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.

This paper proposes a general class of multivariate skew-elliptical distributions. We extend earlier results on the so-called multivariate skew-normal distribution. This family of distributions contains the multivariate normal, Student's t, exponential power, and Pearson type II, but with an extra parameter to regulate skewness. We also obtain the moment generating functions and study some distributional properties. Several examples are provided. Academic PressAMS 1991 subject classifications: 60E05; 62H10.

The covariance matrix formula of the multivariate skew-t distribution for non-null δ was wrong as given on page 137, vol. 31 (2003). The correct expression is:As a result of this correction we now state that in the case of the multivariate skew-t distribution with non-null δ the components will always be correlated. Nothing else changes in the paper since hierarchical Bayesian modelling and its MCMC based computation in Section 5 are performed using the distributional specification and not using their moments.The moment generating function given as Lemma A.3 in the Appendix (page 147) does imply the correct covariance matrix as shown below. The moment generating function has been stated as:where G(w) denotes the cumulative distribution function of (ν/2, ν/2). Note that when W ∼To calculate the covariance we want to evaluateStraightforward calculation yields the following:

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.