On describing multivariate skewed distributions: A directional approach

Most multivariate measures of skewness in the literature measure the overall skewness of a distribution. These measures were designed for testing the hypothesis of distributional symmetry; their relevance for describing skewed distributions is less obvious. In this article, the authors consider the problem of characterizing the skewness of multivariate distributions. They define directional skewness as the skewness along a direction and analyze two parametric classes of skewed distributions using measures based on directional skewness. The analysis brings further insight into the classes, allowing for a more informed selection of classes of distributions for particular applications. The authors use the concept of directional skewness twice in the context of Bayesian linear regression under skewed error: first in the elicitation of a prior on the parameters of the error distribution, and then in the analysis of the skewness of the posterior distribution of the regression residuals.     

Modelling directional dispersion through hyperspherical log-splines

Summary.  We introduce the directionally dispersed class of multivariate distributions, a generalization of the elliptical class. By allowing dispersion of multivariate random variables to vary with direction it is possible to generate a very wide and flexible class of distributions. Directionally dispersed distributions have a simple form for their density, which extends a spherically symmetric density function by including a function D modelling directional dispersion. Under a mild condition, the class of distributions is shown to preserve both unimodality and moment existence. By adequately defining D , it is possible to generate skewed distributions. Using spline models on hyperspheres, we suggest a very flexible, yet practical, implementation for modelling directional dispersion in any dimension. Finally, we use the new class of distributions in a Bayesian regression set-up and analyse the distributions of a set of biomedical measurements and a sample of US manufacturing firms.     

A unified view on skewed distributions arising from selections

Parametric families of multivariate nonnormal distributions have received considerable attention in the past few decades. The authors propose a new definition of a selection distribution that encompasses many existing families of multivariate skewed distributions. Their work is motivated by examples that involve various forms of selection mechanisms and lead to skewed distributions. They give the main properties of selection distributions and show how various families of multivariate skewed distributions, such as the skew‐normal and skew‐elliptical distributions, arise as special cases. The authors further introduce several methods of constructing selection distributions based on linear and nonlinear selection mechanisms.     

Generalized likelihood ratio tests for the structure of semiparametric additive models

Semiparametric additive models (SAMs) are very useful in multivariate nonparametric regression. In this paper, the authors study nonparametric testing problems for the nonparametric components of SAMs. Using the backfitting algorithm and the local polynomial smoothing technique, they extend to SAMs the generalized likelihood ratio tests of Fan &Jiang (2005). The authors show that the proposed tests possess the Wilks‐type property and that they can detect alternatives nearing the null hypothesis with a rate arbitrarily close to root‐n while error distributions are unspecified. They report simulations which demonstrate the Wilks phenomenon and the powers of their tests. They illustrate the performance of their approach by simulation and using the Boston housing data set.     

Bayesian methods for generalized linear models with covariates missing at random

The authors propose methods for Bayesian inference for generalized linear models with missing covariate data. They specify a parametric distribution for the covariates that is written as a sequence of one‐dimensional conditional distributions. They propose an informative class of joint prior distributions for the regression coefficients and the parameters arising from the covariate distributions. They examine the properties of the proposed prior and resulting posterior distributions. They also present a Bayesian criterion for comparing various models, and a calibration is derived for it. A detailed simulation is conducted and two real data sets are examined to demonstrate the methodology.     

Complex elliptically symmetric distributions

In this paper, the authors introduce a class of distributions known as complex elliptically symmetric distributions. The complex multivariate normal and complex multivariate t distributions are members of this class. Various properties of the complex elliptically symmetric distributions are studied. Finally, the robustness of certain test procedures are discussed when the assumption of complex multivariate normality is violated but the underlying distribution still belongs to the class of elliptically symmetric distributions.     

Conjugate analysis of multivariate normal data with incomplete observations

The authors discuss prior distributions that are conjugate to the multivariate normal likelihood when some of the observations are incomplete. They present a general class of priors for incorporating information about unidentified parameters in the covariance matrix. They analyze the special case of monotone patterns of missing data, providing an explicit recursive form for the posterior distribution resulting from a conjugate prior distribution. They develop an importance sampling and a Gibbs sampling approach to sample from a general posterior distribution and compare the two methods.     

