & OS-Boot \\ \hline
\rule[-1ex]{0cm}{4ex} -0.99 & 0.1 & 100 & 96.2 & 98.5 & 89.2 & 90.8 & 0.4457
& 0.7719 & 0.335 & 0.5667 \\
& & 200 & 95.8 & 97.3 & 89.6 & 91.2 & 0.3259 & 0.4585 & 0.2453 & 0.3402 \\
& & 500 & 97.4 & 98.3 & 92 & 92.6 & 0.2135 & 0.2762 & 0.161 & 0.2082 \\
& 0.5 & 100 & 95.5 & 98.2 & 91.3 & 92 & 0.4296 & 0.6582 & 0.3363 & 0.4708 \\
& & 200 & 96.6 & 97.8 & 91.6 & 93.2 & 0.3177 & 0.4209 & 0.2482 & 0.3115 \\
& & 500 & 96 & 97.5 & 89.5 & 90.7 & 0.2155 & 0.3291 & 0.1687 & 0.2505 \\
& 0.9 & 100 & 94.1 & 97.1 & 89.6 & 91.8 & 0.4215 & 0.6159 & 0.342 & 0.4493
\\
& & 200 & 95.4 & 97.2 & 89.5 & 90.7 & 0.3336 & 0.5618 & 0.271 & 0.4244 \\
& & 500 & 96.1 & 97.2 & 93 & 92.8 & 0.2134 & 0.3146 & 0.1723 & 0.244 \\
\rule[-1ex]{0cm}{4ex} -0.5 & 0.1 & 100 & 95.8 & 97.6 & 86.8 & 89.7 & 0.4289
& 0.7199 & 0.3184 & 0.5249 \\
& & 200 & 96.7 & 98.1 & 88.7 & 90.4 & 0.3377 & 0.5376 & 0.2539 & 0.4052 \\
& & 500 & 97.5 & 98.3 & 90.4 & 92.9 & 0.2167 & 0.2894 & 0.163 & 0.2199 \\
& 0.5 & 100 & 94.8 & 97.6 & 88.7 & 89.2 & 0.4201 & 0.7253 & 0.3193 & 0.5077
\\
& & 200 & 95 & 97.4 & 89.1 & 90.5 & 0.3147 & 0.4253 & 0.2398 & 0.3152 \\
& & 500 & 96.3 & 97.6 & 91.2 & 92 & 0.2147 & 0.2837 & 0.1646 & 0.2154 \\
& 0.9 & 100 & 94.5 & 96.5 & 87.3 & 89.7 & 0.445 & 1.0777 & 0.3483 & 0.7591
\\
& & 200 & 97.4 & 98.7 & 92 & 92.8 & 0.3229 & 0.4807 & 0.2522 & 0.3532 \\
& & 500 & 96.5 & 98.2 & 92 & 92 & 0.2124 & 0.2618 & 0.1658 & 0.1986 \\
\rule[-1ex]{0cm}{4ex} 0 & 0.1 & 100 & 96.1 & 98.2 & 86.5 & 90.3 & 0.4306 &
0.7201 & 0.3167 & 0.5286 \\
& & 200 & 97.6 & 99 & 90.2 & 92.4 & 0.3124 & 0.4469 & 0.2303 & 0.3322 \\
& & 500 & 97.1 & 98.2 & 89.7 & 91.9 & 0.2169 & 0.2921 & 0.162 & 0.2181 \\
& 0.5 & 100 & 96.2 & 98.1 & 88.1 & 91 & 0.4337 & 0.6754 & 0.3191 & 0.4905 \\
& & 200 & 96.8 & 98.5 & 89.1 & 89.8 & 0.3242 & 0.4515 & 0.2403 & 0.3344 \\
& & 500 & 97.7 & 98.9 & 90.6 & 92.1 & 0.2143 & 0.3028 & 0.1597 & 0.2302 \\
& 0.9 & 100 & 96.7 & 98.1 & 88.4 & 91.7 & 0.4681 & 1.4358 & 0.3495 & 1.0698
\\
& & 200 & 96.9 & 98.3 & 89.4 & 91.3 & 0.314 & 0.4209 & 0.2317 & 0.3086 \\
& & 500 & 96.9 & 98.2 & 89.6 & 91.7 & 0.2091 & 0.2569 & 0.1552 & 0.1911 \\
\rule[-1ex]{0cm}{4ex} 0.5 & 0.1 & 100 & 96.3 & 98.2 & 87.5 & 91.1 & 0.4336 &
0.6626 & 0.3164 & 0.4803 \\
& & 200 & 96.2 & 98.4 & 86.9 & 88.9 & 0.3224 & 0.4536 & 0.2368 & 0.3373 \\
& & 500 & 98.1 & 98.8 & 89.9 & 92.4 & 0.2217 & 0.3691 & 0.165 & 0.2939 \\
& 0.5 & 100 & 96.8 & 98.3 & 85.8 & 89.3 & 0.4232 & 0.6267 & 0.2963 & 0.4421
\\
& & 200 & 97.6 & 98.9 & 89 & 90.2 & 0.3087 & 0.3954 & 0.2172 & 0.2835 \\
& & 500 & 97.4 & 98.5 & 90.1 & 92.8 & 0.2183 & 0.311 & 0.1574 & 0.2369 \\
& 0.9 & 100 & 96.8 & 98.7 & 85.4 & 89.7 & 0.4224 & 0.653 & 0.2837 & 0.4829
\\
& & 200 & 98.7 & 99.5 & 89.1 & 92 & 0.3118 & 0.4389 & 0.2118 & 0.3215 \\
& & 500 & 97.8 & 98.6 & 90.7 & 92.9 & 0.2143 & 0.2666 & 0.1492 & 0.1953 \\
\rule[-1ex]{0cm}{4ex} 0.99 & 0.1 & 100 & 96.9 & 98.6 & 87.9 & 91.7 & 0.421 &
0.6705 & 0.3021 & 0.4783 \\
& & 200 & 97 & 98.4 & 89.4 & 90.9 & 0.3191 & 0.4711 & 0.2314 & 0.3507 \\
& & 500 & 97.9 & 98.6 & 90.4 & 92.5 & 0.2107 & 0.2683 & 0.1538 & 0.2003 \\
& 0.5 & 100 & 96.8 & 98.3 & 84.9 & 90.1 & 0.4246 & 0.605 & 0.2792 & 0.4362
\\
& & 200 & 98.3 & 98.7 & 87 & 90.6 & 0.3156 & 0.4265 & 0.2107 & 0.3083 \\
& & 500 & 98.9 & 99.2 & 90.4 & 91.5 & 0.2206 & 0.2907 & 0.1518 & 0.2158 \\
& 0.9 & 100 & 98 & 99.2 & 81.9 & 90.2 & 0.4173 & 0.6589 & 0.2451 & 0.5257 \\
& & 200 & 98.9 & 99.3 & 86.1 & 91.3 & 0.3149 & 0.4144 & 0.1912 & 0.2964 \\
& & 500 & 99.2 & 99.5 & 87 & 92.3 & 0.2076 & 0.2586 & 0.1293 & 0.1849 \\
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{adjustbox}
\medskip
\noindent
\footnotesize
Note -- The table provides coverage rates and widths of confidence intervals
of level $95\%$ for the mean difference by asymptotic intersection method
(IM-Asym), bootstrap intersection method (IM-Boot), asymptotic method with
overlapping samples (OS-Asym) and bootstrap method with overlapping samples
(OS-Boot). The data are generated according to DGP I-C for given Gaussian
copula ${\Greekmath 011A}$ in \eqref{eq:DGP-GaussCopula}, overlap portion ${\Greekmath 0115}$ and
sample size $n$. The numbers are based on $1000$ replications with 399
bootstrap repetitions each.
