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Stochastic comparisons of sample mean differences for multivariate random variables

Xuehua Yin, Dan Zhu, Chuancun Yin

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).

Stochastic comparisons of sample mean differences for multivariate random variables

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).

Paper Structure

This paper contains 6 sections, 15 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Usual stochastic order of Gini indexes
  4. Increasing convex order of Gini indexes
  5. Tail asymptotic results for Gini indexes
  6. Concluding remarks

Key Result

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 $C \in {\cal C}$,

Theorems & Definitions (16)

  • Lemma 3.1
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Remark 3.1
  • Proposition 3.5
  • Proposition 3.6
  • Proposition 3.7
  • ...and 6 more