Distributions of the compound and scale mixture of vector and spherical matrix variate elliptical distributions

Several matrix variate hypergeometric type distributions are derived. The compound distributions of left-spherical matrix variate elliptical distributions and inverted hypergeometric type distributions with matrix arguments are then proposed. The scale mixture of left-spherical matrix variate elliptical distributions and univariate inverted hypergeometric type distributions is also derived as a particular case of the compound distribution approach.     

A class of bivariate exponential distributions

We introduce a class of absolutely continuous bivariate exponential distributions, generated from quadratic forms of standard multivariate normal variates.This class is quite flexible and tractable, since it is regulated by two parameters only, derived from the matrices of the quadratic forms: the correlation and the correlation of the squares of marginal components. A simple representation of the whole class is given in terms of 4-dimensional matrices. Integral forms allow evaluating the distribution function and the density function in most of the cases.The class is introduced as a subclass of bivariate distributions with chi-square marginals; bounds for the dimension of the generating normal variable are underlined in the general case.Finally, we sketch the extension to the multivariate case.     

Statistical theory of shape under elliptical models and singular value decompositions

The size-and-shape and shape distributions based on non-central and non-isotropic elliptical distributions are derived in this paper by using the singular value decomposition (SVD). The general densities require the computation of new integrals involving zonal polynomials. The invariance of the central shape distribution is also proved. Finally, some particular densities are applied in a classical data of Biology, and the inference based on exact distributions is performed after choosing the best model by using a modified BIC^∗ criterion.     

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.     

Singular matrix variate beta distribution

In this paper, we determine the symmetrised density of doubly noncentral singular matrix variate beta type I and II distributions under different definitions. As particular cases we obtain the noncentral singular matrix variate beta type I and II distributions and the corresponding joint density of the nonnull eigenvalues. In addition, we propose an alternative approach to find the corresponding nonsymmetrised densities. From the latter, we solve the integral proposed by Constantine [Noncentral distribution problems in multivariate analysis, Ann. Math. Statist. 34 (1963) 1270-1285] and Khatri [A note on Mitra's paper “A density free approach to the matrix variate beta distribution”, Sankhyā A 32 (1970) 311-318] and reconsidered in Farrell [Multivariate Calculation: Use of the Continuous Groups, Springer Series in Statistics, Springer, New York, 1985, p. 191], see also Díaz-García and Gutiérrez-Jáimez [Noncentral matrix variate beta distribution, Comunicación Técnica, No. I-06-06 (PE/CIMAT), Guanajuato, México, 2006, 〈http://www.cimat.mx/biblioteca/RepTec/index.html?m=2〉], for the singular and nonsingular cases.     

Some inequalities for strong mixing random variables with applications to density estimation

In this paper, we establish an inequality of the characteristic functions for strongly mixing random vectors, by which, an upper bound is provided for the supremum of the absolute value of the difference of two multivariate probability density functions based on strongly mixing random vectors. As its application, we consider the consistency and asymptotic normality of a kernel estimate of a density function under strong mixing. Our results generalize some known results in the literature.     

Joint distribution of run statistics in partially exchangeable processes

Let {X_i}_i≥1 be an infinite sequence of recurrent partially exchangeable random variables with two possible outcomes as either “1” (success) or “0” (failure). In this paper we obtain the joint distribution of success and failure run statistics in {X_i}_i≥1. The results can be used to obtain the joint distribution of runs in ordinary Markov chains, exchangeable and independent sequences.     

Minimum and maximum distances between failures in binary sequences

The minimum and maximum distances denoting the extreme numbers of successes between two successive failures in binary sequences, are considered. Exact marginal and joint probability distributions of these statistics are obtained via combinatorial analysis. Applications related to urn models and system reliability are studied and clarify further the theoretical results.     

Dependence structures of multivariate Bernoulli random vectors

In some situations, it is difficult and tedious to check notions of dependence properties and dependence orders for multivariate distributions supported on a finite lattice. The purpose of this paper is to utilize a newly developed tool, majorization with respect to weighted trees, to lay out some general results that can be used to identify some dependence properties and dependence orders for multivariate Bernoulli random vectors. Such a study gives us some new insight into the relations between the concepts of dependence.     

Singular random matrix decompositions: Jacobians

For a singular random matrix X, we find the Jacobians associated to the following decompositions: QR, Polar, Singular Value (SVD), L′U, L′DM and modified QR (QDR). Similarly, for the cross-product matrix S=X′X we find the Jacobians of the Spectral, Cholesky's, L′DL and symmetric nonnegative definite square root decompositions.     

Bivariate distributions characterized by one family of conditionals and conditional percentile or mode functions

It is well known that full knowledge of all conditional distributions will typically serve to completely characterize a bivariate distribution. Partial knowledge will often suffice. For example, knowledge of the conditional distribution of X given Y and the conditional mean of Y given X is often adequate to determine the joint distribution of X and Y. In this paper, we investigate the extent to which a conditional percentile function or a conditional mode function (of Y given X), together with knowledge of the conditional distribution of X given Y will determine the joint distribution. Finally, using this methodology a new characterization of the classical bivariate normal distribution is given.     

On Wishart distribution: Some extensions

This paper proposes a unified approach that enables the Wishart distribution to be studied simultaneously in the real, complex, quaternion and octonion cases under elliptical models. In particular, the matrix multivariate elliptical distribution, the noncentral generalised Wishart distribution, the joint density of the eigenvalues and the distribution of the maximum eigenvalue are obtained for real normed division algebras.     

