Almost Dominance: Inference and Application

bibunit\begin{abstract} This paper proposes a general framework for inference on three types of almost dominances: almost Lorenz dominance, almost inverse stochastic dominance, and almost stochastic dominance. We first generalize almost Lorenz dominance to almost upward and downward Lorenz dominances. We then provide a bootstrap inference procedure for the Lorenz dominance coefficients, which measure the degrees of almost Lorenz dominance. Furthermore, we propose almost upward and downward inverse stochastic dominances and provide inference on the inverse stochastic dominance coefficients. We also show that our results can easily be extended to almost stochastic dominance. Simulation studies demonstrate the finite sample properties of the proposed estimators and the bootstrap confidence intervals. This framework can be applied to economic analysis, particularly in the areas of social welfare, inequality, and decision making under uncertainty. As an empirical example, we apply the methods to the inequality growth in the United Kingdom and find evidence for almost upward inverse stochastic dominance. Keywords: almost Lorenz dominance, almost inverse stochastic dominance, almost stochastic dominance, estimation and inference, bootstrap confidence intervals \end{abstract} \section{Introduction} Suppose that there are two arbitrary cumulative distribution functions (CDFs) $F_{1}:[0,\infty)\to[0,1]$ and $F_{2}:[0,\infty)\to[0,1]$ of two income (or wealth, etc.) distributions in two populations. As introduced in atkinson1970measurement, a Lorenz curve is a function that graphs the cumulative proportion of total income received by the bottom population. The distribution $F_1$ (weakly) Lorenz dominates $F_2$ if the Lorenz curve $L_1$ associated with $F_1$ is everywhere above the Lorenz curve $L_2$ associated with $F_2$. Such Lorenz dominance implies that the wealth is distributed more equally in population $F_1$ compared to $F_2$. Statistical tests of Lorenz dominance can be found in, for example, mcfadden1989testing, bishop1991international, bishop1991lorenz, dardanoni1999inference, davidson2000statistical, barrett2003consistent, BDB14, and Beare2017improved. zheng2018almost introduces the notion of almost Lorenz dominance: When two Lorenz curves cross, $F_1$ almost Lorenz dominates $F_2$ if $L_1$ is above $L_2$ almost everywhere. The degree of almost Lorenz dominance can be measured by a coefficient, Lorenz dominance coefficient (LDC). The LDC was first proposed in zheng2018almost,\footnote{The original definition of LDC in zheng2018almost is slightly different from the proposed one in this paper.} which is similar to the violation ratio in huang2021estimating. zheng2018almost shows that almost Lorenz dominance is highly related to Gini-type measures. Based on the seminal work of aaberge2009ranking, we generalize almost Lorenz dominance to almost upward Lorenz dominance and almost downward Lorenz dominance. We relate the generalized LDCs to a class of inequality measures. It is straightforward to show that the smaller LDCs are, the more inequality measures demonstrate higher distribution equality in the dominant population. Thus, LDCs are of interest to us when we compare distributions in terms of distribution inequality. We then provide estimators for LDCs and establish the asymptotic properties of the estimators.\footnote{The proposed estimators are also different from that proposed by zheng2018almost since the definitions of the proposed LDCs are different.} Based on the asymptotic distributions, we construct bootstrap confidence intervals (CIs) for LDCs. LDCs are nonlinear transformations of the difference functions of two Lorenz curves. We show that this map is Hadamard directionally differentiable, and then we apply the bootstrap method of fang2014inference to approximate the asymptotic distributions of the estimators and obtain the confidence intervals.\footnote{For more discussions on Hadamard directional differentiability and its applications, see dumbgen1993nondifferentiable, andrews2000inconsistency, bickel2012resampling, hirano2012impossibility, beare2015nonparametric, Beare2016global, hansen2017regression, Seo2016tests, Beare2015improved, chen2019improved, Beare2017improved, and sun2018ivvalidity.} The second degree inverse stochastic dominance atkinson1970measurement, also known as generalized Lorenz dominance, is widely used for ranking distribution functions according to social welfare. However, as pointed out by aaberge2021ranking, when the distribution functions intersect, the second degree inverse dominance criterion may have limitations in attaining an unambiguous ranking davies1995making,atkinson2008more. aaberge2021ranking propose an approach for comparing intersecting distribution functions based on high-degree upward and downward inverse stochastic dominance criteria. They provide equivalence results between inverse stochastic dominance and the ranks of social welfare functions. Based on the framework of aaberge2021ranking, we introduce the notions of almost upward inverse stochastic dominance and \emph{almost downward inverse stochastic dominance}. We then show that these types of almost dominances are also related to the ranks of social welfare functions. Similar to LDCs, the degree of almost inverse stochastic dominance can be measured by the \emph{inverse stochastic dominance coefficients} (ISDCs). The smaller the ISDCs are, the more social planner preference functions show higher social welfare for the dominant distribution. We provide inference and construct bootstrap confidence intervals for ISDCs. Almost Lorenz dominance can be viewed as a special but nontrivial case of almost stochastic dominance, which was first introduced by leshno2002preferred. Almost stochastic dominance has been extensively studied by guo2013note, tzeng2013revisiting, denuit2014almost, guo2014moment, tsetlin2015generalized, guo2016almost, and huang2021estimating.\footnote{See a comprehensive review on stochastic dominance in whang2019econometric.} We show that our method for almost Lorenz dominance can be easily generalized to almost stochastic dominance and provide bootstrap confidence intervals for the \emph{stochastic dominance coefficients} (SDCs). We examine the inequality growth in the United Kingdom as an empirical example in the paper. We find that the estimated upward inverse stochastic dominance coefficients are small and the bootstrap confidence intervals are narrow, which implies that there may exist almost inverse stochastic dominance between the years considered in this example. For the sake of simplicity of exposition, we focus on the almost Lorenz dominances in the main text. The results for other types of almost dominances are provided in the supplementary appendix. The paper is organized as follows. Section (ref) introduces the notions of Lorenz dominance coefficients, provides the properties of these coefficients, and relates them to measures of inequality. Section (ref) proposes estimators for LDCs and establishes the asymptotic distributions of the estimators. We then construct the confidence intervals for LDCs based on these asymptotic distributions. Section (ref) provides simulation evidence for the finite sample properties of the proposed method. In Section (ref), we apply our methods to the empirical application. In the supplementary appendix, Sections (ref) and (ref) extend the results to almost inverse stochastic dominance and almost stochastic dominance. Section (ref) provides additional simulation results. Section (ref) contains the proofs of the results in the paper. {\textbf{Notation Beare2017improved:}} Throughout the paper, all the random elements are defined on a probability space $( \Omega, \mathcal{A}, \mathbb{P} )$. For an arbitrary set $A$, let $\ell^\infty(A)$ be the set of bounded real-valued functions on $A$ equipped with the supremum norm $\Vert \cdot \Vert_{\infty}$ such that $\Vert f \Vert_{\infty}=\sup_{x\in A} \vert f(x) \vert$ for every $f\in \ell^\infty(A)$. For a subset $B$ of a metric space, let $C(B)$ be the set of continuous real-valued functions on $B$. Let $\leadsto$ denote the weak convergence defined in van1996weak. We follow the convention in folland2013real and define \begin{align} 0\cdot\infty=0. \end{align} \section{Almost Lorenz Dominance} We suppose that the CDFs $F_{1}$ and $F_{2}$ satisfy the following regularity conditions, which guarantee the weak convergence of the estimated quantile functions K17,Beare2017improved. \begin{assumption} For $j=1,2$, $F_{j}\left( 0\right) =0$ and $F_{j}$ is continuously