A new class of multivariate skew distributions with applications to bayesian regression models

Abstract

The authors develop a new class of distributions by introducing skewness in multivariate elliptically symmetric distributions. The class, which is obtained by using transformation and conditioning, contains many standard families including the multivariate skew-normal and t distributions. The authors obtain analytical forms of the densities and study distributional properties. They give practical applications in Bayesian regression models and results on the existence of the posterior distributions and moments under improper priors for the regression coefficients. They illustrate their methods using practical examples. Une nouvelle classe de lois multivariws asymetriques et ses applications dans le cadre de modeles de rhression bayesiensR6ssud : Les auteurs engendrent une nouvelle classe de lois en introduisant un facteur d'asymttrie dans la famille des distributions multivarites elliptiquement symttriques. La classe, qui est obtenue par transformation et conditionnement, inclut plusieurs familles de lois connues, dont la Student et la normale multivarites asymttriques. Les auteurs donnent la forme explicite de la densitk de ces lois et en examinent les proprittks. Ils en pdsentent des applications pratiques dans le cadre des modkles de dgression baytsiens, oh ils dtmontrent l'existence de lois a posteriori et de leurs moments lorsque les lois a priori des paramktres de la dgression sont impropres. ns illustrent en outre leurs mtthodes dans des cas concrets. SAHU, DEY& BRANCO Vol. 31, No. 2 g(u; k, .I.

Semiparametric Bayesian Latent Variable Regression for Skewed Multivariate Data

“…In Section 3, we explain that this DP( 0 , ) prior process on (⋅) induces a flexible prior on the nonparametric density ℎ (⋅) in (4). When 1 and 2 in (2) are assumed to have parametric multivariate elliptical densities, the resulting model is similar to the "multiple skewed shocks" model of Sahu et al (2003). However, our semiparametric model with nonparametric marginals has two additional crucial differences: (a) a new parametrization of weights ( , * ), and (b) a multivariate copula modeling of 2 as in (3).…”

“…(Chang and Zimmerman, 2016) See Appendix A for details on the expression of̄. In general, similar to the parametric formulation of Sahu et al (2003), the model class in (2) is closed under marginalization, because the marginal density of any subset (1) of = ( (1) , (2) ) is from the same class of semiparametric density SMS(0, ℎ (1) , 11 , (1) ) indexed by (ℎ (1) , 11 , (1) ), where = ( (1) , (2) ), (ℎ 1 , … , ℎ ) = (ℎ (1) , ℎ (2) ) and matrix = ) are partitions of ( , ℎ, ). Unlike the multivariate model of (2) based on multiple "skewing shocks" | 1 1 |, … , | 1 |, the existing multivariate models, such as MSN(0, , Σ) are based on univariate "skewing shock" | * 1 | (common to all components).…”

“…(2) The PGM model, with latent vectors 1 ∼ N 2 (0, 2 ) and 2 ∼ N 2 (0, Σ), where 2 = diag( 2 1 , 2 2 ) and (Σ) 11 = 2 1 , (Σ) 22 = 2 2 , (Σ) 12 = 1 2 . (3) The skew-model (ST), with a Skew- (Sahu et al, 2003) density for ( 1 , 2 ), with degrees of freedom. (4) The Bivariate Normal (BVN) model, with a symmetric N 2 (0, Σ) density for ( 1 , 2 ), ignoring the skewness.…”

“…This paper addresses a number of major challenges and limitations of existing model-based regression analysis of multivariate skewed response data in biostatistical applications. First, many recently developed model-based approaches (Sahu et al, 2003;Genton, 2004;Bandyopadhyay et al, 2012) centers around the restricted parametric family pioneered by Azzalini and Dalla Valle (1996). Our less-restrictive semiparametric proposition with skewed nonparametric marginal error densities will remain attractive under concerns about the validity and consequence of the parametric assumptions on the error density.…”

Linear mixed models for skew‐normal/independent bivariate responses with an application to periodontal disease

“…We assume an SNI distribution for the random effects and a symmetric normal/independent (NI) distribution [21] for the within-subject errors, so that the skew-normal/independent linear mixed model (SNILMM) is defined. The multivariate SNI distributions used in this paper are developed primarily from the multivariate SN density proposed in [14] for Bayesian regression problems and is different from the multivariate SNI densities developed in [22] motivated from the SN version proposed in [15]. However, the differences are only due to the various parameterizations [23] used and an unification of all SN (elliptical) variants is presented in [24].…”