\normalsize
\end{table}
\begin{table}[tb]
\caption{DGP I-C: coverage and width of confidence intervals for the LC
ordinate difference at $p = 0.5$.} \label{tab:CI-dLC-DGP3}
\begin{center}
Table \thetable
DGP I-C: coverage and width of confidence intervals for the LC ordinate
difference at $p = 0.5$.
\end{center}
\begin{adjustbox}{scale={0.9}{0.9}}
\hspace{-0.5\totalhormargin} \begin{minipage}{\paperwidth}
\centering
\small
\begin{center}
\begin{tabular}{ccc|cccc|cccc}
\hline
\rule[-1ex]{0cm}{4ex} ${\Greekmath 011A}$ & ${\Greekmath 0115}$ & $n$ & \multicolumn{4}{c|}{
Coverage $(\%)$} & \multicolumn{4}{c}{Width} \\ \cline{4-11}
\rule[-1ex]{0cm}{4ex} & & & IM-Asym & IM-Boot & OS-Asym & OS-Boot & IM-Asym
& IM-Boot & OS-Asym & OS-Boot \\ \hline
\rule[-1ex]{0cm}{4ex} -0.99 & 0.1 & 100 & 99.3 & 99.8 & 92.4 & 95.3 & 0.1695
& 0.2033 & 0.1034 & 0.1217 \\
& & 200 & 99.9 & 100 & 93 & 95 & 0.1244 & 0.1422 & 0.0761 & 0.0867 \\
& & 500 & 99.8 & 99.9 & 94.3 & 94.9 & 0.0819 & 0.0907 & 0.0504 & 0.0555 \\
& 0.5 & 100 & 99.5 & 99.9 & 91.4 & 95.5 & 0.1685 & 0.201 & 0.0891 & 0.1083
\\
& & 200 & 99.9 & 100 & 93.2 & 94.6 & 0.1245 & 0.1432 & 0.0669 & 0.0778 \\
& & 500 & 99.4 & 99.6 & 92.1 & 92.8 & 0.0817 & 0.0903 & 0.0446 & 0.0511 \\
& 0.9 & 100 & 99.9 & 100 & 90.5 & 94.7 & 0.1679 & 0.2008 & 0.0724 & 0.0938
\\
& & 200 & 99.9 & 100 & 89.8 & 93.5 & 0.1239 & 0.1416 & 0.0557 & 0.0702 \\
& & 500 & 99.9 & 100 & 91 & 92.4 & 0.0814 & 0.0901 & 0.038 & 0.0446 \\
\rule[-1ex]{0cm}{4ex} -0.5 & 0.1 & 100 & 99.5 & 100 & 91.7 & 93.6 & 0.1684 &
0.2007 & 0.1047 & 0.1217 \\
& & 200 & 99.4 & 99.6 & 93.1 & 94.5 & 0.1244 & 0.1421 & 0.078 & 0.0887 \\
& & 500 & 99.6 & 99.7 & 94.1 & 94 & 0.0824 & 0.0912 & 0.0519 & 0.0577 \\
& 0.5 & 100 & 99.4 & 99.6 & 92 & 95.8 & 0.1686 & 0.2015 & 0.1015 & 0.118 \\
& & 200 & 99.8 & 99.8 & 93 & 93.6 & 0.1235 & 0.1408 & 0.0748 & 0.0844 \\
& & 500 & 99.6 & 99.8 & 92.5 & 93.5 & 0.0822 & 0.0912 & 0.0503 & 0.0557 \\
& 0.9 & 100 & 99.7 & 99.9 & 92.3 & 95 & 0.1697 & 0.2034 & 0.0991 & 0.1171 \\
& & 200 & 99.6 & 99.8 & 92.8 & 94.3 & 0.1245 & 0.1432 & 0.0732 & 0.0833 \\
& & 500 & 100 & 99.8 & 92.4 & 93.7 & 0.0824 & 0.0916 & 0.0489 & 0.054 \\
\rule[-1ex]{0cm}{4ex} 0 & 0.1 & 100 & 99.1 & 100 & 90.9 & 93.3 & 0.1694 &
0.2027 & 0.1061 & 0.1242 \\
& & 200 & 99.5 & 99.8 & 90.2 & 92.8 & 0.1227 & 0.1399 & 0.0771 & 0.0864 \\
& & 500 & 99.6 & 100 & 93 & 94 & 0.0825 & 0.0918 & 0.0523 & 0.0582 \\
& 0.5 & 100 & 99.4 & 99.8 & 91.6 & 95 & 0.1687 & 0.2012 & 0.1056 & 0.1236 \\
& & 200 & 99.8 & 99.9 & 93.8 & 95.7 & 0.1246 & 0.1424 & 0.0783 & 0.0891 \\
& & 500 & 99.7 & 99.8 & 93.7 & 94.6 & 0.0819 & 0.0907 & 0.0519 & 0.0574 \\
& 0.9 & 100 & 99 & 99.8 & 91.1 & 94.6 & 0.1687 & 0.2015 & 0.1056 & 0.1246 \\
& & 200 & 99.9 & 99.9 & 93.4 & 94.3 & 0.1236 & 0.1409 & 0.0776 & 0.0875 \\
& & 500 & 99.6 & 99.7 & 93.6 & 94.3 & 0.0817 & 0.0904 & 0.0517 & 0.0567 \\
\rule[-1ex]{0cm}{4ex} 0.5 & 0.1 & 100 & 99.5 & 99.9 & 92.6 & 94.2 & 0.1691 &
0.2021 & 0.1049 & 0.1228 \\
& & 200 & 99.7 & 99.9 & 91.7 & 94 & 0.1242 & 0.1418 & 0.0774 & 0.088 \\
& & 500 & 99.6 & 99.6 & 93 & 94.5 & 0.0821 & 0.0912 & 0.0516 & 0.0577 \\
& 0.5 & 100 & 99.8 & 99.9 & 92.3 & 94.8 & 0.1685 & 0.2016 & 0.1001 & 0.1173
\\
& & 200 & 99.8 & 99.9 & 92.7 & 94.5 & 0.1225 & 0.1391 & 0.0731 & 0.0819 \\
& & 500 & 99.8 & 100 & 92.3 & 93.7 & 0.082 & 0.0908 & 0.0497 & 0.0561 \\
& 0.9 & 100 & 99.9 & 99.8 & 92 & 94.6 & 0.1683 & 0.2009 & 0.0959 & 0.1137 \\
& & 200 & 99.9 & 99.9 & 92.3 & 93.8 & 0.1227 & 0.1391 & 0.0704 & 0.0804 \\
& & 500 & 100 & 100 & 93.5 & 94.1 & 0.0823 & 0.0912 & 0.048 & 0.0536 \\
\rule[-1ex]{0cm}{4ex} 0.99 & 0.1 & 100 & 99.6 & 99.7 & 92.5 & 95 & 0.1676 &
0.2001 & 0.1001 & 0.1165 \\
& & 200 & 99.5 & 99.8 & 94.1 & 95.5 & 0.123 & 0.1401 & 0.0738 & 0.0836 \\
& & 500 & 99.9 & 99.9 & 93.8 & 94.9 & 0.0819 & 0.0904 & 0.0496 & 0.055 \\
& 0.5 & 100 & 100 & 100 & 91.8 & 94.8 & 0.1677 & 0.2001 & 0.08 & 0.098 \\
& & 200 & 100 & 100 & 89.7 & 92.1 & 0.123 & 0.1402 & 0.0597 & 0.0706 \\
& & 500 & 100 & 100 & 92.8 & 94.1 & 0.0827 & 0.0921 & 0.0415 & 0.0489 \\
& 0.9 & 100 & 100 & 100 & 85 & 92.9 & 0.1682 & 0.2008 & 0.0522 & 0.0765 \\
& & 200 & 100 & 100 & 87 & 91.1 & 0.1236 & 0.1411 & 0.0412 & 0.0564 \\
& & 500 & 100 & 100 & 87.2 & 91.4 & 0.0811 & 0.0892 & 0.0291 & 0.0381 \\
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{adjustbox}
\medskip
\noindent
\footnotesize
Note -- The table provides coverage rates and widths of confidence intervals
of level $95\%$ for the Lorenz curve (LC) ordinate difference at percentile $
p = 0.5$ by asymptotic intersection method (IM-Asym), bootstrap intersection
method (IM-Boot), asymptotic method with overlapping samples (OS-Asym) and
bootstrap method with overlapping samples (OS-Boot). The data are generated
according to DGP I-C for given Gaussian copula ${\Greekmath 011A}$ in
\eqref{eq:DGP-GaussCopula}, overlap portion ${\Greekmath 0115}$ and sample size $n$.
