On Bivariate Transformation of Scale Distributions

Elsewhere, I have promoted the notion of (univariate continuous) “transformation of scale” (ToS) distributions which have densities of the form 2g(W(x)) where g is a symmetric distribution and W is a transformation function with a special property. Here, I develop particular bivariate (readily multivariate) ToS distributions. Univariate ToS distributions have a transformation of random variable relationship with Azzalini-type skew-symmetric distributions; the bivariate ToS distribution here arises from marginal variable transformation of a non-standard form of bivariate skewsymmetric distribution. Some examples are given, as are a few basic properties — unimodality, a covariance property, random variate generation — and connections with a particular bivariate inverse Gaussian distribution are pointed out.