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Article title for specific publication (Journal)
\title{
\Huge Article title for specific publication (Journal)
}
\author{
Jean-Marie Dufour \thanks{\ \ William Dow Professor of Economics, McGill University,
Centre interuniversitaire de recherche en analyse des
organisations (CIRANO), et Centre interuniversitaire de recherche en
\'{e}conomie quantitative (CIREQ). Adresse postale:
Department of Economics, McGill University, Leacock Building, Room 519,
855 Sherbrooke Street West, Montr\'{e}al, Qu\'{e}bec H3A 2T7, Canada.
TEL: (1) 514 398 8879; FAX: (1) 514 398 4938; courriel: [email removed] . Web page:
http://www.jeanmariedufour.com } \\
McGill University
}
\date{
D\'ecembre 2025
}
\maketitle
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\textbf{PLAN}
\quad
\end{center}
\begin{enumerate}
\item To do
\begin{enumerate}
\item Abandon the terminology \textquotedblleft decoupling\textquotedblright
: use $D$-representation. Done.
\item Introduce the notion of $D$-statistics
\item The first differences have symmetric distributions centered at zero.
\item Relate to symmetrization inequalities.
\item Relate to the coupling method.
\item Moment decompositions based on Gini representations. Relate to
Samuelson inequality.
\item Sign-skewness statistics.
\item Link with Spearman and Kendall measures of dependence.
\item Improve the results on higher-order moments.
\item Relate to:
\begin{enumerate}
\item $h$-statistics;
\item $U$-statistics.
\end{enumerate}
\item Unbiased estimation of higher-order moments
\begin{enumerate}
\item Document the bias of natural sample moments
\item Relate to Heffernan (1997, JRSS B).
\end{enumerate}
\item Improved procedures based on unbiased estimators of higher-order
moments.
\item Relate to cumulative distributions.
\item Applications
\begin{enumerate}
\item Conceptual: pairwise differences vs. deviations from the mean
\item Categorical data
\item First differences are not contaminated by the location estimator
\item Representations of empirical moments in terms of symmetric random
variables
\item Alternative method for deriving unbiased estimators of moments
\item Decompositions of empirical moments in terms of observations
\item Tests based on pairwise differences
\item Filtered and robust moments
\item Multivariate extensions
\item Lorenz curves in finance
\item Conditioning
\item Ranks as $U$ -statistics
\end{enumerate}
\end{enumerate}
\end{enumerate}
\quad
Potential journal for submission
\begin{enumerate}
\item The American Statistician
\item Statistics and Probability Letters
\item Communications in Statistics
\item Econometrics and Statistics
\item Journal of Statistics and Data Science Education
\item Teaching Statistics
\end{enumerate}
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\begin{center}
\textbf{ABSTRACT}
\quad
\end{center}
\noindent We provide pairwise-difference (Gini-type) representations of
higher-order central moments for both general random variables and empirical
moments. Such representations do not require a measure of location. For
third and fourth moments, this yields pairwise-difference representations of
skewness and kurtosis coefficients. We show that all central moments possess
such representations, so no reference to the mean is needed for moments of
any order. This is done by considering i.i.d. \emph{replications} of the
random variables considered, by observing that central moments can be
interpreted as covariances between a random variable and powers of the same
variable, and by giving recursions which link the pairwise-difference
representation of any moment to lower order ones. Numerical summation
identities are deduced. Through a similar approach, we give analogues of the
Lagrange and Binet-Cauchy identities for general random variables, along
with a simple derivation of the classic Cauchy-Schwarz inequality for
covariances. Finally, an application to unbiased estimation of centered
moments is discussed.
\quad
\noindent \textbf{Key words}: Gini; Covariance; Skewness; Kurtosis; Pairwise
differences; Lagrange identity; Moments; Unbiased estimation.\vspace{0.25pt}
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\subsection*{\huge Liste de d\'efinitions, hypoth\`eses, propositions et th\'eor\`emes
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\subsection{\huge Introduction \label{Sec: Introduction}}
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The sample variance is certainly the most widely used measure of dispersion
among observations in statistics. It is typically defined as the average of
the squared deviations of the observations $x_{1},\ldots ,\,x_{n}$ from
their sample mean:
\begin{equation}
s_{X}^{2}(n):=\frac{1}{n-1}\overset{n}{\underset{i=1}{\sum }}(x_{i}-\bar{x}
)^{2} \label{eq: Sample variance}
\end{equation}
where $\bar{x}=\overset{n}{\underset{i=1}{\sum }}x_{i}/n$. At first sight,
the variance depends crucially on \textquotedblleft
centering\textquotedblright\ the observations with respect to their mean. A
century ago, however, \nocite*{Gini(1912)} noticed that $s_{X}^{2}(n)$ can be
rewritten as the average of pairwise differences between all the
observations:
\begin{equation}
s_{X}^{2}(n)=\frac{1}{2}\frac{1}{n(n-1)}\overset{n}{\underset{i=1}{\sum }}
\overset{n}{\underset{j=1}{\sum }}(x_{i}-x_{j})^{2}=\frac{1}{n(n-1)}\underset
{i=1}{\overset{n}{\sum }}\,\underset{j>i}{\sum }(x_{i}-x_{j})^{2}\,;
\label{eq: Pairwise sample variance}
\end{equation}
see \nocite*{Heffernan(1988)}. The Gini representation underscores that $
s_{X}^{2}(n)$ does not depend on using the sample mean as location
parameter, and thus may be viewed as a measure of \textquotedblleft
intrinsic variability\textquotedblright\ between observations. In
particular, it provides a way of measuring dispersion when there is no
natural location parameter, as with categorical data; see
\citename{Light-Margolin(1971)} (\citeyear*{Light-Margolin(1971)}, \citeyear*{Light-Margolin(1974)})
. Further, by focusing on pairwise differences $x_{i}-x_{j}$, it may be
easier to determine which observations have the greatest influence on $
s_{X}^{2}(n)$, because deviations from the mean $x_{i}-\bar{x}$ depend on
all the observations (through $\bar{x})$. A comprehensive review of
statistical methods based on the Gini approach is presented by \nocite*
{Yitzhaki-Schechtman(2013)}.
If we have bivariate observations $z_{i}:=(x_{i},$\thinspace $y_{i})^{\prime
}$, $i=1,\ldots ,\,n,$ the sample covariance
\begin{equation}
s_{XY}(n):=\frac{1}{n-1}\overset{n}{\underset{i=1}{\sum }}(x_{i}-\bar{x}
)(y_{i}-\bar{y}) \label{eq: Sample covariance}
\end{equation}
is in turn the most widely used \textquotedblleft measure of
association\textquotedblright\ between $x$ and $y.$ For the covariance, a
pairwise representation is provided by the formula
\begin{equation}
s_{XY}(n)=\frac{1}{2}\frac{1}{n(n-1)}\overset{n}{\underset{i=1}{\sum }}
\overset{n}{\underset{j=1}{\sum }}(x_{i}-x_{j})(y_{i}-y_{j})=\frac{1}{n(n-1)}
\overset{n}{\underset{i=1}{\sum }}\underset{j>i}{\sum }
(x_{i}-x_{j})(y_{i}-y_{j})\,; \label{eq: Pairwise sample covariance}
\end{equation}
see \nocite*{Hayes(2011)}. Clearly, (\ref{eq: Pairwise sample variance}) is a
special case of (\ref{eq: Pairwise sample covariance}) obtained by taking $
x_{i}=y_{i},\;i=1,\ldots ,\,n.$ Formula (\ref{eq: Pairwise sample covariance}
) shows that the sample covariance measures the tendency of $x$ and $y$ to
move in the same direction, without reference to sample means. Similar
expressions for the variances and covariances of general random variables
have also been used in the literature on $U$-statistics; see \nocite*[Chapter 1]
{LeeAJ(1990)}. Since centering is not needed, formula (\ref{eq: Pairwise
sample covariance}) provides a basis for developing alternative measures of
dependence, such as Kendall's and Spearman's measures of dependence [see
\nocite*{Lehmann(1966)}]. Again, pairwise differences $x_{i}-x_{j}$ and $
y_{i}-y_{j}$ may be easier to interpret than $x_{i}-\bar{x}$ and $y_{i}-\bar{
y}$, because $\bar{x}$ and $\bar{y}$ depend on all the observations.
Another similar result we consider here is the Lagrange identity:
\begin{gather}
\Big(\overset{n}{\underset{i=1}{\sum }}x_{i}^{2}\Big)\Big(\overset{n}{
\underset{i=1}{\sum }}y_{i}^{2}\Big)-\Big(\overset{n}{\underset{i=1}{\sum }}
x_{i}y_{i}\Big)^{2}=\underset{1\leq i<j\leq n}{\sum }
(x_{i}y_{j}-x_{j}y_{i})^{2} \notag \\
=\frac{1}{2}\overset{n}{\underset{i=1}{\sum }}\overset{n}{\underset{j=1}{
\sum }}(x_{i}y_{j}-x_{j}y_{i})^{2}=\frac{1}{2}\overset{n}{\underset{i=1}{
\sum }}\overset{n}{\underset{j=1}{\sum }}(\det [z_{i},\,z_{j}])^{2}\,;\;
\label{eq: Lagrange identity}
\end{gather}
see \nocite*{Wright(1992)} and \nocite*[Chapter 3]{Steele(2004)}. An interesting
feature of this identity is that it involves pairwise comparisons between
cross-products of the components of $x:=(x_{1},\ldots ,\,x_{n})^{\prime }$
and\ $y:=(y_{1},\ldots ,\,y_{n})^{\prime }$. Since the right-hand side is
non-negative, the Lagrange identity provides a simple way of proving the
discrete Cauchy-Schwarz inequality:
\begin{equation}
\Big(\overset{n}{\underset{i=1}{\sum }}x_{i}y_{i}\Big)^{2}\leq \Big(\overset{
n}{\underset{i=1}{\sum }}x_{i}^{2}\Big)\Big(\overset{n}{\underset{i=1}{\sum }
}y_{i}^{2}\Big)\,.
\end{equation}
Interestingly, (\ref{eq: Lagrange identity}) directly shows when the bound
is an equality $(x_{i}y_{j}=x_{j}y_{i}\;$for all $i$ and $j)$ and what
determines its tightness. The Lagrange identity is itself a special case of
the Binet-Cauchy identity
\begin{equation}
\Big(\overset{n}{\underset{i=1}{\sum }}a_{i}c_{i}\Big)\Big(\overset{n}{
\underset{j=1}{\sum }}b_{j}d_{j}\Big)-\Big(\overset{n}{\underset{i=1}{\sum }}
a_{i}d_{i}\Big)\Big(\overset{n}{\underset{j=1}{\sum }}b_{j}c_{j}\Big)=
\underset{1\leq i<j\leq n}{\sum }
(a_{i}b_{j}-a_{j}b_{i})(c_{i}d_{j}-c_{j}d_{i})\,
\label{eq: Binet-Cauchy identity}
\end{equation}
where $a_{i},\,b_{i},\,c_{i},\,d_{i},$ $i=1,\ldots ,\,n$, are arbitrary real
constants [see \nocite*[p. 49]{Steele(2004)}]. Formula (\ref{eq: Binet-Cauchy
identity}) relates several inner products to pairwise comparisons between
the components of the vectors considered.