Finite mixtures of multivariate Poisson distributions with application

In the present paper we examine finite mixtures of multivariate Poisson distributions as an alternative class of models for multivariate count data. The proposed models allow for both overdispersion in the marginal distributions and negative correlation, while they are computationally tractable using standard ideas from finite mixture modelling. An EM type algorithm for maximum likelihood (ML) estimation of the parameters is developed. The identifiability of this class of mixtures is proved. Properties of ML estimators are derived. A real data application concerning model based clustering for multivariate count data related to different types of crime is presented to illustrate the practical potential of the proposed class of models.     

Bayesian inference for bivariate generalized linear models in diagnosing renal arterial obstruction

Generalized linear models are well-established generalizations of the linear models used for regression and analysis of variance. They allow flexible mean structures and general distributions, other than the linear link and normal response assumed in regression. Further enhancements using ideas from multivariate analysis improve power and precision by modelling dependencies between response variables. This paper focuses on the specific case of regression models for bivariate Bernoulli responses and investigates their analysis using a Bayesian approach. The important problem of renal arterial obstruction is considered, as a medical application of these models.     

Shannon Entropy and Mutual Information for Multivariate Skew‐Elliptical Distributions

Abstract. The entropy and mutual information index are important concepts developed by Shannon in the context of information theory. They have been widely studied in the case of the multivariate normal distribution. We first extend these tools to the full symmetric class of multivariate elliptical distributions and then to the more flexible families of multivariate skew‐elliptical distributions. We study in detail the cases of the multivariate skew‐normal and skew‐t distributions. We implement our findings to the application of the optimal design of an ozone monitoring station network in Santiago de Chile.     

The estimating function bootstrap

The authors propose a bootstrap procedure which estimates the distribution of an estimating function by resampling its terms using bootstrap techniques. Studentized versions of this so‐called estimating function (EF) bootstrap yield methods which are invariant under reparametrizations. This approach often has substantial advantage, both in computation and accuracy, over more traditional bootstrap methods and it applies to a wide class of practical problems where the data are independent but not necessarily identically distributed. The methods allow for simultaneous estimation of vector parameters and their components. The authors use simulations to compare the EF bootstrap with competing methods in several examples including the common means problem and nonlinear regression. They also prove symptotic results showing that the studentized EF bootstrap yields higher order approximations for the whole vector parameter in a wide class of problems.     

On the multivariate spatial median for clustered data

The authors consider the multivariate one-sample location problem with clustered data from a nonparametric viewpoint. They propose the spatial median and its affine equivariant version as companion estimators to the affine invariant sign test of Larocque (2003). They extend the asymptotics of the proposed estimators to cluster dependent data and explore the limiting as well as finite-sample efficiencies for multivariate Student distributions. They demonstrate that the efficiency of the spatial median suffers less from intracluster correlation than the mean vector. They use data on the well-being of pupils in Finnish schools to illustrate their work.     

A multivariate von mises distribution with applications to bioinformatics

Motivated by problems of modelling torsional angles in molecules, Singh, Hnizdo & Demchuk (2002) proposed a bivariate circular model which is a natural torus analogue of the bivariate normal distribution and a natural extension of the univariate von Mises distribution to the bivariate case. The authors present here a multivariate extension of the bivariate model of Singh, Hnizdo & Demchuk (2002). They study the conditional distributions and investigate the shapes of marginal distributions for a special case. The methods of moments and pseudo‐likelihood are considered for the estimation of parameters of the new distribution. The authors investigate the efficiency of the pseudo‐likelihood approach in three dimensions. They illustrate their methods with protein data of conformational angles     

Some characterizations of multivariate distributions using products of the hazard gradient and mean residual life components

We give a general procedure to characterize multivariate distributions by using products of the hazard gradient and mean residual life components. This procedure is applied to characterize multivariate distributions as Gumbel exponential, Lomax, Burr, Pareto and generalized Pareto multivariate distributions. Our results extend the results of several authors and can be used to study how to extend univariate models to the multivariate set-up.     