The numbers are based on $1000$ replications with 399 bootstrap repetitions
each.
\normalsize
\end{table}
To sum up, the asymptotic CIs with overlapping samples (OS-Asym) perform
well, unless the tail is heavy. The bootstrap inference with overlapping
samples (OS-Boot) can effectively address the issue of OS-Asym, except that
the tail is too heavy. Such a finding is consistent with the literature;
see, for example,
\citename{Davidson2009} (\citeyear*{Davidson2009}, \citeyear*{Davidson2012})
. The intersection methods (IMs) tend to be conservative, provided the tail
is not too heavy so that the variance does not exist; see a numerical
exercise in the appendix. However, they can yield reliable results when both
OS-Asym and OS-Boot fail, particularly when overlapping samples (Assumption
3.2) or the limiting joint distribution (Assumption 3.3) are in question, as
we will illustrate below.
\FloatBarrier
\subsection{Beyond overlapping samples}
\label{sec:MC-BeyondOS}
This section examines scenarios where the overlapping samples framework in
Assumption \ref{assump:OS} may not be applicable. We take the mean
difference $\Delta{\Greekmath 0116} = {\Greekmath 0116}_{1} - {\Greekmath 0116}_{2}$ as an example for two reasons.
First, ${\Greekmath 0120}$ in \eqref{eq:ALG} has a simple expression of ${\Greekmath 0120}(x) = x -
{\Greekmath 0116} $. As ${\Greekmath 0120}(x)$ is increasing in $x$, we can compute ${\Greekmath 011A}_{{\Greekmath 0112}}$ in
\eqref{eq:AVar-dIndex-rho} as ${\Greekmath 011A}_{{\Greekmath 0112}} = {\Greekmath 011A}$. Second, the population
moments are usually easier to calculate than the welfare indices.
We demonstrate the robustness of intersection methods (IM) by comparing them
to approaches based on overlapping samples. Specifically, we aim to
investigate the performance of various CIs when Assumption \ref{assump:OS}
fails, while Assumption \ref{assump:DS} remains valid.
First, we consider samples that violate Assumption \ref{assump:OS} $(\mathbf{
A2})$. Suppose that pairs of unobserved observations, denoted by $
\{(U_{1i},U_{2i})\}_{i=1}^{m}$ are iid data from the joint distribution $
(F_{1},F_{2},{\Greekmath 011A} )$, where $F_{1}=\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}(1,1.6971,8.3679)$ and $F_{2}=
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}(0.4,2.8,1.7)$. The correlation coefficient ${\Greekmath 011A} $ in
\eqref{eq:DGP-GaussCopula} is either ${\Greekmath 011A} =\pm 0.95$ or selected randomly
from a uniform distribution over $(-1,1)$, \emph{i.e.}, $U(-1,1)$. Suppose
that we only observe the ordered values of $\{(U_{1i},U_{2i})\}_{i=1}^{m}$.
Let $X_{k,i}=U_{k(i)}$ for $k=1,2$ and $U_{k(1)}\leq \cdots ,\leq U_{k(n)}$.
The overlap of samples are then $(X_{1i},X_{2i})_{i=1}^{m}$. The condition $(
\mathbf{A2})$ is violated in this case. The unmatched observations are iid
data from $F_{1}$ and $F_{2}$. The sample sizes are $n=100,200,500,1000$
with overlap portions ${\Greekmath 0115} =0.5,1$. So, the DGP is as follows.
\medskip \textbf{DGP II-A}: (Violate $(\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{A}2)$) $\{X_{k, i}\}_{i=1}^{n}
= \{U_{k(i)}\}_{i=1}^{n}$ for $k = 1, 2$, where $U_{k(i)}$ is the $i$-th
order statistic. $\{U_{1, i}\}_{i=1}^{n} \overset{iid}{\sim} \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}(1,
1.6971, 8.3679)$, $\{U_{2, i}\}_{i=1}^{n} \overset{iid}{\sim} \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}(0.4,
2.8, 1.7)$, ${\Greekmath 011A} = 0.95$, $-0.95$ or ${\Greekmath 011A} \sim U(-1, 1)$. The sample sizes
are $n = 100, 200, 500, 1000$. The overlap portions are ${\Greekmath 0115} = 0.5, 1$.
\medskip
Second, we conduct an experiment where condition Assumption $(\mathbf{A3})$
fails. We assume that the second sample consists of $2n$ iid observations
from $\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}(1,1.6971,8.3679)$. Given the sample $\{X_{2,i}\}_{i=1}^{2n}$
, define $X_{1,i}=X_{2,2i-1}-X_{2,2i}$ for $i=1,\ldots ,\,n$. Let $
\{(X_{1,i},X_{2,2i-1})\}_{i=1}^{n}$ be the matched pairs. By construction, $
\{X_{1,i}\}_{i=1}^{n}$ and $\{X_{2,2i-1}\}_{i=1}^{n}$ are dependent, which
violates condition $(\mathbf{A3})$. The DGP is summarized as below.
\medskip \textbf{DGP II-B}: (Violate $(\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{A}3)$) $\{X_{2, i}\}_{i=1}^{2n}
\overset{iid}{\sim} \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SM}(0.4, 2.8, 1.7)$ and $\{X_{1, i}\}_{i=1}^{n}$
where $X_{1, i} = X_{2, 2i - 1} - X_{2, 2i}$. $n = 100, 200, 500, 1000$.
\medskip
Table \ref{tab:CI-dMean-DGP-B1} provides simulation results for DGP II-A.
Table \ref{tab:CI-dMean-DGP-B2} provides simulation results for DGP II-B.
Both tables demonstrate that intersection methods are effective, but
approaches based on overlapping samples are invalid.
\begin{table}[tb]
\caption{DGP II-A: coverage and width of confidence intervals for
mean difference.} \label{tab:CI-dMean-DGP-B1}
\begin{center}
Table \thetable
DGP II-A: coverage and width of confidence intervals for mean difference.