Since the representations in (\ref{eq: Pairwise sample variance}) and (\ref
{eq: Pairwise sample covariance}) only depend on \emph{differences} between
observations, we will call such representations (and similar ones) $D$\emph{
-representations }(or \emph{Gini-type representations}). In this paper, we
provide $D$\emph{-}representations of higher-order central moments for both
general random variables and empirical moments. For third and fourth
moments, this yields new representations of skewness and kurtosis
coefficients, where reference to a location parameter is eliminated. We show
that all central moments possess $D$\emph{-}representations, so again no
reference to the mean is needed for moments of any order. This is done by
considering i.i.d. \emph{replications} (or realizations) of the random
variables considered, by observing that central moments can be interpreted
as covariances between a random variable and powers of the same variable (a
\emph{nonlinear} relation), and by giving \emph{recursions} which link the $
D $-representation of any moment to lower order ones. Numerical summation
identities similar to (\ref{eq: Pairwise sample variance})\thinspace
-\thinspace (\ref{eq: Pairwise sample covariance}) are also deduced.
Finally, through a similar approach, we give the analogues of the Lagrange
and Binet-Cauchy identities for general random variables, along with a
simple derivation of the classic Cauchy-Schwarz inequality for covariances.
The paper is organized as follows. In Section \ref{Sec: Gini representation
of covariances}, we discuss $D$-representations for covariances and
regression coefficients, and we show how corresponding numerical identities
can be derived from these. In Section \ref{Sec: Skewness and kurtosis}, we
provide $D$-representations for third and fourth moments, along with
corresponding expressions for skewness and kurtosis coefficients. In Section
\ref{Sec: Higher-order moments}, we show that all moments possess such
representations, and we give recursive formulae for computing a $D$
-representation for any moment. Generalized Lagrange and Binet-Cauchy
identities are given in Section \ref{Sec: Generalized Lagrange identities}.
In Section \ref{Sec: Applications}, we discuss some applications of the
proposed moment representations. We conclude in Section \ref{Sec: Conclusion}
. Proofs are given in an online Appendix.
\subsection{$D$-representation of Covariances \label{Sec: Gini representation
of covariances}}
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In order to generalize the expressions (\ref{eq: Pairwise sample variance}
)\thinspace -\thinspace (\ref{eq: Pairwise sample covariance}) to
higher-order moments, it will be useful to discuss the pairwise difference
representation of the covariance between general random variables. Let $X$
and $Y$ be random variables with finite second moments, and means ${\Greekmath 0116} _{X}=
\mathbb{E}\{X\}$, ${\Greekmath 0116} _{Y}=\mathbb{E}\{Y\}.$ We denote by $\mathsf{C}
[X,\,Y]:=\mathbb{E}\{(X-{\Greekmath 0116} _{X})(Y-{\Greekmath 0116} _{Y})\}$ the covariance between $X$
and $Y$. Consider two independent and identically distributed replications
of the random vector $(X,\,Y)^{\prime }$: $\;(X_{1},\,Y_{1})^{\prime }$ and $
(X_{2},\,Y_{2})^{\prime }$. On observing
\begin{eqnarray}
\mathbb{E}\{(X_{1}-X_{2})(Y_{1}-Y_{2})\} &=&\mathbb{E}\{X_{1}Y_{1}\}+\mathbb{
E}\{X_{2}Y_{2}\}-\mathbb{E}\{X_{1}Y_{2}\}-\mathbb{E}\{X_{2}Y_{1}\} \notag \\
&=&\mathbb{E}\{X_{1}Y_{1}\}+\mathbb{E}\{X_{2}Y_{2}\}-\mathbb{E}\{X_{1}\}
\mathbb{E}\{Y_{2}\}-\mathbb{E}\{X_{2}\}\mathbb{E}\{Y_{1}\} \notag \\
&=&2(\mathbb{E}\{XY\}-\mathbb{E}\{X\}\mathbb{E}\{Y\})=2\mathsf{C}[X,\,Y]\,,
\label{eq: E[(X_1-X_2)(Y_1-Y_2)]}
\end{eqnarray}
it follows that
\begin{equation}
\mathsf{C}[X,\,Y]=\frac{1}{2}\mathbb{E}\{(X_{1}-X_{2})(Y_{1}-Y_{2})\}\,.
\label{eq: Cov}
\end{equation}
It is also easy to see that
\begin{eqnarray}
\mathsf{C}[X,\,Y] &=&\mathbb{E}\{X_{1}(Y_{1}-Y_{2})\}=-\mathbb{E}
\{X_{2}(Y_{1}-Y_{2})\} \notag \\
&=&\mathbb{E}\{(X_{1}-X_{2})Y_{1}\}=-\mathbb{E}\{(X_{1}-X_{2})Y_{2}\}\,.
\label{eq: Cov One diff}
\end{eqnarray}
In other words, only one of the variables in $(X,$\thinspace $Y)^{\prime }$
need be replicated to avoid centering with respect to the means (${\Greekmath 0116} _{X}$
and ${\Greekmath 0116} _{Y}$). (\ref{eq: E[(X_1-X_2)(Y_1-Y_2)]})\thinspace -\thinspace (
\ref{eq: Cov One diff}) also hold under the weaker assumption (without
identical distributions) that $(X_{1},\,Y_{1})^{\prime }$ and $
(X_{2},\,Y_{2})^{\prime }$ have the same first and second moments as $
(X,\,Y) $ with
\begin{equation}
\mathbb{E}\{X_{1}Y_{2}\}=\mathbb{E}\{X_{2}Y_{1}\}=\mathbb{E}\{X\}\mathbb{E}
\{Y\}\,.
\end{equation}
For $X_{1}=Y_{1}$ and $X_{2}=Y_{2},$ we obtain the following formulae for
the variance of $X:$
\begin{eqnarray}
{\Greekmath 011B} _{X}^{2} &=&\mathrm{Var}(X)=\frac{1}{2}\mathbb{E}\{(X_{1}-X_{2})^{2}\}
\notag \\
&=&\mathbb{E}\{X_{1}(X_{1}-X_{2})\}=-\mathbb{E}\{X_{2}(X_{1}-X_{2})\}\,.
\label{eq: Var}
\end{eqnarray}
If $X_{3}$ is a third replication of $X$ [so that $X_{1}$, $X_{2}$, $X_{3}$
are i.i.d.], it is also easy to check that
\begin{equation}
{\Greekmath 011B} _{X}^{2}=\mathbb{E}\{(X_{1}-X_{3})(X_{1}-X_{2})\}\,.
\label{eq: Var 3 rep}
\end{equation}
The uncentered second moment of $X$ can be written as
\begin{eqnarray}
\mathbb{E}\{X^{2}\} &=&\frac{1}{2}\mathbb{E}\{(X_{1}-X_{2})^{2}\}+{\Greekmath 0116}
_{X}^{2} \notag \\
&=&\mathbb{E}\{X_{1}(X_{1}-X_{2})\}+{\Greekmath 0116} _{X}^{2}=-\mathbb{E}
\{X_{2}(X_{1}-X_{2})\}+{\Greekmath 0116} _{X}^{2}
\end{eqnarray}
and the linear regression coefficient of $Y$ on $X$ is
\begin{eqnarray}
{\Greekmath 010C} _{Y\cdot X} &=&\frac{{\Greekmath 011B} _{XY}}{{\Greekmath 011B} _{X}^{2}}=\frac{\mathbb{E}
\{(X_{1}-X_{2})(Y_{1}-Y_{2})\}}{\mathbb{E}\{(X_{1}-X_{2})^{2}\}} \notag \\
&=&\frac{\mathbb{E}\{(X_{1}-X_{2})Y_{1}\}}{\mathbb{E}\{(X_{1}-X_{2})X_{1}\}}=
\frac{\mathbb{E}\{X_{1}(Y_{1}-Y_{2})\}}{\mathbb{E}\{X_{1}(X_{1}-X_{2})\}}\,.
\end{eqnarray}
In view of the above discussion on the covariance and the variance, we will
define more formally what we mean by the $D$-representation of a parameter.
\begin{definition}
A parameter ${\Greekmath 010D} $ has a $D$-representation for the variables $
X_{1},\ldots ,\,X_{n}$ if there is a function $g(x_{1},\ldots
,\,x_{n})=h(x_{2}-x_{1},\ldots ,\,x_{n}-x_{n-1})$ such that
\begin{equation}
{\Greekmath 010D} =\mathbb{E}\{h(X_{2}-X_{1},\ldots ,\,X_{n}-X_{n-1})\}\,.
\end{equation}
\end{definition}
In other words, ${\Greekmath 010D} $ has a $D$-representation for the variables $
X_{1},\ldots ,\,X_{n}$ if it can be written as the expected value of a
function $g(X_{1},\ldots ,\,X_{n})$ which depends on $X_{1},\ldots ,\,X_{n}$
only through the differences between the variables $X_{1},\ldots ,\,X_{n}$.
Equivalently, this means that
\begin{equation}
{\Greekmath 010D} =\mathbb{E}\{g(X_{1},\ldots ,\,X_{n})\}
\end{equation}
where $g(x_{1},\ldots ,\,x_{n})$ is invariant to uniform location changes,
\emph{i.e.}
\begin{equation}
g(x_{1}+c,\ldots ,\,x_{n}+c)=g(x_{1},\ldots ,\,x_{n})\quad \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for any }c.
\end{equation}
A potentially useful feature of the representation (\ref{eq: Cov}) follows
from the fact that the variables $(X_{1}-X_{2})$ and $(Y_{1}-Y_{2})$ have
marginal distributions symmetric around zero. Consequently, the odd moments
of $(X_{1}-X_{2})$ and $(Y_{1}-Y_{2})$ are all zero (when they exist). More
generally, the distribution of the vector $(X_{1}-X_{2},\,Y_{1}-Y_{2})^{
\prime }$ is jointly symmetric around zero in the sense that
\begin{equation}
(X_{1}-X_{2},\,Y_{1}-Y_{2})^{\prime }\sim
-(X_{1}-X_{2},\,Y_{1}-Y_{2})^{\prime }.
\end{equation}
This type of symmetry can be useful for proving unbiasedness results in
various setups; see \citename{Dufour(1984)} (\citeyear*{Dufour(1984)}, \citeyear*{Dufour(1985)}). This property is
shared by all $D$-representations when the paired differences are based on
i.i.d. observations.
It is of interest to note that the numerical identity (\ref{eq: Pairwise
sample covariance}) can be derived from the general moment formula (\ref{eq:
Cov}). Let $(I,$\thinspace $J)$ be a pair of random integers such that
\begin{equation}
\mathbb{P}[(I,\,J)=(i,\,j)\}=(\frac{1}{n})^{2}\,,\;i,\,j=1,...,n,
\end{equation}
and consider the random vector
\begin{equation}
(X^{\ast },\,Y^{\ast })=(x_{I},\,y_{J})\,.
