Multivariate extreme value distributions for stationary Gaussian sequences

Under weak regularity conditions of the covariance sequence, it is shown that the joint limiting distribution of the maxima on each coordinate of a stationary Gaussian multivariate sequence is that of independent random variables with marginal Gumbel distributions.     

A multivariate semi-logistic autoregressive process and its characterization

An autoregressive multivariate stochastic model is constructed which yields a stationary Markov process with a marginal invariant distribution as a multivariate semi-logistic distribution. This model is denoted as an MSL-AR(1) process. Some properties of the MSL-AR(1) process are studied and its characterization is also derived.     

Characterizations of symmetric distributions based on Rényi entropy

It is proved that the equality of Rényi entropies of upper and lower order statistics as well as upper and lower k-records is a characteristic property of symmetric distributions. Also, for Farlie-Gumbel-Morgenstern (FGM) family, it is shown that under some conditions the equality of entropies of concomitants of upper and lower order statistics as well as concomitants of upper and lower record values is a characteristic property for the uniform distribution.     

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.     

Multivariate quadratic forms of random vectors

We obtain the distribution of the sum of n random vectors and the distribution of their quadratic forms: their densities are expanded in series of Hermite and Laguerre polynomials. We do not suppose that these vectors are independent. In particular, we apply these results to multivariate quadratic forms of Gaussian vectors. We obtain also their densities expanded in Mac Laurin series or in the form of an integral. By this last result, we introduce a new method of computation which can be much simpler than the previously known techniques. In particular, we introduce a new method in the very classical univariate case. We remark that we do not assume the independence of normal variables.     

Characteristic functions of scale mixtures of multivariate skew-normal distributions

We obtain the characteristic function of scale mixtures of skew-normal distributions both in the univariate and multivariate cases. The derivation uses the simple stochastic relationship between skew-normal distributions and scale mixtures of skew-normal distributions. In particular, we describe the characteristic function of skew-normal, skew-t, and other related distributions.     

Multivariate order statistics via multivariate concomitants

Let denote a set of n independent identically distributed k-dimensional absolutely continuous random variables. A general class of complete orderings of such random vectors is supplied by viewing them as concomitants of an auxiliary random variable. The resulting definitions of multivariate order statistics subsume and extend orderings that have been previously proposed such as norm ordering and N-conditional ordering. Analogous concepts of multivariate record values and multivariate generalized order statistics are also described.     

Limit distributions of generalized order statistics in a stationary Gaussian sequence

In this paper we study the limit distributions of extreme, intermediate and central m-generalized order statistics, as well as m-dual generalized order statistics, of a stationary Gaussian sequence under equi-correlated set up. Moreover, the result of extremes is extended to a wide subclass of generalized order statistics, as well as dual generalized order statistics, when the parameters γ_1,n, γ_2,n,?…?, γ_n,n are assumed to be pairwise different.     

Empirical distributions in marked point processes

We study the asymptotic behaviour of the empirical distribution function derived from a stationary marked point process when a convex sampling window is expanding without bounds in all directions. We consider a random field model which assumes that the marks and the points are independent and admits dependencies between the marks. The main result is the weak convergence of the empirical process under strong mixing conditions on both independent components of the model. Applying an approximation principle weak convergence can be also shown for appropriately weighted empirical process defined from a stationary d-dimensional germ-grain process with dependent grains.     

On the dependence structure of order statistics

Given a random sample from a continuous variable, it is observed that the copula linking any pair of order statistics is independent of the parent distribution. To compare the degree of association between two such pairs of ordered random variables, a notion of relative monotone regression dependence (or stochastic increasingness) is considered. Using this concept, it is proved that for i<j, the dependence of the jth order statistic on the ith order statistic decreases as i and j draw apart. This extends earlier results of Tukey (Ann. Math. Statist. 29 (1958) 588) and Kim and David (J. Statist. Plann. Inference 24 (1990) 363). The effect of the sample size on this type of dependence is also investigated, and an explicit expression is given for the population value of Kendall's coefficient of concordance between two arbitrary order statistics of a random sample.     

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.     

A new dependence ordering with applications

In this paper, we introduce a new copula-based dependence order to compare the relative degree of dependence between two pairs of random variables. Relationship of the new order to the existing dependence orders is investigated. In particular, the new ordering is stronger than the partial ordering, more monotone regression dependence as developed by Avérous et al. [J. Avérous, C. Genest, S.C. Kochar, On dependence structure of order statistics, Journal of Multivariate Analysis 94 (2005) 159-171]. Applications of this partial order to order statistics, k-record values and frailty models are given.     