differentiable on the interior of its support with $F_{j}^{\prime}\left( x\right) >0$ for all $x\in\left( 0,\infty\right) $. In addition, $F_{j}$ has finite $\left( 2+\epsilon\right) $th absolute moment for some $\epsilon>0$. \end{assumption} With Assumption (ref), we introduce the Lorenz curves. Let $Q_1$ and $Q_2$ denote the quantile functions corresponding to $F_1$ and $F_2$, respectively, that is, \begin{align} Q_j(p)&=\inf\left\{x\in [0,\infty):F_j(x)\ge p \right\},\quad p\in[0,1]. \end{align} When $F_j$ has finite first moment $\mu_j$ as implied by Assumption (ref), the quantile function $Q_j$ is integrable with $\int_0^1 Q_j(p)\mathrm{d}p=\mu_j$. Under Assumption (ref) with $\mu_j>0$, $Q_j(p)=F_j^{-1}(p)$ for all $p\in(0,1)$, and the Lorenz curve $L_j$ corresponding to $F_j$ can then be defined as \begin{align} L_{j}(p)&=\frac{1}{\mu_{j}}\int_{0}^{p}Q_{j}(t)\mathrm{d}t,\quad p\in[0,1]. \end{align} We say $F_1$ (weakly) Lorenz dominates $F_2$ if $L_1(p)\ge L_2(p)$ for all $p\in[0,1]$. We now follow zheng2018almost and introduce the notion of almost Lorenz dominance. For all arbitrary CDFs $F_{1}:[0,\infty)\to[0,1]$ and $F_{2}:[0,\infty)\to[0,1]$ such that the corresponding Lorenz curves exist, define \[ S\left( F_{1},F_{2}\right) =\left\{ p\in\left[ 0,1\right] :L_{1}\left( p\right) <L_{2}\left( p\right) \right\} . \] \begin{definition} For every $\varepsilon\in[ 0,1/2) $, \emph{the CDF $F_{1}$ $\varepsilon$-almost Lorenz dominates the CDF $F_{2}$} ($F_{1}$ $\varepsilon $-ALD $F_{2}$), if \begin{align} \int_{S\left( F_{1},F_{2}\right) }\left( L_{2}\left( p\right) -L_{1}\left( p\right) \right) \mathrm{d}p\leq\varepsilon\int_{0} ^{1}\left\vert L_{2}\left( p\right) -L_{1}\left( p\right) \right\vert \mathrm{d}p. \end{align} \end{definition} \begin{remark} zheng2018almost defines the almost Lorenz dominance for all $\varepsilon\in(0,1/2)$, while we allow $\varepsilon=0$. The inequality (ref) holds with $\varepsilon=0$ if and only if $F_1$ Lorenz dominates $F_2$. Thus, by Definition (ref), almost Lorenz dominance is a weaker and generalized version of Lorenz dominance. \end{remark} \begin{lemma} For every $\varepsilon\in [0,1/2)$, $F_{1}$ $\varepsilon$-ALD $F_{2}$ if and only if there is $c\in\left[ 0,\varepsilon\right] $ such that \begin{align} \int_{S\left( F_{1},F_{2}\right) }\left( L_{2}\left( p\right) -L_{1}\left( p\right) \right) \mathrm{d}p=c\int_{0}^{1}\left\vert L_{2}\left( p\right) -L_{1}\left( p\right) \right\vert \mathrm{d}p. \end{align} \end{lemma} We define the distance function of the two Lorenz curves by \begin{align} \phi\left( p\right) =L_{2}\left( p\right) -L_{1}\left( p\right),\quad p\in[0,1]. \end{align} Based on the equality in (ref), we follow zheng2018almost and define the \emph{$F_1$-$F_2$ Lorenz Dominance Coefficient} as follows.\footnote{In Section 2.3 of zheng2018almost, the Lorenz dominance coefficient is defined in a slightly different way.} \begin{definition} Suppose that there are two Lorenz curves $L_1$ and $L_2$ corresponding to the CDFs $F_1$ and $F_2$, respectively. The \emph{$F_1$-$F_2$ Lorenz dominance coefficient ($F_1$-$F_2$ LDC)}, denoted by $c(L_1,L_2)$, is defined as \begin{align} c(L_1,L_2)=\inf\left\{\varepsilon\in[0,1]:\int_{S\left( F_{1},F_{2}\right) }\left( L_{2}\left( p\right) -L_{1}\left( p\right) \right) \mathrm{d}p\leq\varepsilon\int_{0} ^{1}\left\vert L_{2}\left( p\right) -L_{1}\left( p\right) \right\vert \mathrm{d}p\right\}. \end{align} \end{definition} The following lemma summarizes the properties of the LDC $c(L_1,L_2)$. \begin{lemma} The LDC $c(L_1,L_2)$ is the smallest $\varepsilon$ in $[0,1]$ such that (ref) holds, $c(L_1,L_2)=0$ if and only if $F_1$ Lorenz dominates $F_2$, and $c(L_1,L_2)=1$ implies that $F_2$ Lorenz dominates $F_1$. With (ref), it follows that \begin{align} c(L_1,L_2)=\frac{\int_{S\left( F_{1},F_{2}\right) }\left( L_{2}\left( p\right) -L_{1}\left( p\right) \right) \mathrm{d}p}{\int_{0}^{1}\left\vert L_{2}\left( p\right) -L_{1}\left( p\right) \right\vert \mathrm{d}p}=\frac{\int_{0}^{1}\max\left\{ \phi\left( p\right) ,0\right\} \mathrm{d}p}{\int_{0}^{1}\max\left\{ \phi\left( p\right) ,0\right\} \mathrm{d}p+\int_{0}^{1}\max\left\{ -\phi\left( p\right) ,0\right\} \mathrm{d}p}. \end{align} In addition, if $c(L_1,L_2)\in(0,1]$, then $c(L_2,L_1)=1-c(L_1,L_2)$. \end{lemma} According to Lemma (ref), $F_1$ $\varepsilon$-ALD $F_2$ for all $\varepsilon\in[c(L_1,L_2),1/2)$ if $c(L_1,L_2)<1/2$. On the other hand, $c(L_1,L_2)>1/2$ implies that $F_2$ $\varepsilon$-ALD $F_1$ for all $\varepsilon\in[1-c(L_1,L_2),1/2)$. Thus, $c(L_1,L_2)$ presents the almost Lorenz dominance relationship between $F_1$ and $F_2$, and provides all $\varepsilon$ for which $\varepsilon$-ALD holds. When $L_1$ and $L_2$ cross, given some $\varepsilon\in(0,1/2)$, one interesting hypothesis is \begin{align*} \mathrm{H}_0: \int_{S\left( F_{1},F_{2}\right) }\left( L_{2}\left( p\right) -L_{1}\left( p\right) \right) \mathrm{d}p\leq\varepsilon\int_{0} ^{1}\left\vert L_{2}\left( p\right) -L_{1}\left( p\right) \right\vert \mathrm{d}p. \end{align*} Lemma (ref) shows that it is equivalent to test the following hypothesis on $c(L_1,L_2)$: \begin{align*} \mathrm{H}_0: c(L_1,L_2)\le \varepsilon. \end{align*} The LDC provided in Definition (ref) is closely related to Gini-type measures that are weighted averages of the area between the diagonal line and Lorenz curves. Following shorrocks2002approximating, for every possible nonnegative weighting function $\theta$, a Gini-type measure for distribution $F$ can be defined as \begin{align} I(F,\theta)=\frac{\int_0^1 [p-L_F(p)]\theta(p)\mathrm{d}p}{\int_0^1 p\theta(p)\mathrm{d}p}, \quad p\in[0,1], \end{align} where $L_F$ denotes the Lorenz curve associated with the distribution $F$. We follow zheng2018almost and denote $\mathscr{B}$ the set of all Gini-type measures for all possible weighting functions $\theta$. Define \[ \tilde{\theta}\left( p\right) =\theta\left( p\right) /\int_{0}^{1} p\theta\left( p\right) \mathrm{d}p \] for every possible weighting function $\theta$. It then follows that \[ I\left( F,\theta\right) =\int_{0}^{1}\left[ p-L_{F}\left( p\right) \right] \tilde{\theta}\left( p\right) \mathrm{d}p. \] For every $0<\varepsilon<1/2$, we define \begin{align} \mathscr{B}^{\ast}\left( \varepsilon\right) =\left\{ I\left( \cdot ,\theta\right) :\tilde{\theta}>0\text{ and }\sup_{p\in\left[ 0,1\right] }\left\{ \tilde{\theta}\left( p\right) \right\} \leq\inf_{p\in\left[ 0,1\right] }\left\{ \tilde{\theta}\left( p\right) \right\} \cdot\left( \frac{1}{\varepsilon}-1\right) \right\} . \end{align} As discussed in zheng2018almost, the condition in (ref) essentially requires that the largest weight in weighting the distance $p-L_F(p)$ cannot be larger than $(1/\varepsilon-1)$ times the smallest weight. Proposition 1 of zheng2018almost shows that for every $\varepsilon\in\left( 0,1/2\right) $, $F$ $\varepsilon$-ALD $G$ if and only if $I\left( F,\theta\right) \leq I\left( G,\theta\right) $ for all $I\left( \cdot,\theta\right) \in\mathscr{B}^{\ast}\left( \varepsilon\right) $. According to Lemma (ref), for every $\varepsilon\geq c\left( L_{1},L_{2}\right) $ with $c\left( L_{1},L_{2}\right) \in(0,1/2)$, $F$ $\varepsilon$-ALD $G$ and thus $I\left( F,\theta\right) \leq I\left( G,\theta\right) $ for all $I\left( \cdot,\theta\right) \in\mathscr{B}^{\ast }\left( \varepsilon\right) $. Therefore, we have the following proposition. \begin{proposition} If $c\left( L_{1},L_{2}\right) \in(0,1/2)$, it follows that $I\left( F_1,\theta\right) \leq I\left( F_2,\theta\right)$ \text{ for all Gini-type measures} $I\left(\cdot,\theta\right) \in\cup_{\varepsilon\in[c\left( L_{1},L_{2}\right),1/2)}\mathscr{B}^{\ast}\left( \varepsilon\right)=\mathscr{B}^{\ast}\left( c\left( L_{1},L_{2}\right) \right)$. If $c\left( L_{1},L_{2}\right)=0$, it follows that $ I\left( F_1,\theta\right) \leq I\left( F_2,\theta\right) \text{ for all } I\left(\cdot,\theta\right) \in\cup_{\varepsilon\in(0,1/2)}\mathscr{B}^{\ast}\left( \varepsilon\right)$. \end{proposition} Proposition (ref) highlights the importance of LDC in relation to Gini-type measures. The smaller the LDC $c(L_1,L_2)$ is, the more Gini-type measures show higher equality in the distribution $F_1$ compared to $F_2$. With the knowledge of $c(L_1,L_2)$, we can infer the relationship between $F_1$ and $F_2$ based on a class of Gini-type measures. Let $P:[0,1]\to[0,1]$ be a continuous, differentiable, and concave preference function employed by a social decision maker that assigns weights to individual incomes according to their ranks in the income distribution yaari1988controversial,aaberge2001axiomatic,aaberge2009ranking. Following