\end{equation}
The values $x_{1},\ldots ,\,x_{n}$ and $y_{1},\ldots ,\,y_{n}$ need not be
distinct. It is then easy to see that:
\begin{equation}
\mathbb{E}\{X^{\ast }\}=\frac{1}{n^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}x_{i}=
\bar{x}\,,\quad \mathbb{E}\{Y^{\ast }\}=\frac{1}{n^{2}}\sum_{i=1}^{n}
\sum_{j=1}^{n}y_{j}=\bar{y}\,,
\end{equation}
\begin{equation}
\mathbb{E}\{X^{\ast }Y^{\ast }\}=\frac{1}{n^{2}}\sum_{i=1}^{n}
\sum_{j=1}^{n}x_{i}y_{j}\,, \label{eq: E(X*Y*)}
\end{equation}
and by the definition of the covariance,
\begin{equation}
{\Greekmath 011B} _{X^{\ast }Y^{\ast }}=\mathbb{E}\{X^{\ast }Y^{\ast }\}-\mathbb{E}
\{X^{\ast }\}\mathbb{E}\{Y^{\ast }\}=\frac{1}{n^{2}}\sum_{i=1}^{n}
\sum_{j=1}^{n}x_{i}y_{j}-\bar{x}\bar{y}\,.
\end{equation}
Consider now two i.i.d. replications of $(X^{\ast },\,Y^{\ast })$:\ \ $
(X_{1}^{\ast },\,Y_{1}^{\ast })$ and $(X_{2}^{\ast },\,Y_{2}^{\ast })$. By
applying (\ref{eq: Cov}) and (\ref{eq: E(X*Y*)}) to $(X^{\ast },$\thinspace $
Y^{\ast })$, the covariance between $X^{\ast }$ and $Y^{\ast }$ is given by:
\begin{equation}
{\Greekmath 011B} _{X^{\ast }Y^{\ast }}=\frac{1}{2}\mathbb{E}\{(X_{1}^{\ast
}-X_{2}^{\ast })(Y_{1}^{\ast }-Y_{2}^{\ast })\}=\frac{1}{2}\frac{1}{n^{2}}
\sum_{i=1}^{n}\sum_{j=1}^{n}(x_{i}-x_{j})(y_{i}-y_{j})\,.
\end{equation}
Since the above sum contains a zero whenever $i=j$, it is natural to divide
by $(n^{2}-n)\;$rather than $n^{2}$. This leads to:
\begin{equation}
s_{X^{\ast }Y^{\ast }}:=(\frac{n}{n-1}){\Greekmath 011B} _{X^{\ast }Y^{\ast }}=\frac{1}{
2}\frac{1}{n(n-1)}\sum_{i=1}^{n}
\sum_{j=1}^{n}(x_{i}-x_{j})(y_{i}-y_{j})=s_{XY}(n)\,.
\end{equation}
In the special case where $X^{\ast }=Y^{\ast },$ we have
\begin{equation}
{\Greekmath 011B} _{X^{\ast }}^{2}=\frac{1}{2}\mathbb{E}\{(X_{1}^{\ast }-X_{2}^{\ast
})^{2}\}=\frac{1}{2}\frac{1}{n^{2}}\sum_{i=1}^{n}
\sum_{j=1}^{n}(x_{i}-x_{j})^{2}
\end{equation}
hence, on setting $s_{X^{\ast }}^{2}:=(\frac{n}{n-1}){\Greekmath 011B} _{X^{\ast }}^{2}$
, the Gini formula (\ref{eq: Pairwise sample variance}) for the variance is:
\begin{equation}
s_{X^{\ast }}^{2}=\frac{1}{2}\frac{1}{n(n-1)}\sum_{i=1}^{n}
\sum_{j=1}^{n}(x_{i}-x_{j})^{2}=s_{X}^{2}(n)\,.
\end{equation}
\subsection{Skewness and Kurtosis \label{Sec: Skewness and kurtosis}}
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In this section, we express skewness and kurtosis coefficients in terms of
pairwise differences. Such representations can be written as functions of
differences of the form $X_{i}-X_{j}$ or differences between powers of
variables [$X_{i}^{2}-X_{j}^{2}$]. We first underscore that the third and
fourth central moments of a random variable $X$ can be interpreted as
covariances, which in turn can be represented in terms of pairwise
differences between replicated random variables. In each case, we give
several expressions which may have independent interest.
\begin{proposition}
\label{Th: Covariance representation of third and fourth central moments}
\textsc{ Covariance representation of third and
fourth central moments}.
\addcontentsline{lth}{theorem}{{\bf Proposition \hspace{0em} \protect\numberline{\theproposition}}
{\hspace{-3.5em} : \hspace{0em} Covariance representation of third and
fourth central moments}} Let $X$ be a random variable with finite mean ${\Greekmath 0116}
_{X}$ and variance ${\Greekmath 011B} _{X}^{2}$. Let $X_{1}$, $X_{2}$ two \emph{i.i.d.}
replications of $X$. If\ $X$ has finite third moment, then
\begin{eqnarray}
\mathbb{E}\{(X-{\Greekmath 0116} _{X})^{3}\} &=&\mathsf{C}[X,\,(X-{\Greekmath 0116} _{X})^{2}]=\mathsf{C}
[X,\,X^{2}]-2{\Greekmath 0116} _{X}{\Greekmath 011B} _{X}^{2} \notag \\
&=&\mathbb{E}\{(X_{1}-X_{2})(X_{1}-{\Greekmath 0116} _{X})^{2}\}=\mathbb{E}
\{(X_{1}-X_{2})X_{1}^{2}\}-2{\Greekmath 0116} _{X}{\Greekmath 011B} _{X}^{2} \notag \\
&=&\frac{1}{2}\mathbb{E}\{(X_{1}-X_{2})[(X_{1}-{\Greekmath 0116} _{X})^{2}-(X_{2}-{\Greekmath 0116}
_{X})^{2}]\} \notag \\
&=&\frac{1}{2}\mathbb{E}\{(X_{1}-X_{2})(X_{1}^{2}-X_{2}^{2})\}-{\Greekmath 0116} _{X}
\mathbb{E}\{(X_{1}-X_{2})^{2}\}\,. \label{eq: Cov Third moment}
\end{eqnarray}
If $X$ has finite fourth moment, then
\begin{eqnarray}
\mathbb{E}\{(X-{\Greekmath 0116} _{X})^{4}\} &=&\mathsf{C}[X,\,(X-{\Greekmath 0116} _{X})^{3}]=\mathsf{C}
[X,\,X^{3}]-3{\Greekmath 0116} _{X}\mathsf{C}[X,\,X^{2}]+3{\Greekmath 0116} _{X}^{2}{\Greekmath 011B} _{X}^{2}
\notag \\
&=&\mathbb{E}\{(X_{1}-X_{2})(X_{1}-{\Greekmath 0116} _{X})^{3}\}=\mathbb{E}
\{(X_{1}-X_{2})(X_{1}^{2}-2{\Greekmath 0116} _{X}X_{1})\} \notag \\
&=&\frac{1}{2}\mathbb{E}\{(X_{1}-X_{2})[(X_{1}-{\Greekmath 0116} _{X})^{3}-(X_{2}-{\Greekmath 0116}
_{X})^{3}]\} \notag \\
&=&\frac{1}{2}\mathbb{E}\{(X_{1}-X_{2})(X_{1}^{3}-X_{2}^{3})\}-3{\Greekmath 0116} _{X}
\mathbb{E}\{(X-{\Greekmath 0116} _{X})^{3}\}-3{\Greekmath 0116} _{X}^{2}{\Greekmath 011B} _{X}^{2}\,.
\label{eq: Cov Fourth moment}
\end{eqnarray}
\end{proposition}
To interpret Proposition \ref{Th: Covariance representation of third and
fourth central moments}, consider first the case where the mean is zero ($
{\Greekmath 0116} _{X}=0$), so the third moment does not depend on the mean or the
variance. When the distribution is skewed to the right [$\mathbb{E}\{(X-{\Greekmath 0116}
_{X})^{3}\}>0$\}, equation (\ref{eq: Cov Third moment}) shows that an
increase in the value of $X$ ($X_{1}-X_{2}>0$) is associated with an
increase in the absolute value of $X$ [$X_{1}^{2}-X_{2}^{2}>0$], while a
shift to the left would go with a decrease in the absolute value of $X$.
More generally, $\mathbb{E}\{(X-{\Greekmath 0116} _{X})^{3}\}$ can be written as
\begin{equation}
\mathbb{E}\{(X-{\Greekmath 0116} _{X})^{3}\}=\frac{1}{2}\mathbb{E}\{[(X_{1}-{\Greekmath 0116}
_{X})-(X_{2}-{\Greekmath 0116} _{X})][(X_{1}-{\Greekmath 0116} _{X})^{2}-(X_{2}-{\Greekmath 0116} _{X})^{2}]\}\,
\end{equation}
so that $\mathbb{E}\{(X-{\Greekmath 0116} _{X})^{3}\}$ measures the association between a
change in the deviation from the mean and the corresponding change in the
absolute values of the deviations. A positive (negative) association entails
positive (negative) skewness, which is clearly exhibited by pairwise
differences.
The formulae given by Proposition \ref{Th: Covariance representation of
third and fourth central moments} depend on unknown parameters (${\Greekmath 0116} _{X}$
and ${\Greekmath 011B} _{X}^{2}$). It is possible to avoid this by using a larger
number of replications. For the third moment, only one additional
replication of $X$ is needed. If $X_{1}$, $X_{2}$, $X_{3}$ are three i.i.d.
replications of $X$, we see [from (\ref{eq: Cov Third moment})] that
\begin{eqnarray}
\mathbb{E}\{(X-{\Greekmath 0116} _{X})^{3}\} &=&\frac{1}{2}\mathbb{E}
\{(X_{1}-X_{2})(X_{1}^{2}-X_{2}^{2})\}-\mathbb{E}\{X_{3}\}\mathbb{E}
\{(X_{1}-X_{2})^{2}\} \notag \\
&=&\frac{1}{2}\mathbb{E}
\{(X_{1}-X_{2})[(X_{1}^{2}-X_{2}^{2})-2X_{3}(X_{1}-X_{2})]\}\,.
\label{eq: Cov Third moment 3 rep}
\end{eqnarray}
For the fourth moment, this can be done with four replications. If $X_{1}$, $
X_{2}$, $X_{3}$, $X_{4}$ are four i.i.d. replications of $X$, we have:
\begin{align*}
\mathbb{E}\{(X-{\Greekmath 0116} _{X})^{4}\}=& \frac{1}{2}\mathbb{E}
\{(X_{1}-X_{2})(X_{1}^{3}-X_{2}^{3})\}-\frac{3}{2}\mathbb{E}\{X_{3}\}\mathbb{
E}\{(X_{1}-X_{2})(X_{1}^{2}-X_{2}^{2})\} \\
& +\frac{3}{2}\mathbb{E}\{X_{3}\}\mathbb{E}\{X_{4}\}\mathbb{E}
\{(X_{1}-X_{2})^{2}\}
\end{align*}
\vspace{-1.5\baselineskip}
\begin{equation}
\quad\quad\quad =\frac{1}{2}\mathbb{E}
\{(X_{1}-X_{2})(X_{1}^{3}-X_{2}^{3})-3X_{3}(X_{1}-X_{2})(X_{1}^{2}-X_{2}^{2})+3X_{3}X_{4}(X_{1}-X_{2})^{2}\}.
