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Distribution of a Ratio of Correlated Gamma Random Variables

Ru-Ying Lee, Burt S. Holland and John A. Flueck
SIAM Journal on Applied Mathematics
Vol. 36, No. 2 (Apr., 1979), pp. 304-320
Stable URL: http://www.jstor.org/stable/2100936
Page Count: 17
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Distribution of a Ratio of Correlated Gamma Random Variables
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Abstract

Relatively little is known about the distributional properties of ratios of correlated random variables. This paper presents exact distributional results for a ratio, R, of correlated gamma random variables under the Cherian-David-Fix bivariate gamma distributional structure. The p.d.f. of R is expressed in terms of hypergeometric functions, and the parameters of the ratio distribution characterize the shape of the p.d.f. Some moments of R may be infinite and the criterion for a finite mth moment is given. Graphs of the functional forms are presented in addition to a catalogue of the parametric conditions which delineate the family members. It is shown that naive application of the central limit theorem to the distribution of R is potentially dangerous, particularly for small sample sizes. However, the large sample distribution of R approaches normality as all parameters increase.

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