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Four Applications of a Bivariate Pareto Distribution
Author(s) -
Hutchinson T. P.
Publication year - 1979
Publication title -
biometrical journal
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.108
H-Index - 63
eISSN - 1521-4036
pISSN - 0323-3847
DOI - 10.1002/bimj.4710210605
Subject(s) - bivariate analysis , mathematics , gamma distribution , scale parameter , statistics , exponential family , hypergeometric distribution , shape parameter , exponential function , exponential distribution , contingency table , distribution (mathematics) , scale (ratio) , pareto distribution , generalized pareto distribution , natural exponential family , population , pareto principle , order statistic , mathematical analysis , extreme value theory , physics , demography , quantum mechanics , sociology
The following model, of “latent structure” type, is considered: in each subpopulation, X and Y are random variables drawn independently from the same exponential distribution, and the parameter of the exponential distribution varies between subpopulations with a Gamma density. Over the whole population, X and Y are then positively correlated, and jointly have a bivariate P ARETO distribution. Four examples show how this distribution is useful in analysing ordered contingency tables in which the two dimensions can be regarded as alternative measures of the same thing: the injuries to the two drivers in a road accident, or the severity of a lesion present in a patient as assessed by two physicians, for instance. Two extensions are considered: (a) allowing X and Y to have Gamma distributions, with each subpopulation having the same shape parameter but different scale parameters; (b) allowing the scale parameter for Y to be correlated with the scale parameter for X , rather than being identical to it. A new bivariate distribution with three shape parameters is derived, expressed in terms of a generalised hypergeometric function.