\label{eq: Cov Fourth moment 4 rep}
\end{equation}
(\ref{eq: Cov Third moment 3 rep}) - (\ref{eq: Cov Fourth moment 4 rep})
still do not provide $D$-representations, because $X_{3}$ and $X_{4}$ are in
levels. This is done in the following propositions by using additional
replicates of $X$. A number of alternative expressions are provided in the
following proposition.
\begin{proposition}
\label{Th: D-representation of third central moment}
\textsc{ $D$-representation of third central moment}.
\addcontentsline{lth}{theorem}{{\bf Proposition \hspace{0em} \protect\numberline{\theproposition}}
{\hspace{-3.5em} : \hspace{0em} $D$-representation of third central moment}}
Let $X$ be a random variable with mean ${\Greekmath 0116} _{X}$, variance ${\Greekmath 011B} _{X}^{2}$
, and finite third moment. If $X_{1}$, $X_{2}$, $X_{3}$ are three i.i.d.
replications of $X$, then
\begin{eqnarray}
\mathbb{E}\{(X-{\Greekmath 0116} _{X})^{3}\} &=&\mathbb{E}\{(X_{1}-X_{3})(X_{1}-X_{2})^{2}
\}=\frac{1}{2}\mathbb{E}\{(X_{1}-X_{2})[(X_{1}-X_{3})^{2}-(X_{2}-X_{3})^{2}]
\} \notag \\
&=&\frac{1}{2}\mathbb{E}\{(X_{1}-X_{2})^{2}[(X_{1}-X_{3})+(X_{2}-X_{3})]\}=
\frac{1}{6}\mathbb{E}\{D_{1}^{(3)}D_{2}^{(3)}D_{3}^{(3)}\} \notag \\
&=&\frac{9}{2}\mathbb{E}\{(X_{1}-\bar{X}^{(3)})(X_{2}-\bar{X}^{(3)})(X_{3}-
\bar{X}^{(3)})\}\quad \quad \label{eq: D-representation third moment}
\end{eqnarray}
where\ \ $\bar{X}^{(3)}:=\,\underset{j=1}{\overset{3}{\sum }}X_{j}/3\;$and
\begin{equation}
D_{i}^{(3)}:=\,\underset{j\neq i}{\sum }(X_{i}-X_{j})=3(X_{i}-\bar{X}
^{(3)})\,,\quad i=1,\,2,\,3. \label{eq: Sum of differences}
\end{equation}
\end{proposition}
As the variables $D_{i}^{(3)}$ are sums of pairwise differences, (\ref{eq:
D-representation third moment}) provides simple $D$-representations of the
third central moment of $X$. Note also that $\mathbb{E}\{(X-{\Greekmath 0116} _{X})^{3}\}$
can be rewritten as a function of deviations with respect to a
\textquotedblleft reference\textquotedblright\ value $X_{3}$:
\begin{eqnarray}
\mathbb{E}\{(X-{\Greekmath 0116} _{X})^{3}\} &=&\frac{1}{2}\mathbb{E}
\{[(X_{1}-X_{3})-(X_{2}-X_{3})\}[(X_{1}-X_{3})^{2}-(X_{2}-X_{3})^{2}]\}
\notag \\
&=&\frac{1}{2}\mathbb{E}
\{[(X_{1}-X_{3})-(X_{2}-X_{3})]^{2}[(X_{1}-X_{3})+(X_{2}-X_{3})]\}\,.
\end{eqnarray}
We now consider the fourth central moment.
\begin{proposition}
\label{Th: D-representation of fourth central moment}
\textsc{ $D$-representation of fourth central
moment}.
\addcontentsline{lth}{theorem}{{\bf Proposition \hspace{0em} \protect\numberline{\theproposition}}
{\hspace{-3.5em} : \hspace{0em} $D$-representation of fourth central
moment}} Let $X$ be a random variable with mean ${\Greekmath 0116} _{X}$, variance ${\Greekmath 011B}
_{X}^{2}$, and finite fourth moment. If $X_{1}$, $X_{2}$, $X_{3}$, $X_{4}$
four i.i.d. replications of $X$, then
\begin{eqnarray}
\mathbb{E}\{(X-{\Greekmath 0116} _{X})^{4}\} &=&\frac{1}{2}\mathbb{E}\{(X_{1}-X_{2})^{4}
\}-3{\Greekmath 011B} _{X}^{4}=\frac{1}{2}\mathbb{E}\{(X_{1}-X_{2})^{4}\}-\frac{3}{4}(
\mathbb{E}\{(X_{1}-X_{2})^{2}\})^{2} \notag \\
&=&\mathbb{E}\big\{\frac{1}{2}(X_{1}-X_{2})^{4}-\frac{3}{4}
(X_{1}-X_{2})^{2}(X_{3}-X_{4})^{2}\big\}\,.
\label{eq: D-representation fourth moment}
\end{eqnarray}
\end{proposition}
The last two identities in (\ref{eq: D-representation fourth moment})
provide simple $D$-representations of the fourth central moment of $X$. We
now apply the above results to skewness, kurtosis and excess kurtosis
coefficients:
\begin{equation}
\mathrm{Sk}(X):=\frac{\mathbb{E}\{(X-{\Greekmath 0116} _{X})^{3}\}}{{\Greekmath 011B} _{X}^{3}}\,,\;\;
\mathrm{Kur}(X):=\frac{\mathbb{E}\{(X-{\Greekmath 0116} _{X})^{4}\}}{{\Greekmath 011B} _{X}^{4}}
\,,\;\;\mathrm{EKur}(X)=\mathrm{Kur}(X)-3\,.
\label{eq: Def Skewness Kurtosis}
\end{equation}
$\mathrm{Kur}(X)$ is also called the \textquotedblleft Pearson
kurtosis\textquotedblright\ coefficient, while the excess kurtosis $\mathrm{
EKur}(X)$ is the \textquotedblleft Fisher kurtosis\textquotedblright .
\begin{proposition}
\label{Th: D-representation of skewness and kurtosis}
\textsc{ $D$-representation of skewness and kurtosis}.
\addcontentsline{lth}{theorem}{{\bf Proposition \hspace{0em} \protect\numberline{\theproposition}}
{\hspace{-3.5em} : \hspace{0em} $D$-representation of skewness and kurtosis}}
Let $X$ be a random variable with mean ${\Greekmath 0116} _{X}$ and variance ${\Greekmath 011B}
_{X}^{2}$, and $X_{1}$, $X_{2}$, $X_{3}$, $X_{4}$\ four i.i.d. replications
of $X$. If $X$ has finite third moment, then
\begin{eqnarray}
\mathrm{Sk}(X) &=&\frac{\mathbb{E}\{(X_{1}-X_{2})(X_{1}^{2}-X_{2}^{2})\}}{
2{\Greekmath 011B} _{X}^{3}}-2\frac{{\Greekmath 0116} _{X}}{{\Greekmath 011B} _{X}}=\frac{\mathbb{E}
\{(X_{1}-X_{3})(X_{1}-X_{2})^{2}\}}{{\Greekmath 011B} _{X}^{3}} \notag \\
&=&\frac{\sqrt{8}\,\mathbb{E}\{(X_{1}-X_{3})(X_{1}-X_{2})^{2}\}}{(\mathbb{E}
\{(X_{1}-X_{2})^{2}\})^{^{3/2}}}=\frac{\sqrt{2}}{3}\frac{\mathbb{E}
\{D_{1}^{(3)}D_{2}^{(3)}D_{3}^{(3)}\}}{(\mathbb{E}\{(X_{1}-X_{2})^{2}
\})^{^{3/2}}}
\end{eqnarray}
where $D_{i}^{(3)}$ is defined by $(\ref{eq: Sum of differences})$. If $X$
has finite fourth moment, then
\begin{equation}
\mathrm{Kur}(X)=\frac{2\mathbb{E}\{(X_{1}-X_{2})^{4}\}}{(\mathbb{E}
\{(X_{1}-X_{2})^{2}\})^{2}}-3\,.
\end{equation}
\end{proposition}
If $X_{1}$ and $X_{2}$ are i.i.d. normal, it is easy to see that
\begin{equation}
\frac{\mathbb{E}\{(X_{1}-X_{2})^{4}\}}{(\mathbb{E}\{(X_{1}-X_{2})^{2}\})^{2}}
=\frac{12{\Greekmath 011B} _{X}^{4}}{4{\Greekmath 011B} _{X}^{4}}=3
\end{equation}
so that $\mathrm{EKur}(X)=0$.
\begin{Tentative}
\begin{comment}
*******************
Discrete analogues.
*******************
\begin{Tentative}
\end{comment}
\end{Tentative}
\subsection{Higher-order Moments \label{Sec: Higher-order moments}}
\renewcommand{\Alph{section}}{\Alph{section}}
\renewcommand{\Alph{section}}{\Alph{section}}
\setcounter{theorem}{0} \setcounter{definition}{0} \setcounter{equation}{0}
In this section, we study how higher-order central moments can be
represented in terms of pairwise differences between i.i.d. realizations of
a random variable. Consider a random variable $X$ with mean ${\Greekmath 0116} _{X}$ and
finite moments up to order $n\geq 1$. We denote the $k$-th central moment of
$X$ by
\begin{equation}
{\Greekmath 0116} _{k}:=\mathbb{E}\{(X-{\Greekmath 0116} _{X})^{k}\},\quad k=1,\ldots ,\,n\,.
\end{equation}
Suppose that $X_{0},$ $X_{1},\ldots ,\,X_{n}$\ are i.i.d. replications of $X$
, and define
\begin{equation}
P_{k}:=\underset{i=1}{\overset{k}{\prod }}(X_{0}-X_{i})\,,\quad \tilde{X}
_{i}:=X_{i}-{\Greekmath 0116} _{X}\,,\quad i=1,\ldots ,\,n\,. \label{eq: P_j}
\end{equation}
Since $\mathbb{E}\{\tilde{X}_{i}\}={\Greekmath 0116} _{1}=0$, for $i=0,\ldots ,\,n$, the
independence of $\tilde{X}_{0},\ldots ,\,\tilde{X}_{n}$ entails:
\begin{equation}
\mathbb{E}\{P_{k}\}=\mathbb{E}\big\{\underset{i=1}{\overset{k}{\prod }}(
\tilde{X}_{0}-\tilde{X}_{i})\big\}=\mathbb{E}\{\tilde{X}_{0}^{k}\}={\Greekmath 0116}
_{k}\,,\quad k=1,\ldots ,\,n\,. \label{eq: E(P_j)}
\end{equation}
In other words, $P_{k}$ can be viewed as an unbiased estimator of ${\Greekmath 0116} _{k}$
, for $k=1,\ldots ,\,n$ . This immediately shows that all the central
moments of $X$ have a $D$-representation for moments up to order $n$,
provided $X$ has moments up to order $n$.
