A family of kurtosis orderings for multivariate distributions

In this paper, a family of kurtosis orderings for multivariate distributions is proposed and studied. Each ordering characterizes in an affine invariant sense the movement of probability mass from the “shoulders” of a distribution to either the center or the tails or both. All even moments of the Mahalanobis distance of a random vector from its mean (if exists) preserve a subfamily of the orderings. For elliptically symmetric distributions, each ordering determines the distributions up to affine equivalence. As applications, the orderings are used to study elliptically symmetric distributions. Ordering results are established for three important families of elliptically symmetric distributions: Kotz type distributions, Pearson Type VII distributions, and Pearson Type II distributions.     

On the theory of elliptically contoured distributions

The theory of elliptically contoured distributions is presented in an unrestricted setting, with no moment restrictions or assumptions of absolute continuity. These distributions are defined parametrically through their characteristic functions and then studied primarily through the use of stochastic representations which naturally follow from the work of Schoenberg [5] on spherically symmetric distributions. It is shown that the conditional distributions of elliptically contoured distributions are elliptically contoured, and the conditional distributions are precisely identified. In addition, a number of the properties of normal distributions (which constitute a type of elliptically contoured distributions) are shown, in fact, to characterize normality.     

从SPVⅡ分布生成的新的多元偏态t分布的有关性质

随机向量的t分布属于椭球等高分布族,然而,它是对称分布.在许多诸如经济学、生理学、社会学等领域中,有时回归模型中的随机误差不再满足对称性,通常表现出高度的偏态性(skewness).于是就有了偏态椭球等高分布族.本文在已有的多元偏态t分布的基础上,着重研究它的分布性质,包括线性组合分布、边缘分布、条件分布及各阶矩.     

球对称分布一些子类的特征函数

本文用统一的方法给出了对称Kotz型分布,多元PearsonⅡ分布和椭球logistic对称分布.我们的结果是Iyenger and Tong的结果的推广,另外我们得到了PearsonⅡ型的特征函数的另一种形式.同时,我们讨论了相关的问题.     

Approximation of the Power of Kurtosis Test for Multinormality

In this paper we investigate performances of the test of multinormality introduced by Malkovich and Afifi. An approximation formula of the power of the test against elliptically symmetric distributions is derived. Examples which illustrate the present results are also discussed.     

On Pearson-Kotz Dirichlet distributions

In this paper, we discuss some basic distributional and asymptotic properties of the Pearson-Kotz Dirichlet multivariate distributions. These distributions, which appear as the limit of conditional Dirichlet random vectors, possess many appealing properties and are interesting from theoretical as well as applied points of view. We illustrate an application concerning the approximation of the joint conditional excess distribution of elliptically symmetric random vectors.     

Mean–variance–skewness efficient surfaces,Stein’s lemma and the multivariate extended skew-Student distribution

Recent advances in Stein’s lemma imply that under elliptically symmetric distributions all rational investors will select a portfolio which lies on Markowitz’ mean–variance efficient frontier. This paper describes extensions to Stein’s lemma for the case when a random vector has the multivariate extended skew-Student distribution. Under this distribution, rational investors will select a portfolio which lies on a single mean–variance–skewness efficient hyper-surface. The same hyper-surface arises under a broad class of models in which returns are defined by the convolution of a multivariate elliptically symmetric distribution and a multivariate distribution of non-negative random variables. Efficient portfolios on the efficient surface may be computed using quadratic programming.     

Smooth depth contours characterize the underlying distribution

The Tukey depth is an innovative concept in multivariate data analysis. It can be utilized to extend the univariate order concept and advantages to a multivariate setting. While it is still an open question as to whether the depth contours uniquely determine the underlying distribution, some positive answers have been provided. We extend these results to distributions with smooth depth contours, with elliptically symmetric distributions as special cases. The key ingredient of our proofs is the well-known Cramér-Wold theorem.     

Singular random matrix decompositions: distributions

Assuming that Y has a singular matrix variate elliptically contoured distribution with respect to the Hausdorff measure, the distributions of several matrices associated to QR, modified QR, SV and polar decompositions of matrix Y are determined, for central and non-central, non-singular and singular cases, as well as their relationship to the Wishart and pseudo-Wishart generalized singular and non-singular distributions. Some of these results are also applied to two particular subfamilies of elliptical distributions, the singular matrix variate normal distribution and the singular matrix variate symmetric Pearson type VII distribution.     

Locally minimax tests in symmetrical distributions

In this paper we give an extension of the theory of local minimax property of Giri and Kiefer (1964, Ann. Math. Statist., 35, 21–35) to the family of elliptically symmetric distributions which contains the multivariate normal distribution as a member.This work was partially supported by the Canadian N.S.E.R.C. grant     

Influence functions and efficiencies of the canonical correlation and vector estimates based on scatter and shape matrices

In this paper, the influence functions and limiting distributions of the canonical correlations and coefficients based on affine equivariant scatter matrices are developed for elliptically symmetric distributions. General formulas for limiting variances and covariances of the canonical correlations and canonical vectors based on scatter matrices are obtained. Also the use of the so-called shape matrices in canonical analysis is investigated. The scatter and shape matrices based on the affine equivariant Sign Covariance Matrix as well as the Tyler's shape matrix serve as examples. Their finite sample and limiting efficiencies are compared to those of the Minimum Covariance Determinant estimators and S-estimator through theoretical and simulation studies. The theory is illustrated by an example.     