aaberge2009ranking, we introduce the family of rank-dependent measures of inequality: \begin{align} J_{P}(L_{j})=1-\int_{0}^{1}P^{\prime}\left( t\right) \mathrm{d}L_{j}\left( t\right) =1-\frac{1}{\mu_{j}}\int_{0}^{1}P^{\prime}\left( t\right) Q_{j}\left( t\right) \mathrm{d}t, \end{align} where $L_{j}$ is the Lorenz curve of the distribution $F_{j}$ with mean $\mu_{j}$ and the weighting function $P^{\prime}$ is the derivative of the preference function $P$. As illustrated by aaberge2001axiomatic, $J_P(L)$ measures the degree of inequality within an income distribution, as represented by the Lorenz curve $L$. We follow aaberge2009ranking and formally {define} the set $\mathrm{P}$ of preference functions by \[ \mathrm{P}=\left\{ \begin{array} [c]{c} P\in C([0,1]):P\left( 0\right) =0,P\left( 1\right) =1,P^{\prime}\left( 1\right) =0,\\ P^{\prime}\left( t\right) >0\text{ and }P^{\left( 2\right) }\left( t\right) <0\text{ for all }t\in\left( 0,1\right) \end{array} \right\} . \] \subsection{Almost Upward Lorenz Dominance and Inequality Measure} To unify notation, we let $L_j^1=L_j$ for $j=1,2$. For $m\ge 2$, define function \[ L_{j}^{m}\left( p\right) =\int_{0}^{p}L_j^{m-1}\left( t\right) \mathrm{d}t,\quad p\in\left[ 0,1\right] . \] aaberge2009ranking introduces the $m$th-degree upward Lorenz dominance for $m\ge2$ as illustrated in the following. \begin{definition} A distribution $F_{1}$ $m$th-degree upward Lorenz dominates a distribution $F_{2}$ if $L_{1}^{m}\left( p\right) \geq L_{2}^{m}\left( p\right) $ for all $p\in\left[ 0,1\right] $. \end{definition} We generalize upward Lorenz dominance in aaberge2009ranking to almost upward Lorenz dominance. \begin{definition} For every $\varepsilon_m\in [ 0,1/2) $, the distribution $F_{1}$ $\varepsilon_{m}$-almost $m$th-degree upward Lorenz dominates the distribution $F_{2}$ ($F_{1}$ $\varepsilon_{m}$-A$m$ULD $F_{2}$), if \begin{align} \int_{0}^{1}\max\left( L_{2}^{m}\left( p\right) -L_{1}^{m}\left( p\right) ,0\right) \mathrm{d}p\leq\varepsilon_{m}\int_{0}^{1}\left\vert L_{2} ^{m}\left( p\right) -L_{1}^{m}\left( p\right) \right\vert \mathrm{d}p. \end{align} \end{definition} \begin{remark} In Definition (ref), if $\varepsilon_m=0$, then $F_1$ $m$th-degree upward Lorenz dominates $F_2$. \end{remark} For every $\varepsilon_{m}\in\left( 0,1/2\right) $, we define two sets of preference functions \[ \mathrm{P}_{m}\left( \varepsilon_{m}\right) =\left\{ \begin{array} [c]{c} P\in\mathrm{P}:\left( -1\right) ^{m}P^{\left( m+1\right) }\geq0,P^{\left( j\right) }\left( 1\right) =0\text{ for }j\in\{2,\ldots, m\},\\ \sup_{t}\left\{ \left( -1\right) ^{m}P^{\left( m+1\right) }\left( t\right) \right\} \leq\inf_{t}\left\{ \left( -1\right) ^{m}P^{\left( m+1\right) }\left( t\right) \right\} \cdot\left( \frac{1}{\varepsilon _{m}}-1\right) \end{array} \right\} \] and \[ \mathrm{P}_{m}^{\prime}\left( \varepsilon_{m}\right) =\left\{ \begin{array} [c]{c} P\in\mathrm{P}:\left( -1\right) ^{m}P^{\left( m+1\right) }\geq0,\left( -1\right) ^{j-1}P^{\left( j\right) }\left( 1\right) \geq0\text{ for }j\in\{2,\ldots, m\},\\ \sup_{t}\left\{ \left( -1\right) ^{m}P^{\left( m+1\right) }\left( t\right) \right\} \leq\inf_{t}\left\{ \left( -1\right) ^{m}P^{\left( m+1\right) }\left( t\right) \right\} \cdot\left( \frac{1}{\varepsilon _{m}}-1\right) \end{array} \right\} . \] The sets $\mathrm{P}_m(\varepsilon_m)$ and $\mathrm{P}'_m(\varepsilon_m)$ are constructed based on aaberge2009ranking. As discussed in aaberge2001axiomatic, risk aversion is equivalent to second degree stochastic dominance and requires the concavity of the utility function. Theorem 2.1 of aaberge2009ranking shows that the concavity on $P$ is related to first degree Lorenz dominance and “inequality aversion” aaberge2009ranking. The requirement $(-1)^mP^{(m+1)}\ge0$ in $\mathrm{P}_m(\varepsilon_m)$ and $\mathrm{P}'_m(\varepsilon_m)$ may be viewed as an extension of the concavity condition to higher degree upward dominance. Moreover, as shown in the proof of Proposition 2.2 aaberge2009ranking, when $J_P(L_2)-J_P(L_1)$ is written as a function of $L^m_1-L^m_2$, $(-1)^mP^{(m+1)}$ plays a role of the weighting function. We require it to be nonnegative in $\mathrm{P}_m(\varepsilon_m)$ and $\mathrm{P}'_m(\varepsilon_m)$. The following proposition relates almost upward Lorenz dominance to the inequality measures $J_P$ in (ref) for $P$ in $\mathrm{P}_m(\varepsilon_m)$ and $\mathrm{P}'_m(\varepsilon_m)$. \begin{proposition} For every $\varepsilon_{m}\in\left( 0,1/2\right) $, if $F_{1}$ $\varepsilon_{m}$-A$m$ULD $F_{2}$, then $J_{P}\left( L_{1}\right) \leq J_{P}\left( L_{2}\right) $ for every $P\in\mathrm{P}_{m}\left( \varepsilon_{m}\right) $. If $F_{1}$ $\varepsilon_{m}$-A$m$ULD $F_{2}$ and $L_{1}^{j}\left( 1\right) \geq L_{2}^{j}\left( 1\right) $ for all $j\in\left\{ 2,\ldots,m\right\} $, then $J_{P}\left( L_{1}\right) \leq J_{P}\left( L_{2}\right) $ for every $P\in\mathrm{P}_{m}^{\prime}\left( \varepsilon_{m}\right) $. \end{proposition} For $m\ge 2$, following Lemma (ref), it is straightforward to show that $F_{1}$ $\varepsilon_{m}$-A$m$ULD $F_{2}$ if and only if there is $c_{m}\in\left[ 0,\varepsilon_{m}\right] $ such that \[ \int_{0}^{1}\max\left( L_{2}^{m}\left( p\right) -L_{1}^{m}\left( p\right) ,0\right) \mathrm{d}p= c_{m}\int_{0}^{1}\left\vert L_{2}^{m}\left( p\right) -L_{1}^{m}\left( p\right) \right\vert \mathrm{d}p. \] Similar to (ref), we define the $m$th-degree upward Lorenz dominance coefficient $c_m^u(L_1,L_2)$, where the superscript “$u$” represents “upward”. \begin{definition} For $m\ge2$, the \emph{$F_1$-$F_2$ $m$th-degree upward Lorenz dominance coefficient ($F_1$-$F_2$ $m$ULDC)} $c_m^u(L_1,L_2)$ is defined as \begin{align} &c_m^u(L_1,L_2)=\notag\\ &\inf\left\{\varepsilon_m\in[0,1]:\int_0^1\max\left( L_{2}^{m}\left( p\right) -L_{1}^{m }\left( p\right),0 \right) \mathrm{d}p\leq\varepsilon_{m}\int_{0}^{1}\left\vert L_{2}^{m}\left(p \right) -L_{1}^{m}\left( p\right) \right\vert \mathrm{d}p\right\}. \end{align} \end{definition} For $m\ge2$, define the difference function \[ \phi_m^u\left( p\right) =L_{2}^{m}\left( p\right) -L_{1}^{m}\left( p\right),\quad p\in[0,1]. \] The following lemma summarizes the properties of the $m$ULDC $c_m^u(L_1,L_2)$. \begin{lemma} The $m$ULDC $c_m^u(L_1,L_2)$ is the smallest $\varepsilon_m$ in $[0,1]$ such that (ref) holds. With (ref), it follows that \begin{align} c_m^u(L_1,L_2)&=\frac{\int_0^1\max\left( L_{2}^{m}\left( p\right) -L_{1}^{m}\left( p\right) ,0 \right) \mathrm{d}p}{\int_{0}^{1}\left\vert L_{2}^{m}\left( p\right) -L_{1}^{m}\left( p\right) \right\vert \mathrm{d}p}\notag\\ &=\frac{\int_{0}^{1}\max\left\{ \phi_m^u\left( p\right) ,0\right\} \mathrm{d}p}{\int_{0}^{1}\max\left\{ \phi_m^u\left( p\right) ,0\right\} \mathrm{d}p+\int_{0}^{1}\max\left\{ -\phi_m^u\left( p\right) ,0\right\} \mathrm{d}p}. \end{align} In addition, if $c_m^u(L_1,L_2)\in(0,1]$, then $c_m^u(L_2,L_1)=1-c_m^u(L_1,L_2)$. \end{lemma} According to Lemma (ref), $F_1$ $\varepsilon_m$-A$m$ULD $F_2$ for all $\varepsilon_m\in[c_m^u(L_1,L_2),1/2)$ if $c_m^u(L_1,L_2)<1/2$. On the other hand, $c_m^u(L_1,L_2)>1/2$ implies that $F_2$ $\varepsilon_m$-A$m$ULD $F_1$ for all $\varepsilon_m\in[1-c_m^u(L_1,L_2),1/2)$. Thus, $c_m^u(L_1,L_2)$ presents the degree of almost upward Lorenz dominance relationship between $F_1$ and $F_2$, and provides all $\varepsilon_m$ such that the $\varepsilon_m$-A$m$ULD holds. \begin{proposition} If $c_m^u(L_1,L_2)\in(0,1/2)$, it then follows that $J_P(L_1)\le J_P(L_2)$ for all preference functions $P\in \cup_{\varepsilon_m\in[c_m^u(L_1,L_2),1/2)}\mathrm{P}_m(\varepsilon_m)$. If, in addition, $L_{1}^{j}\left( 1\right) \geq L_{2}^{j}\left( 1\right) $ for all $j\in\left\{ 2,\ldots,m\right\} $, then $J_{P}\left( L_{1}\right) \leq J_{P}\left( L_{2}\right) $ for all $P\in\cup_{\varepsilon_m\in[c_m^u(L_1,L_2),1/2)}\mathrm{P}_{m}^{\prime}\left( \varepsilon_{m}\right) $. \end{proposition} Similar to Proposition (ref), Proposition (ref) shows the importance of ULDC concerning inequality measures $J_P$. The smaller the ULDC $c_m^u(L_1,L_2)$ is, the more inequality measures show higher equality in the distribution $F_1$ compared to $F_2$. With the knowledge of $c_m^u(L_1,L_2)$, we can infer the relationship between $F_1$ and $F_2$ based on a class of inequality measures. \subsection{Almost Downward Lorenz Dominance and Inequality Measure} Define function \[ \tilde{L}_{j}^{2}\left( p\right) =\int_{p}^{1}\left( 1-L_{j}\left( t\right) \right) \mathrm{d}t=\frac{1}{\mu_{j}}\int_{p}^{1}\left( t-p\right) Q_j\left( t\right) \mathrm{d}t,\quad p\in\left[ 0,1\right], \] and for $m\geq3$, \[ \tilde{L}_{j}^{m}\left( p\right) =\int_{p}^{1}\tilde{L}_j^{m-1}\left( t\right) \mathrm{d}t,\quad