Stochastic comparisons of sample mean differences for multivariate random variables

TL;DR

This paper extends stochastic ordering results of the Gini index from multivariate normal risks to more general multivariate elliptical risks, analyzing their dispersion matrices and tail probabilities.

Contribution

It generalizes existing stochastic ordering results for Gini indexes to multivariate elliptical risks and revises large deviation results for these risks.

Findings

Conditions on dispersion matrices ensure monotonicity of Gini index

Established stochastic ordering for multivariate elliptical risks

Revised large deviation results for Gini indexes

Abstract

In this paper, we establish the stochastic ordering of the Gini indexes for multivariate elliptical risks which generalized the corresponding results for multivariate normal risks. It is shown that several conditions on dispersion matrices and the components of dispersion matrices of multivariate normal risks for the monotonicity of the Gini index in the usual stochastic order proposed by Samanthi, Wei and Brazauskas (2016) and Kim and Kim (2019) are also suitable for multivariate elliptical risks. We also study the tail probability of Gini index for multivariate elliptical risks and revised a large deviation result for the Gini indexes of multivariate normal risks in Kim and Kim (2019).

Equations

$$G_n(\mathbf{X})=\frac{1}{n^2}\sum _{1\le i,j\le n}|X_i-X_j|.$$

$$G_n(\mathbf{X})=\sum _{i=1}^n(4i-2n-2)X_{(i)},$$

$$f(\mathbf{x})+f(\mathbf{y})\le f(\mathbf{x}\wedge \mathbf{y})+f(\mathbf{x}\vee \mathbf{y}),$$

$$\frac{\partial f(\mathbf{x})}{\partial x_i\partial x_j}\ge 0$$

$${\mathbf{\Psi }}_n\subset {\mathbf{\Psi }}_{n-1}\cdots \subset {\mathbf{\Psi }}_1.$$

$$\mathbf{X}=\mathbf{\mu }+R{\mathbf{A}}^{\prime }{\mathbf{U}}^{(n)},$$

$$f(\mathbf{x})=c_n|\mathbf{\Sigma }{|}^{-\frac{1}{2}}g((\mathbf{x}-\mathbf{\mu }{)}^T{\mathbf{\Sigma }}^{-1}(\mathbf{x}-\mathbf{\mu })),\mathbf{x}\in {\mathbb{R}}^n,$$

$${\int }_0^{\infty }{z}^{\frac{n}{2}-1}g(z)dz<\infty ,$$

$$c_n=\frac{\Gamma (\frac{n}{2})}{{\pi }^{\frac{n}{2}}}{\left({\int }_0^{\infty }{z}^{\frac{n}{2}-1}g(z)dz\right)}^{-1}.$$

$$h_R(v)=c_n\frac{2{\pi }^{\frac{n}{2}}}{\Gamma (\frac{n}{2})}{v}^{n-1}g(v^2),v\ge 0.$$

$$P(\mathbf{X}\in C)\ge P(\mathbf{Y}\in C).$$

$$G_n(\mathbf{X}-\mathbf{\mu }){\le }_{st}{G}_n(\mathbf{Y}-\mathbf{\mu }).$$

$$G_n(\mathbf{X}){\le }_{st}{G}_n(\mathbf{Y}).$$

$$\mathbf{X}=\mathbf{\mu }+\sqrt{V}{\mathbf{\Sigma }}^{\frac{1}{2}}\mathbf{Z},$$

$$G_n(\mathbf{X}-\mathbf{\mu }){\le }_{st}{G}_n(\mathbf{Y}-\mathbf{\mu }).$$

$$G_n(\mathbf{X}){\le }_{st}{G}_n(\mathbf{Y}).$$

$$G_n(\mathbf{X}-\mathbf{\mu }){\le }_{st}{G}_n(\mathbf{Y}-\mathbf{\mu }).$$

$$G_n(\sqrt{v}{\mathbf{\Sigma }}_x^{\frac{1}{2}}\mathbf{Z}){\le }_{st}{G}_n(\sqrt{v}{\mathbf{\Sigma }}_y^{\frac{1}{2}}\mathbf{Z}).$$

$$P(G_n(\mathbf{X}-\mathbf{\mu })>t)$$

$${\bf A}=\left(\begin{array}[]{cccc}\frac{n-1}{n}&-\frac{1}{n}&\cdots&-\frac{1}{n}\\ -\frac{1}{n}&\frac{n-1}{n}&\cdots&-\frac{1}{n}\\ \vdots&\vdots&\ddots&\vdots\\ -\frac{1}{n}&-\frac{1}{n}&\cdots&\frac{n-1}{n}\\ \end{array}\right).$$

$$G_n(\mathbf{AX}-\mathbf{A}\mathbf{\mu }){\le }_{st}{G}_n(\mathbf{AY}-\mathbf{A}\mathbf{\mu }).$$

$$G_n(\mathbf{X}){\le }_{st}{G}_n(\mathbf{Y}).$$

$$\mathbf{AX}\sim EL{L}_n(\mathbf{A}\mathbf{\mu },\mathbf{A}{\mathbf{\Sigma }}_x{\mathbf{A}}^{\prime },\varphi )$$

$$\mathbf{AY}\sim EL{L}_n(\mathbf{A}\mathbf{\mu },\mathbf{A}{\mathbf{\Sigma }}_y{\mathbf{A}}^{\prime },\varphi ).$$

$$G_n(\mathbf{AX}-\mathbf{A}\mathbf{\mu }){\le }_{st}{G}_n(\mathbf{AY}-\mathbf{A}\mathbf{\mu }).$$