The function $P_{n}$ requires $n+1$ replications of $X$ to represent the $n$
-th central moment of $X$. However, for $n=2$, $3,$ $4$, we have given
representations which only require $n$ replications (see Section \ref{Sec:
Skewness and kurtosis}). We will now show this is indeed feasible for $n\geq
5$. We first state some useful recursions.
\begin{proposition}
\label{Th: Recursions for higher-order moments}
\textsc{ Recursions for higher-order moments}.
\addcontentsline{lth}{theorem}{{\bf Proposition \hspace{0em} \protect\numberline{\theproposition}}
{\hspace{-3.5em} : \hspace{0em} Recursions for higher-order moments}} Let $X$
be a random variable with mean ${\Greekmath 0116} _{X}$, and finite central moments ${\Greekmath 0116}
_{j}$ for $j=1,\ldots ,\,n+1$, where $n\geq 1.$ Then
\begin{equation}
{\Greekmath 0116} _{n+1}=\mathsf{C}[X,\,(X-{\Greekmath 0116} _{X})^{n}]=\mathbb{E}\{X(X-{\Greekmath 0116}
_{X})^{n}\}-{\Greekmath 0116} _{X}{\Greekmath 0116} _{n}\,. \label{eq: mu(n+1) cov}
\end{equation}
If $X_{1}$, $X_{2}$, $X_{3}\;$are i.i.d. replications of $X$, then
\begin{eqnarray}
{\Greekmath 0116} _{n+1} &=&\mathbb{E}\{(X_{1}-X_{2})(X_{1}-{\Greekmath 0116} _{X})^{n}\} \notag \\
&=&\mathbb{E}\{(X_{1}-X_{3})(X_{1}-X_{2})^{n}\}-
\sum_{j=2}^{n-1}(-1)^{j}C_{n}^{j}\,{\Greekmath 0116} _{j}{\Greekmath 0116} _{n+1-j}
\label{eq: recusion mu_n}
\end{eqnarray}
where $C_{n}^{j}=n!/[j!(n-j)!]$ and, for $n+1$ even,
\begin{equation}
{\Greekmath 0116} _{n+1}=\frac{1}{2}\big[\mathbb{E}\{(X_{1}-X_{2})^{n+1}\}-
\sum_{j=2}^{n-1}(-1)^{j}C_{n+1}^{j}\,{\Greekmath 0116} _{j}{\Greekmath 0116} _{n+1-j}\big]\,.
\label{eq: recusion mu_n n even}
\end{equation}
\end{proposition}
When the summations in (\ref{eq: recusion mu_n})\thinspace -\thinspace (\ref
{eq: recusion mu_n n even}) are empty (for $n=1$ and $n=2$), the
corresponding formulae yield [as expected from (\ref{eq: Var 3 rep}) and (
\ref{eq: D-representation third moment})]:
\begin{equation}
{\Greekmath 0116} _{2}=\mathbb{E}\{(X_{1}-X_{3})(X_{1}-X_{2})\}=\frac{1}{2}\mathbb{E}
\{(X_{1}-X_{2})^{2}\}\,,\quad {\Greekmath 0116} _{3}=\mathbb{E}
\{(X_{1}-X_{3})(X_{1}-X_{2})^{2}\}\,.
\end{equation}
Now, let $X_{1},\ldots ,\,X_{n}$ be i.i.d. replications of $X$ with $n\geq 5$
, and consider the following sequence of functions:
\begin{equation}
\bar{{\Greekmath 0116}}_{1}=0\,,\quad \bar{{\Greekmath 0116}}_{2}(x_{1},\,x_{2})=\frac{1}{2}
(x_{1}-x_{2})^{2}\,, \label{eq: muhat(2)}
\end{equation}
\begin{equation}
\bar{{\Greekmath 0116}}_{3}(x_{1},\,x_{2},\,x_{3})=(x_{1}-x_{3})(x_{1}-x_{2})^{2}\,,
\label{eq: muhat(3)}
\end{equation}
\begin{equation}
\bar{{\Greekmath 0116}}_{4}(x_{1},\,x_{2},\,x_{3},\,x_{4})=\frac{1}{2}(x_{1}-x_{2})^{4}-
\frac{3}{4}(x_{1}-x_{2})^{2}(x_{3}-x_{4})^{2}, \label{eq: mubar(4)}
\end{equation}
and for $k\geq 4$,
\begin{equation}
\bar{{\Greekmath 0116}}_{k+1}(x_{1},\ldots
,\,x_{k+1})=(x_{1}-x_{3})(x_{1}-x_{2})^{k}-\sum_{j=2}^{k-1}(-1)^{j}C_{k}^{j}
\bar{{\Greekmath 0116}}_{j}(x_{1},\ldots ,\,x_{j})\bar{{\Greekmath 0116}}_{k+1-j}(x_{j+1},\ldots
,\,x_{k+1})\,. \label{eq: mubar(k+1)}
\end{equation}
If $k+1$ is even (with $k\geq 1$), we can also consider the functions
\begin{eqnarray}
\tilde{{\Greekmath 0116}}_{k+1}(x_{1},\ldots ,\,x_{k+1}) &=&\frac{1}{2}\big[
(x_{1}-x_{2})^{k+1} \notag \\
&&-\sum_{j=2}^{k-1}(-1)^{j}C_{k+1}^{j}\bar{{\Greekmath 0116}}_{j}(x_{1},\ldots ,\,x_{j})
\bar{{\Greekmath 0116}}_{k+1-j}(x_{j+1},\ldots ,\,x_{k+1})\big]\,.\quad \quad
\label{eq: mutilde(k)}
\end{eqnarray}
From the above definitions, it is clear that the functions $\bar{{\Greekmath 0116}}
_{k}(x_{1},\ldots ,\,x_{k})$ and $\tilde{{\Greekmath 0116}}_{k}(x_{1},\ldots ,\,x_{k})$
depend on their arguments only through differences $x_{i}-x_{j}$ ($
i,j=1,\ldots ,\,k$). Note also that $\bar{{\Greekmath 0116}}_{j}$ can be replaced by $
\tilde{{\Greekmath 0116}}_{j}$ in the summations of (\ref{eq: mubar(k+1)})\thinspace
-\thinspace (\ref{eq: mutilde(k)}) whenever $j$ is even.
\begin{proposition}
\label{Th: Recursive D-representations for higher-order moments}
\textsc{ Recursive $D$-representations for
higher-order moments}.
\addcontentsline{lth}{theorem}{{\bf Proposition \hspace{0em} \protect\numberline{\theproposition}}
{\hspace{-3.5em} : \hspace{0em} Recursive $D$-representations for
higher-order moments}} Let $X$ be a random variable with mean ${\Greekmath 0116} _{X}$, and
finite moments up to order $n+1$, where $n\geq 1$, and let $\bar{{\Greekmath 0116}}
_{k}(x_{1},\ldots ,\,x_{k})$ and $\tilde{{\Greekmath 0116}}_{k}(x_{1},\ldots ,\,x_{k})$ be
defined by $(\ref{eq: muhat(2)})\,$-\thinspace $\,(\ref{eq: mutilde(k)}),$
for $k\geq 1.$ If $X_{1},\ldots ,\,X_{n+1}$ are i.i.d. replications of $X$,
then
\begin{equation}
\mathbb{E}\{\bar{{\Greekmath 0116}}_{k+1}(X_{1},\ldots ,\,X_{k+1})\}={\Greekmath 0116} _{k+1}\,,\quad
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for }k=1,\ldots ,\,n\,, \label{eq: E(mubar_k)}
\end{equation}
and, if $k+1$ is even,
\begin{equation}
\mathbb{E}\{\tilde{{\Greekmath 0116}}_{k+1}(X_{1},\ldots ,\,X_{k+1})\}={\Greekmath 0116} _{k+1}\,,\quad
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for\ }1\leq k\leq n\,. \label{eq: E(mutilde_k)}
\end{equation}
\end{proposition}
The above proposition shows that the $k$-th central moment of $X$ has a $D$
-representation which only requires $k$ replications of $X$ (provided ${\Greekmath 0116}
_{k}$ is finite).
\begin{Tentative}
\begin{comment}
*******************
Discrete analogues.
*******************
\begin{Tentative}
\end{comment}
\end{Tentative}
\subsection{Generalized Lagrange Identities \label{Sec: Generalized Lagrange
identities}}
\renewcommand{\Alph{section}}{\Alph{section}}
\renewcommand{\Alph{section}}{\Alph{section}}
\setcounter{theorem}{0} \setcounter{definition}{0} \setcounter{equation}{0}
In this section, we give some generalizations of the Lagrange identity. We
first give a moment-based Lagrange-type identity using independent random
vectors.
\begin{proposition}
\label{Th: Generalized Lagrange identity}
\textsc{ Generalized Lagrange identity}.
\addcontentsline{lth}{theorem}{{\bf Proposition \hspace{0em} \protect\numberline{\theproposition}}
{\hspace{-3.5em} : \hspace{0em} Generalized Lagrange identity}} Let $(X_{1},$
\thinspace $Y_{1})^{\prime }$ and $(X_{2},$\thinspace $Y_{2})^{\prime }$ be
two independent random vectors with finite second moments. Then
\begin{equation}
\frac{1}{2}\big(\mathbb{E}\{X_{1}^{2}\}\mathbb{E}\{Y_{2}^{2}\}+\mathbb{E}
\{X_{2}^{2}\}\mathbb{E}\{Y_{1}^{2}\}\big)-\mathbb{E}\{X_{1}Y_{1}\}\mathbb{E}
\{X_{2}Y_{2}\}=\frac{1}{2}\mathbb{E}\{(X_{1}Y_{2}-X_{2}Y_{1})^{2}\}\,.
\label{eq: General Lagrange identity}
\end{equation}
\end{proposition}
In Proposition \ref{Th: Generalized Lagrange identity}, $(X_{1},$\thinspace $
Y_{1})^{\prime }$ and $(X_{2},$\thinspace $Y_{2})^{\prime }$ need not be
identically distributed. Since the right-hand side of (\ref{eq: General
Lagrange identity}) is nonnegative, it follows that
\begin{equation}
\mathbb{E}\{X_{1}Y_{1}\}\mathbb{E}\{X_{2}Y_{2}\}\leq \frac{1}{2}(\mathbb{E}
\{X_{1}^{2}\}\mathbb{E}\{Y_{2}^{2}\}+\mathbb{E}\{X_{2}^{2}\}\mathbb{E}
\{Y_{1}^{2}\}) \label{eq: Generalized CS}
\end{equation}
and, on replacing each variable $X_{1}$ , $X_{2}$ , $Y_{1}$ , $Y_{2}$ by its
deviation from the mean ($X_{1}-\mathbb{E}\{X_{1}\},$ etc.),
\begin{equation}
\mathsf{C}[X_{1},\,Y_{1}]\mathsf{C}[X_{2},\,Y_{2}]\leq \frac{1}{2}({\Greekmath 011B}
_{X_{1}}^{2}{\Greekmath 011B} _{Y_{2}}^{2}+{\Greekmath 011B} _{X_{2}}^{2}{\Greekmath 011B} _{Y_{1}}^{2})\,.