\end{center}
\begin{adjustbox}{scale={0.9}{0.9}}
\hspace{-0.5\totalhormargin} \begin{minipage}{\paperwidth}
\centering
\small
\begin{center}
\begin{tabular}{ccc|cccc|cccc}
\hline
\rule[-1ex]{0cm}{4ex} ${\Greekmath 011A} $ & ${\Greekmath 0115} $ & $n$ & \multicolumn{4}{c|}{
Coverage $(\%)$} & \multicolumn{4}{c}{Width} \\ \cline{4-11}
\rule[-1ex]{0cm}{4ex} & & & IM-Asym & IM-Boot & OS-Asym & OS-Boot & IM-Asym
& IM-Boot & OS-Asym & OS-Boot \\ \hline
\rule[-1ex]{0cm}{4ex} 0.95 & 0.5 & 100 & 100 & 100 & 69.6 & 66.7 & 0.1686 &
0.1799 & 0.0228 & 0.0349 \\
& & 200 & 100 & 100 & 67.6 & 65.3 & 0.1189 & 0.1235 & 0.015 & 0.0233 \\
& & 500 & 100 & 100 & 67.1 & 68.4 & 0.0755 & 0.0775 & 0.0094 & 0.0144 \\
& & 1000 & 100 & 100 & 63.6 & 64.9 & 0.0534 & 0.0545 & 0.0062 & 0.0082 \\
\rule[-1ex]{0cm}{4ex} & 1 & 100 & 100 & 100 & 65 & 63.6 & 0.167 & 0.1772 &
0.0151 & 0.022 \\
& & 200 & 100 & 100 & 61.8 & 60.3 & 0.1182 & 0.123 & 0.01 & 0.0159 \\
& & 500 & 100 & 100 & 63.9 & 65.8 & 0.0754 & 0.0775 & 0.0064 & 0.009 \\
& & 1000 & 100 & 100 & 66 & 67.9 & 0.0534 & 0.0545 & 0.0045 & 0.0058 \\
\rule[-1ex]{0cm}{4ex} -0.95 & 0.5 & 100 & 99.6 & 99.6 & 25.7 & 26.9 & 0.1664
& 0.1759 & 0.0365 & 0.0544 \\
& & 200 & 98.9 & 99 & 22 & 24 & 0.1192 & 0.1239 & 0.0206 & 0.0292 \\
& & 500 & 99.4 & 99.5 & 18.5 & 21.1 & 0.0752 & 0.0772 & 0.0112 & 0.0159 \\
& & 1000 & 99.1 & 99.1 & 16.9 & 20.3 & 0.0535 & 0.0545 & 0.0073 & 0.0099 \\
\rule[-1ex]{0cm}{4ex} & 1 & 100 & 98.6 & 98.6 & 20.6 & 23.5 & 0.1676 & 0.1778
& 0.0216 & 0.034 \\
& & 200 & 98.4 & 98.5 & 17.9 & 21 & 0.1186 & 0.1235 & 0.0126 & 0.0178 \\
& & 500 & 98.7 & 98.8 & 18.2 & 22.3 & 0.0755 & 0.0774 & 0.0074 & 0.0102 \\
& & 1000 & 98 & 98.3 & 16.5 & 18.9 & 0.0533 & 0.0545 & 0.0048 & 0.006 \\
$\rule[-1ex]{0cm}{4ex}U(-1,1)$ & 0.5 & 100 & 100 & 99.9 & 32.2 & 32.5 &
0.1672 & 0.177 & 0.0348 & 0.0499 \\
& & 200 & 99.8 & 99.8 & 27.8 & 27.6 & 0.1187 & 0.1233 & 0.0208 & 0.029 \\
& & 500 & 99.6 & 99.7 & 25.8 & 27.9 & 0.0755 & 0.0776 & 0.0114 & 0.0162 \\
& & 1000 & 99.8 & 99.8 & 22.9 & 25.7 & 0.0535 & 0.0546 & 0.0073 & 0.01 \\
\rule[-1ex]{0cm}{4ex} & 1 & 100 & 99.7 & 99.7 & 26.5 & 27.6 & 0.1672 & 0.1779
& 0.0209 & 0.0307 \\
& & 200 & 100 & 100 & 22.2 & 24.8 & 0.1185 & 0.1229 & 0.0126 & 0.0171 \\
& & 500 & 99.9 & 99.9 & 21.5 & 23 & 0.0755 & 0.0774 & 0.0071 & 0.0092 \\
& & 1000 & 99.9 & 99.9 & 22.2 & 24.8 & 0.0535 & 0.0544 & 0.0049 & 0.0061 \\
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{adjustbox}
\medskip
\noindent
\footnotesize
Note -- The table provides coverage rates and widths of confidence intervals
of level $95\%$ for mean difference by asymptotic intersection method
(IM-Asym), bootstrap intersection method (IM-Boot), asymptotic method with
overlapping samples (OS-Asym) and bootstrap method with overlapping samples
(OS-Boot). The data are generated according to DGP II-A for given sample
size $n$, overlap portion ${\Greekmath 0115}$ and ${\Greekmath 011A}$ of Gaussian copula in
\eqref{eq:DGP-GaussCopula}. The numbers are based on $1000$ replications
with 399 bootstrap repetitions each.
\normalsize
\end{table}
\begin{table}[tb]
\caption{DGP II-B: coverage and width of confidence intervals for
mean difference.} \label{tab:CI-dMean-DGP-B2}
\begin{center}
Table \thetable
DGP II-B: coverage and width of confidence intervals for mean difference.
\end{center}
\begin{adjustbox}{scale={0.9}{0.9}}
\hspace{-0.5\totalhormargin} \begin{minipage}{\paperwidth}
\centering
\small
\begin{center}
\begin{tabular}{c|cc|cc|cc|cc}
\hline
\rule[-1ex]{0cm}{4ex} n & \multicolumn{4}{c|}{Coverage $(\%)$} &
\multicolumn{4}{c}{Width} \\ \cline{2-5}\cline{6-9}
\rule[-1ex]{0cm}{4ex} & IM-Asym & IM-Boot & OS-Asym & OS-Boot & IM-Asym &
IM-Boot & OS-Asym & OS-Boot \\ \hline
\rule[-1ex]{0cm}{4ex} 100 & 99.9 & 99.9 & 86 & 76.6 & 0.1686 & 0.1738 &
0.0852 & 0.0869 \\
200 & 99.7 & 99.8 & 88.8 & 79.4 & 0.1198 & 0.1227 & 0.0605 & 0.0614 \\
500 & 99.8 & 99.8 & 86.4 & 79.7 & 0.0758 & 0.0771 & 0.0382 & 0.0386 \\
1000 & 99.7 & 99.7 & 86.2 & 77.2 & 0.0536 & 0.0545 & 0.0271 & 0.0273 \\
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{adjustbox}
\medskip
\noindent
\footnotesize
Note -- The table provides coverage rates and widths of confidence intervals
of level $95\%$ for mean difference by asymptotic intersection method
(IM-Asym), bootstrap intersection method (IM-Boot), asymptotic method with
overlapping samples (OS-Asym) and bootstrap method with overlapping samples
(OS-Boot). The data are generated according to DGP II-B for given sample
size $n$. The numbers are based on $1000$ replications with 399 bootstrap
repetitions each.
\normalsize
\end{table}
\newpage
\FloatBarrier
\subsection{\huge Application to the SHIW Data of the Bank of Italy
\label{sec:App-SHIW}}
\renewcommand{\Alph{section}}{\Alph{section}}
\renewcommand{\Alph{section}}{\Alph{section}}
\setcounter{theorem}{0} \setcounter{definition}{0} \setcounter{equation}{0}
This section examines the dynamic change in household financial inequality
over time in Italy. We utilize the data from the Survey of Household Income
and Wealth (SHIW) conducted by the Bank of Italy. This data has been used by
several studies, including \nocite*{Raffinetti2015}, \nocite*{Raffinetti2017},
\nocite*{Bouezmarni2024} and \nocite*{Dufour2024}.