$$G_n(\mathbf{AX}){\le }_{st}{G}_n(\mathbf{AY}),$$

$$G_n(\mathbf{AX}-\mathbf{A}\mathbf{\mu }){\le }_{st}{G}_n(\mathbf{AY}-\mathbf{A}\mathbf{\mu }),$$

$$G_n(\mathbf{X}){\le }_{st}{G}_n(\mathbf{Y}).$$

$$\mathbf{A}{\mathbf{\Sigma }}_y{\mathbf{A}}^{\prime }-\mathbf{A}{\mathbf{\Sigma }}_x{\mathbf{A}}^{\prime }=\mathbf{A}({\mathbf{\Sigma }}_y-{\mathbf{\Sigma }}_x+\epsilon 1_{n\times n}){\mathbf{A}}^{\prime }⪰{\mathbf{O}}_{n\times n},$$

$$G_n(\mathbf{AX}-\mathbf{A}\mathbf{\mu }){\le }_{st}{G}_n(\mathbf{AY}-\mathbf{A}\mathbf{\mu }).$$

Full text

Stochastic ordering of Gini indexes for multivariate elliptical random variables

Chuancun Yin  

*School of Statistics, Qufu Normal University

Shandong 273165, China

*e-mail: ccyin@qfnu.edu.cn

Abstract

In this paper, we establish the stochastic ordering of the Gini indexes for multivariate elliptical risks which generalized the corresponding results for multivariate normal risks. It is shown that several conditions on dispersion matrices and the components of dispersion matrices of multivariate normal risks for the monotonicity of the Gini index in the usual stochastic order and increasing convex order proposed by Samanthi, Wei and Brazauskas (2016) and Kim and Kim (2019) are also suitable for multivariate elliptical risks. We also study the tail probability of Gini index for multivariate elliptical risks and revised a large deviation result for the Gini indexes of multivariate normal risks in Kim and Kim (2019).

Keywords: Gini index; increasing convex order; large deviation; multivariate elliptical risk; multivariate normal risk; usual stochastic order

1 Introduction

Since Corrado Gini introduced an index to measure concentration or inequality of incomes (see Gini (1936), for English translation of the original article), it has been studied extensively because of its importance in many fields such as economics, actuarial science, finance, operations research, queuing theory and statistics; see, for example, Denuit et al. (2005), McNeil et al. (2005), Brazauskas et al. (2007), Goovaerts et al. (2010), Frees et al. (2011, 2014), Samanthi et al. (2017), to name but a few. Recently, there is a growing interest in the study and applications of the Gini index; See e.g. Samanthi, Wei and Brazauskas (2016), Kim and Kim (2019) and the references therein. Let ${\bf X}=(X_{1},...,X_{n})$ be $n$ portfolios of risks or random variables, these portfolios can be independent or dependent. To check whether or not the $n$ risk measures $X_{i}$’s are all equal. Brazauskas et al. (2007) and Samanthi et al. (2017) proposed a nonparametric test statistic based on the following Gini index:

[TABLE]

It is easy to see that $G_{n}({\bf X})$ can be rewritten in terms of order statistics:

[TABLE]

where $X_{(i)}$ denotes the $i$th smallest component of ${X_{1},\cdots,X_{n}}$, i.e. $X_{(1)}\leq X_{(2)}\leq\cdots\leq X_{(n)}$. Note that the expression (2.2) in Samanthi et al. (2016) lost a negative sign before summation sign.

The comparison of Gini indexes of multivariate elliptical risks also shows its own independent interest. For example, for the ordering of Gini indexes of multivariate normal risks, Samanthi et al. (2016) proposed the following conjecture.

Conjecture 1. Let random vector ${\bf X}=(X_{1},X_{2},\cdots,X_{n})^{\prime}$ follow a multivariate normal distribution $N_{n}({\bf 0},{\bf\Sigma})$. Then its Gini index $G_{n}({\bf X})$ decreases in the sense of usual stochastic order as the covariance matrix ${\bf\Sigma}$ increases componentwise with diagonal elements remaining unchanged.

Samanthi et al. (2016) pointed out that proving Conjecture 1 is a challenging task. They partially completes this task and claim that generalizes the conclusion to elliptical distributions, yet still leaves some open problems. Recent paper of Kim and Kim (2019) shows that this conjecture is true when $n=2$. However, this conjecture is not true when $n\geq 3$. By using the positive semidefinite ordering of covariance matrices, they obtain the usual stochastic order of the Gini indexes for multivariate normal risks and generalized to the scale mixture of multivariate normal risks. In this paper we generalize the main results in Samanthi et al. (2016) and Kim and Kim (2019) from multivariate normal risks and scale mixture of multivariate normal risks to multivariate elliptical risks and scale mixture of multivariate elliptical risks.

The rest of the paper is organized as follows. In the next section, we introduce some basic notation and review the definition and properties of stochastic orders and elliptical distributions. Sections 3 and 4 establish the usual stochastic orders and increasing convex orders between Gini indexes for multivariate elliptical risks. We study the tail probability of Gini index for multivariate elliptical risks and a large deviation result for the Gini indexes of multivariate normal risks in Section 5. Section 6 provides concluding remarks of the paper.

