An indexed multivariate dispersion ordering based on the Hausdorff distance

A new multivariate dispersion ordering based on the Hausdorff distance between nonempty convex compact sets is proposed. This dispersion ordering depends on an index, whose purpose is to blur for each random vector the ball centered at its expected value, and with a radius equal to the index. So, on the basis of such an index, we consider a random set associated with each random vector and dispersion comparisons are established by means of the Hausdorff distance associated with the random sets. Different properties of the new dispersion ordering are stated as well as some characterization theorems. Possible relationships with other dispersion orderings are also studied. Finally, several examples are developed.     

The simplex dispersion ordering and its application to the evaluation of human corneal endothelia

A multivariate dispersion ordering based on random simplices is proposed in this paper. Given a R^d-valued random vector, we consider two random simplices determined by the convex hulls of two independent random samples of sizes d+1 of the vector. By means of the stochastic comparison of the Hausdorff distances between such simplices, a multivariate dispersion ordering is introduced. Main properties of the new ordering are studied. Relationships with other dispersion orderings are considered, placing emphasis on the univariate version. Some statistical tests for the new order are proposed. An application of such ordering to the clinical evaluation of human corneal endothelia is provided. Different analyses are included using an image database of human corneal endothelia.     

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.     

Regression models for multivariate ordered responses via the Plackett distribution

We investigate the properties of a class of discrete multivariate distributions whose univariate marginals have ordered categories, all the bivariate marginals, like in the Plackett distribution, have log-odds ratios which do not depend on cut points and all higher-order interactions are constrained to 0. We show that this class of distributions may be interpreted as a discretized version of a multivariate continuous distribution having univariate logistic marginals. Convenient features of this class relative to the class of ordered probit models (the discretized version of the multivariate normal) are highlighted. Relevant properties of this distribution like quadratic log-linear expansion, invariance to collapsing of adjacent categories, properties related to positive dependence, marginalization and conditioning are discussed briefly. When continuous explanatory variables are available, regression models may be fitted to relate the univariate logits (as in a proportional odds model) and the log-odds ratios to covariates.     

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.     

On the singularity of multivariate skew-symmetric models

In recent years, the skew-normal models introduced by Azzalini (1985) [1]-and their multivariate generalizations from Azzalini and Dalla Valle (1996) [4]-have enjoyed an amazing success, although an important literature has reported that they exhibit, in the vicinity of symmetry, singular Fisher information matrices and stationary points in the profile log-likelihood function for skewness, with the usual unpleasant consequences for inference. It has been shown (DiCiccio and Monti (2004) [23], DiCiccio and Monti (2009) [24] and Gómez et al. (2007) [25]) that these singularities, in some specific parametric extensions of skew-normal models (such as the classes of skew-t or skew-exponential power distributions), appear at skew-normal distributions only. Yet, an important question remains open: in broader semiparametric models of skewed distributions (such as the general skew-symmetric and skew-elliptical ones), which symmetric kernels lead to such singularities? The present paper provides an answer to this question. In very general (possibly multivariate) skew-symmetric models, we characterize, for each possible value of the rank of Fisher information matrices, the class of symmetric kernels achieving the corresponding rank. Our results show that, for strictly multivariate skew-symmetric models, not only Gaussian kernels yield singular Fisher information matrices. In contrast, we prove that systematic stationary points in the profile log-likelihood functions are obtained for (multi)normal kernels only. Finally, we also discuss the implications of such singularities on inference.     

Preserving multivariate dispersion: An application to the Wishart distribution

Univariate dispersive ordering has been extensively characterized by many authors over the last two decades. However, the multivariate version lacks extensive analysis. In this paper, sufficient and necessary conditions are given to preserve the strong multivariate dispersion order through properties of the corresponding transformation. Finally, these results are applied to the Wishart distribution which can be viewed as “the spread of the dispersion”.     

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.     

High dimensional data analysis using multivariate generalized spatial quantiles

High dimensional data routinely arises in image analysis, genetic experiments, network analysis, and various other research areas. Many such datasets do not correspond to well-studied probability distributions, and in several applications the data-cloud prominently displays non-symmetric and non-convex shape features. We propose using spatial quantiles and their generalizations, in particular, the projection quantile, for describing, analyzing and conducting inference with multivariate data. Minimal assumptions are made about the nature and shape characteristics of the underlying probability distribution, and we do not require the sample size to be as high as the data-dimension. We present theoretical properties of the generalized spatial quantiles, and an algorithm to compute them quickly. Our quantiles may be used to obtain multidimensional confidence or credible regions that are not required to conform to a pre-determined shape. We also propose a new notion of multidimensional order statistics, which may be used to obtain multidimensional outliers. Many of the features revealed using a generalized spatial quantile-based analysis would be missed if the data was shoehorned into a well-known probabilistic configuration.     