For simplicity, we assume that each sample consists of iid observations. To
address the heterogeneity in household structures, we normalize data by the
equivalence scale proposed by \nocite*{Kakwani1998}:
\begin{equation}
hhscale_{i}=(ad_{i}+0.2ch_{1,i}+0.4ch_{2,i}+0.7ch_{3,i})^{0.8}+0.1w_{i}
\end{equation}
where, for family $i$, $ad_{i}$ is the number of adults within a family, $
\{ch_{j,i}\}_{j=1}^{3}$ denote the number of children with age in $(0,5]$, $
[6,14]$ and $[15,17]$, respectively, and $w_{i}$ is the number of employees
or self-employed individuals within the families. Table \ref
{tab:SHIW-sum-stat} presents some summary statistics, and Figure \ref
{fig:SHIW-12-14-hist-cdf} plots the estimated densities and distribution
functions.
\begin{table}[tb]
\caption[Descriptive statistics for household financial income.]{} \label
{tab:SHIW-sum-stat}
\begin{center}
Table \thetable
Descriptive statistics for household financial income.
\end{center}
\hspace{-0.5\totalhormargin} \begin{minipage}{\paperwidth}
\centering
\small
\begin{center}
\begin{tabular}{cccc}
\hline
\rule[-1ex]{0cm}{4ex} Statistic & 2012 & & 2014 \\ \hline
\rule[-1ex]{0cm}{4ex} $n$ & 8151 & & 8156 \\
$m$ & 4459 & & 4459 \\
$\Pr (X<0)$ & 0.1002 & & 0.0858 \\
$\Pr (X=0)$ & 0.187 & & 0.1629 \\
$\Pr (X>0)$ & 0.7128 & & 0.7512 \\
Min & -17123.2 & & -10067 \\
$Q(0.25)$ & 0 & & 0.3183 \\
Mean & 189.5418 & & 108.1888 \\
Median & 29.6702 & & 24.8268 \\
$Q(0.75)$ & 154.7083 & & 125.8219 \\
Max & 73127.16 & & 24280.84 \\
Skew & 15.703 & & 7.6478 \\
Kurt & 528.6745 & & 157.134 \\
$\mathbb{E}(X|X>0)$ & 458.921 & & 285.0303 \\
$\mathbb{E}(X|X<0)$ & -1372.55 & & -1234.28 \\
$\mathbb{E}(|X|)$ & 464.6487 & & 320.0156 \\ \hline
\end{tabular}
\end{center}
\end{minipage}
\medskip
\noindent
\footnotesize
Note -- $n$ is sample size. $m$ is the size of overlap. $Q(p)$ is p-th
quantile. Skew stands for skewness. Kurt stands for kurtosis.
\normalsize
\end{table}
\begin{figure}
\caption[2014 vs 2012: distributions of financial income.]{} \label
{fig:SHIW-12-14-hist-cdf}
\begin{center}
\begin{minipage}{\paperwidth}
\includegraphics[width=0.35\linewidth]{figures/SHIW_Application/hist_12vs14_Legn.eps}
\includegraphics[width=0.35\linewidth]{figures/SHIW_Application/ECDF_12vs14_Legn.eps}
\end{minipage}
Figure \thefigure. 2014 vs 2012: distributions of financial income.
\end{center}
{\footnotesize {\ } }
\end{figure}
\begin{comment}
\footnote{
\nocite*{Dufour2024a} compared the financial inequality levels between two areas
where independent sampling is assumed.
}
\end{comment}
To analyze the changes in financial inequality in Italy from 2012 to 2014,
we must address two key issues. First, there is an overlap in the samples,
as nearly half of the observations appear in both waves. Therefore, we must
take the sample dependence into account. Second, financial income can be
negative, indicating that some households experienced losses. According to
Table \ref{tab:SHIW-sum-stat}, $10\%$ of households suffered losses in 2012,
while $8.5\%$ faced losses in 2014, respectively. We therefore consider a
family of extended inequality measures proposed by \nocite*{Bouezmarni2024} for
possibly negative variables. For convenience, we focus on two types of
specific measures.
\begin{definition}
\label{def:Ext-Ineq}
\textsc{ Extended Lorenz curve}.
\addcontentsline{lth}{theorem}{{\bf Definition \hspace{0em} \protect\numberline{\thedefinition}}
{\hspace{-3.5em} : \hspace{0em} Extended Lorenz curve}}
\textbf{\ } Consider a general variable $X$ with distribution $F$. Let ${\Greekmath 0116}
(F)=\int x\,dF$ be the mean, $Q(p;F)=\inf \{x:F(x)\geq p\}$ be the p-th
quantile, and ${\Greekmath 0116} ^{p}(F)=\int |x|\,dF$ be the absolute mean.
\begin{enumerate}[$(1)$]
\item
Let $F^{\oplus }(x)=\Pr (X\leq x\mid X>0)$ and $F^{\ominus }(x)=\Pr (X\leq
x\mid X<0)$. Then the sign-conditional Lorenz curves are
\begin{equation}
L^{\oplus }(p;F^{\oplus })=\dfrac{\int_{0}^{p}Q(u;F^{\oplus })\,du}{{\Greekmath 0116}
(F^{\oplus })}\,,\quad L^{\ominus }(p;F^{\ominus })=\dfrac{
\int_{0}^{p}Q(u;F^{\ominus })\,du}{{\Greekmath 0116} (F^{\ominus })}.
\label{eq:def-trun-LC}
\end{equation}
\item
The signed Lorenz curve is
\begin{equation}
L^{s}(p;F)=\dfrac{\int_{0}^{p}Q(u;F)\,du}{{\Greekmath 0116} ^{p}(F)}.
\label{eq:def-sgn-LC}
\end{equation}
\item
Let $X_{1}^{\oplus }$ and $X_{2}^{\oplus }$ (resp. $X_{1}^{\ominus }$ and $
X_{2}^{\ominus }$) be independent copies of variable $X^{\oplus }$ (resp. $
X^{\ominus }$) with CDF $F^{\oplus }$ (resp. $F^{\ominus }$). Then the
sign-conditional Gini indices are
\begin{equation}
I^{\oplus }(F^{\oplus })=\frac{\mathbb{E}(\lvert X_{1}^{\oplus
}-X_{2}^{\oplus }\rvert )}{2{\Greekmath 0116} (F^{\oplus })}\,,\quad I^{\ominus
}(F^{\ominus })=\frac{\mathbb{E}(\lvert X_{1}^{\ominus }-X_{2}^{\ominus
}\rvert )}{2\lvert {\Greekmath 0116} (F^{\ominus })\rvert }. \label{eq:trun-Gini-def}
\end{equation}
\item
Let $X_{1}$ and $X_{2}$ are independent copies of variable $X$ with CDF $F$.
The positive Gini index is
\begin{equation}
I^{p}(F)=\frac{\mathbb{E}(\lvert X_{1}-X_{2}\rvert )}{2{\Greekmath 0116} ^{p}(F)}.
\label{eq:pos-Gini-def}
\end{equation}
\end{enumerate}
\end{definition}
The sign-conditional LCs and Gini indices are tools for analyzing internal
inequality levels inside the group of winners or losers. The signed LC and
positive Gini index measure the overall level of inequality. See \nocite*
{Bouezmarni2024} and \nocite*{Dufour2024} for more discussion. To simplify our
discussion, we will call $L^{\oplus}$ (resp. $L^{\ominus}$) the positive
(resp. negative) truncated LC. We call $I^{\oplus}$ and $I^{\ominus}$
similarly.