\label{eq: Generalized CS cov}
\end{equation}
The latter inequality holds even if the random variables $X_{1}$ , $X_{2}$ ,
$Y_{1}$ , $Y_{2}$ have different means and variances. When $(X_{1},$
\thinspace $Y_{1})^{\prime }$ and $(X_{2},$\thinspace $Y_{2})^{\prime }$
have the same covariance matrix (though possibly different means), (\ref{eq:
Generalized CS}) yields the usual Cauchy-Schwarz inequality for covariances:
\begin{equation}
\mathsf{C}[X_{1},\,Y_{1}]^{2}\leq {\Greekmath 011B} _{X_{1}}^{2}{\Greekmath 011B} _{Y_{1}}^{2}\,.
\label{eq: Cauchy-Shwarz}
\end{equation}
Interestingly, (\ref{eq: General Lagrange identity}) provides a remarkably
simple way of proving the Cauchy-Schwarz inequality for covariances.
The distance $\mathbb{E}\{(X_{1}Y_{2}-X_{2}Y_{1})^{2}\}$ can be viewed as a
measure of non-proportionality between $Y$ and $X$ : if $Y_{1}=b_{1}X_{1}$
and $Y_{2}=b_{2}X_{2}$, then
\begin{equation}
\mathbb{E}\{(X_{1}Y_{2}-X_{2}Y_{1})^{2}\}=\mathbb{E}
\{(b_{2}X_{1}X_{2}-b_{1}X_{2}X_{1})^{2}\}=(b_{2}-b_{1})^{2}\mathbb{E}
\{(X_{1}X_{2})^{2}\}\, \label{eq: General Lagrange identity Zero}
\end{equation}
with $\mathbb{E}\{(X_{1}Y_{2}-X_{2}Y_{1})^{2}\}=0$ when $b_{1}=b_{2}$. When $
\mathbb{E}\{(X_{1}X_{2})^{2}\}\neq 0$, the condition $b_{1}=b_{2}$ is
necessary and sufficient for $\mathbb{E}\{(X_{1}Y_{2}-X_{2}Y_{1})^{2}\}=0$.
The identity (\ref{eq: General Lagrange identity Zero}) holds even if $X_{1}$
and $X_{2}$ do not have the same distribution.
When additional homogeneity restrictions are imposed, we get simpler
identities which are spelled out in the following corollary.
\begin{corollary}
\label{Th: Covariance generalized Lagrange identity}
\textsc{ Covariance generalized Lagrange identity}.
\addcontentsline{lth}{theorem}{{\bf Proposition \hspace{0em} \protect\numberline{\theproposition}}
{\hspace{-3.5em} : \hspace{0em} Covariance generalized Lagrange identity}}
Let $(X_{1},$\thinspace $Y_{1})^{\prime }$ and $(X_{2},$\thinspace $
Y_{2})^{\prime }$ be two independent random vectors with finite second
moments, $\tilde{X}_{i}:=X_{i}-\mathbb{E}\{X_{i}\}$ and $\tilde{Y}
_{i}:=Y_{i}-\mathbb{E}\{Y_{i}\}$, $i=1,2$. Then
\begin{equation}
\frac{1}{2}({\Greekmath 011B} _{X_{1}}^{2}{\Greekmath 011B} _{Y_{2}}^{2}+{\Greekmath 011B} _{X_{2}}^{2}{\Greekmath 011B}
_{Y_{1}}^{2})-\mathsf{C}[X_{1},\,Y_{1}]\mathsf{C}[X_{2},\,Y_{2}]=\frac{1}{2}
\mathbb{E}\{(\tilde{X}_{1}\tilde{Y}_{2}-\tilde{X}_{2}\tilde{Y}_{1})^{2}\}\,.
\label{eq: General Lagrange identity order 2}
\end{equation}
If ${\Greekmath 011B} _{X_{1}}^{2}={\Greekmath 011B} _{X_{2}}^{2}$ and ${\Greekmath 011B} _{Y_{1}}^{2}={\Greekmath 011B}
_{Y_{2}}^{2}$, then
\begin{equation}
{\Greekmath 011B} _{X_{1}}^{2}{\Greekmath 011B} _{Y_{1}}^{2}-\mathsf{C}[X_{1},\,Y_{1}]\mathsf{C}
[X_{2},\,Y_{2}]=\frac{1}{2}\mathbb{E}\{(\tilde{X}_{1}\tilde{Y}_{2}-\tilde{X}
_{2}\tilde{Y}_{1})^{2}\}\,.
\label{eq: General Lagrange identity order 2 equal var}
\end{equation}
If the two random vectors $(X_{1},$\thinspace $Y_{1})^{\prime }$ and $
(X_{2}, $\thinspace $Y_{2})^{\prime }$ have the same covariance matrix, then
\begin{equation}
{\Greekmath 011B} _{X_{1}}^{2}{\Greekmath 011B} _{Y_{1}}^{2}-\mathsf{C}[X_{1},\,Y_{1}]^{2}=\frac{1
}{2}\mathbb{E}\{(\tilde{X}_{1}\tilde{Y}_{2}-\tilde{X}_{2}\tilde{Y}
_{1})^{2}\}\,. \label{eq: General Lagrange identity order 2 iid}
\end{equation}
\end{corollary}
The identity (\ref{eq: General Lagrange identity order 2 iid}) holds \emph{a
fortiori} when $(X_{1},$\thinspace $Y_{1})^{\prime }$ and $(X_{2},$
\thinspace $Y_{2})^{\prime }$ are i.i.d. (with finite second moments). When $
(X_{1},$\thinspace $Y_{1})^{\prime }$ and $(X_{2},$\thinspace $
Y_{2})^{\prime }$ have the same covariance matrix, the correlation ${\Greekmath 011A}
(X_{1},\,Y_{1})$ between $X_{1}$ and $Y_{1}$ satisfies:
\begin{equation}
{\Greekmath 011A} (X_{1},\,Y_{1})^{2}={\Greekmath 011A} (X_{2},\,Y_{2})^{2}=1-\frac{\mathbb{E}\{(
\tilde{X}_{1}\tilde{Y}_{2}-\tilde{X}_{2}\tilde{Y}_{1})^{2}\}}{2{\Greekmath 011B}
_{X_{1}}^{2}{\Greekmath 011B} _{Y_{1}}^{2}}\,.
\end{equation}
Thus, if $(X_{1},$\thinspace $Y_{1})^{\prime }$ and $(X_{2},$\thinspace $
Y_{2})^{\prime }$ are i.i.d. replications of the random vector $(X,$
\thinspace $Y)^{\prime },$ ${\Greekmath 011A} (X_{1},\,Y_{1})^{2}$ measures how close two
independent replications of $(X,$\thinspace $Y)^{\prime }$ are according to
the distance $\mathbb{E}\{(\tilde{X}_{1}\tilde{Y}_{2}-\tilde{X}_{2}\tilde{Y}
_{1})^{2}\}$.
The following proposition extends in a similar way the Binet-Cauchy identity
(\ref{eq: Binet-Cauchy identity}).
\begin{proposition}
\label{Th: Generalized Binet-Cauchy identity}
\textsc{ Generalized Binet-Cauchy identity}.
\addcontentsline{lth}{theorem}{{\bf Proposition \hspace{0em} \protect\numberline{\theproposition}}
{\hspace{-3.5em} : \hspace{0em} Generalized Binet-Cauchy identity}}
Let $Z_{i}:=(A_{i},$\thinspace $B_{i},\,C_{i},\,D_{i})^{\prime }\;$and\ $
\tilde{Z}_{i}:=Z_{i}-\mathbb{E}\{Z_{i}\}=(\tilde{A}_{i},$\thinspace $\tilde{B
}_{i},\,\tilde{C}_{i},\,\tilde{D}_{i})^{\prime }$, $i=1,2$, be independent
random vectors with finite fourth moments. Then
\begin{eqnarray}
\mathbb{E}\{(A_{1}B_{2}-A_{2}B_{1})(C_{1}D_{2}-C_{2}D_{1})\} &=&\mathbb{E}
\{A_{1}C_{1}\}\mathbb{E}\{B_{2}D_{2}\}+\mathbb{E}\{A_{2}C_{2}\}\mathbb{E}
\{B_{1}D_{1}\} \notag \\
&-&[\mathbb{E}\{A_{1}D_{1}\}\mathbb{E}\{B_{2}C_{2}\}+\mathbb{E}\{A_{2}D_{2}\}
\mathbb{E}\{B_{1}C_{1}\}]\,.\quad \quad \label{eq: Generalized Binet-Cauchy}
\end{eqnarray}
If $(A_{1},$\thinspace $B_{1},\,C_{1},\,D_{1})^{\prime }$ and $(A_{2},$
\thinspace $B_{2},\,C_{2},\,D_{2})^{\prime }$ have the same covariance
matrix, then
\begin{equation}
\mathsf{C}[A_{1},\,C_{1}]\mathsf{C}[B_{1},\,D_{1}]-\mathsf{C}[A_{1},\,D_{1}]
\mathsf{C}[B_{1},\,C_{1}]=\frac{1}{2}\mathbb{E}\{(\tilde{A}_{1}\tilde{B}_{2}-
\tilde{A}_{2}\tilde{B}_{1})(\tilde{C}_{1}\tilde{D}_{2}-\tilde{C}_{2}\tilde{D}
_{1})\}\,. \label{eq: Generalized Binet-Cauchy Cov}
\end{equation}
\end{proposition}
The above proposition yields restrictions on the covariances of a
four-dimensional random vector. From (\ref{eq: Generalized Binet-Cauchy Cov}
), we see that $A_{1}B_{2}-A_{2}B_{1}=0$ (a.s.) or $C_{1}D_{2}-C_{2}D_{1}=0$
(a.s.) entails
\begin{equation}
\mathsf{C}[A_{1},\,C_{1}]\mathsf{C}[B_{1},\,D_{1}]=\mathsf{C}[A_{1},\,D_{1}]
\mathsf{C}[B_{1},\,C_{1}]\,.
\end{equation}
On taking\ $A_{i}=C_{i}$\ and\ $B_{i}=D_{i}\;$for $i=1,$\thinspace $2$, we
get the Lagrange identity:
\begin{equation}
{\Greekmath 011B} _{A_{1}}^{2}{\Greekmath 011B} _{B_{1}}^{2}-(\mathsf{C}[A_{1},\,B_{1}])^{2}=\frac{
1}{2}\mathbb{E}\{(\tilde{A}_{1}\tilde{B}_{2}-\tilde{A}_{2}\tilde{B}
_{1})^{2}\}\,.