Likelihood ratio orderings of spacings of heterogeneous exponential random variables

Let X₁,X₂,…,X_n be independent exponential random variables such that X_i has failure rate λ for i=1,…,p and X_j has failure rate λ^* for j=p+1,…,n, where p≥1 and q=n-p≥1. Denote by D_i:n(p,q)=X_i:n-X_i-1:n the ith spacing of the order statistics , where X_0:n≡0. It is shown that D_i:n(p,q)?_lrD_i+1:n(p,q) for i=1,…,n-1, and that if λ?λ^* then , and for i=1,…,n, where ?_lr denotes the likelihood ratio order. The main results are used to establish the dispersive orderings between spacings.     

Multivariate measures of skewness for the skew-normal distribution

The main objective of this work is to calculate and compare different measures of multivariate skewness for the skew-normal family of distributions. For this purpose, we consider the Mardia (1970) [10], Malkovich and Afifi (1973) [9], Isogai (1982) [17], Srivastava (1984) [15], Song (2001) [14], Móri et al. (1993) [11], Balakrishnan et al. (2007) [3] and Kollo (2008) [7] measures of skewness. The exact expressions of all measures of skewness, except for Song’s, are derived for the family of skew-normal distributions, while Song’s measure of shape is approximated by the use of delta method. The behavior of these measures, their similarities and differences, possible interpretations, and their practical use in testing for multivariate normal are studied by evaluating their power in the case of some specific members of the multivariate skew-normal family of distributions.     

Nonparametric tests of independence between random vectors

A nonparametric test of the mutual independence between many numerical random vectors is proposed. This test is based on a characterization of mutual independence defined from probabilities of half-spaces in a combinatorial formula of Möbius. As such, it is a natural generalization of tests of independence between univariate random variables using the empirical distribution function. If the number of vectors is p and there are n observations, the test is defined from a collection of processes R_n,A, where A is a subset of {1,…,p} of cardinality |A|>1, which are asymptotically independent and Gaussian. Without the assumption that each vector is one-dimensional with a continuous cumulative distribution function, any test of independence cannot be distribution free. The critical values of the proposed test are thus computed with the bootstrap which is shown to be consistent. Another similar test, with the same asymptotic properties, for the serial independence of a multivariate stationary sequence is also proposed. The proposed test works when some or all of the marginal distributions are singular with respect to Lebesgue measure. Moreover, in singular cases described in Section 4, the test inherits useful invariance properties from the general affine invariance property.     

Limit distributions of V- and U-statistics in terms of multiple stochastic Wiener-type integrals

We describe the limit distribution of V- and U-statistics in a new fashion. In the case of V-statistics the limit variable is a multiple stochastic integral with respect to an abstract Brownian bridge G_Q. This extends the pioneer work of Filippova (1961) [8]. In the case of U-statistics we obtain a linear combination of G_Q-integrals with coefficients stemming from Hermite Polynomials. This is an alternative representation of the limit distribution as given by Dynkin and Mandelbaum (1983) [7] or Rubin and Vitale (1980) [13]. It is in total accordance with their results for product kernels.     

A connection between supermodular ordering and positive/negative association

In this paper, we show that a vector of positively/negatively associated random variables is larger/smaller than the vector of their independent duplicates with respect to the supermodular order. In that way, we solve an open problem posed by Hu (Chinese J. Appl. Probab. Statist. 16 (2000) 133) refering to whether negative association implies negative superadditive dependence, and at the same time to an open problem stated in Müller and Stoyan (Comparison Methods for Stochastic Modes and Risks, Wiley, Chichester, 2002) whether association implies positive supermodular dependence. Therefore, some well-known results concerning sums and maximum partial sums of positively/negatively associated random variables are obtained as an immediate consequence. The aforementioned result can be exploited to give useful probability inequalities. Consequently, as an application we provide an improvement of the Kolmogorov-type inequality of Matula (Statist. Probab. Lett. 15 (1992) 209) for negatively associated random variables. Moreover, a Rosenthal-type inequality for associated random variables is presented.     

Applications of average and projected systems to the study of coherent systems

In this paper, we introduce the concepts of average and projected systems associated to a coherent (parent) system. We analyze several aspects of these notions and show that they can be useful tools in studying the performance of coherent systems with non-exchangeable components. We show that the average and projected systems are especially useful in studying the tail behavior of reliability, hazard rate and mean residual life functions of the parent system and also in obtaining the tail best systems (under different criteria) by permuting the components at the system structure. Moreover, they can be useful in assessing how the asymmetry of the joint distribution of the component lifetimes (with respect to permutations of the components in the system structure) affects the system performance.     

Stochastic comparisons of order statistics from gamma distributions

Let (X₁,X₂,…,X_n) and (Y₁,Y₂,…,Y_n) be gamma random vectors with common shape parameter α(0<α?1) and scale parameters (λ₁,λ₂,…,λ_n), (μ₁,μ₂,…,μ_n), respectively. Let X₍₎=(X₍₁₎,X₍₂₎,…,X_(n)), Y₍₎=(Y₍₁₎,Y₍₂₎,…,Y_(n)) be the order statistics of (X₁,X₂,…,X_n) and (Y₁,Y₂,…,Y_n). Then (λ₁,λ₂,…,λ_n) majorizes (μ₁,μ₂,…,μ_n) implies that X₍₎ is stochastically larger than Y₍₎. However if the common shape parameter α>1, we can only compare the the first- and last-order statistics. Some earlier results on stochastically comparing proportional hazard functions are shown to be special cases of our results.