Inference in multivariate linear regression models with elliptically distributed errors

In this study we investigate the problem of estimation and testing of hypotheses in multivariate linear regression models when the errors involved are assumed to be non-normally distributed. We consider the class of heavy-tailed distributions for this purpose. Although our method is applicable for any distribution in this class, we take the multivariate t-distribution for illustration. This distribution has applications in many fields of applied research such as Economics, Business, and Finance. For estimation purpose, we use the modified maximum likelihood method in order to get the so-called modified maximum likelihood estimates that are obtained in a closed form. We show that these estimates are substantially more efficient than least-square estimates. They are also found to be robust to reasonable deviations from the assumed distribution and also many data anomalies such as the presence of outliers in the sample, etc. We further provide test statistics for testing the relevant hypothesis regarding the regression coefficients.     

A nonparametric test for interaction in two‐way layouts

The authors present a new nonparametric approach to test for interaction in two‐way layouts. Based on the concept of composite linear rank statistics, they combine the correlated row and column ranking information to construct the test statistic. They determine the limiting distributions of the proposed test statistic under the null hypothesis and Pitman alternatives. They also propose consistent estimators for the limiting covariance matrices associated with the test. They illustrate the application of their test in practical settings using a microarray data set.     

Estimation and inference of the joint conditional distribution for multivariate longitudinal data using nonparametric copulas

In this paper we study estimating the joint conditional distributions of multivariate longitudinal outcomes using regression models and copulas. For the estimation of marginal models, we consider a class of time-varying transformation models and combine the two marginal models using nonparametric empirical copulas. Our models and estimation method can be applied in many situations where the conditional mean-based models are not good enough. Empirical copulas combined with time-varying transformation models may allow quite flexible modelling for the joint conditional distributions for multivariate longitudinal data. We derive the asymptotic properties for the copula-based estimators of the joint conditional distribution functions. For illustration we apply our estimation method to an epidemiological study of childhood growth and blood pressure.     

Multivariate distributions with generalized inverse gaussian marginals,and associated poisson mixtures

Several types of multivariate extensions of the inverse Gaussian (IG) distribution and the reciprocal inverse Gaussian (RIG) distribution are proposed. Some of these types are obtained as random-additive-effect models by means of well-known convolution properties of the IG and RIG distributions, and they have one-dimensional IG or RIG marginals. They are used to define a flexible class of multivariate Poisson mixtures.     

Change-point detection in a linear model by adaptive fused quantile method

A novel approach to quantile estimation in multivariate linear regression models with change-points is proposed: the change-point detection and the model estimation are both performed automatically, by adopting either the quantile-fused penalty or the adaptive version of the quantile-fused penalty. These two methods combine the idea of the check function used for the quantile estimation and the L₁ penalization principle known from the signal processing and, unlike some standard approaches, the presented methods go beyond typical assumptions usually required for the model errors, such as sub-Gaussian or normal distribution. They can effectively handle heavy-tailed random error distributions, and, in general, they offer a more complex view on the data as one can obtain any conditional quantile of the target distribution, not just the conditional mean. The consistency of detection is proved and proper convergence rates for the parameter estimates are derived. The empirical performance is investigated via an extensive comparative simulation study and practical utilization is demonstrated using a real data example.     

Modeling with a large class of unimodal multivariate distributions

In this paper we introduce a new class of multivariate unimodal distributions, motivated by Khintchine's representation for unimodal densities on the real line. We start by introducing a new class of unimodal distributions which can then be naturally extended to higher dimensions, using the multivariate Gaussian copula. Under both univariate and multivariate settings, we provide MCMC algorithms to perform inference about the model parameters and predictive densities. The methodology is illustrated with univariate and bivariate examples, and with variables taken from a real data set.