Hierarchical orbital decompositions and extended decomposable distributions

Elliptically contoured distributions can be considered to be the distributions for which the contours of the density functions are proportional ellipsoids. Kamiya, Takemura and Kuriki [Star-shaped distributions and their generalizations, J. Statist. Plann. Inference, 2006, available at 〈http://arxiv.org/abs/math.ST/0605600〉, to appear] generalized the elliptically contoured distributions to star-shaped distributions, for which the contours are allowed to be arbitrary proportional star-shaped sets. This was achieved by considering the so-called orbital decomposition of the sample space in the general framework of group invariance. In the present paper, we extend their results by conducting the orbital decompositions in steps and obtaining a further, hierarchical decomposition of the sample space. This allows us to construct probability models and distributions with further independence structures. The general results are applied to the star-shaped distributions with a certain symmetric structure, the distributions related to the two-sample Wishart problem and the distributions of preference rankings.     

椭球等高矩阵分布的条件分布

本文给出了椭球等高矩阵分布的条件分布的随机表示,证明了椭球等高矩阵分布的条件分布仍是椭球等高分布。     

On Mardia's and Song's measures of kurtosis in elliptical distributions

The main objective of this paper is the calculation and the comparative study of two general measures of multivariate kurtosis, namely Mardia's measure β_2,p and Song's measure S(f). In this context, general formulas for the said measures are derived for the broad family of the elliptically contoured symmetric distributions and also for specific members of this family, like the multivariate t-distribution, the multivariate Pearson type II, the multivariate Pearson type VII, the multivariate symmetric Kotz type distribution and the uniform distribution in the unit sphere. Analytic expressions for computing Shannon and Rényi entropies are obtained under the elliptic family. The behaviour of Mardia's and Song's measures, their similarities and differences, possible interpretations and uses in practice are investigated by comparing them in specific members of the elliptic family of multivariate distributions. An empirical estimator of Song's measure is moreover proposed and its asymptotic distribution is investigated under the elliptic family of multivariate distributions.     

Testing for affine equivalence of elliptically symmetric distributions

Let X and Y be d-dimensional random vectors having elliptically symmetric distributions. Call X and Y affinely equivalent if Y has the same distribution as AX+b for some nonsingular d×d-matrix A and some . This paper studies a class of affine invariant tests for affine equivalence under certain moment restrictions. The test statistics are measures of discrepancy between the empirical distributions of the norm of suitably standardized data.     

Data Sphering: Some Properties and Applications

Centring-then-sphering is a very important pretreatment in data analysis. The purpose of this paper is to study the asymptotic behavior of the sphering matrix based on the square root decomposition (SRD for short) and its applications. A sufficient condition is given under which SRD has nondegenerate asymptotic distribution. As examples, some commonly used and affine equivariant estimates of the dispersion matrix are shown to satisfy this condition. The case when the population dispersion matrix varies is also treated. Applications to projection pursuit (PP) are presented. It is shown that for elliptically symmetric distributions the PP index after centring-then-sphering is independent of the underlying population location and dispersion.     

Tests for sphericity under correlated multivariate regression equations model

Summary In this paper, the authors investigated the asymptotic distribution theory connected with the likelihood ratio test (LRT)-like test statistic for sphericity under correlated multivariate regression equations (CMRE) model. An asymptotic expression is obtained for the null distribution of the above test statistic. Asymptotic nonnull distribution of the above test statistic under fixed alternatives is also derived. The above results are derived when the underlying distribution is multivariate normal. It was also shown that the above results are valid even when the joint distribution of the observations is elliptically symmetric. The authors also derived the asymptotic null distribution of the LRT-like test statistic when the observations on each variable are elliptically symmetric. This work was supported by the Air Force Office of Scientific Research under Contract F49620-82-K-0001. Reproduction in whole or in part is permitted for any purpose of the United States Government.     

Some optima of parameter tests for elliptically contoured distribution class

This paper proves that for elliptically contoured distributions many likelihood ratio tests are uniformly most powerful or unbiased.     

A Characterization of Nash Equilibrium for the Games with Random Payoffs

We consider a two-player random bimatrix game where each player is interested in the payoffs which can be obtained with certain confidence. The payoff function of each player is defined using a chance constraint. We consider the case where the entries of the random payoff matrix of each player jointly follow a multivariate elliptically symmetric distribution. We show an equivalence between the Nash equilibrium problem and the global maximization of a certain mathematical program. The case where the entries of the payoff matrices are independent normal/Cauchy random variables is also considered. The case of independent normally distributed random payoffs can be viewed as a special case of a multivariate elliptically symmetric distributed random payoffs. As for Cauchy distribution, we show that the Nash equilibrium problem is equivalent to the global maximization of a certain quadratic program. Our theoretical results are illustrated by considering randomly generated instances of the game.     

The Probability Content of Cones in Isotropic Random Fields

This paper provides computable representations for the evaluation of the probability content of cones in isotropic random fields. A decomposition of quadratic forms in spherically symmetric random vectors is obtained and a representation of their moments is derived in terms of finite sums. These results are combined to obtain the distribution function of quadratic forms in spherically symmetric or central elliptically contoured random vectors. Some numerical examples involving the sample serial covariance are provided. Ratios of quadratic forms are also discussed.