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.

In this paper many convergence issues concerning the implementation of the Gibbs sampler are investigated. Exact computable rates of convergence for Gaussian target distributions are obtained. Different random and non-random updating strategies and blocking combinations are compared using the rates. The effect of dimensionality and correlation structure on the convergence rates are studied. Some examples are considered to demonstrate the results. For a Gaussian image analysis problem several updating strategies are described and compared. For problems in Bayesian linear models several possible parameterizations are analysed in terms of their convergence rates characterizing the optimal choice.

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:

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