\end{equation}
\subsection{Applications \label{Sec: Applications}}
\renewcommand{\Alph{section}}{\Alph{section}}
\renewcommand{\Alph{section}}{\Alph{section}}
\setcounter{theorem}{0} \setcounter{definition}{0} \setcounter{equation}{0}
The $D$-representations given above show that the central moments of a real
random variable $X$ can all be represented as expected values of polynomial
functions in differences between a small number of replications of $X$. For $
k\geq 2,$ the $k$-th central moment of $X$ only requires $k$ replications $
X_{1},\ldots ,\,X_{k}$. More precisely, we can write:
\begin{equation}
{\Greekmath 0116} _{k}:=\mathbb{E}\{(X-{\Greekmath 0116} _{X})^{k}\}=\mathbb{E}\{h_{k}(X_{1},\ldots
,\,X_{k})\}\,,\quad \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for }k\geq 2,
\end{equation}
where
\begin{equation}
h_{2}(X_{1},\,X_{2})=\frac{1}{2}(X_{1}-X_{2})^{2},
\end{equation}
\begin{equation}
h_{3}(X_{1},\,X_{2},\,X_{3})=\frac{1}{2}
(X_{1}-X_{2})^{2}[(X_{1}-X_{3})+(X_{2}-X_{3})]\,,
\end{equation}
\begin{equation}
h_{4}(X_{1},\,X_{2},\,X_{3},\,X_{4})=\frac{1}{2}(X_{1}-X_{2})^{4}-\frac{3}{4}
(X_{1}-X_{2})^{2}(X_{3}-X_{4})^{2},
\end{equation}
\begin{align}
h_{k+1}(X_{1},\ldots ,\,X_{k+1})& =(X_{1}-X_{3})(X_{1}-X_{2})^{k} \notag \\
-\sum_{j=2}^{k-1}(-1)^{j}& C_{k}^{j}\,h_{j}(X_{1},\ldots
,\,X_{j})h_{k+1-j}(X_{j+1},\ldots ,\,X_{k+1})\,,\quad k\geq 4\,.
\end{align}
In other words, $h_{k}(X_{1},\ldots ,\,X_{k})$ is an unbiased estimator of $
{\Greekmath 0116} _{k}$ based on $k$ observations.
If $n$ observations $X_{1},\ldots ,\,X_{n}$ are available, any (fixed or
randomly selected) subset of $k$ observations from $\{X_{1},\ldots
,\,X_{n}\} $ yields an unbiased estimator of ${\Greekmath 0116} _{k},$ and similarly for
the arithmetic (or a weighted) average of such subsets. In particular, the
arithmetic average of $h_{k}(X_{i_{1}},\ldots ,\,X_{i_{k}})$ over $N$
randomly selected $\{i_{1},\ldots ,\,i_{k}\}$ of $\{1,\ldots ,\,n\}$ is an
unbiased estimator of ${\Greekmath 0116} _{k}$. \emph{A fortiori}, this will also hold if
the average is taken over all subsets of $k$ observations from $
\{X_{1},\ldots ,\,X_{n}\}$, but the large number of such subsets can easily
make this approach computationally prohibitive. We will call such estimators
based on observation differences $D$-estimators.
It is of interest to compare $D$-estimators of ${\Greekmath 0116} _{k}$ with the
\textquotedblleft natural\textquotedblright\ estimators based on deviations
from the sample mean:
\begin{equation}
\hat{m}_{k}(n)=\frac{1}{n-1}\sum_{i=i}^{n}(X_{i}-\bar{X}_{n})^{k}\,,\quad
\bar{X}_{n}=\frac{1}{n}\sum_{i=i}^{n}X_{i}\,.
\end{equation}
For $k=2,$ $\hat{m}_{k}(n)$ is unbiased, but it is not generally unbiased
for $k\geq 3$. To see how big the bias difference can be, we take $n=k$ and
compare by simulation $\hat{{\Greekmath 0116}}_{k}(k)$ with the corresponding $D$
-estimator with $h_{k}(X_{1},\ldots ,\,X_{k});$ see Table \ref{tab:
Simulation Comparison of Bias for Exponential(2) when n=k}. We use the
Exponential distribution as an example because its central moments are
nonzero at all orders, which helps us assess estimator bias without
interference from sign-canceling or offsetting effects that occur when the
true moment is zero. We see from these results that $\hat{m}_{k}(k)$ is
heavily biased, while the corresponding $D$-estimators exhibit no bias as
predicted by the results presented above.
\begin{table}[tbph]
\caption{Add caption}
\label{tab: Simulation Comparison of Bias for Exponential(2) when n=k}
\centering
\resizebox{1\columnwidth}{!}{ \begin{tabular}{rlrrrrrrr}
\toprule
\multicolumn{9}{c}{\textbf{ Table 1. Simulation Comparison of Bias for Exponential(2) when n=k }} \\
\midrule
& \textbf{Central Moment} & \multicolumn{1}{l}{\textbf{Ord. 2}} & \multicolumn{1}{l}{\textbf{Ord. 3}} & \multicolumn{1}{l}{\textbf{Ord. 4}} & \multicolumn{1}{l}{\textbf{Ord. 5}} & \multicolumn{1}{l}{\textbf{Ord. 6}} & \multicolumn{1}{l}{\textbf{Ord. 7}} & \multicolumn{1}{l}{\textbf{Ord. 8}} \\
& \textbf{Number of Observations} & \textbf{2} & \textbf{3} & \textbf{4} & \textbf{5} & \textbf{6} & \textbf{7} & \textbf{8} \\
\midrule
& \textit{True Value} & \textit{0.25} & \textit{0.25} & \textit{0.56} & \textit{1.38} & \textit{4.00} & \textit{14.48} & \textit{57.94} \\
\cmidrule{2-9} & \textbf{Natural Estimator $\hat{m}_k(k)$} & \textbf{0.000} & -0.167 & -0.258 & -0.768 & -2.171 & -8.321 & -33.739 \\
&\textbf{D-estimators $\hat{{\Greekmath 0116}}_k(k)$} & \textbf{0.000} & \textbf{0.000} & \textbf{0.000} & \textbf{0.003} & \textbf{0.094} & \textbf{0.131} & \textbf{0.053} \\
\bottomrule
\bottomrule
\multicolumn{9}{l}{\textit{* Number of Simulation = 20000000}}
\end{tabular} }
\end{table}
Of course, if $n>k$, more efficient unbiased estimators can be obtained by
averaging several minimal estimators. This can be easily implemented through
computer programming by selecting the index tuples $(i_{1},\ldots ,i_{k+1})$
independently and identically distributed (i.i.d.) from a uniform
distribution over the index set $I$. Due to its stochastic nature, we refer
to this estimator as a Monte Carlo approximation of the $D$-estimator, $\hat{
{\Greekmath 0116}}_k(n)^{MC}$. Table~\ref{tab: Simulation Comparison of Bias for
Exponential(2) when n>k} presents simulation results comparing the bias of
the \textquotedblleft natural\textquotedblright\ estimators $\hat{m}_k(n)$
with the proposed $D$-estimator Monte Carlo approximation for central
moments of orders $k=2,\ldots ,8$ under an $\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{Exponential}(2)$
distribution. For each replication, we generate 30,000 Monte Carlo samples.
\begin{table}[htbp]
\caption{Add caption}
\label{tab: Simulation Comparison of Bias for Exponential(2) when n>k}
\centering
\resizebox{1\columnwidth}{!}{ \begin{tabular}{rlrrrrrrr}
\toprule
\multicolumn{9}{c}{\textbf{Table 2. Simulation Comparison of Bias for Exponential(2) when n>k (\# of Simulation = 1000)}} \\
\midrule
& \textbf{Central Moment} & \multicolumn{1}{l}{\textbf{Ord.2}} & \multicolumn{1}{l}{\textbf{Ord.3}} & \multicolumn{1}{l}{\textbf{Ord.4}} & \multicolumn{1}{l}{\textbf{Ord.5}} & \multicolumn{1}{l}{\textbf{Ord.6}} & \multicolumn{1}{l}{\textbf{Ord.7}} & \multicolumn{1}{l}{\textbf{Ord.8}} \\
\midrule
& \textit{True Value} & \textit{0.25} & \textit{0.25} & \textit{0.56} & \textit{1.38} & \textit{4.00} & \textit{14.48} & \textit{57.94} \\
\cmidrule{1-9} \multicolumn{1}{l}{\textbf{n=50}} & \textbf{Natural Estimator $\hat{m}_k(n)$} & -0.004 & -0.018 & -0.059 & -0.240 & -0.921 & -5.203 & -27.483 \\
& \textbf{D-Estimator MC $\hat{{\Greekmath 0116}}_k(n)^{MC}$} & \textbf{-0.004} & \textbf{-0.009} & \textbf{-0.035} & \textbf{-0.162} & \textbf{-0.592} & \textbf{-4.048} & \textbf{-22.484} \\
\cmidrule{2-9} \multicolumn{1}{l}{\textbf{n=100}} & \textbf{Natural Estimator $\hat{m}_k(n)$} & \textbf{-0.001} & -0.008 & -0.020 & -0.084 & -0.295 & -2.621 & -16.574 \\
& \textbf{D-Estimator MC $\hat{{\Greekmath 0116}}_k(n)^{MC}$} & -0.001 & \textbf{-0.003} & \textbf{-0.005} & \textbf{-0.035} & \textbf{-0.133} & \textbf{-1.991} & \textbf{-13.020} \\
\bottomrule
\bottomrule
\multicolumn{9}{l}{\textit{* Number of Simulation = 1000}}
\end{tabular} }
\end{table}
The simulation results demonstrate that the proposed $D$-estimator Monte
Carlo approximation substantially reduces bias across all orders of the
central moment compared to the natural estimator. This improvement is
particularly notable for higher-order moments where the naive estimator
exhibits larger negative bias. Moreover, increasing the sample size from 50
to 100 decreases the bias for both estimators, but the $D$-estimator
approach consistently outperforms the natural estimator at each sample size.
These findings highlight the effectiveness of the Monte Carlo approximation
in providing more accurate and unbiased estimates of central moments,
especially for higher-order moments that are typically more challenging to
estimate.
\begin{Tentative}
\begin{comment}
It is of interest to compare $D$-estimators of ${\Greekmath 0116} _{k}$ with the
\textquotedblleft natural\textquotedblright\ estimators based on deviations
from the sample mean:
\begin{equation}
\hat{m}_{k}(n)=\frac{1}{n-1}\sum_{i=i}^{n}(X_{i}-\bar{X}_{n})^{k}\,,\quad
\bar{X}_{n}=\frac{1}{n}\sum_{i=i}^{n}X_{i}\,.