p\in\left[ 0,1\right] . \] aaberge2009ranking introduces the $m$th-degree downward Lorenz dominance for $m\ge2$ as illustrated in the following. \begin{definition} A distribution $F_{1}$ $m$th-degree downward Lorenz dominates a distribution $F_{2}$ if $\tilde{L}_{1}^{m}\left( p\right) \le \tilde{L}_{2}^{m}\left( p\right) $ for all $p\in\left[ 0,1\right] $. \end{definition} We generalize downward Lorenz dominance in aaberge2009ranking to almost downward Lorenz dominance. \begin{definition} For every $\varepsilon_m\in[ 0,1/2) $, the distribution $F_{1}$ $\varepsilon_{m}$-almost $m$th-degree downward Lorenz dominates the distribution $F_{2}$ ($F_{1}$ $\varepsilon_{m}$-A$m$DLD $F_{2}$), if \begin{align} \int_{0}^{1}\max\left( \tilde{L}_{1}^{m}\left( p\right) -\tilde{L}_{2} ^{m}\left( p\right) ,0\right) \mathrm{d}p\leq\varepsilon_{m}\int_{0} ^{1}\left\vert \tilde{L}_{1}^{m}\left( p\right) -\tilde{L}_{2}^{m}\left( p\right) \right\vert \mathrm{d}p. \end{align} \end{definition} \begin{remark} In Definition (ref), if $\varepsilon_m=0$, then $F_1$ $m$th-degree downward Lorenz dominates $F_2$. \end{remark} For every $\varepsilon_{m}\in\left( 0,1/2\right) $, we define \[ \mathrm{\tilde{P}}_{m}\left( \varepsilon_{m}\right) =\left\{ \begin{array} [c]{c} P\in\mathrm{P}:P^{\left( m+1\right) }\leq0,P^{\left( j\right) }\left( 0\right) =0\text{ for }j\in\{2,\ldots, m\},\\ \sup_{t}\left\{ -P^{\left( m+1\right) }\left( t\right) \right\} \leq \inf_{t}\left\{ -P^{\left( m+1\right) }\left( t\right) \right\} \cdot\left( \frac{1}{\varepsilon_{m}}-1\right) \end{array} \right\} \] and \[ \mathrm{\tilde{P}}_{m}^{\prime}\left( \varepsilon_{m}\right) =\left\{ \begin{array} [c]{c} P\in\mathrm{P}:P^{\left( m+1\right) }\leq0,P^{\left( j\right) }\left( 0\right) \leq0\text{ for }j\in\{2,\ldots, m\},\\ \sup_{t}\left\{ -P^{\left( m+1\right) }\left( t\right) \right\} \leq \inf_{t}\left\{ -P^{\left( m+1\right) }\left( t\right) \right\} \cdot\left( \frac{1}{\varepsilon_{m}}-1\right) \end{array} \right\} . \] The sets $\tilde{\mathrm{P}}_m(\varepsilon_m)$ and $\tilde{\mathrm{P}}'_m(\varepsilon_m)$ are constructed based on aaberge2009ranking. The requirement $P^{(m+1)}\le0$ in $\tilde{\mathrm{P}}_m(\varepsilon_m)$ and $\tilde{\mathrm{P}}'_m(\varepsilon_m)$ may be viewed as an extension of the concavity condition to higher degree downward dominance. Moreover, as shown in the proof of Proposition (ref), when $J_P(L_2)-J_P(L_1)$ is written as a function of $\tilde{L}^m_2-\tilde{L}^m_1$, $-P^{(m+1)}$ plays a role of the weighting function. We require it to be nonnegative in $\tilde{\mathrm{P}}_m(\varepsilon_m)$ and $\tilde{\mathrm{P}}'_m(\varepsilon_m)$. The following proposition relates almost downward Lorenz dominance to the inequality measures $J_P$ in (ref) for $P$ in $\tilde{\mathrm{P}}_m(\varepsilon_m)$ and $\tilde{\mathrm{P}}'_m(\varepsilon_m)$. \begin{proposition} For every $\varepsilon_{m}\in\left( 0,1/2\right) $, if $F_{1}$ $\varepsilon_{m}$-A$m$DLD $F_{2}$, then $J_{P}\left( L_{1}\right) \leq J_{P}\left( L_{2}\right) $ for every $P\in\mathrm{\tilde{P}}_{m}\left( \varepsilon_{m}\right) $. If $F_{1}$ $\varepsilon_{m}$-A$m$DLD $F_{2}$ and $\tilde{L}_{1}^{j}\left( 0\right) \leq\tilde{L}_{2}^{j}\left( 0\right) $ for all $j\in\left\{ 2,\ldots,m\right\} $, then $J_{P}\left( L_{1}\right) \leq J_{P}\left( L_{2}\right) $ for every $P\in\mathrm{\tilde{P}} _{m}^{\prime}\left( \varepsilon_{m}\right) $. \end{proposition} For $m\ge 2$, following Lemma (ref), it is straightforward to show that $F_{1}$ $\varepsilon_{m}$-A$m$DLD $F_{2}$ if and only if there is $c_{m}\in\left[ 0,\varepsilon_{m}\right] $ such that \[ \int_{0}^{1}\max\left( \tilde L_{1}^{m}\left( p\right) -\tilde L_{2}^{m}\left( p\right) ,0\right) \mathrm{d}p= c_{m}\int_{0}^{1}\left\vert \tilde L_{1}^{m}\left( p\right) -\tilde L_{2}^{m}\left( p\right) \right\vert \mathrm{d}p. \] Similar to (ref), we now define the $m$th-degree downward Lorenz dominance coefficient $c_m^d(L_1,L_2)$, where the superscript “$d$” represents “downward”. \begin{definition} For $m\ge2$, the \emph{$F_1$-$F_2$ $m$th-degree downward Lorenz dominance coefficient ($F_1$-$F_2$ $m$DLDC)} $c_m^d(L_1,L_2)$ is defined as \begin{align} &c_m^d(L_1,L_2)=\notag\\ &\inf\left\{\varepsilon_m\in[0,1]:\int_0^1\max\left( \tilde L_{1}^{m}\left( p\right) -\tilde L_{2}^{m }\left( p\right),0 \right) \mathrm{d}p\leq\varepsilon_{m}\int_{0}^{1}\left\vert \tilde L_{1}^{m}\left(p \right) -\tilde L_{2}^{m}\left( p\right) \right\vert \mathrm{d}p\right\}. \end{align} \end{definition} For $m\ge2$, we define the difference function \[ \phi_m^d\left( p\right) =\tilde L_{1}^{m}\left( p\right) -\tilde L_{2}^{m}\left( p\right),\quad p\in[0,1]. \] The following lemma summarizes the properties of the $m$DLDC $c_m^d(L_1,L_2)$. \begin{lemma} The $m$DLDC $c_m^d(L_1,L_2)$ is the smallest $\varepsilon_m$ in $[0,1]$ such that (ref) holds. With (ref), it follows that \begin{align} c_m^d(L_1,L_2)&=\frac{\int_0^1\max\left( \tilde L_{1}^{m}\left( p\right) -\tilde L_{2}^{m}\left( p\right) ,0 \right) \mathrm{d}p}{\int_{0}^{1}\left\vert\tilde L_{1}^{m}\left( p\right) -\tilde L_{2}^{m}\left( p\right) \right\vert \mathrm{d}p}\notag\\ &=\frac{\int_{0}^{1}\max\left\{ \phi_m^d\left( p\right) ,0\right\} \mathrm{d}p}{\int_{0}^{1}\max\left\{ \phi_m^d\left( p\right) ,0\right\} \mathrm{d}p+\int_{0}^{1}\max\left\{ -\phi_m^d\left( p\right) ,0\right\} \mathrm{d}p}. \end{align} In addition, if $c_m^d(L_1,L_2)\in(0,1]$, then $c_m^d(L_2,L_1)=1-c_m^d(L_1,L_2)$. \end{lemma} According to Lemma (ref), $F_1$ $\varepsilon_m$-A$m$DLD $F_2$ for all $\varepsilon_m\in[c_m^d(L_1,L_2),1/2)$ if $c_m^d(L_1,L_2)<1/2$. On the other hand, $c_m^d(L_1,L_2)>1/2$ implies that $F_2$ $\varepsilon_m$-A$m$DLD $F_1$ for all $\varepsilon_m\in[1-c_m^d(L_1,L_2),1/2)$. Thus, $c_m^d(L_1,L_2)$ presents the degree of almost downward Lorenz dominance relationship between $F_1$ and $F_2$, and provides all $\varepsilon_m$ such that the $\varepsilon_m$-A$m$DLD holds. \begin{proposition} If $c_m^d(L_1,L_2)\in(0,1/2)$, it then follows that $J_P(L_1)\le J_P(L_2)$ for all preference functions $P\in \cup_{\varepsilon_m\in[c_m^d(L_1,L_2),1/2)}\tilde{\mathrm{P}}_m(\varepsilon_m)$. If, in addition, $\tilde L_{1}^{j}\left( 0\right) \leq \tilde L_{2}^{j}\left( 0\right) $ for all $j\in\left\{ 2,\ldots,m\right\} $, then $J_{P}\left( L_{1}\right) \leq J_{P}\left( L_{2}\right) $ for all $P\in\cup_{\varepsilon_m\in[c_m^d(L_1,L_2),1/2)}\tilde{\mathrm{P}}_{m}^{\prime}\left( \varepsilon_{m}\right) $. \end{proposition} Similar to Proposition (ref), Proposition (ref) shows the importance of DLDC concerning inequality measures $J_P$. The smaller the DLDC $c_m^d(L_1,L_2)$ is, the more inequality measures show higher equality in the distribution $F_1$ compared to $F_2$. With the knowledge of $c_m^d(L_1,L_2)$, we can infer the relationship between $F_1$ and $F_2$ based on a class of inequality measures. \subsection{Estimation and Inference} As discussed in the previous sections, LDCs play an important role in comparing the equality of income (or wealth) distributions between two populations. In this section, we consider the estimation and inference of LDCs. \subsubsection{Sampling Frameworks} Following BDB14 and Beare2017improved, we consider two alternative frameworks for sampling from $F_1$ and $F_2$. From $F_1$ and $F_2$, we draw identically and independently distributed (iid) samples $\{X_i^j\}_{i=1}^{n_j}$ ($j=1,2$) that satisfy the following assumption. \begin{assumption} BDB14,Beare2017improved The iid samples $\{X_i^1\}_{i=1}^{n_1}$ and $\{X_i^2\}_{i=1}^{n_2}$ drawn from $F_1$ and $F_2$ satisfy one of the following conditions. \begin{enumerate}[label=(\roman*)] • \emph{Independent samples:} $\{X_i^1\}_{i=1}^{n_1}$ and $\{X_i^2\}_{i=1}^{n_2}$ are mutually independent, and the sample sizes $n_1$ and $n_2$ are treated as functions of an underlying index $n\in\mathbb N$ such that as $n\to\infty$, \begin{align} \frac{n_1n_2}{n_1+n_2}\to\infty\quad\text{and}\quad\frac{n_1}{n_1+n_2}\to\lambda\in[0,1]. \end{align} • \emph{Matched pairs:} The sample sizes $n_1$ and $n_2$ satisfy $n_1=n_2=n$, the pairs $\{(X_i^1,X_i^2)\}_{i=1}^{n}$ are iid, and the bivariate copula for those pairs has maximal correlation strictly less than one beare2010copulas. \end{enumerate} \end{assumption} \subsubsection{Construction of Estimator} We define maps $\mathcal{F}_{1}:\ell^{\infty}\left( \left[ 0,1\right] \right) \rightarrow\mathbb{R}$, $\mathcal{F}_{2}:\ell^{\infty}\left( \left[ 0,1\right] \right) \rightarrow\mathbb{R}$, and $\mathcal{F}:\ell^{\infty }\left( \left[ 0,1\right] \right) \rightarrow\mathbb{R}$ by \begin{align} &\mathcal{F}_{1}\left( \psi\right) =\int_{0}^{1}\max\left\{ \psi\left( p\right) ,0\right\} \mathrm{d}p,\quad \mathcal{F}_{2}\left( \psi\right) =\int_{0}^{1}\max\left\{ -\psi\left( p\right) ,0\right\} \mathrm{d}p,\notag\\ &\text{and }\mathcal{F}\left( \psi\right) =\frac{\mathcal{F}_{1}\left( \psi\right) }{\mathcal{F}_{1}\left( \psi\right) +\mathcal{F}_{2}\left( \psi\right) },\quad\psi\in\ell^{\infty}\left( \left[ 0,1\right] \right) . \end{align} To unify notation, for $w\in\{u,d\}$, we let $\phi_1^w=\phi$ and $c_1^w(L_1,L_2)=c(L_1,L_2)$ defined in (ref) and (ref), respectively. Clearly, for $m\ge 1$, we have $c_m^w\left( L_{1},L_{2}\right) =\mathcal{F}\left( \phi_m^w\right)$ by (ref), (ref), and (ref). Following BDB14 and Beare2017improved, for $j=1,2$, define the empirical CDF \begin{align} \hat{F}_j(x)=\frac{1}{n_j}\sum_{i=1}^{n_j} {1}(X_i^j\leq x),\quad x\in[0,\infty), \end{align} the empirical quantile function \begin{align} \hat{Q}_j(p)=\inf\{x\in[0,\infty):\hat{F}_j(x)\geq p\},\quad p\in[0,1], \end{align} and the empirical Lorenz curve \begin{align*} \hat{L}_j(p)=\frac{1}{\hat{\mu}_j}\int_0^p\hat{Q}_j(t)\mathrm{d}t,\quad p\in[0,1], \end{align*} where $\hat{\mu}_j=n_j^{-1}\sum_{i=1}^{n_j}X_i^j$ is the sample mean of $\{X_i^j\}_{i=1}^{n_j}$. Let $\hat{L}_j^1=\hat{L}_j$. For $m\ge 2$, define \[ \hat L_{j}^{m}\left( p\right) =\int_{0}^{p}\hat L_j^{m-1}\left( t\right) \mathrm{d}t,\quad p\in\left[ 0,1\right] . \] Define \[ \hat{\tilde{L}}_{j}^{2}\left( p\right) =\int_{p}^{1}\left( 1-\hat L_{j}\left( t\right) \right) \mathrm{d}t\text{ and for }m\ge 3, \hat{\tilde{L}}_{j}^{m}\left( p\right) =\int_{p}^{1}\hat{\tilde{L}}_j^{m-1}\left( t\right) \mathrm{d}t,\quad p\in\left[ 0,1\right] . \] The difference function between empirical Lorenz curves is defined by \begin{align*} \hat{\phi}(p)=\hat{L}_2(p)-\hat{L}_1(p),\quad p\in[0,1]. \end{align*} For $w\in\{u,d\}$, let $\hat{\phi}_1^w=\hat{\phi}$ for unified notation. For $m\ge 2$, define \[ \hat{\phi}_m^u\left( p\right) =\hat L_{2}^{m}\left( p\right) -\hat L_{1}^{m}\left( p\right)\text{ and }\hat{\phi}_m^d\left( p\right) =\hat{\tilde{L}}_{1}^{m}\left( p\right) -\hat{\tilde{L}}_{2}^{m}\left( p\right),\quad p\in[0,1]. \] The estimator of $c_m^w(L_1,L_2)$ is given by $\hat{c}_m^w(L_1,L_2)=\mathcal{F}(\hat{\phi}_m^w)$. \subsubsection{Asymptotic Analysis} We now establish the consistency of $\hat{c}_m^w(L_1,L_2)$ and derive the asymptotic distribution of \begin{align} \sqrt{T_n}(\hat{c}_m^w(L_1,L_2)-{c}_m^w(L_1,L_2))=\sqrt{T_n}(\mathcal{F}(\hat{\phi}_m^w)-\mathcal{F}(\phi_m^w)) \end{align} for all $m\ge1$ and $w\in\{u,d\}$, where $T_n=n_1n_2/(n_1+n_2)$. We follow fang2014inference and introduce the following definition of Hadamard directional differentiability. \begin{definition} Let $\mathbb{D}$ and $\mathbb{E}$ be normed spaces. A map $\mathcal{G}:\mathbb{D}\to\mathbb{E}$ is said to be Hadamard directionally differentiable at $\psi\in\mathbb{D}$ tangentially to $\mathbb{D}_0\subset\mathbb{D}$ if there is a continuous map $\mathcal{G}_{\psi}^{\prime}:\mathbb{D}_0\rightarrow \mathbb{E}$ such that \begin{align} \lim_{n\rightarrow\infty}\left\Vert\frac{\mathcal{G}(\psi+t_{n}h_{n} )-\mathcal{G}(\psi)}{t_{n}}-\mathcal{G}_{\psi}^{\prime}(h)\right\Vert _{\mathbb{E}}=0, \end{align} for all sequences $\{h_{n}\}\subset\mathbb{D}$ and $\{t_{n}\}\subset \mathbb{R}_{+}$ such that $t_{n}\downarrow0$ and $h_{n}\rightarrow h\in\mathbb{D}_0$. \end{definition} For every $\psi\in\ell^{\infty}\left( \left[ 0,1\right] \right) $, define \begin{align} B_{0}\left( \psi\right) =\left\{ p\in\left[ 0,1\right] :\psi\left( p\right) =0\right\} \text{ and }B_{+}\left( \psi\right) =\left\{ p\in\left[ 0,1\right] :\psi\left( p\right) >0\right\} . \end{align} Here, $B_0(\phi_m^w)$ is the contact set we need to estimate later. For more discussions on the estimation of contact sets, see, for example, linton2010improved for an improved bootstrap test of stochastic dominance and lee2018testing for testing functional inequalities. By Lemma S.4.5 of fang2014inference, we have that $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ are both Hadamard directionally differentiable at $\phi_m^w$ tangentially to $\ell^{\infty}\left( \left[ 0,1\right] \right) $ with \[ \mathcal{F}_{1\phi_m^w}^{\prime}\left( h\right) =\int_{B_{+}\left( \phi_m^w\right) }h\left( p\right) \mathrm{d}p+\int_{B_{0}\left( \phi_m^w\right) }\max\left\{ h\left( p\right) ,0\right\} \mathrm{d}p \] and \[ \mathcal{F}_{2\phi_m^w}^{\prime}\left( h\right) =\int_{B_{+}\left( -\phi_m^w\right) }-h\left( p\right) \mathrm{d}p+\int_{B_{0}\left( \phi_m^w\right) }\max\left\{ -h\left( p\right) ,0\right\} \mathrm{d}p \] for all $h\in\ell^{\infty}\left( \left[ 0,1\right] \right) $. Following Beare2017improved, we let $\mathcal{B}$ be a centered Gaussian random element in $C(\left[0,1\right] ^{2})$ with covariance kernel \[ Cov\left( \mathcal{B}\left( u,v\right) ,\mathcal{B(}u^{\prime},v^{\prime })\right) =\mathbf{C}\left( u\wedge u^{\prime},v\wedge v^{\prime}\right) -\mathbf{C}\left( u,v\right) \mathbf{C}(u^{\prime},v^{\prime}), \] where $\mathbf{C}(u,v)=uv$ under Assumption (ref)(i) and $\mathbf C$ is the unique copula function for the pair $(X_{i}^{1},X_{i}^{2}) $ under Assumption (ref)(ii). Let $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ be the centered Gaussian random elements in $C(\left[ 0,1\right] )$ such that $\mathcal{B}_{1}\left( u\right) =\mathcal{B}\left( u,1\right) $ and $\mathcal{B}_{2}\left( v\right) =\mathcal{B}\left( 1,v\right) $. Under Assumptions (ref) and (ref), Beare2017improved show that \begin{align} \sqrt{T_n}(\hat{\phi}-\phi)\leadsto \mathbb{G} \end{align} in $C([0,1])$ for some random element $\mathbb{G}=\lambda^{1/2}\mathcal{L}_{2}-\left( 1-\lambda\right) ^{1/2}\mathcal{L}_{1}$, where $\mathcal{L}_{j}$ is a centered random element of $C([0,1])$ given by \begin{align} \mathcal L_j(p)=-\int_0^pL_j”(t)\mathcal B_j(t)\mathrm{d}t+L_j(p)\int_0^1L_j”(t)\mathcal B_j(t)\mathrm{d}t,\quad p\in[0,1]. \end{align} Beare2017improved show that \begin{align} Var(\mathbb{G}(p))= Var\left( \frac{(1-\lambda)^{1/2}}{\mu_1}(L_1(p)X_i^1-Q_1(p)\wedge X_i^1)-\frac{\lambda^{1/2}}{\mu_2}(L_2(p)X_i^2-Q_2(p)\wedge X_i^2) \right) \end{align} for every $p\in [0,1]$. We follow Beare2017improved and provide the following lemma. \begin{lemma} For all $p,p'\in[0,1]$, it follows that \begin{align*} E\left[ \mathbb{G}\left( p\right) \mathbb{G}( p^{\prime}) \right] =&\,\lambda E\left[ \mathcal{L}_{2}\left( p\right) \mathcal{L}_{2}( p^{\prime}) \right] -\mathbb{\lambda}^{1/2}\left( 1-\lambda\right) ^{1/2}E\left[ \mathcal{L}_{2}\left( p\right) \mathcal{L}_{1}( p^{\prime}) \right] \& -\mathbb{\lambda}^{1/2}\left( 1-\lambda\right) ^{1/2}E\left[ \mathcal{L}_{1}\left( p\right) \mathcal{L}_{2}( p^{\prime}) \right] +\left( 1-\lambda\right) E\left[ \mathcal{L}_{1}\left( p\right) \mathcal{L}_{1}( p^{\prime}) \right], \end{align*} where for $j,j^{\prime}\in\left\{ 1,2\right\} $ and $p,p^{\prime}\in\left[ 0,1\right] $, \begin{align*} &E\left[ \mathcal{L}_{j}\left( p\right) \mathcal{L}_{j^{\prime}}( p^{\prime}) \right]= \\ & Cov\left( \frac{1}{\mu_{j}}\left\{ L_{j}\left( p\right) X_i^{j} -Q_{j}\left( p\right) \wedge X_i^{j}\right\} ,\frac{1}{\mu_{j^{\prime}} }\left\{ L_{j'}( p^{\prime}) X_i^{j^{\prime}}-Q_{j'}( p^{\prime}) \wedge X_i^{j^{\prime}}\right\} \right) . \end{align*} \end{lemma} For $m\geq2$ and $w\in\{u,d\}$, define $\mathcal{I}_{m}^{w}:\ell^{\infty}\left( \left[ 0,1\right] \right) \rightarrow\ell^{\infty}\left( \left[ 0,1\right] \right) $ such that \begin{align*} &\mathcal{I}_{m}^{u}\left( f\right) \left( p\right) =\int_{0}^{p}\cdots \int_{0}^{t_{3}}\int_{0}^{t_{2}}f\left( t_{1}\right) \mathrm{d}t_{1}\mathrm{d}t_{2}\cdots\mathrm{d} t_{m-1} \text{ and } \\ &\mathcal{I}_{m}^{d}\left( f\right) \left( p\right) =\int_{p}^{1}\cdots \int_{t_{3}}^{1}\int_{t_{2}}^{1}f\left( t_{1}\right) \mathrm{d}t_{1}\mathrm{d}t_{2}\cdots\mathrm{d} t_{m-1},\quad p\in\left[ 0,1\right] . \end{align*} To unify notation, we let $\mathcal{I}_{1}^{w}:\ell^{\infty}\left( \left[ 0,1\right] \right) \rightarrow\ell^{\infty}\left( \left[ 0,1\right] \right) $ such that $\mathcal{I}_{1}^{w}(f)=f$ for $w\in\{u,d\}$. Then we obtain the following lemma. \begin{lemma} For $m\ge1$ and $w\in\{u,d\}$, under Assumptions (ref) and (ref), it follows that \begin{align} \sqrt{T_n}(\hat{\phi}_m^w-\phi_m^w)=\mathcal{I}_{m}^{w}\left(\sqrt{T_n}(\hat{\phi}-\phi)\right)\leadsto \mathcal{I}_{m}^{w}(\mathbb{G}) \end{align} with \begin{align} &Var(\mathcal{I}_{m}^{u}(\mathbb{G})(p))=\notag\\ &\int_{0}^{p}\cdots\int_{0}^{t_{3}^{\prime}}\int_{0}^{t_{2}^{\prime}}\left( \int_{0}^{p}\cdots\int_{0}^{t_{3}}\int_{0}^{t_{2}}E\left[ \mathbb{G}( t_{1}) \mathbb{G}( t_{1}^{\prime }) \right] \mathrm{d}t_{1}\mathrm{d}t_{2}\cdots\mathrm{d}t_{m-1}\right) \mathrm{d}t_{1}^{\prime}\mathrm{d}t_{2}^{\prime}\cdots\mathrm{d}t_{m-1}^{\prime} \end{align} and \begin{align} &Var(\mathcal{I}_{m}^{d}(\mathbb{G})(p))=\notag\\ &\int_{p}^{1}\cdots\int_{t_{3}^{\prime}}^1\int_{t_{2}^{\prime}}^1\left( \int_{p}^{1}\cdots\int_{t_{3}}^1\int_{t_{2}}^1E\left[ \mathbb{G}( t_{1}) \mathbb{G}( t_{1}^{\prime }) \right] \mathrm{d}t_{1}\mathrm{d}t_{2}\cdots\mathrm{d}t_{m-1}\right) \mathrm{d}t_{1}^{\prime}\mathrm{d}t_{2}^{\prime}\cdots\mathrm{d}t_{m-1}^{\prime} \end{align} for every $p\in[0,1]$. \end{lemma} We then introduce the following proposition for the asymptotic properties of the estimator $\hat{c}_m^w(L_1,L_2)$. Recall that for $w\in\{u,d\}$, $\phi_1^w=L_2-L_1$; for all $m\ge2$, $\phi_m^u=L_2^m-L_1^m$ and $\phi_m^d=\tilde L_1^m-\tilde L_2^m$. \begin{proposition} Suppose that Assumptions (ref) and (ref) hold. For $m\ge 1$ and $w\in\{u,d\}$, if $\phi_m^w(p)\neq 0$ for some $p\in[0,1]$, then $\hat{c}_m^w(L_1,L_2)\to c_m^w(L_1,L_2)$ almost surely (a.s.), and \begin{align} \sqrt{T_n}(\hat{c}_m^w(L_1,L_2)-{c}_m^w(L_1,L_2))=\sqrt{T_n}(\mathcal{F}(\hat{\phi}_m^w)-\mathcal{F}(\phi_m^w))\leadsto \mathcal{F}'_{\phi_m^w}(\mathcal{I}_m^w(\mathbb{G})), \end{align} where for every $h\in\ell^{\infty}([0,1])$, \begin{align*} \mathcal{F}_{\phi_m^w}^{\prime}\left( h\right) =\frac{\mathcal{F}_{1\phi_m^w}^{\prime}\left( h\right) \mathcal{F}_{2}\left( \phi_m^w\right) -\mathcal{F}_{1}\left( \phi_m^w\right) \mathcal{F}_{2\phi_m^w}^{\prime }\left( h\right) }{\left( \mathcal{F}_{1}\left( \phi_m^w\right) +\mathcal{F}_{2}\left( \phi_m^w\right) \right) ^{2}}. \end{align*} \end{proposition} Proposition (ref) provides the asymptotic distribution of $\hat{c}_m^w(L_1,L_2)$ when $\phi_m^w(p)\neq0$ for some $p\in[0,1]$. We next construct a bootstrap confidence interval for $c_m^w(L_1,L_2)$ based on this asymptotic distribution. \subsubsection{Bootstrap Confidence Interval} Since the distribution of $\mathcal{F}'_{\phi_m^w}(\mathcal{I}_m^w(\mathbb{G}))$ in (ref) is unknown and depends on the underlying data generating processes (DGPs), we use the bootstrap method to approximate this distribution. Because $\mathcal{F}'_{\phi_m^w}$ is nonlinear, the standard bootstrap method may not consistently approximate the distribution of $\mathcal{F}'_{\phi_m^w}(\mathcal{I}_m^w(\mathbb{G}))$ andrews2000inconsistency,bickel2012resampling,fang2014inference. We next employ the bootstrap method of fang2014inference to obtain a consistent approximation of the asymptotic distribution and construct valid critical values. By Lemma (ref), for $j,j^{\prime}\in\left\{ 1,2\right\} $ and $p,p^{\prime}\in\left[ 0,1\right] $, the estimator for $E\left[ \mathcal{L}_{j}\left( p\right) \mathcal{L}_{j^{\prime}}\left( p^{\prime}\right) \right] $, denoted by $\hat{E}\left[ \mathcal{L}_{j}\left( p\right) \mathcal{L}_{j^{\prime}}\left( p^{\prime}\right) \right] $, is defined as the sample covariance of the two samples \[ \left\{ \frac{1}{\hat{\mu}_{j}}\left( \hat{L}_{j}(p)X_{i}^{j}-\hat{Q} _{j}(p)\wedge X_{i}^{j}\right) \right\} _{i=1}^{n_{j}} \text{ and } \left\{ \frac{1}{\hat{\mu}_{j^{\prime}}}\left( \hat{L}_{j^{\prime} }(p^{\prime})X_{i}^{j^{\prime}}-\hat{Q}_{j^{\prime}}(p^{\prime})\wedge X_{i}^{j^{\prime}}\right) \right\} _{i=1}^{n_{j^{\prime}}}. \] Let $\hat{\lambda}=n_{1}/(n_{1}+n_{2})$. By Lemma (ref), for independent samples, we estimate $E\left[ \mathbb{G}\left( p\right) \mathbb{G}\left( p^{\prime}\right) \right] $ by \begin{align} \hat{E}\left[ \mathbb{G}\left( p\right) \mathbb{G}(p^{\prime})\right] =( 1-\hat{\lambda}) \hat{E}\left[ \mathcal{L}_{1}\left( p\right) \mathcal{L}_{1}(p^{\prime})\right] +\hat{\lambda}\hat{E}\left[ \mathcal{L}_{2}\left( p\right) \mathcal{L}_{2}(p^{\prime})\right] ,\end{align} and for matched pairs, we estimate $E\left[ \mathbb{G}\left( p\right) \mathbb{G}\left( p^{\prime}\right) \right] $ by \begin{align} \hat{E}\left[ \mathbb{G}\left( p\right) \mathbb{G}(p^{\prime})\right] =&\,( 1-\hat{\lambda}) \hat{E}\left[ \mathcal{L}_{1}\left( p\right) \mathcal{L}_{1}(p^{\prime})\right] -\sqrt{\hat{\lambda}( 1-\hat{\lambda}) }\hat{E}\left[ \mathcal{L}_{1}\left( p\right) \mathcal{L}_{2}(p^{\prime})\right] \notag \\ & -\sqrt{\hat{\lambda}( 1-\hat{\lambda}) }\hat{E}\left[ \mathcal{L}_{2}\left( p\right) \mathcal{L}_{1}(p^{\prime})\right] +\hat{\lambda}\hat{E}\left[ \mathcal{L}_{2}\left( p\right) \mathcal{L} _{2}(p^{\prime})\right] . \end{align} We let $\hat\sigma(p)^2=\hat{E}[ \mathbb{G}\left( p\right) ^2]$ for all $p\in[0,1]$, and for $w\in\{u,d\}$, we let $\hat{\sigma}_{1}^{w}=\hat{\sigma}$. Based on (ref) and (ref) with the estimates in (ref) and (ref), for $m\ge2$, we estimate ${Var}(\mathcal{I}_{m}^{u}(\mathbb{G})(p))$ and ${Var}(\mathcal{I}_{m}^{d}(\mathbb{G})(p))$ by \begin{align} \hat{\sigma}_{m}^{u}(p)^2=\int_{0}^{p}\cdots\int_{0}^{t_{3}^{\prime}}\int_{0}^{t_{2}^{\prime}}\left( \int_{0}^{p}\cdots\int_{0}^{t_{3}}\int_{0}^{t_{2}}\hat{E}\left[ \mathbb{G}( t_{1}) \mathbb{G}( t_{1}^{\prime }) \right] \mathrm{d}t_{1}\mathrm{d}t_{2}\cdots\mathrm{d}t_{m-1}\right) \mathrm{d}t_{1}^{\prime}\mathrm{d}t_{2}^{\prime}\cdots\mathrm{d}t_{m-1}^{\prime} \end{align} and \begin{align} \hat{\sigma}_m^{d}(p)^2=\int_{p}^{1}\cdots\int_{t_{3}^{\prime}}^1\int_{t_{2}^{\prime}}^1\left( \int_{p}^{1}\cdots\int_{t_{3}}^1\int_{t_{2}}^1\hat{E}\left[ \mathbb{G}( t_{1}) \mathbb{G}( t_{1}^{\prime }) \right] \mathrm{d}t_{1}\mathrm{d}t_{2}\cdots\mathrm{d}t_{m-1}\right) \mathrm{d}t_{1}^{\prime}\mathrm{d}t_{2}^{\prime}\cdots\mathrm{d}t_{m-1}^{\prime}, \end{align} respectively. Following Beare2017improved, we construct the estimators of $B_{+}\left( \phi_m^w\right) $, $B_{+}\left(-\phi_m^w\right) $, and $B_{0}\left( \phi_m^w\right) $ by \begin{align*} &\widehat{B_{+}\left( \phi_m^w\right) } =\left\{ p\in\left[ 0,1\right] :\frac{\sqrt{T_{n}}\hat{\phi}_m^w\left( p\right) }{\xi_{0}\vee\hat{\sigma }_m^w\left( p\right) }>t_{n}\right\} , \widehat{B_{+}\left( -\phi_m^w\right) }=\left\{ p\in\left[ 0,1\right] :\frac{\sqrt{T_{n}}\hat{\phi}_m^w\left( p\right) }{\xi_{0}\vee\hat{\sigma}_m^w\left( p\right) }<-t_{n}\right\} ,\\ &\text{and }\widehat{B_{0}\left( \phi_m^w\right) } =\left\{ p\in\left[ 0,1\right] :\left\vert \frac{\sqrt{T_{n}}\hat{\phi}_m^w\left( p\right) } {\xi_{0}\vee\hat{\sigma}_m^w\left( p\right) }\right\vert \leq t_{n}\right\} , \end{align*} where $t_n$ is some tuning parameter such that $t_{n}\rightarrow\infty$ and $t_{n}/\sqrt{T_{n}}\rightarrow0$ as $n\rightarrow\infty$ and $\xi_0$ is a small trimming parameter that bounds $\hat{\sigma}_m^w$ away from zero. In our simulations and application, $\xi_0$ is set to $0.001$. We then construct the estimators of $\mathcal{F}_{1\phi_m^w }^{\prime}$ and $\mathcal{F}_{2\phi_m^w}^{\prime}$ by \begin{align*} &\mathcal{\hat{F}}_{1\phi_m^w}^{\prime}\left( h\right) =\int_{\widehat {B_{+}\left( \phi_m^w\right) }}h\left( p\right) \mathrm{d}p+\int _{\widehat{B_{0}\left( \phi_m^w\right) }}\max\left\{ h\left( p\right) ,0\right\} \mathrm{d}p\\ &\text{and } \mathcal{\hat{F}}_{2\phi_m^w}^{\prime}\left( h\right) =\int_{\widehat{B_{+}\left( -\phi_m^w\right) }}-h\left( p\right) \mathrm{d}p+\int_{\widehat{B_{0}\left( \phi_m^w\right) }}\max\left\{ -h\left( p\right) ,0\right\} \mathrm{d}p \end{align*} for every $h$ $\in\ell^{\infty}\left( \left[ 0,1\right] \right) $. The estimator of $\mathcal{F}_{\phi_m^w}^{\prime}$ is defined by \[ \mathcal{\hat{F}}_{\phi_m^w}^{\prime}\left( h\right) =\frac{\mathcal{\hat{F} }_{1\phi_m^w}^{\prime}\left( h\right) \mathcal{F}_{2}( \hat{\phi}_m^w) -\mathcal{F}_{1}( \hat{\phi}_m^w) \mathcal{\hat{F}}_{2\phi_m^w}^{\prime }\left( h\right) }{\left( \mathcal{F}_{1}( \hat{\phi}_m^w) +\mathcal{F}_{2}( \hat{\phi}_m^w) \right) ^{2}} \] for every $h$ $\in\ell^{\infty}\left( \left[ 0,1\right] \right) $. We follow BDB14 and Beare2017improved to draw bootstrap samples: For independent samples, we draw a bootstrap sample $\{\hat{X}_{i}^{j} \}_{i=1}^{n_j}$ identically and independently from $\{X_{i}^{j}\}_{i=1}^{n_j}$ for $j=1,2$, where $\{\hat{X}_{i}^{1}\}_{i=1}^{n_1}$ is jointly independent of $\{\hat{X}_{i}^{2}\}_{i=1}^{n_2}$; for matched pairs, we draw a bootstrap sample $\{(\hat{X}_{i}^{1},\hat{X}_{i}^{2})\}_{i=1}^{n}$ identically and independently from $\{(X_{i}^{1},X_{i}^{2})\}_{i=1}^{n}$. Then following BDB14 and Beare2017improved, for $j=1,2$, we define the bootstrap empirical CDF \begin{align*} \hat{F}_j^*(x)=\frac{1}{n_j}\sum_{i=1}^{n_j} {1}(\hat{X}_i^j\leq x),\quad x\in[0,\infty), \end{align*} the bootstrap empirical quantile function \begin{align} \hat{Q}^*_j(p)=\inf\{x\in[0,\infty):\hat{F}^*_j(x)\geq