On the Gaussian representation of the Riesz probability distribution on symmetric matrices

AStA Advances in Statistical Analysis - The Riesz probability distribution on symmetric matrices represents an important extension of the Wishart distribution. It is defined by its Laplace...     

Mixture of the Riesz Distribution with Respect to a Multivariate Poisson

The aim of this article is to study a statistical model obtained by the mixture of the Riesz probability distribution on symmetric matrices with respect to a multivariate Poisson distribution. We show that this distribution is related to the modified Bessel function of the first kind. We then determine the domain of the means and the variance function of the generated natural exponential family.     

On the Dirichlet distributions on symmetric matrices

In this paper, we introduce a generalization of the Dirichlet distribution on symmetric matrices which represents the multivariate version of the Connor and Mosimann generalized real Dirichlet distribution. We establish some properties concerning this generalized distribution. We also extend to the matrix Dirichlet distribution a remarkable characterization established in the real case by Darroch and Ratcliff.     

Bhattacharyya matrices and cubic exponential families

This paper introduces a notion of semi-diagonality for a Bhattacharyya matrix to give a characterization of the Letac–Mora class of real natural exponential families having cubic variance function.     

Multivariate stable exponential families and Tweedie scale

In this paper, we characterize the multivariate stable natural exponential families by a property of homogeneity of the cumulant function of some basis, and by a property of homogeneity of the variance function. We also extend the definition of a Tweedie scale to a finite dimensional space and we give a class of natural exponential families belonging to this scale on the space of symmetric matrices.     

On the Tukey depth of an atomic measure

This paper gives a relation between the convex Tukey trimmed region (see [J.C. Massé, R. Theodorescu, Halfplane trimming for bivariate distributions, J. Multivariate Anal. 48(2) (1994) 188–202]) of an atomic measure and the support of the measure. It is shown that an atomic measure is concentrated on the extreme points of its Tukey trimmed region. A property concerning the extreme points which have 0 mass is given. As a corollary, we give a new method of proof of the Koshevoy characterization result.     

Avoiding Ranking Contradictions in Human Development Index Using Goal Programming

This paper builds on the extensive literature of the rank reversal issue in multi-criteria decision making (MCDM) techniques. It is a continuation of the study of Sayed et al. (Soc Indic Res 123(1):1–27, 2015) that exhibited this problem in the human development index (HDI) framework. The proposed methodology, the Goal Programming Benefit-of-the-Doubt (GP-BOD), aims to overcome this problem and obtain consistent and stable rankings. For investigating the credibility of the proposed method in solving this issue, it has been applied to the HDI dataset in 2012. The resulted HDI rankings are compared with those evaluated from eleven overlapping sub-groups that are internationally categorized based on geographic regions and income levels. The results show a solution to the ranking contradictions problem. Among other merits, the results prove two additional features of the proposed GP-BOD model. First, the resulted countries’ rankings are distinguishable and absolutely tie-free. This enhances the discriminating power of the proposed rank preservation model. Second, the GP-BOD weights are evaluated on a common base to compare all countries on the same scale. Moreover, a lower bound is endogenously imposed on these weights to avoid the problem of zero weights. Finally, the validity of the proposed GP-BOD technique has been thoroughly examined using sensitivity tests. The results show stability in the rankings when different methods of normalization and weighting are applied.     

Riesz inverse Gaussian distributions

The notion of generalized power of a positive definite symmetric matrix and a related notion of generalized Bessel function are used to introduce an extension of the class of matrix generalized inverse Gaussian distributions. The new distributions are shown to arise as conditional distributions of Peirce components of Riesz random matrices. Things are explained in the modern framework of symmetric cones and simple Euclidean Jordan algebra.