Construction of multivariate dispersion models

We consider methods for constructing multivariate dispersion models, illustrated by examples. Such methods are motivated by the need for good regression modelling of multivariate nonnormal correlated data, which requires multivariate distributions with a flexible correlation structure. We first review existing methods for constructing multivariate proper dispersion models, involving quadratic forms of deviance residuals in the style of the multivariate normal density, which we illustrate by a multivariate hyperbola distribution. We develop an extended convolution method for constructing multivariate exponential dispersion models, designed to create a fully flexible covariance structure, which we illustrate by two bivariate gamma distributions. We develop a similar technique for constructing multivariate extreme dispersion models for extremes and survival data, and introduce new bivariate logistic and Gumbel distributions.