2 Preliminaries

In this section we fix the notation that will be used in the sequel and we recall some well known results about stochastic orders of random variables and elliptical distributions. Throughout the paper, we use bold letters to denote vectors or matrices. For example, ${\bf X}^{\prime}=(X_{1},\cdots,X_{n})$ is a row vector and ${\bf\Sigma}=(\sigma_{ij})_{n\times n}$ is an $n\times n$ matrix. In particular, the symbol ${\bf 0}_{n}$ denotes the $n$-dimensional column vector with all entries equal to 0, ${\bf 1}_{n}$ denotes the $n$-dimensional column vector with all components equal to 1, and ${\bf 1}_{n\times n}$ denotes the ${n\times n}$ matrix with all entries equal to 1. Denote as ${\bf O}_{n\times n}$ the $n\times n$ matrix having all components equal to 0 and ${\bf I}_{n}$ denotes the $n\times n$ identity matrix. For symmetric matrices $A$ and $B$ of the same size, the notion $A\preceq B$ or $B-A\succeq{\bf O}$ means that $B-A$ is positive semidefinite.

In order to compare Gini indexes, we recall definitions of some stochastic orders, see, Denuit et al. (2005) and Shaked and Shanthikumar (2007). Let $X$ and $Y$ be two random variables, $X$ is said to be smaller than $Y$ in usual stochastic order, denoted as $X\leq_{st}Y$, if $P(X>t)\leq P(Y>t)$ for all real numbers $t$. Random vector ${\bf X}$ is said to be smaller than random vector ${\bf Y}$ in increasing convex order (written ${\bf X}\leq_{icx}{\bf Y}$), if $E[f({\bf X})]\leq E[f({\bf Y})]$ for all increasing convex functions $f:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}$ such that the expectations exist. A function $f:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}$ is said to be supermodular if for any ${\bf x,y}\in{\mathbb{R}}^{n}$ it holds that

[TABLE]

where the operators $\wedge$ and $\vee$ denote coordinatewise minimum and maximum respectively. Supermodular functions are also called quasimonotone or $L$-superadditive. Note that if $f$ is twice differentiable, then $f$ is supermodular if and only if

[TABLE]

for all $i\neq j$ and ${\bf x}\in\mathbb{R}^{n}$. If $-f$ is supermodular, then $f$ is called submodular. Random vector ${\bf X}$ is said to be smaller than random vector ${\bf Y}$ in the supermodular order, denoted as ${\bf X}\leq_{sm}{\bf Y}$, if $E[f({\bf X})]\leq E[f({\bf Y})]$ for any supermodular function $f$ such that the expectations exist.

We next state some basics about elliptical distributions. Elliptical distributions have been used widely in insurance, finance and multicriteria decision theory; see, for example, Owen and Rabinovitch (1983), Landsman and Valdez (2003), Hamada and Valdez (2008), Landsman, Makov and Shushi (2018), Sha et al. (2019) and Kim and Kim (2019). We follow the notation of Cambanis, Huang and Simons (1981) and Fang, Kotz and Ng (1990). Let ${\bf\Psi}_{n}$ be a class of functions $\psi:[0,\infty)\rightarrow\mathbb{R}$ such that function $\psi(|\bf t|^{2}),t\in\mathbb{R}^{n}$ is an $n$-dimensional characteristic function. It is clear that

[TABLE]

Denote by ${\bf\Psi}_{\infty}$ the set of characteristic generators that generate an $n$-dimensional elliptical distribution for arbitrary $n\geq 1$. That is ${\bf\Psi}_{\infty}=\cap_{n=1}^{\infty}{\bf\Psi}_{n}.$

An $n\times 1$ random vector $X=(X_{1},X_{2},\cdots,X_{n})^{\prime}$ is said to have an elliptically symmetric distribution if its characteristic function has the form $e^{i{\bf t}^{\prime}{\boldsymbol{\mu}}}\phi({\bf t}^{\prime}{\bf\Sigma}{\bf t})$ for all ${\bf t}\in\mathbb{R}^{n}$, where $\phi\in{\bf\Psi}_{n}$ is called the characteristic generator satisfying $\phi(0)=1$, $\boldsymbol{\mu}$ ($n$-dimensional vector) is its location parameter and $\bf{\Sigma}$ ($n\times n$ matrix with $\bf{\Sigma}\succeq{\bf O}$) is its dispersion matrix (or scale matrix). The mean vector $E({\bf X})$ (if it exists) coincides with the location vector and the covariance matrix Cov$({\bf X})$ (if it exists), being $-2\phi^{\prime}(0){\bf\Sigma}$. We shall write ${\bf{X}}\sim ELL_{n}({\boldsymbol{\mu}},{\bf\Sigma},\phi)$. It is well known that $\bf X$ admits the stochastic representation

[TABLE]

where ${\bf A}$ is a square matrix such that ${\bf A}^{\prime}{\bf A}={\bf\Sigma}$, ${\bf U}^{(n)}$ is uniformly distributed on the unit sphere ${\cal S}^{n-1}=\{{\bf u}\in\mathbb{R}^{n}:{\bf u}^{\prime}{\bf u}=1\}$, $R\geq 0$ is the random variable with $R\sim F$ in $[0,\infty)$ called the generating variate and $F$ is called the generating distribution function, $R$ and ${\bf U}^{(n)}$ are independent. In general, an elliptically distributed random vector ${\bf{X}}\sim ELL_{n}({\boldsymbol{\mu}},{\bf\Sigma},\phi)$ does not necessarily possess a density. However, if density of $X$ exists it must be of the form

[TABLE]

for some non-negative function $g$ satisfying the condition

[TABLE]

and a normalizing constant $c_{n}$ given by

[TABLE]

