Marginal Replacement in Multivariate Densities, with Application to Skewing Spherically Symmetric Distributions

We consider a simple general construct, that of marginal replacement in a multivariate distribution, that provides, in particular, an interesting way of producing distributions that have a skew marginal from spherically symmetric starting points. Particular examples include a new multivariate beta distribution and a new multivariate t/skew t distribution, along with Azzalini and colleaguesʹ multivariate skew normal distribution.