A test of uniformity on shape spaces

A test of uniformity on the shape space Σ_m^k is presented, together with modifications of the test statistic which bring its null distribution close to the large-sample asymptotic distribution. The asymptotic distribution under suitable local alternatives to uniformity is given. A family of distributions on Σ_m^k is proposed, which is suitable for modelling shapes given by landmarks which are almost collinear.     

Dependence structure of conditional Archimedean copulas

In this article, copulas associated to multivariate conditional distributions in an Archimedean model are characterized. It is shown that this popular class of dependence structures is closed under the operation of conditioning, but that the associated conditional copula has a different analytical form in general. It is also demonstrated that the extremal copula for conditional Archimedean distributions is no longer the Fréchet upper bound, but rather a member of the Clayton family. Properties of these conditional distributions as well as conditional versions of tail dependence indices are also considered.     

A class of multivariate symmetric stable distributions

Multivariate symmetric stable characteristic functions and their properties, as well as conditions for independence and an analogue of the correlation coefficient in bivariate symmetric stable distributions, are discussed.     

The multivariate Behrens-Fisher distribution

The main purpose of this paper is the study of the multivariate Behrens-Fisher distribution. It is defined as the convolution of two independent multivariate Student t distributions. Some representations of this distribution as the mixture of known distributions are shown. An important result presented in the paper is the elliptical condition of this distribution in the special case of proportional scale matrices of the Student t distributions in the defining convolution. For the bivariate Behrens-Fisher problem, the authors propose a non-informative prior distribution leading to highest posterior density (H.P.D.) regions for the difference of the mean vectors whose coverage probability matches the frequentist coverage probability more accurately than that obtained using the independence-Jeffreys prior distribution, even with small samples.     

Some remarks on multivariate stable distributions

This paper deals with multivariate stable distributions. Press has given an explicit algebraic representation of characteristic functions of such distributions [J. Multivariate Analysis2 (1972), 444–462]. We present counter-examples and correct proofs of some of the statements of Press. The properties of multivariate stable distributions, connected with the spectral measure Γ, present in the expression of the characteristic function, are studied.     

Total positivity order and the normal distribution

Unlike the usual stochastic order, total positivity order is closed under conditioning. Here we provide a general formulation of the preservation properties of the order under conditioning; we study certain properties of the order including translation properties and the implications of having equality in the inequality defining the order. Specializing to the multivariate normal distribution, the study of total positivity order leads to new cones defined in terms of covariance M-matrices related to positive dependence, whose properties we study.     

On the asymptotic power of some k-sample statistics based on the multivariate empirical process

Some k-sample Kolmogorov-Smirnov and Cramér-von Mises-type statistics, based on the multivariate empirical process, are studied. Expressions for their asymptotic power are obtained against various classes of alternative distribution functions.     

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.     

Bounds for functions of multivariate risks

Li et al. [Distributions with Fixed Marginals and Related Topics, vol. 28, Institute of Mathematics and Statistics, Hayward, CA, 1996, pp. 198-212] provide bounds on the distribution and on the tail for functions of dependent random vectors having fixed multivariate marginals. In this paper, we correct a result stated in the above article and we give improved bounds in the case of the sum of identically distributed random vectors. Moreover, we provide the dependence structures meeting the bounds when the fixed marginals are uniformly distributed on the k-dimensional hypercube. Finally, a definition of a multivariate risk measure is given along with actuarial/financial applications.     

A trivariate chi-squared distribution derived from the complex Wishart distribution

The joint density for a particular trivariate chi-squared distribution given by the diagonal elements of a complex Wishart matrix is derived. This distribution has applications in the processing of multilook synthetic aperture radar data. The expression for the density is in the form of an infinite series that converges rapidly and is simple and fast to compute. The expression is shown to reduce to known forms for a number of special cases and is validated by simulation. The characteristic function is also derived and used to relate joint moments of the trivariate distribution to the parameters of the density function.