The function $g$ is called the density generator. One sometimes writes $X\sim ELL_{n}({\boldsymbol{\mu}},{\bf\Sigma},g)$ for the $n$-dimensional elliptical distributions generated from the function $g$. In this case $R$ in (1.1) has the pdf given by

[TABLE]

Theorem 2.21 in Fang, Kotz and Ng (1990) shows that $\psi\in{\bf\Psi}_{\infty}$ if and only if ${\bf{X}}\sim{ELL}_{n}({\boldsymbol{\mu}},{\bf\Sigma},{\bf\psi})$ is a mixture of normal distributions. Some such elliptical distributions are the multivariate normal distribution, the multivariate $T$-distribution, the multivariate Cauchy distribution and the exponential power distribution $EP_{n}({\boldsymbol{\mu}},{\bf\Sigma},\beta)$ with $\beta\in(0,1]$. Some elliptical distributions like logistic distribution and Kotz type distribution are not mixture of normal distributions. A comprehensive review of the properties and characterizations of elliptical distributions can be found in Cambanis et al. (1981) and Fang et al. (1990).

3 Usual stochastic order of Gini indexes

In this section, we extended the results of multivariate normal risks and scale mixture multivariate normal risks in Samanthi et al. (2016) and Kim and Kim (2019) to scale mixture multivariate elliptical risks. To compare the usual stochastic orders between Gini indexes for multivariate elliptical risks, we use the following result due to Fefferman, Jodeit, and Perlman (1972); see Eaton and Erlman (1991) for a different proof. In the case of Gaussian distribution, this was proved by Anderson (1955). Let ${\cal{C}}$ denote the class of all convex, centrally symmetric (i.e., $C=-C$) subsets $C$ of ${\mathbb{R}^{n}}$.

Lemma 3.1**.**

Suppose that ${\bf{X}}\sim ELL_{n}({\bf 0},{\bf\Sigma}_{x},\phi)$ and ${\bf{Y}}\sim ELL_{n}({\bf 0},{\bf\Sigma}_{y},\phi)$. If ${\bf\Sigma}_{x}\preceq{\bf\Sigma}_{y}$, then for every $C\in{\cal C}$,

[TABLE]

The following result generalized Proposition 2 in Kim and Kim (2019) in which they only considered a special class of multivariate elliptical risks with zero mean vector, i.e., scale mixture of multivariate normal risks with zero mean vector.

Proposition 3.1**.**

Let ${\bf{X}}\sim ELL_{n}({\boldsymbol{\mu}},{\bf\Sigma}_{x},\phi)$ and ${\bf{Y}}\sim ELL_{n}({\boldsymbol{\mu}},{\bf\Sigma}_{y},\phi)$. If ${\bf\Sigma}_{x}\preceq{\bf\Sigma}_{y}$, then

[TABLE]

In particular, if ${\boldsymbol{\mu}}={\bf 0}$, or $\mu{\bf 1}_{n}$, then

[TABLE]

Proof.  If ${\bf{X}}\sim ELL_{n}({\boldsymbol{\mu}},{\bf\Sigma}_{x},\phi)$ and ${\bf{Y}}\sim ELL_{n}({\boldsymbol{\mu}},{\bf\Sigma}_{y},\phi)$, then ${\bf{X}}-{\boldsymbol{\mu}}\sim ELL_{n}({\bf 0},{\bf\Sigma}_{x},\phi)$ and ${\bf{Y}}-{\boldsymbol{\mu}}\sim ELL_{n}({\bf 0},{\bf\Sigma}_{y},\phi)$, and thus (3.1) follows from Lemma 3.1 by taking $C_{t}=\{{\bf x}\in{\mathbb{R}}^{n}:G_{n}({\bf X})\leq t\}$ for $t>0$ as in Kim and Kim (2019). It is easy to check that the Gini index $G_{n}(\cdot)$ is invariant under drift $\mu{\bf 1}_{n}$, i.e., $G_{n}({\bf X}+\mu{\bf 1}_{n})=G_{n}({\bf X})$ for all $n$-dimensional random vector ${\bf X}$. Therefore, (3.2) follows. $\hfill\square$

We will extend the result of Proposition 3.1 to the scale mixture of multivariate elliptical risks.

Definition 2.1 A $n$-dimensional random variable ${\bf X}$ is said to have a scale mixture of elliptical distributions with the parameters ${\boldsymbol{\mu}}$ and ${\bf\Sigma}$, if

[TABLE]

where ${\bf Z}\sim ELL_{n}({\bf 0},{\bf I}_{n},\phi)$, $V$ is a nonnegative, scalar-valued random variable with the distribution $F$, ${\bf Z}$ and $V$ are independent, ${\boldsymbol{\mu}}\in\mathbb{R}^{n}$, ${\bf\Sigma}\in\mathbb{R}^{n\times n}$ with ${\bf\Sigma}\succeq{\bf O}$, and ${\bf\Sigma}^{\frac{1}{2}}$ is the square root of ${\bf\Sigma}$. Here ${\bf 0}$ is an $n\times 1$ vector of zeros, and ${\bf I}_{n}$ is an $n\times n$ identity matrix. We will use the notation ${\bf Y}\sim SMELL_{n}({\boldsymbol{\mu}},{\bf\Sigma},\phi;F)$.

Note that when ${\bf Z}\sim N_{n}(0,{\bf I_{n}})$ we get the multivariate normal variance mixture distribution (see, e.g., McNeil et al., 2005); When ${\bf Z}\sim KTD_{n}(0,{\bf I_{n}},N,\frac{1}{2},\beta)$ we have the variance mixture of the Kotz-type distribution introduced by Arslan (2009).

