A bivariate F distribution with marginals on arbitrary numerator and denominator degrees of freedom, and related bivariate beta and t distributions

摘要

The classical bivariate distribution arises from ratios of chi-squared random variables with common denominators. A consequent disadvantage is that its univariate marginal distributions have one degree of freedom parameter in common. In this paper, we add a further independent chi-squared random variable to the denominator of one of the ratios and explore the extended bivariate distribution, with marginals on arbitrary degrees of freedom, that results. Transformations linking , beta and skew distributions are then applied componentwise to produce bivariate beta and skew distributions which also afford marginal (beta and skew ) distributions with arbitrary parameter values. We explore a variety of properties of these distributions and give an example of a potential application of the bivariate beta distribution in Bayesian analysis.