This article develops a new class of distributions by introducing skewness in the multivariate elliptically symmetric distributions. The class is obtained by using transformation and conditioning. The class contains many standard families including the multivariate skew normal and t distributions. Analytical forms of the densities are obtained and distributional properties are studied. These developments are followed by practical examples in Bayesian regression models. Results on the existence of the posterior distributions and moments under improper priors for the regression coefficients are obtained. The methods are illustrated using practical examples. <br/
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