Proposition 3.1 can be generalized to scale mixture of multivariate elliptical risks.

Proposition 3.2**.**

Let ${\bf{X}}\sim SMELL_{n}({\boldsymbol{\mu}},{\bf\Sigma}_{x},\phi;F)$ and ${\bf{Y}}\sim SMELL_{n}({\boldsymbol{\mu}},{\bf\Sigma}_{y},\phi;F)$. If ${\bf\Sigma}_{x}\preceq{\bf\Sigma}_{y}$, then

[TABLE]

In particular, if ${\boldsymbol{\mu}}={\bf 0}$, or $\mu{\bf 1}_{n}$, then

[TABLE]

Proof.  It can be easily seen that for any $v>0$, ${\bf X}|V=v\sim ELL_{n}({\boldsymbol{\mu}},v{\bf\Sigma}_{x},\phi)$ and ${\bf Y}|V=v\sim ELL_{n}({\boldsymbol{\mu}},v{\bf\Sigma}_{y},\phi)$. Since ${\bf\Sigma}_{x}\preceq{\bf\Sigma}_{y}$, one has $v{\bf\Sigma}_{x}\preceq v{\bf\Sigma}_{y}$. By Proposition 3.1, given $V=v$, we get

[TABLE]

Or, equivalently,

[TABLE]

Therefore, for all $t\in{\mathbb{R}}$,

[TABLE]

which is (3.4). In particular, if ${\boldsymbol{\mu}}={\bf 0}$, or $\mu{\bf 1}_{n}$, then $G_{n}({\bf X}-{\boldsymbol{\mu}})=G_{n}({\bf X})$ and $G_{n}({\bf Y}-{\boldsymbol{\mu}})=G_{n}({\bf Y})$, and (3.5) follows. $\hfill\square$

An important property of elliptical distributions is that linear transformations of elliptical vectors are also ellipticals, with the same characteristic generator. Specifically

Lemma 3.2**.**

Suppose that ${\bf{X}}\sim ELL_{n}({\boldsymbol{\mu}},{\bf\Sigma},\phi)$, ${\bf B}$ is an $m\times n$ matrix of rank $m\leq n$, and ${\bf b}$ is an $m\times 1$ vector, then ${\bf BX+b}\sim ELL_{m}({\bf B}{\boldsymbol{\mu}}+{\bf b},{\bf B}{\bf\Sigma}{\bf B}^{\prime},\phi)$.

We will give a weaker sufficient condition for stochastic ordering of Gini indexes for multivariate elliptical risks.

Proposition 3.3**.**

Let ${\bf{X}}\sim ELL_{n}({\boldsymbol{\mu}},{\bf\Sigma}_{x},\phi)$ and ${\bf{Y}}\sim ELL_{n}({\boldsymbol{\mu}},{\bf\Sigma}_{y},\phi)$. If ${\bf A}{\bf\Sigma}_{x}{\bf A}^{\prime}\preceq{\bf A}{\bf\Sigma}_{y}{\bf A}^{\prime}$, where ${\bf A}$ is an $n\times n$ matrix defined as

[TABLE]

Then

[TABLE]

In particular, if ${\boldsymbol{\mu}}={\bf 0}$, or $\mu{\bf 1}_{n}$, then

[TABLE]

Proof.  If ${\bf{X}}\sim ELL_{n}({\boldsymbol{\mu}},{\bf\Sigma}_{x},\phi)$ and ${\bf{Y}}\sim ELL_{n}({\boldsymbol{\mu}},{\bf\Sigma}_{y},\phi)$, by Lemma 3.2 we get

[TABLE]

and

[TABLE]

According to Proposition 3.1, if ${\bf A}{\bf\Sigma}_{x}{\bf A}^{\prime}\preceq{\bf A}{\bf\Sigma}_{y}{\bf A}^{\prime}$, then

[TABLE]

In particular, if ${\boldsymbol{\mu}}={\bf 0}$, or $\mu{\bf 1}_{n}$, then ${\bf A\boldsymbol{\mu}}={\bf 0}$. Thus

[TABLE]

and (3.7) follows since $G_{n}({\bf AX})=G_{n}({\bf X})$ and $G_{n}({\bf AY})=G_{n}({\bf Y})$. $\hfill\square$

The following proposition generalized the result of Proposition 4.4 in Samanthi et al. (2016) in which only the multivariate normal risks with zero mean vectors were considered. Moreover, we provide a short proof.

Proposition 3.4**.**

Let ${\bf{X}}\sim ELL_{n}({\boldsymbol{\mu}},{\bf\Sigma}_{x},\phi)$ and ${\bf{Y}}\sim ELL_{n}({\boldsymbol{\mu}},{\bf\Sigma}_{y},\phi)$. If there exists $\varepsilon\in{\mathbb{R}}$ such that ${\bf\Sigma}_{y}-{\bf\Sigma}_{x}+\varepsilon{\bf 1}_{n\times n}\succeq{\bf O}_{n\times n}$, then

[TABLE]

where ${\bf A}$ is defined in Proposition 3.3. In particular, if ${\boldsymbol{\mu}}={\bf 0}$, or $\mu{\bf 1}_{n}$, then

[TABLE]

Proof.  By Proposition 3.3, it suffices to show that ${\bf A}{\bf\Sigma}_{x}{\bf A}^{\prime}\preceq{\bf A}{\bf\Sigma}_{y}{\bf A}^{\prime}$. In fact,

[TABLE]

since ${\bf A}{\bf 1}_{n\times n}{\bf A}^{\prime}={\bf 0},$ as desired. $\hfill\square$

Remark 3.1**.**

If ${\bf\Sigma}_{x}\preceq{\bf\Sigma}_{y}$, then for any $\varepsilon\geq 0$, then ${\bf\Sigma}_{y}-{\bf\Sigma}_{x}+\varepsilon{\bf 1}_{n\times n}\succeq{\bf O}_{n\times n}$. But conversely is not true in general.