To facilitate our discussion, we use the superscript $\dagger$ (or $*$) to
simplify notations to indicate a specific object from a collection. For
example, $(L^{\dagger}, F^{\dagger})\in \{(L^{\oplus}, F^{\oplus}),
(L^{\ominus}, F^{\ominus})\}$ means that $(L^{\dagger}, F^{\dagger})$ can be
either the sign-conditional LCs with their corresponding distributions. We
will omit the collection if no confusion arises.
We estimate the extended inequality measures by plugging the associated
EDFs. The estimated extended LCs are plotted in the left panels of Figure
\ref{fig:SHIW-ppLCs}, \ref{fig:SHIW-nnLCs}, and \ref{fig:SHIW-sLCs}. The
estimates for extended Gini indices are displayed in Table \ref
{tab:SHIW-dExtI-Est}. We will then conduct statistical inference for these
estimates. As shown in \nocite*{Dufour2024}, the estimators are ALG functionals
under specific conditions (outlined below).
\begin{lemma}
\label{lmm:ExtIneq-ALG}
\textsc{ Some properties of $H^{p}(x)$}.
\addcontentsline{lth}{theorem}{{\bf Lemma \hspace{0em} \protect\numberline{\thelemma}}
{\hspace{-3.5em} : \hspace{0em} Some properties of $H^{p}(x)$}}
Suppose $\{X_{i}\}_{i=1}^{n}\overset{iid}{\sim }F$ where distribution
function $F$ has a finite variance.
\begin{enumerate}[$(1)$]
\item
(Extended LC ordinate) Let $(L^{\dagger },F^{\dagger })\in \{(L^{\oplus
},F^{\oplus }),(L^{\ominus },F^{\ominus }),(L^{s},F)\}$. If $F^{\dagger }$
is differentiable at its $p$-th quantile, then $\hat{L^{\dagger }}(p)$
satisfies Definition \ref{def:ALG} with
\begin{equation}
{\Greekmath 0120} ^{\dagger }(x;L^{\dagger },F^{\dagger },p)=W^{\dagger }(x;L^{\dagger
},F^{\dagger },p)-\mathbb{E}[W^{\dagger }(X^{\dagger };L^{\dagger
},F^{\dagger },p)]
\end{equation}
where $(W^{\dagger },X^{\dagger })\in \{(W^{\oplus },X^{\oplus
}),(W^{\ominus },X^{\ominus }),(W^{s},X)\}$
\begin{equation}
\begin{split}
W^{\oplus }(x;L^{\oplus },F^{\oplus },p)& =\frac{\big[x-Q(p;F^{\oplus })\big]
\mathbf{1}\big(x\leq Q(p;F^{\oplus })\big)-x\,L^{\oplus }(p;F^{\oplus })}{
{\Greekmath 0116} (F^{\oplus })}, \\
W^{\ominus }(x;L^{\ominus },F^{\ominus },p)& =\frac{\big[x-Q(p;F^{\ominus })
\big]\mathbf{1}\big(x\leq Q(p;F^{\ominus })\big)-x\,L^{\ominus
}(p;F^{\ominus })}{{\Greekmath 0116} (F^{\ominus })}, \\
W^{s}(x;L^{s},F,p)& =\frac{[x-Q(p;F)]\mathbf{1}(x\leq Q(p;F))-|x|\,L^{s}(p;F)
}{{\Greekmath 0116} ^{p}(F)}.
\end{split}
\label{eq:ExtLC-ALG}
\end{equation}
\item
(Extended Gini index) Let $(I^{\dagger },F^{\dagger })\in \{(I^{\oplus
},F^{\oplus }),(I^{\ominus },I^{\ominus }),(I^{p},F)\}$. If $F^{\dagger }$
is continuous, then $\hat{I^{\dagger }}$ satisfies the Definition \ref
{def:ALG} with
\begin{equation}
{\Greekmath 0120} ^{\dagger }(x;I^{\dagger },F^{\dagger })=K^{\dagger }(x;I^{\dagger
},F^{\dagger })-\mathbb{E}[K^{\dagger }(X^{\dagger };I^{\dagger },F^{\dagger
})]
\end{equation}
where $(K^{\dagger },X^{\dagger })\in \{(K^{\oplus },X^{\oplus
}),(K^{\ominus },X^{\ominus }),(K^{p},X)\}$
\begin{equation}
\begin{split}
K^{\oplus }(x;I^{\oplus },F^{\oplus })& =\frac{2xF^{\dagger
}(x)-2\int^{x}t\,dF^{\oplus }(t)-[I^{\oplus }(F^{\oplus })+1]x}{{\Greekmath 0116}
(F^{\oplus })}, \\
K^{\ominus }(x;I^{\ominus },F^{\ominus })& =\frac{2xF^{\ominus
}(x)-2\int^{x}t\,dF^{\ominus }(t)-[I^{\ominus }(F^{\ominus })+1]x}{{\Greekmath 0116}
(F^{\ominus })}, \\
K^{p}(x;I^{p},F)& =\frac{2xF(x)-2\int^{x}t\,dF(t)-I^{p}(F)\,|x|-x}{{\Greekmath 0116}
^{p}(F)}.
\end{split}
\label{eq:ExtI-ALG}
\end{equation}
\end{enumerate}
\end{lemma}
Under Assumption \ref{assump:OS}, it is not difficult to show that the above
estimates fulfill Assumption \ref{assump:JointNormal} by multivariate
central limit theorem. So, we obtain the following corollary.
\begin{corollary}
\label{coro:AsyN-dExtIneq}
\textsc{ Convergence in distribution for the extended inequality difference}.
\addcontentsline{lth}{theorem}{{\bf Proposition \hspace{0em} \protect\numberline{\theproposition}}
{\hspace{-3.5em} : \hspace{0em} Convergence in distribution for the extended inequality difference}}
Suppose that the Assumption \ref{assump:OS} holds with finite variance
matrix.
\begin{enumerate}[$(1)$]
\item
(Extended LC ordinate) If the distribution $F_{k}^{\dagger }\in
\{F_{k}^{\oplus },F_{k}^{\ominus },F_{k}\}$, $k=1,2$, is differentiable at
its $p$-th quantile, then
\begin{equation}
\sqrt{N}\big(\Delta \hat{L^{\dagger }}(p)-\Delta L^{\dagger }(p)\big)\overset
{d}{\longrightarrow }\mathrm{N}\big(0,\Sigma _{\Delta }^{\dagger }(p)\big)
\end{equation}
where $\Delta \,L^{\dagger }\in \{\Delta \,L^{\oplus },\Delta \,L^{\ominus
},\Delta \,L^{s}\}$, and $\Sigma _{\Delta }^{\dagger }\in \{\Sigma _{\Delta
}^{\oplus },\Sigma _{\Delta }^{\ominus },\Sigma _{\Delta }^{s}\}$ is such
that
\begin{equation}
\begin{split}
\Sigma _{\Delta }^{\dagger }(p)=& (1-{\Greekmath 0111} _{1})\mathrm{Var}\big[
W_{1}^{\dagger }\big(X_{1}^{\dagger };p\big)\big]+{\Greekmath 0111} _{1}\mathrm{Var}\big[
W_{2}^{\dagger }\big(X_{2}^{\dagger };p\big)\big] \\
& -2\sqrt{{\Greekmath 0111} _{1}(1-{\Greekmath 0111} _{1}){\Greekmath 0115} _{1}{\Greekmath 0115} _{2}}\mathrm{Cov}\big[
W_{1}^{\dagger }\big(X_{1}^{\dagger };p\big),W_{2}^{\dagger }\big(
X_{2}^{\dagger };p\big)\big],
\end{split}
\end{equation}
with $W_{k}^{\dagger }(x;p)=W^{\dagger }(x;L^{\dagger },F_{k}^{\dagger },p)$
given by \eqref{eq:ExtLC-ALG}.