Exact distributions of MLEs of regression coefficients in GMANOVA-MANOVA model

This paper studies the exact distributions of the MLEs of the regression coefficient matrices in a GMANOVA-MANOVA model with normal error. The unique conditions for linear functions of the MLEs of regression coefficient matrices are presented, and the exact density functions or characteristic functions for these linear functions are derived.     

A class of weighted multivariate normal distributions and its properties

This article proposes a class of weighted multivariate normal distributions whose probability density function has the form of a product of a multivariate normal density and a weighting function. The class is obtained from marginal distributions of various doubly truncated multivariate normal distributions. The class strictly includes the multivariate normal and multivariate skew-normal. It is useful for selection modeling and inequality constrained normal mean vector analysis. We report on a study of some distributional properties and the Bayesian perspective of the class. A probabilistic representation of the distributions is also given. The representation is shown to be straightforward to specify the distribution and to implement computation, with output readily adapted for the required analysis. Necessary theories and illustrative examples are provided.     

Exponential families of mixed Poisson distributions

If I=(I₁,…,I_d) is a random variable on [0,∞)^d with distribution μ(dλ₁,…,dλ_d), the mixed Poisson distribution MP(μ) on N^d is the distribution of (N₁(I₁),…,N_d(I_d)) where N₁,…,N_d are ordinary independent Poisson processes which are also independent of I. The paper proves that if F is a natural exponential family on [0,∞)^d then MP(F) is also a natural exponential family if and only if a generating probability of F is the distribution of v₀+v₁Y₁+?+v_qY_q for some q?d, for some vectors v₀,…,v_q of [0,∞)^d with disjoint supports and for independent standard real gamma random variables Y₁,…,Y_q.     

Multivariate dependence of spacings of generalized order statistics

Multivariate dependence of spacings of generalized order statistics is studied. It is shown that spacings of generalized order statistics from DFR (IFR) distributions have the CIS (CDS) property. By restricting the choice of the model parameters and strengthening the assumptions on the underlying distribution, stronger dependence relations are established. For instance, if the model parameters are decreasingly ordered and the underlying distribution has a log-convex decreasing (log-concave) hazard rate, then the spacings satisfy the MTP₂ (S- MRR₂) property. Some consequences of the results are given. In particular, conditions for non-negativity of the best linear unbiased estimator of the scale parameter in a location-scale family are obtained. By applying a result for dual generalized order statistics, we show that in the particular situation of usual order statistics the assumptions can be weakened.     

Multivariate maximum entropy identification, transformation, and dependence

This paper shows that multivariate distributions can be characterized as maximum entropy (ME) models based on the well-known general representation of density function of the ME distribution subject to moment constraints. In this approach, the problem of ME characterization simplifies to the problem of representing the multivariate density in the ME form, hence there is no need for case-by-case proofs by calculus of variations or other methods. The main vehicle for this ME characterization approach is the information distinguishability relationship, which extends to the multivariate case. Results are also formulated that encapsulate implications of the multiplication rule of probability and the entropy transformation formula for ME characterization. The dependence structure of multivariate ME distribution in terms of the moments and the support of distribution is studied. The relationships of ME distributions with the exponential family and with bivariate distributions having exponential family conditionals are explored. Applications include new ME characterizations of many bivariate distributions, including some singular distributions.     

Spatial autoregression and related spatio-temporal models

We propose a spatial autoregressive random field of order p on the spatial domain for p?2 in this paper, whose univariate margins are the continuous-time autoregression of order p on the real line, and introduce a class of semiparametric spatio-temporal covariance models stationary in space with the spatial autoregressive margin.     

Slepian’s inequality with respect to majorization

In this paper, we obtain some sufficient conditions for Slepian’s inequality with respect to majorization for two Gaussian random vectors.     

On polynomial variance functions

Summary Let be a natural exponential family on and (V, ) be its variance function. Here, is the mean domain of andV, defined on , is the variance of . A problem of increasing interest in the literature is the following: Given an open interval and a functionV defined on , is the pair (V, ) a variance function of some natural exponential family? Here, we consider the case whereV is a polynomial. We develop a complex-analytic approach to this problem and provide necessary conditions for (V, ) to be such a variance function. These conditions are also sufficient for the class of third degree polynomials and certain subclasses of polynomials of higher degree.