Proposition 3.5 can be generalized to scale mixture of multivariate elliptical risks. The proof is very similar to that used in extending Propositions 3.1 to 3.2 and hence is omitted.

Proposition 3.5**.**

Let ${\bf{X}}\sim SMELL_{n}({\boldsymbol{\mu}},{\bf\Sigma}_{x},\phi;F)$ and ${\bf{Y}}\sim SMELL_{n}({\boldsymbol{\mu}},{\bf\Sigma}_{y},\phi;F)$. If ${\bf A}{\bf\Sigma}_{x}{\bf A}^{\prime}\preceq{\bf A}{\bf\Sigma}_{y}{\bf A}^{\prime}$, where ${\bf A}$ is defined in Proposition 3.3, then

[TABLE]

In particular, if ${\boldsymbol{\mu}}={\bf 0}$, or $\mu{\bf 1}_{n}$, then

[TABLE]

Proposition 3.6**.**

Let ${\bf{X}}\sim SMELL_{n}({\boldsymbol{\mu}},{\bf\Sigma}_{x},\phi;F)$ and ${\bf{Y}}\sim SMELL_{n}({\boldsymbol{\mu}},{\bf\Sigma}_{y},\phi;F)$. If there exists $\varepsilon\in{\mathbb{R}}$ such that ${\bf\Sigma}_{y}-{\bf\Sigma}_{x}+\varepsilon{\bf 1}_{n\times n}\succeq{\bf O}_{n\times n}$, then

[TABLE]

In particular, if ${\boldsymbol{\mu}}={\bf 0}$, or $\mu{\bf 1}_{n}$, then

[TABLE]

where ${\bf A}$ is defined in Proposition 3.3.

Proof.  It is an immediate consequence of Proposition 3.5 since the condition ${\bf\Sigma}_{y}-{\bf\Sigma}_{x}+\varepsilon{\bf 1}_{n\times n}\succeq{\bf O}_{n\times n}$ implies ${\bf A}{\bf\Sigma}_{x}{\bf A}^{\prime}\preceq{\bf A}{\bf\Sigma}_{y}{\bf A}^{\prime}$ as shown in the proof of Proposition 3.4. $\hfill\square$

The condition on the components of dispersion matrix of multivariate normal risk or scale mixture of multivariate normal risk ${\bf X}$ for the monotonicity of the Gini index $G_{n}({\bf X})$ in the usual stochastic order proposed by Kim and Kim (2019) also suitable for general multivariate elliptical risk or scale mixture of multivariate elliptical risk, as shown below.

The following result generalized Propositions 3 and 4 in Kim and Kim (2019) in which they only considered a special class of multivariate elliptical risks with zero mean vector, i.e., the multivariate normal risks and scale mixture of multivariate normal risks with zero mean vector.

Proposition 3.7**.**

Let ${\bf{X}}\sim SMELL_{n}({\boldsymbol{\mu}},{\bf\Sigma}_{x},\phi;F)$ and ${\bf{Y}}\sim SMELL_{n}({\boldsymbol{\mu}},{\bf\Sigma}_{y},\phi;F)$ with ${\bf\Sigma}_{x}=(\sigma^{x}_{ij})_{n\times n}$ and ${\bf\Sigma}_{y}=(\sigma^{y}_{ij})_{n\times n}$. Let $\varepsilon>0$, if $\sigma^{y}_{ij}=\sigma^{x}_{1j}+\varepsilon,j=2,\cdots,n$, $\sigma^{y}_{i1}=\sigma^{x}_{i1}+\varepsilon,i=2,\cdots,n$ and for other $1\leq i,j\leq n$, $\sigma^{y}_{ij}=\sigma^{x}_{ij}$. Then

[TABLE]

In particular, if ${\boldsymbol{\mu}}={\bf 0}$, or $\mu{\bf 1}_{n}$, then

[TABLE]

Proof.  According to Kim and Kim (2019), under the assumed condition, we know that,

[TABLE]

from which we get

[TABLE]

We conclude that the latter matrix is positive semidefinite. In fact, for any ${\bf x}=(x_{1},\cdots,x_{n})^{\prime}\in\mathbb{R}^{n}$,

[TABLE]

Therefore, ${\bf\Sigma}_{x}-{\bf\Sigma}_{y}+\varepsilon{\bf 1}_{n\times n}\succeq{\bf O}_{n\times n}.$ Then Proposition 3.6 implies the desired results. $\hfill\square$

The following result generalized Propositions 5 and 6 in Kim and Kim (2019) in which they only considered the multivariate normal risks and scale mixture of multivariate normal risks with zero mean vector.