\item
(Extended Gini index) If the distribution $F_{k}$, $k=1,2$, is continuous
with a possible jump at zero, then
\begin{equation}
\sqrt{N}\big(\Delta \hat{I^{\dagger }}-\Delta I^{\dagger }\big)\overset{d}{
\longrightarrow }\mathrm{N}\big(0,V_{\Delta }^{\dagger }\big)
\end{equation}
where $\Delta \,I^{\dagger }\in \{\Delta \,I^{\oplus },\Delta \,I^{\ominus
},\Delta \,I^{p}\}$, and $V_{\Delta }^{\dagger }\in \{V_{\Delta }^{\oplus
},V_{\Delta }^{\ominus },V_{\Delta }^{p}\}$ is such that
\begin{equation}
\begin{split}
V_{\Delta }^{\dagger }=& (1-{\Greekmath 0111} _{1})\mathrm{Var}\big[K_{1}^{\dagger }\big(
X_{1}^{\dagger }\big)\big]+{\Greekmath 0111} _{1}\mathrm{Var}\big[K_{2}^{\dagger }\big(
X_{2}^{\dagger }\big)\big] \\
& -2\sqrt{{\Greekmath 0111} _{1}(1-{\Greekmath 0111} _{1}){\Greekmath 0115} _{1}{\Greekmath 0115} _{2}}\mathrm{Cov}\big[
K_{1}^{\dagger }\big(X_{1}^{\dagger }\big),K_{2}^{\dagger }\big(
X_{2}^{\dagger }\big)\big]
\end{split}
\end{equation}
with $K_{k}^{\dagger }(x)=K^{\dagger }(x;I^{\dagger },F_{k}^{\dagger })$
given by \eqref{eq:ExtI-ALG}.
\end{enumerate}
\end{corollary}
According to \eqref{eq:AVar-dIndex-Est}, we estimate the asymptotic variance
by the sample variance of the estimated influence function. We then obtain
CIs for the inequality changes at the level of $95\%$ using four proposed
methods, including the asymptotic intersection method (IM-Asym), bootstrap
intersection method (IM-Boot), asymptotic inference with overlapping samples
(OS-Asym), and bootstrap inference with overlapping samples (OS-Boot).
We first examine the internal inequality among winners. Figure \ref
{fig:SHIW-ppLCs} plots the change in positive LC from 2012 to 2014, \emph{
i.e.}, $\Delta L^{\oplus }=L_{2014}^{\oplus }-L_{2012}^{\oplus }$, and $95\%$
CIs. We find that $L_{2014}^{\oplus }$ is closer to the diagonal line, which
suggests a decrease in inequality level inside the group of winners.
Asymptotic and bootstrap CIs based on overlapping samples (OS-Asym and
OS-Boot) confirm this result, as zero falls outside the CIs. However, the
intersection methods (IM-Asym and IM-Boot) give an opposite conclusion.
Given the data structure, we believe that overlapping samples are a suitable
assumption and thus prefer the findings from OS-Asym and OS-Boot, as IM can
be conservative under such a scenario based on simulation results in the
previous section. We observe similar results for $\Delta I^{\oplus }$ in
Table \ref{tab:SHIW-dExtI-CI}. The estimate for $\Delta I^{\oplus }$ in
Table \ref{tab:SHIW-dExtI-Est} is only $-0.0246$, suggesting a relatively
small decrease in inequality level.
\begin{figure}
\caption[2014 vs 2012: positively truncated Lorenz curves.]{} \label
{fig:SHIW-ppLCs}
\begin{center}
\begin{minipage}{\paperwidth}
\includegraphics[width=0.35\linewidth]{figures/SHIW_Application/ppLC_2014_2012.eps}
\includegraphics[width=0.35\linewidth]{figures/SHIW_Application/CIs_dppLC_2014_2012.eps}
\end{minipage}
Figure \thefigure. 2014 vs 2012: positively truncated Lorenz curves
\end{center}
\end{figure}
We next study the internal inequality among the losers, as measured by $
\Delta L^{\ominus}$ in Figure \ref{fig:SHIW-nnLCs} and $\Delta I^{\ominus}$
in Table \ref{tab:SHIW-dExtI-CI}. We find that only the OS-Asym CI for $
\Delta I^{\ominus}$ excludes zero, while all the other CIs contain zero.
Given the simulation evidence in section \ref{sec:MC}, OS-Boot CI is
generally more reliable than OS-Asym CI. Therefore, we cannot conclude that
there is a significant change in internal inequality within the subgroup of
losers.
\begin{figure}
\caption[2014 vs 2012: negatively truncated Lorenz curves.]{} \label
{fig:SHIW-nnLCs}
\begin{center}
\begin{minipage}{\paperwidth}
\includegraphics[width=0.35\linewidth]{figures/SHIW_Application/nnLC_2014_2012.eps}
\includegraphics[width=0.35\linewidth]{figures/SHIW_Application/CIs_dnnLC_2014_2012.eps}
\end{minipage}
Figure \thefigure. 2014 vs 2012: negatively truncated Lorenz curves
\end{center}
\end{figure}
Finally, we analyze the overall inequality based on $\Delta L^{s}$ in Figure
\ref{fig:SHIW-sLCs} and $\Delta I^{p}$ in Table \ref{tab:SHIW-dExtI-CI}. The
estimated signed LC displays a flat segment (in the left panel of Figure \ref
{fig:SHIW-sLCs}), implying that the underlying distribution is not
differentiable at the corresponding quantile. So, we should be cautious when
interpreting inference results for the flat segments of $\Delta L^{s}$.
For $p < 0.1$, we note that asymptotic and bootstrap CIs with overlapping
samples (OS-Asym and OS-Boot) exclude zero, indicating a change over time.
We observe similar findings about $\Delta I^{p}$ by OS-Asym and OS-Boot CIs.
In contrast, the CIs based on IMs (IM-Asym and IM-Boot) include zero,
possibly due to the method's conservativeness. Note that, however, the
magnitude of the change in inequality is economically small.
\begin{figure}
\caption[2014 vs 2012: signed Lorenz curves.]{} \label{fig:SHIW-sLCs}
\begin{center}
\begin{minipage}{\paperwidth}
\includegraphics[width=0.35\linewidth]{figures/SHIW_Application/sLC_2014_2012.eps}
\includegraphics[width=0.35\linewidth]{figures/SHIW_Application/CIs_dsLC_2014_2012.eps}
\end{minipage}
Figure \thefigure. 2014 vs 2012: signed Lorenz curves
\end{center}
\end{figure}
\begin{table}[tb]
\caption{2014 vs 2012: estimates for extended Gini indices.} \label
{tab:SHIW-dExtI-Est}
\begin{center}
Table \thetable
2014 vs 2012: estimates for extended Gini indices
\end{center}
\hspace{-0.5\totalhormargin} \begin{minipage}{\paperwidth}
\centering
\small
\begin{center}
\begin{tabular}{cccc}
\hline
\rule[-1ex]{0cm}{4ex} Type $(I^{*})$ & $I^{*}_{2012}$ & $I^{*}_{2014}$ & $
\Delta I^{*}$ \\ \hline
\rule[-1ex]{0cm}{4ex} $I^{\oplus}$ & 0.8110 & 0.7864 & -0.0246 \\
$I^{\ominus}$ & 0.4836 & 0.4699 & -0.0137 \\
$I^{+}$ & 0.8653 & 0.8395 & -0.0258 \\
$I^{-}$ & 0.9483 & 0.9546 & 0.0063 \\
$I^{p}$ & 0.8899 & 0.8776 & -0.0123 \\ \hline
\end{tabular}
\end{center}
\end{minipage}
\medskip
\noindent
\footnotesize
Note -- $\Delta I^{*} = I^{*}_{2014} - I^{*}_{2012}$, where $I^{*}$ is one
of extended Gini indices $\{I^{\oplus}, I^{\ominus}, I^{p}\}$ in Definition
\ref{def:Ext-Ineq}.
