On multivariate selection scale-mixtures of normal distributions

In this paper, we establish some results for multivariate selection scale-mixtures of normal distributions with arbitrary mixing variable. First, we discuss their stochastic representation in terms of multivariate selection normal distributions. Next, the conditional distributions as well as the first two moments of multivariate selection scale-mixtures of normal distributions are obtained when the selection set is an arbitrary rectangle in the q-dimensional Euclidean space of R. The unified skew-scale mixture of normal (SUSMN) distributions are subsequently discussed as a special case. As a subclass of SUSMN distributions, the class of unified skew-symmetric generalized hyperbolic (SUSGH) distributions are studied in detail. Finally, we show that our results can be used to obtain moments of L-statistics and of multivariate concomitants from multivariate scale-mixtures of normal distributions.