Proposition 3.8**.**

Let ${\bf{X}}\sim SMELL_{n}({\boldsymbol{\mu}},{\bf\Sigma}_{x},\phi;F)$ and ${\bf{Y}}\sim SMELL_{n}({\boldsymbol{\mu}},{\bf\Sigma}_{y},\phi;F)$ with ${\bf\Sigma}_{x}=(\sigma^{x}_{ij})_{n\times n}$ and ${\bf\Sigma}_{y}=(\sigma^{y}_{ij})_{n\times n}$. Let $\varepsilon>0$, if

[TABLE]

then

[TABLE]

In particular, if ${\boldsymbol{\mu}}={\bf 0}$, or $\mu{\bf 1}_{n}$, then

[TABLE]

Proof.  Under the assumed condition, Kim and Kim (2019) found that,

[TABLE]

from which we get

[TABLE]

where ${\bf A}$ is defined in Proposition 3.3. We find that ${\bf A}{\bf\Sigma}_{x}{\bf A}^{\prime}\succeq{\bf A}{\bf\Sigma}_{y}{\bf A}^{\prime}$. Therefore, Proposition 3.5 provides the desired result. $\hfill\square$

4 Increasing convex order of Gini indexes

Samanthi, Wei and Brazauskas (2016) established a sufficient and necessary condition for the supermodular order between two scale mixture of multivariate normal risks. Based on this result they found a sufficient condition for the increasing convex order of Gini indexes for two 3-dimensional elliptical random variables. In addition, they remarked that ordering $G_{n}({\bf X})$ in the increasing convex order for higher dimensional risk is still an open problem. However, Proposition 3.5 in Samanthi, Wei and Brazauskas (2016) is not true due to their Lemma 3.2 is not true. For example, $-G^{2}(x_{1},x_{2},x_{3})$ is not a supermodular function. The following theorem solves this open problem.

Theorem 4.1**.**

Let ${\bf{X}}\sim ELL_{n}({\boldsymbol{\mu}}^{x},{\bf\Sigma}^{x},\phi)$ and ${\bf{Y}}\sim ELL_{n}({\boldsymbol{\mu}}^{x},{\bf\Sigma}^{y},\phi)$ with $\phi\in{\bf\Psi}_{\infty}$, ${\bf\Sigma}^{x}=(\sigma^{x}_{ij})_{n\times n}$ and ${\bf\Sigma}^{y}=(\sigma^{y}_{ij})_{n\times n}$. If ${\bf X}$ and ${\bf Y}$ have the same marginals and $\sigma^{x}_{ij}\leq\sigma^{y}_{ij}$ for all $1\leq i<j\leq n$, then $-G_{n}({\bf X})\leq_{icx}-G_{n}({\bf Y})$.

Proof.  According to Lemma 3.1 in Samanthi, Wei and Brazauskas (2016) $-G_{n}({\bf X})$ is supermodular and componentwise monotone. For any convex and increasing function $\psi:{\mathbb{R}}\rightarrow{\mathbb{R}}$, a result of Topkis (1968) shows that the composition $\psi(-G_{n}({\bf X}))$ is componentwise monotone and supermodular; See also Marshall, Olkin and Arnold (2011). This, together with Proposition 3.4 in Samanthi, Wei and Brazauskas (2016) yields the desired result.

The following result is a direct consequence of Theorem 4.1, which generalizes Proposition 3.6 in Samanthi, Wei and Brazauskas (2016) for the case of zero mean vector.

Corollary 4.1**.**

Let ${\bf{X}}\sim ELL_{n}({\boldsymbol{\mu}}^{x},{\bf\Sigma}^{x},\phi)$ and ${\bf{Y}}\sim ELL_{n}({\boldsymbol{\mu}}^{y},{\bf\Sigma}^{y},\phi)$ with $\phi\in{\bf\Psi}_{\infty}$, ${\bf\Sigma}^{x}=(\sigma^{x}_{ij})_{n\times n}$ and ${\bf\Sigma}^{y}=(\sigma^{y}_{ij})_{n\times n}$. If ${\bf X}$ and ${\bf Y}$ have the same marginals and $\sigma^{x}_{ij}\leq\sigma^{y}_{ij}$ for all $1\leq i<j\leq n$, then $EG_{n}({\bf X})\geq EG_{n}({\bf Y})$ given the expectations exist.

5 Tail asymptotic results for Gini indexes

In this section, we will discuss the asymptotic result for the tail probability of Gini index $G_{n}({\bf X})$ when ${\bf{X}}\sim{ELL}_{n}({\boldsymbol{0}},{\bf\Sigma},{\bf\psi})$. The following lemma can be found in Fang and Liang (1989), see also Tong (1990).

Lemma 5.1**.**

For all real vectors ${\bf X}=(X_{1},\cdots,X_{n})^{\prime}$ and all given real numbers $0\leq C_{1}\leq\cdots\leq C_{n}$, we have

[TABLE]

where $X_{(1)}\leq X_{(2)}\leq\cdots\leq X_{(n)}$ is the rearranged order of $(X_{1},\cdots,X_{n})^{\prime}$ and $\Pi$ is the $n!$ vectors of permutations of $(C_{1},\cdots,C_{n})^{\prime}$.

We first give a counterexample demonstrating that the large deviation result for Gini indexes in Kim and Kim (2019) is not true.