p\},\quad p\in[0,1], \end{align} and the bootstrap empirical Lorenz curve \begin{align*} \hat{L}^*_j(p)=\frac{1}{\hat{\mu}^*_j}\int_0^p\hat{Q}^*_j(t)\mathrm{d}t,\quad p\in[0,1], \end{align*} where $\hat{\mu}^*_j$ is the sample mean of $\{\hat{X}_i^j\}_{i=1}^{n_j}$. Let $\hat{L}_j^{1*}=\hat{L}_j^*$. For $m\ge 2$, define \[ \hat L_{j}^{m*}\left( p\right) =\int_{0}^{p}\hat L_j^{m-1*}\left( t\right) \mathrm{d}t,\quad p\in\left[ 0,1\right] . \] Define \[ \hat{\tilde{L}}_{j}^{2*}\left( p\right) =\int_{p}^{1}\left( 1-\hat L_{j}^*\left( t\right) \right) \mathrm{d}t,\quad p\in\left[ 0,1\right], \] and for $m\geq3$, \[ \hat{\tilde{L}}_{j}^{m*}\left( p\right) =\int_{p}^{1}\hat{\tilde{L}}^{m-1*}\left( t\right) \mathrm{d}t,\quad p\in\left[ 0,1\right] . \] The bootstrap approximation of $\phi$ is defined by \begin{align*} \hat{\phi}^*(p)=\hat{L}^*_2(p)-\hat{L}^*_1(p),\quad p\in[0,1]. \end{align*} For $w\in\{u,d\}$, let $\hat{\phi}_1^{w*}=\hat{\phi}^*$. For $m\ge 2$, define \[ \hat{\phi}_m^{u*}\left( p\right) =\hat L_{2}^{m*}\left( p\right) -\hat L_{1}^{m*}\left( p\right)\text{ and }\hat{\phi}_m^{d*}\left( p\right) =\hat{\tilde{L}}_{1}^{m*}\left( p\right) -\hat{\tilde{L}}_{2}^{m*}\left( p\right),\quad p\in[0,1]. \] We define the bootstrap approximation of $c_m^w(L_1,L_2)$ by \begin{align} \hat{c}_m^{w*}(L_1,L_2)=\mathcal{\hat{F}}_{\phi_m^{w}}^{\prime}(\sqrt{T_n}(\hat{\phi}_m^{w*}-\hat{\phi}_m^w)). \end{align} For every $\beta\in(0,1)$, let $c_{m,\beta}^w$ denote the $\beta$ quantile of the distribution of $\mathcal{F}'_{\phi_m^w}(\mathcal{I}_m^w(\mathbb{G}))$. We construct the bootstrap approximation of $c_{m,\beta}^w$ by \begin{align} \hat{c}_{m,\beta}^w=\inf\left\{c: \mathbb{P}\left(\hat{c}_m^{w*}(L_1,L_2)\le c|\{X^1_i\}_{i=1}^{n_1},\{X^2_i\}_{i=1}^{n_2}\right)\ge \beta\right\}. \end{align} In practice, we approximate $\hat{c}_{m,\beta}^w$ by computing the $\beta$ quantile of the $n_B$ independently generated $\hat{c}_m^{w*}(L_1,L_2)$, where $n_B$ is chosen as large as is computationally convenient. For a nominal significance level $\alpha\in(0,1/2)$, we construct the $1-\alpha$ bootstrap confidence interval of $c_m^w(L_1,L_2)$ by \begin{align} \mathrm{CI}_{m,1-\alpha}^w=[\hat{c}_m^w(L_1,L_2)-T_n^{-1/2}\hat{c}_{m,1-\alpha/2}^w,\hat{c}_m^w(L_1,L_2)-T_n^{-1/2}\hat{c}_{m,\alpha/2}^w]. \end{align} \begin{proposition} Suppose that Assumptions (ref) and (ref) hold. For $m\ge1$ and $w\in\{u,d\}$, if $c_m^w(L_1,L_2)\in(0,1)$ and the CDF of $\mathcal{F}'_{\phi_m^w}(\mathcal{I}_m^w(\mathbb{G}))$ is continuous and increasing at $c_{m,\alpha/2}^w$ and $c_{m,1-\alpha/2}^w$, then it follows that \begin{align} \lim_{n\rightarrow\infty}\mathbb{P}(c_m^w(L_1,L_2)\in\mathrm{CI}_{m,1-\alpha}^w)= 1-\alpha. \end{align} \end{proposition} Proposition (ref) provides the two-sided confidence interval for $c_m^w(L_1,L_2)$. It is straightforward to extend the result to one-sided confidence intervals. \begin{remark} In Proposition (ref), it is required that $c_m^w(L_1,L_2)\in(0,1)$, which excludes some cases where Lorenz dominance holds. In these cases, $\mathcal{F}'_{\phi_m^w}(\mathcal{I}_m^w(\mathbb{G}))=0$ or $c_m^w(L_1,L_2)$ may incur the zero denominator issue (in the case where $F_1=F_2$). As a consequence, the asymptotic results may not hold. In practice, we suggest first performing tests for Lorenz dominance and proceeding with the proposed method if the dominance relationship is rejected. \end{remark} \begin{remark} The asymptotic results in the paper are established under a fixed DGP, which may not hold uniformly over all possible DGPs in general. For example, if $c_m^w(L_1,L_2)$ drifts to zero as $T_n\to\infty$, the results in Proposition (ref) may not hold. Then the proposed method may not work well in finite samples. In practice, when $c_m^w(L_1,L_2)$ is close to $0$ or $1$ and the sample size is relatively small, we may use the bounds of $c_m^w(L_1,L_2)$ to improve the confidence interval, that is, we may use $\mathrm{CI}_{m,1-\alpha}^w\cap [0,1]$ as the confidence interval. \end{remark} \section{Simulation Evidence} We run Monte Carlo simulations to provide evidence for the finite sample properties of the estimators and the bootstrap confidence intervals. In this section, we focus on the Lorenz dominance coefficient. Additional simulations are provided in Section (ref) in the appendix. The number of bootstrap samples, $n_B$, is $1000$. The number of Monte Carlo iterations is $1000$. The nominal significance level is set to $0.05$. We present the mean (Mean), the bias (Bias), the standard error (SE), and the root mean square error (RMSE) of the estimators, as well as the coverage rate (CR) of the bootstrap confidence intervals in $1000$ iterations. As discussed in Beare2017improved, reed2001pareto,reed2003pareto and toda2012double show that income distributions can be well approximated by members of the double Pareto parametric family. We use distributions from this family to construct the simulations. The density function for the double Pareto family is \[ f\left( x\right) =\left\{ \begin{array} [c]{c} \frac{\alpha\beta}{\alpha+\beta}M^{\alpha}x^{-\alpha-1}\\ \frac{\alpha\beta}{\alpha+\beta}M^{-\beta}x^{\beta-1} \end{array} \right. \begin{array} [c]{c} x\geq M,\\ 0\leq x<M, \end{array} \] where $M>0$ is a scale parameter. For $\alpha,\beta>0$, we write $X\sim \mathrm{dP}(\alpha,\beta)$ to denote that the random variable $X$ has the double Pareto distribution with $M$ normalized to one and shape parameters $\alpha,\beta$. Assumption (ref) is satisfied when $\alpha>2$. We let $X^1\sim \mathrm{dP}(3,1.5)$ and $X^2_{(\beta)}\sim \mathrm{dP}(2.1,\beta)$ with $\beta\in\{2,3,4,5\}$. Figure (ref) displays the Lorenz curves $L_1$ and $L_{2(\beta)}$ corresponding to the above DGPs. The LDC $c(L_1,L_2)=$ $0.04703$, $0.31489$, $0.45198$, and $0.51960$, respectively, for $\beta=$ $2$, $3$, $4$, and $5$. We choose the tuning parameter from $t_n\in (0,20]$. In Section (ref), we show how to choose $t_n$ empirically to construct the bootstrap confidence intervals. For independent samples, we let $$(n_1,n_2)\in\{(200,200),(200,500),(200,1000),(1000,2000),(10000,10000)\}.$$ For matched pairs, we let $$(n_1,n_2)\in\{(200,200),(500,500),(1000,1000),(2000,2000),(10000,10000)\}.$$ Tables (ref) and (ref) show the simulation results for independent samples and matched pairs. For all the DGPs, as $n_1$ and $n_2$ increase, Mean gets close to $c(L_1,L_2)$, Bias decreases to $0$, and both SE and RMSE decrease; under appropriate choices of $t_n$, CR approaches $95\%$. \begin{figure} [ht!] \caption{Lorenz Curves for Four DGPs} \begin{subfigure}[b]{0.45\textwidth} \scalebox{0.2} \subcaption{$L_1$ (solid) and $L_{2(2)}$ (dashed)} \end{subfigure} \begin{subfigure}[b]{0.45\textwidth} \scalebox{0.2} \subcaption{$L_1$ (solid) and $L_{2(3)}$ (dashed)} \end{subfigure} \begin{subfigure}[b]{0.45\textwidth} \scalebox{0.2} \subcaption{$L_1$ (solid) and $L_{2(4)}$ (dashed)} \end{subfigure} \begin{subfigure}[b]{0.45\textwidth} \scalebox{0.2} \subcaption{$L_1$ (solid) and $L_{2(5)}$ (dashed)} \end{subfigure} \end{figure} \begin{table}[ht!] \caption{Estimations and Coverage Rates for LDC (Independent Samples)} \scalebox{0.95}{ \begin{tabular}{cccccccccc} \hline\hline {DGP} & $c(L_1,L_2)$ & {$n_1 $} & {$n_2 $} & Mean & Bias & SE & RMSE & $t_n$ & CR \\ \hline \multirow{5}{*}{(a)} & \multirow{5}{*}{0.04703} & 200 & 200 & 0.2305 & 0.1834 & 0.3088 & 0.3592 & 0.001 & 0.7040 \\ & & 200 & 500 & 0.1529 & 0.1058 & 0.2420 & 0.2641 & 0.001 & 0.7270 \\ & & 200 & 1000 & 0.1258 & 0.0788 & 0.2144 & 0.2284 & 0.001 & 0.7200 \\ & & 1000 & 2000 & 0.0733 & 0.0263 & 0.0754 & 0.0798 & 0.001 & 0.8630 \\ & & 10000& 10000& 0.0523 & 0.0053 & 0.0233 & 0.0239 & 0.001 & 0.9330\\ \hline \multirow{5}{*}{(b)} & \multirow{5}{*}{0.31489} & 200 & 200 & 0.4753 & 0.1604 & 0.3341 & 0.3706 & 0.001 & 0.6970 \\ & & 200 & 500 & 0.3986 & 0.0837 & 0.2865 & 0.2985 & 0.001 & 0.7840 \\ & & 200 & 1000 & 0.3695 & 0.0546 & 0.2650 & 0.2706 & 0.001 & 0.7900 \\ & & 1000 & 2000 & 0.3515 & 0.0366 & 0.1612 & 0.1653 & 0.001 & 0.9140 \\ & & 10000& 10000& 0.3260 & 0.0111 & 0.0718 & 0.0727 & 0.101 & 0.9340\\ \hline \multirow{5}{*}{(c)} & \multirow{5}{*}{0.45198} & 200 & 200 & 0.5680 & 0.1160 & 0.3182 & 0.3387 & 0.001 & 0.6720 \\ & & 200 & 500 & 0.5056 & 0.0537 & 0.2770 & 0.2821 & 0.001 & 0.7800 \\ & & 200 & 1000 & 0.4811 & 0.0292 & 0.2574 & 0.2590 & 0.001 & 0.7940 \\ & & 1000 & 2000 & 0.4804 & 0.0284 & 0.1678 & 0.1702 & 0.001 & 0.9100 \\ & & 10000& 10000& 0.4619 & 0.0099 & 0.0799 & 0.0805 & 0.001 & 0.9260\\ \hline \multirow{5}{*}{(d)} & \multirow{5}{*}{0.51960} & 200 & 200 & 0.6114 & 0.0918 & 0.3060 & 0.3195 & 0.001 & 0.6690 \\ & & 200 & 500 & 0.5575 & 0.0379 & 0.2674 & 0.2701 & 0.001 & 0.7740 \\ & & 200 & 1000 & 0.5360 & 0.0164 & 0.2484 & 0.2489 & 0.001 & 0.7850 \\ & & 1000 & 2000 & 0.5427 & 0.0231 & 0.1655 & 0.1671 & 0.001 & 0.9030 \\ & & 10000& 10000& 0.5283 & 0.0087 & 0.0806 & 0.0811 & 0.001 & 0.9280\\ \hline\hline \end{tabular} } \end{table} \begin{table}[ht!] \caption{Estimations and Coverage Rates for LDC (Matched Pairs)} \scalebox{0.95}{ \begin{tabular}{cccccccccc} \hline\hline {DGP} & $c(L_1,L_2)$ & {$n_1 $} & {$n_2 $} & Mean & Bias & SE & RMSE & $t_n$ & CR \\ \hline \multirow{5}{*}{(a)} & \multirow{5}{*}{0.04703} & 200 & 200 & 0.1733 & 0.1262 & 0.2616 & 0.2904 & 0.001 & 0.7330 \\ & & 500 & 500 & 0.1173 & 0.0703 & 0.1783 & 0.1917 & 0.001 & 0.8280 \\ & & 1000 & 1000 & 0.0914 & 0.0444 & 0.1004 & 0.1097 & 0.001 & 0.9010 \\ & & 2000 & 2000 & 0.0692 & 0.0222 & 0.0585 & 0.0626 & 0.001 & 0.9070 \\ & & 10000& 10000& 0.0491 & 0.0021 & 0.0179 & 0.0181 & 0.501 & 0.9480\\ \hline \multirow{5}{*}{(b)} & \multirow{5}{*}{0.31489} & 200 & 200 & 0.4158 & 0.1009 & 0.3068 & 0.3230 & 0.001 & 0.7620 \\ & & 500 & 500 & 0.4162 & 0.1013 & 0.2334 & 0.2544 & 0.001 & 0.8770 \\ & & 1000 & 1000 & 0.3907 & 0.0758 & 0.1808 & 0.1961 & 0.001 & 0.9270 \\ & & 2000 & 2000 & 0.3568 & 0.0419 & 0.1434 & 0.1494 & 0.001 & 0.9250 \\ & & 10000& 10000& 0.3196 & 0.0047 & 0.0602 & 0.0604 & 7.7 & 0.9500\\ \hline \multirow{5}{*}{(c)} & \multirow{5}{*}{0.45198} & 200 & 200 & 0.5185 & 0.0665 & 0.2986 & 0.3059 & 0.001 & 0.7560 \\ & & 500 & 500 & 0.5473 & 0.0953 & 0.2249 & 0.2443 & 0.001 & 0.8590 \\ & & 1000 & 1000 & 0.5168 & 0.0648 & 0.1811 & 0.1923 & 0.001 & 0.9060 \\ & & 2000 & 2000 & 0.4903 & 0.0383 & 0.1546 & 0.1593 & 0.001 & 0.9170 \\ & & 10000& 10000& 0.4561 & 0.0041 & 0.0686 & 0.0687 & 4.4 & 0.9500\\ \hline \multirow{5}{*}{(d)} & \multirow{5}{*}{0.51960} & 200 & 200 & 0.5682 & 0.0486 & 0.2898 & 0.2938 & 0.001 & 0.7520 \\ & & 500 & 500 & 0.6094 & 0.0898 & 0.2144 & 0.2325 & 0.001 & 0.8510 \\ & & 1000 & 1000 & 0.5752 & 0.0556 & 0.1760 & 0.1845 & 0.001 & 0.9000 \\ & & 2000 & 2000 & 0.5547 & 0.0351 & 0.1549 & 0.1589 & 0.001 & 0.9060 \\ & & 10000& 10000& 0.5232 & 0.0036 & 0.0700 & 0.0701 & 3.5 & 0.9500 \\ \hline\hline \end{tabular} } \end{table} \section{Empirical Application} In this section, we revisit the example of aaberge2021ranking regarding the inequality growth in the United Kingdom over the past few decades, which may be related to the business cycle blundell2010consumption. We use the replication data of aaberge2021ranking, which initially came from the European Community Household Panel (ECHP) for 1995–2001, and from the European Union Statistics on Income and Living Conditions (EU-SILC) for 2005–2010. As described by aaberge2021ranking, the dataset is limited to households with a male aged 25–64, and focuses on the distribution of individual equivalent income, adjusted for inflation as well as household size and composition. Tables 3 and 4 of aaberge2021ranking report the successful rankings based on the 3rd-degree upward and downward inverse stochastic dominances. They show that the 3rd-degree upward inverse stochastic dominance does not provide a complete ranking for several years. We apply the proposed method to estimate the third degree upward inverse stochastic dominance coefficients (3UISDCs) and construct confidence intervals for these coefficients for specified years.\footnote{See theoretical and simulation results for upward and downward inverse stochastic dominance coefficients in Sections (ref) and (ref) in the appendix.} We compare the latter years ($F_1$) with the former years ($F_2$) shown in Table (ref), and find that all the estimated 3UISDCs are small. Furthermore, the bootstrap confidence intervals are narrow. These facts provide evidence that the income distributions of the latter years may almost upward inverse stochastically dominate the income distributions of the former years, which complements the results of aaberge2021ranking. \begin{table}[H] \caption{Inequality Growth in the United Kingdom} \scalebox{1}{ \begin{tabular}{cccccc} \hline\hline Year & $3$UISDC & CI & Year & $3$UISDC & CI \\ \hline 94--98&3.2067e-08&[0, 3.0653e-07]&96--98&2.9089e-06&[0, 1.6521e-05]\\ 94--99&1.287e-07&[0, 8.9855e-07]&96--99&5.9121e-06&[0, 3.0879e-05]\\ 94--00&3.0482e-07&[0, 1.7727e-06]&96--00&3.8907e-06&[0, 1.6773e-05]\\ 94--01&3.9715e-07&[0, 2.5549e-06]&96--01&1.2211e-05&[0, 5.0282e-05]\\ 95--98&1.0184e-07&[0, 8.0492e-07]&97--98&0.0063743&[0, 0.056701]\\ 95--99&3.6099e-07&[0, 2.5261e-06]&97--99&0.0091885&[0, 0.069839]\\ 95--00&6.5108e-07&[0, 3.5034e-06]&97--00&8.1315e-05&[0, 0.0004528]\\ 95--01&1.7317e-06&[0, 9.5748e-06]&97--01&0.00027027&[0, 0.0011134]\\ \hline\hline \end{tabular} } \end{table} In Table (ref), we provide the estimated $F_1$-$F_2$ LDCs and their corresponding confidence intervals, using the same data as in Table (ref). For most of the comparisons in Table (ref), the conclusions are similar to those in Table (ref), while the results for 97--98, 97--99, 97--00, and 97--01 are different. In these four groups of comparisons, the LDCs are all estimated to be $1$ and the confidence intervals all shrink to $[1,1]$. This may indicate Lorenz dominance between the Year 1997 and the other years. We then run the Lorenz dominance tests of Beare2017improved for the null hypothesis that $F_2$ Lorenz dominates $F_1$, and the null can not be rejected, which is consistent with the estimations in Table (ref). It is worth noting that, although the asymptotic results in the paper may not hold in cases where $c_m^w(L_1,L_2)\in\{0,1\}$, the results in Table (ref) demonstrate that the proposed method performs well in practice. \begin{table}[H] \caption{$F_1$-$F_2$ Lorenz Dominance Coefficients} \scalebox{1}{ \begin{tabular}{cccccc} \hline\hline Year & LDC & CI & Year & LDC & CI \\ \hline 94--98 & 1.1151e-05 & [0, 1.3027e-05] & 96--98 & 1.0914e-02 & [0, 9.0150e-02] \\ 94--99 & 3.7292e-05 & [0, 6.4719e-05] & 96--99 & 9.1456e-02 & [0, 9.1430e-01] \\ 94--00 & 9.0555e-05 & [0, 1.4074e-04] & 96--00 & 2.3386e-02 & [0, 1.8231e-01] \\ 94--01 & 2.2390e-04 & [0, 3.4754e-04] & 96--01 & 5.4596e-02 & [0, 2.9047e-01] \\ 95--98 & 8.3603e-05 & [0, 1.3646e-04] & 97--98 & 1 & [1, 1] \\ 95--99 & 4.1599e-04 & [0, 3.0698e-03] & 97--99 & 1 & [1, 1] \\ 95--00 & 7.3659e-04 & [0, 4.5734e-03] & 97--00 & 1 & [1, 1] \\ 95--01 & 4.6061e-03 & [0, 2.4259e-02] & 97--01 & 1 & [1, 1] \\ \hline\hline \end{tabular} } \end{table} Note that Lorenz dominance is related to the inequality measure $J_P(L_F)$ aaberge2009ranking, while inverse stochastic dominance is related to the social welfare function $W_P(F)=\mu_F(1-J_P(L_F))$ aaberge2021ranking, where $L_F$ is the Lorenz curve for the distribution $F$, and $\mu_F$ is the mean for $F$. For illustration, we consider a preference function $P(t)=t^3-3t^2+3t$ with $t\in[0,1]$ such that $P\in\cap_{\varepsilon_3\in(0,1)}\mathcal{P}_3(\varepsilon_3)$, where $\mathcal{P}_3(\varepsilon_3)$ is defined as in Section (ref). The values of $\mu_F$, $W_P(F)$, and $J_P(L_F)$ for Year 1997, 1998, 1999, 2000, and 2001 are listed in Table (ref), which are consistent with the results in Tables (ref) and (ref). \begin{table}[h] \caption{Values of $\mu_F$, $W_P(F)$, and $J_P(L_F)$} \begin{tabular}{c c c c c c} \hline \hline & 1997 & 1998 & 1999 & 2000 & 2001 \\ \hline $\mu_F$ & 22625.08788 & 23927.0261 & 24276.49701 & 27607.35936 & 28253.11169 \\ $W_P(F)$ & 13164.54128 & 13596.19132 & 13574.86124 & 15645.45898 & 15940.32593 \\ $J_P(L_F)$ & 0.418144082 & 0.431764262 & 0.440822898 & 0.433286655 & 0.43580282 \\ \hline \hline \end{tabular} \end{table} \section{Conclusion} In this paper, we develop a general framework for inferring three types of almost dominances: almost Lorenz dominance, almost inverse stochastic dominance, and almost stochastic dominance. A bootstrap inference procedure is introduced for the corresponding coefficients, which quantify the extent of almost dominances. Simulation studies evaluate the finite sample performance of the proposed estimators and the bootstrap confidence intervals. The proposed methods are applied to analyze the growth of inequality in the United Kingdom, revealing evidence of almost upward inverse stochastic dominance. While traditional dominance concepts are powerful, strict requirements can limit their applicability. The almost dominance framework allows for small violations of dominance rules, and provides a more nuanced and flexible approach to conducting robust social welfare and inequality comparisons, generalizing investor preferences, and gaining a deeper understanding of income distributional changes. \putbib