\end{equation}
For $k=2,$ $\hat{m}_{k}(n)$ is unbiased, but it is not generally unbiased
for $k\geq 3$. To see how big the bias difference can be, we take $n=k$ and
compare by simulation $\hat{{\Greekmath 0116}}_{k}(k)$ with the corresponding $D$
-estimator with $h_{k}(X_{1},\ldots ,\,X_{k});$ see Table \ref{tab:
Simulation Comparison of Bias for Two Distributions when n=k}. We see from
these results that $\hat{m}_{k}(k)$ is heavily biased, especially for
non-zero central moments, while the corresponding $D$-estimators exhibit no
bias as predicted by the results presented above.
\begin{table}[tbph]
\caption{Add caption}
\label{tab: Simulation Comparison of Bias for Two Distributions when n=k}
\centering
\resizebox{\columnwidth}{!}{ \begin{tabular}{rlrrrrrrr}
\toprule
\multicolumn{9}{c}{\textbf{ Table 1. Simulation Comparison of Bias for Two Distributions when n=k (\# of Simulation = 20000000) }} \\
\midrule
& \textbf{Central Moment} & \multicolumn{1}{l}{\textbf{Ord. 2}} & \multicolumn{1}{l}{\textbf{Ord. 3}} & \multicolumn{1}{l}{\textbf{Ord. 4}} & \multicolumn{1}{l}{\textbf{Ord. 5}} & \multicolumn{1}{l}{\textbf{Ord. 6}} & \multicolumn{1}{l}{\textbf{Ord. 7}} & \multicolumn{1}{l}{\textbf{Ord. 8}} \\
& \textbf{Number of Observations} & \textbf{2} & \textbf{3} & \textbf{4} & \textbf{5} & \textbf{6} & \textbf{7} & \textbf{8} \\
\midrule
\multicolumn{1}{l}{\textbf{Exponential(2)}} & \textit{True Value} & \textit{0.25} & \textit{0.25} & \textit{0.56} & \textit{1.38} & \textit{4.00} & \textit{14.48} & \textit{57.94} \\
\cmidrule{2-9} & \textbf{Natural Estimator $\hat{m}_k(k)$} & \textbf{0.000} & -0.167 & -0.258 & -0.768 & -2.171 & -8.321 & -33.739 \\
&\textbf{D-estimators $\hat{{\Greekmath 0116}}_k(k)$} & \textbf{0.000} & \textbf{0.000} & \textbf{0.000} & \textbf{0.003} & \textbf{0.094} & \textbf{0.131} & \textbf{0.053} \\
\midrule
\multicolumn{1}{l}{\textbf{Normal(0,1)}} & \textit{True Value} & \textit{1} & \textit{0} & \textit{3} & \textit{0} & \textit{15} & \textit{0} & \textit{105} \\
\cmidrule{2-9} &\textbf{Natural Estimator $\hat{m}_k(k)$} & \textbf{0.000} & \textbf{0.000} & -0.748 & \textbf{0.001} & -4.582 & \textbf{0.009} & -34.650 \\
& \textbf{D-estimators $\hat{{\Greekmath 0116}}_k(k)$} & \textbf{0.000} & -0.001 & \textbf{-0.003} & -0.061 & \textbf{0.246} & -0.292 & \textbf{0.493} \\
\bottomrule
\bottomrule
\end{tabular} }
\end{table}
Of course, if $n>k$, more efficient unbiased estimators can be obtained by
averaging several minimal estimators. This can be easily implemented through
computer programming by selecting the index tuples $(i_{1},\ldots ,i_{k+1})$
independently and identically distributed (i.i.d.) from a uniform
distribution over the index set $I$. Due to its stochastic nature, we refer
to this estimator as a Monte Carlo approximation of the $D$-estimator, $\hat{
{\Greekmath 0116}}_k(n)^{MC}$. Table~\ref{tab: Simulation Comparison of Bias for Two
distributions when n>k} presents simulation results comparing the bias of
the \textquotedblleft natural\textquotedblright\ estimators $\hat{m}_k(n)$
with the proposed $D$-estimator Monte Carlo approximation for central
moments of orders $k=2,\ldots ,8$ under an $\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{Exponential}(2)$
distribution and a $\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{Normal}(0,1)$ distribution. For each replication,
we generate 30,000 Monte Carlo samples.
\begin{table}[htbp]
\caption{Add caption}
\label{tab: Simulation Comparison of Bias for Two distributions when n>k}
\centering
\resizebox{\columnwidth}{!}{ \begin{tabular}{rlrrrrrrr}
\toprule
\multicolumn{9}{c}{\textbf{Table 2. Simulation Comparison of Bias for Two Distributions when n>k (\# of Simulation = 1000)}} \\
\midrule
& \textbf{Central Moment} & \multicolumn{1}{l}{\textbf{Ord.2}} & \multicolumn{1}{l}{\textbf{Ord.3}} & \multicolumn{1}{l}{\textbf{Ord.4}} & \multicolumn{1}{l}{\textbf{Ord.5}} & \multicolumn{1}{l}{\textbf{Ord.6}} & \multicolumn{1}{l}{\textbf{Ord.7}} & \multicolumn{1}{l}{\textbf{Ord.8}} \\
\midrule
\multicolumn{1}{l}{\textbf{Exponential (2)}} & \textit{True Value} & \textit{0.25} & \textit{0.25} & \textit{0.56} & \textit{1.38} & \textit{4.00} & \textit{14.48} & \textit{57.94} \\
\cmidrule{1-9} \multicolumn{1}{l}{\textbf{n=50}} & \textbf{Natural Estimator $\hat{m}_k(n)$} & -0.004 & -0.018 & -0.059 & -0.240 & -0.921 & -5.203 & -27.483 \\
& \textbf{D-Estimator MC $\hat{{\Greekmath 0116}}_k(n)^{MC}$} & \textbf{-0.004} & \textbf{-0.009} & \textbf{-0.035} & \textbf{-0.162} & \textbf{-0.592} & \textbf{-4.048} & \textbf{-22.484} \\
\cmidrule{2-9} \multicolumn{1}{l}{\textbf{n=100}} & \textbf{Natural Estimator $\hat{m}_k(n)$} & \textbf{-0.001} & -0.008 & -0.020 & -0.084 & -0.295 & -2.621 & -16.574 \\
& \textbf{D-Estimator MC $\hat{{\Greekmath 0116}}_k(n)^{MC}$} & -0.001 & \textbf{-0.003} & \textbf{-0.005} & \textbf{-0.035} & \textbf{-0.133} & \textbf{-1.991} & \textbf{-13.020} \\
\midrule
& & & & & & & & \\
\midrule
\multicolumn{1}{l}{\textbf{N(0,1)}} & \textit{True Value} & \textit{1} & \textit{0} & \textit{3} & \textit{0} & \textit{15} & \textit{0} & \textit{105} \\
\cmidrule{1-9} \multicolumn{1}{l}{\textbf{n=50}} & \textbf{Natural Estimator $\hat{m}_k(n)$} & 0.003 & -0.003 & -0.055 & \textbf{0.014} & -0.767 & \textbf{0.403} & -11.345 \\
& \textbf{D-Estimator MC $\hat{{\Greekmath 0116}}_k(n)^{MC}$} & \textbf{0.002} & \textbf{-0.003} & \textbf{0.000} & 0.041 & \textbf{-0.287} & 0.516 & \textbf{5.071} \\
\cmidrule{2-9} \multicolumn{1}{l}{\textbf{n=100}} & \textbf{Natural Estimator $\hat{m}_k(n)$} & -0.002 & \textbf{0.002} & -0.036 & \textbf{0.081} & -0.381 & \textbf{1.355} & -5.460 \\
& \textbf{D-Estimator MC $\hat{{\Greekmath 0116}}_k(n)^{MC}$} & \textbf{-0.002} & 0.003 & \textbf{-0.007} & 0.118 & \textbf{-0.228} & 2.115 & \textbf{-3.893} \\
\bottomrule
\bottomrule
\end{tabular} }
\end{table}
The simulation results demonstrate that the proposed $D$-estimator Monte
Carlo approximation substantially reduces bias across all orders of the
central moment compared to the natural estimator in Exponential
distributions. This improvement is particularly notable for higher-order
moments where the natural estimator exhibits larger negative bias. Moreover,
increasing the sample size from 50 to 100 decreases the bias for both
estimators, but the $D$-estimator approach consistently outperforms the
natural estimator at each sample size. For Normal(0,1) distributions where
the odd order central moments are zero, the performance of estimator may be
partially masked by sign calcellation in the simulation data. In cases of
non-zero central moments, the $D$-estimator approach outperforms the natural
estimator. These findings highlight the effectiveness of the Monte Carlo
approximation in providing more accurate and unbiased estimates of central
moments, especially for higher-order moments that are typically more
challenging to estimate.
\begin{Tentative}
\end{comment}
\end{Tentative}
From (\ref{eq: D-representation third moment}), one can see that several
choices of minimal estimator $h_{3}$ are possible (and similarly for
higher-order moments), which may have different efficiency properties. The
above results can also be used to build unbiased estimators of cumulants as
well as to analyze which observations deviate from the rest of the sample.
The latter application can be based on analyzing the population of subgroups
of observations $(X_{i_{1}},\ldots ,\,X_{i_{k}})$ along with the
corresponding unbiased estimates $h_{k}(X_{i_{1}},\ldots ,\,X_{i_{k}})$.
Discussing in detail variants of $D$-estimators, as well as other
applications, goes beyond the scope of this paper.
\subsection{Conclusion \label{Sec: Conclusion}}
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In this paper, following an approach (apparently) initiated by \nocite*
{Gini(1912)} for the variance, we have studied the general problem of
representing the central moments of a random variable in terms of pairwise
differences between i.i.d. replications, without reference to a location
parameter. We first gave pairwise-difference representations (defined
formally as $D$\emph{-representations}) for the third and fourth central
moments of a general random variable. These yield intuitive interpretations
of the familiar skewness and kurtosis coefficients. For general moments, we
have observed that central moments can be interpreted as covariances between
a random variable and powers of the same variable (a \emph{nonlinear}
relation). These then provide \emph{recursions} which link the $D$
-representation of any moment to lower order ones, hence $D$-representations
for all central moments (as long as they are finite). Through a similar
approach, analogues of Lagrange and Binet-Cauchy identities were established
for general random variables. These provide a simple derivation of the
classic Cauchy-Schwarz inequality for covariances. Finally, it is of
interest to note that the formulas given in this paper can be interpreted as
applications of the \textquotedblleft coupling\textquotedblright\ method
[see \nocite*{Thorisson(2000)}], which opens the way to additional
applications.
\noindent \textbf{Acknowledgements}
\noindent
Cette recherche a b\'{e}n\'{e}fici\'{e}
du support financier de la Chaire William Dow (McGill University),
de la Banque du Canada (Bourse de recherche),
d'une Bourse Guggenheim, de la Bourse Konrad-Adenauer
(Fondation Alexander von Humboldt, Allemagne),
du Conseil de recherche en sciences humaines du Canada, du
Conseil de recherche en sciences naturelles et en g\'{e}nie du
Canada, du R\'{e}seau canadien de centres d'excellence (projet MITACS), et du
Fonds de recherche sur la soci\'{e}t\'{e} et la culture
(Qu\'{e}bec).
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