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

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Abstract

The classical bivariate F distribution arises from ratios of chi-squared random variables with common denominators. A consequent disadvantage is that its univariate F 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 F distribution, with marginals on arbitrary degrees of freedom, that results. Transformations linking F, beta and skew t distributions are then applied componentwise to produce bivariate beta and skew t distributions which also afford marginal (beta and skew t) 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.

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Authors and Affiliations

  1. Department of Mathematics, College of Science, Mansoura University, Mansoura, 35516, Egypt

    A. H. El-Bassiouny

  2. Department of Statistics, The Open University, Walton Hall, Milton Keynes, MK7 6AA, UK

    M. C. Jones

Authors
  1. A. H. El-Bassiouny
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  2. M. C. Jones
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Correspondence to M. C. Jones.

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El-Bassiouny, A.H., Jones, M.C. A bivariate F distribution with marginals on arbitrary numerator and denominator degrees of freedom, and related bivariate beta and t distributions. Stat Methods Appl 18, 465–481 (2009). https://doi.org/10.1007/s10260-008-0103-y

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  • DOI: https://doi.org/10.1007/s10260-008-0103-y

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