Example 4.1. Let ${\bf X}\sim N_{2}({\bf 0},{\bf I}_{2})$ and ${\bf Y}=(Y_{1},Y_{2})^{\prime}={\bf C_{2}}^{\prime}{\bf X}$, where

[TABLE]

Then ${\bf Y}\sim N_{2}({\bf 0},{\bf C_{2}}{\bf C_{2}}^{\prime})$ with

[TABLE]

Recall that

[TABLE]

and note that the Pearson correlation coefficient of $Y_{1}$ and $Y_{2}$ is $-1$, so we get $Y_{1}=-Y_{2}$ and hence $G_{2}({\bf X})=|Y_{1}|$. It follows that for $t>0$,

[TABLE]

We have

[TABLE]

due to the fact that $Y_{1}\sim N_{1}({0},8)$. Therefore,

[TABLE]

This, together with (5.1) implies that

[TABLE]

However, in the case of $n=2$ and ${\bf\Sigma}={\bf I_{2}}$, the limit in Theorem 5 of Kim and Kim (2019) is $-\beta({\bf\Sigma})=-\frac{2}{5}.$

To discuss the asymptotic result for the tail probability of Gini index $G_{n}({\bf X})$, we first recall some results for elliptical distributions. Consider the linear transformation ${\bf Y}={\bf CX}$, where

[TABLE]

where $m=n!$ and ${\bf C}$ is an $m\times n$ matrix such that ${\bf C_{r}}$ is a permutation of $(C_{1},\cdots,C_{n})^{\prime}$. If ${\bf{X}}\sim ELL_{n}({\boldsymbol{\mu}},{\bf\Sigma},\phi)$, then by Lemma 3.2, ${\bf{Y}}=(Y_{1},\cdots,Y_{m})^{\prime}\sim ELL_{m}({\bf C}{\boldsymbol{\mu}},{\bf C}{\bf\Sigma}{\bf C}^{\prime},\phi)$. This, together with Lemma 5.1 implies that

[TABLE]

This result in the multinormal case can be found in Tong (1990). In particular, taking $C_{i}=4i-2n-2,i=1,2,\cdots,n$ leads to the following result.

Proposition 5.1**.**

Assume that ${\bf{X}}\sim ELL_{n}({\boldsymbol{\mu}},{\bf\Sigma},\phi)$, then for any $x>0$,

[TABLE]

Or, equivalently,

[TABLE]

where ${\bf{Y}}=(Y_{1},\cdots,Y_{m})^{\prime}\sim ELL_{m}({\bf C}{\boldsymbol{\mu}},{\bf C}{\bf\Sigma}{\bf C}^{\prime},\phi)$ with $m=n!$, $C_{i}=4i-2n-2,i=1,2,\cdots,n$ and

[TABLE]

Here ${\bf C}_{r}$ is a permutation of $(C_{1},\cdots,C_{n})^{\prime}$, $r=1,2,\cdots,m$.

The next theorem establishes a large deviation result for the Gini indexes of multivariate normal risks which corrected the result of Theorem 5 in Kim and Kim (2019).

Theorem 5.1**.**

Let ${\bf{X}}\sim N_{n}({\boldsymbol{\mu}},{\bf\Sigma})$, the we have

[TABLE]

where $a^{2}_{ii},i=1,2,\cdots,n!$ are diagonal elements of matrix ${\bf C}{\bf\Sigma}{\bf C}^{\prime}$ in Proposition 4.1 with $C_{i}=4i-2n-2,i=1,2,\cdots,n$.

Proof  By Proposition 5.1, if ${\bf{X}}\sim N_{n}({\boldsymbol{\mu}},{\bf\Sigma})$, then ${\bf{Y}}=(Y_{1},\cdots,Y_{m})^{\prime}\sim N_{m}({\bf C}{\boldsymbol{\mu}},{\bf C}{\bf\Sigma}{\bf C}^{\prime})$. In particular, $Y_{i}\sim N_{1}(\nu_{i},a^{2}_{ii}),i=1,2,\cdots,m,$ where $\nu_{i}$’s are the elements of vector ${\bf C}{\boldsymbol{\mu}}$ and $a^{2}_{ii}$’s are diagonal elements of matrix ${\bf C}{\bf\Sigma}{\bf C}^{\prime}$. Using the well known fact

[TABLE]

we obtain, for all $1\leq i\leq n!$,

[TABLE]

or, equivalently,

[TABLE]

Using (5.4) we get

[TABLE]

Without loss of generality we assume that $a^{2}_{11}>a^{2}_{ii},i=2,\cdots,m$. Then (5.6) can be rewritten as

[TABLE]

where

[TABLE]

One easily obtains

[TABLE]

It follows from (5.5), (5.7) and (5.8) that

[TABLE]

as desired.

6 Concluding remarks

In this paper, we have considered usual stochastic order and increasing convex order problems about the Gini indexes for multivariate elliptical random variables. The related issues for multivariate normal risks and scale mixture of multivariate normal risks have been studied by Samanthi et al. (2016) and Kim and Kim (2019). Here, we have investigated the issues for multivariate elliptical risks and scale mixture of multivariate elliptical risks. This paper also answered the following open problems proposed in the Concluding Remarks in Samanthi et al. (2016): To what extent can Gini indexes of multivariate elliptical risks be ordered in the sense of usual stochastic order? Does the conclusion still hold for high dimensional risks with general elliptical distribution? We also solve another open problem in Samanthi et al. (2016) about the increasing convex order of Gini indexes for higher dimensional risks. In addition, we found the tail probability of Gini index $G_{n}({\bf X})$ when ${\bf{X}}\sim{ELL}_{n}({\boldsymbol{\mu}},{\bf\Sigma},{\bf\phi})$. Especially, a large deviation result for the Gini indexes of multivariate normal risks was established which revised the corresponding result in Kim and Kim (2019).

Acknowledgements  The author is grateful to Managing Editor Rob Kaas and the referees for their suggestions which helped to improve this article. The research was supported by the National Natural Science Foundation of China (No. 11171179, 11571198, 11701319).