Some properties of bivariate Gumbel Type A distributions with proportional hazard rates
We call a set of univariate distributions with the same mathematical form but different parameter values a family . Consider a bivariate Gumbel Type A survival distribution, S12(x1, x2), defined in (2.1), for which both marginal distributions, S1(x1), S2(x2), belong to the same family, of distributions. It is proved in this paper that subject to weak conditions, the crude hazard rates, h1(t) and h2(t), are proportional if and only if the marginal hazard rates, [lambda]1(t) and [lambda]2(t), are proportional (Theorem 1). It is also shown that the survival functions of W = min(X1, X2), and of the identified minimum, Wi = Xi, for Xi < Xj, j [not equal to] i, belong to the same family as do S1(x1), S2(x2) (Corollary 1). Counter-examples of distributions other than Gumbel Type A, for which these properties do not hold, are given. Some applications to the analysis of competing risks, using a family of Gompertz distributions, are discussed.
Year of publication: |
1978
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Authors: | Elandt-Johnson, Regina C. |
Published in: |
Journal of Multivariate Analysis. - Elsevier, ISSN 0047-259X. - Vol. 8.1978, 2, p. 244-254
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Publisher: |
Elsevier |
Keywords: | Survival functions marginal and crude hazard rates proportional hazard rates extreme value distributions Gumbel Type A distributions Gompertz distributions |
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