\normalsize
\end{table}
\begin{table}[tb]
\caption{2014 vs 2012: $95\%$ CIs for difference in extended Gini index.}
\label{tab:SHIW-dExtI-CI}
\begin{center}
Table \thetable
2014 vs 2012: $95\%$ CIs for difference in extended Gini index
\end{center}
\begin{adjustbox}{scale={0.75}{0.8}}
\hspace{-0.5\totalhormargin} \begin{minipage}{\paperwidth}
\centering
\small
\begin{center}
\begin{tabular}{crlrlrlrl}
\hline
\rule[-1ex]{0cm}{4ex} Type $(I^{*})$ & \multicolumn{2}{c}{$95\%$ IM-Asym CI}
& \multicolumn{2}{c}{$95\%$ IM-Boot CI} & \multicolumn{2}{c}{$95\%$ OS-Asym
CI} & \multicolumn{2}{c}{$95\%$ OS-Boot CI} \\ \hline
\rule[-1ex]{0cm}{4ex} $I^{\oplus}$ & $[ -0.0584 $, & $0.0091]$ & $[ -0.0642 $
, & $0.0096]$ & $[ -0.0411 $, & $-0.0082]$ & $[ -0.0409 $, & $-0.0092]$ \\
$I^{\ominus}$ & $[ -0.0685 $, & $0.0410]$ & $[ -0.0750 $, & $0.0456]$ & $[
-0.0231 $, & $-0.0044]$ & $[ -0.0429 $, & $0.0157]$ \\
$I^{p}$ & $[ -0.0271 $, & $0.0026]$ & $[ -0.0292 $, & $0.0037]$ & $[ -0.0201
$, & $-0.0045]$ & $[ -0.0201 $, & $-0.0053]$ \\ \hline
\end{tabular}
\end{center}
\end{minipage}
\end{adjustbox}
\medskip
\noindent
\footnotesize
Note -- $\Delta I^{*} = I^{*}_{2014} - I^{*}_{2012}$, where $I^{*}$ is one
of extended Gini indices $\{I^{\oplus}, I^{\ominus}, I^{p}\}$ in Definition
\ref{def:Ext-Ineq}.
\normalsize
\end{table}
\FloatBarrier
\newpage
\subsection{\huge Extension to complex survey data \label{sec:clusters}}
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So far, we have assumed that researchers have a sample of micro-level data
drawn independently and identically distributed from a population
distribution. However, many large-scale household surveys by design involve
stratification and clustering. In this section, we briefly sketch the
potential extension of our methods to complex survey data.
In most surveys, the strata are chosen in advance and thus fixed. The
correction is straightforward; see \nocite*{Cochran1977} for a textbook-level
treatment. Therefore, we mainly discuss how to extend our methods,
especially the intersection approach, to clustered data. Cluster-robust
inference has become increasingly popular over the past decade and is now
employed routinely in many empirical microeconomics studies; see \nocite*
{MacKinnon2023d} for a recent survey. There are at least two ways to
incorporate clusters into our framework.
\textit{First}, both samples in Assumption \ref{assump:DS} can be
generalized to clustered data. In general, the intra-cluster correlation is
positive, so when clusters are large, incorrectly assuming IID observations
can lead to smaller asymptotic variances and thus unreliable inference
(over-rejection of tests or insufficient coverage of CIs) [\nocite*
{MacKinnon2016a}].
Two distinct types of assumptions are available for the asymptotic theory of
cluster-robust inference. The most common method is the "large number of
clusters" approach, where the number of clusters, often denoted by $G$,
tends to infinity [\nocite*{Djogbenou2019} and \nocite*{Hansen2019}]. The other is
the fixed-$G$ or "small number of large clusters" method [\nocite*
{Ibragimov2010} and \nocite*{Ibragimov2016}]. The bootstrap methods, especially
the wild cluster restricted (WCR) bootstrap, often yield much more reliable
inferences than asymptotic procedures [\nocite*{MacKinnon2016} and \nocite*
{MacKinnon2023c}]. The WCR bootstrap is usually based on the Rademacher
distribution [\nocite*{Davidson2008a} and \nocite*{Djogbenou2019}]. Nevertheless,
when the number of clusters is small, a six-point distribution is preferred [
\nocite*{Webb2023}].
We may further extend to a multi-stage sampling design using techniques in
\nocite*{Zheng2002} or \nocite*{Bhattacharya2005}.
\textit{Second}, the clustered data can generate dependence between the two
samples. To see it, suppose that there are $G$ groups, with group $g$
containing a sample of $n_{g}$ pairs of observations $(X_{1, gi}, X_{2, gi})$
. One may conduct cluster-robust inference using either the "large number of
clusters" or the "small number of large clusters" approach. However, the
intersection methods (IMs) are also applicable. Furthermore, we can still
employ IMs even when asymptotic inference can fail.
One example is cluster heterogeneity, which we can demonstrate by the
simulation experiment under \textbf{DGP II-A}. We may view such a DGP for
the "large number of clusters" approach with $G = n$. That is, each group
contains only one pair of observations. Note that, due to sorting, the
correlation within each pair varies dramatically across groups, except for $
{\Greekmath 011A} = 0.95$, the highly positively correlated case. As reported in Table
\ref{tab:CI-dMean-DGP-B1}, the OS-based approach, which coincides trivially
with the cluster-robust inference, fails to provide reliable CIs. However,
the IM approach remains valid.
\FloatBarrier
\newpage
\subsection{\huge Conclusion \label{sec:conclusion}}
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Traditional inference methods for comparing welfare indices typically assume
that samples are independent. However, this assumption can be problematic
because income samples are often dependent due to the overlap between
consecutive periods.
This paper considers a broad family of indices expressed as asymptotically
linear Gaussian functionals and proposes inference methods for index
differences with dependent samples. We first study generic dependent samples
and provide asymptotic and bootstrap intersection methods that are robust to
arbitrary sample dependence. We then explore the overlapping samples, a
specific type of dependent samples where the dependence arises solely from
the matched pairs, and propose asymptotic and bootstrap confidence intervals
for changes in indices. We additionally justify using influence functions to
estimate the asymptotic variance and covariance consistently.
We evaluate the performance of the proposed methods through a series of
simulation experiments. First, we find that sample dependence can
substantially impact asymptotic variance. Therefore, our proposed methods,
which utilize overlapping samples, can be more efficient than conventional
procedures based on independent samples. Next, we analyze the effects of
heavy tails and find that the asymptotic inference performs well for
realistic distributions but poorly for distributions with heavy tails. The
bootstrap method can alleviate this issue, except when the variance is
substantial or nonexistent. Finally, the intersection methods prove to be
reliable under generic dependent samples, particularly when the assumption
of overlapping samples is inappropriate. The intersection methods are also
reasonably efficient in certain instances, such as those involving extremely
heavy-tailed distributions.
To illustrate our methods, we applied them to analyze the changes in
financial inequality among Italian households from $2012$ to $2014$. The
application also highlights the practical importance of using extended
inequality measures in the presence of negative values.
We finally briefly discuss potential extension to clustered data and
multi-stage survey sampling. Moreover, we demonstrate that the intersection
methods are also robust to some cases where cluster-robust inference fails.
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