Characterizations of multivariate exponential distributions

Abstract

In summary, it should be emphasized that the development of the theory of multivariate exponential distributions has proceeded in the last decade mainly along the lines of constructing multivariate models which are based on a suitably chosen specific physical model rather than on what would seem to be a more general approach—that of extending the characterizing properties of univariate exponential distributions. Characterization theorems—with occasional exceptions—are formulated and proved after a model has been properly described and analyzed and do not as yet serve, in general, as a stimulus and guiding light for new multivariate distributions. Moreover, again with few exceptions, there are no satisfactory characterization theorems for the several very ingenious multivariate distributions which were developed recently. This applies for example, to the generalization of the Freund-Weinman multivariate exponential distribution proposed by Block (1977) and that of Friday and Patil (1977). It is hoped that the unification and clarification of characterizations of univariate exponential distributions presented in the preceding chapters will aid researchers to extend these characterizations to the multivariate case resulting in sound models and variants of multivariate exponential distributions which are sufficiently wide and at the same time well-defined and the realm of their applicability can be conveniently checked.

To the best of the authors' knowledge, apart from a rather well developed area of multivariate extreme-value distributions (see, Galambos, 1978, Chapter 5 for a comprehensive and up-to-date treatment of this topic), there are as yet very few characterization theorems available for multivariate distributions formed by monotone transforms of multivariate exponential distributions except for direct exponential transforms on the marginals leading to multivariate Pareto distributions (Mardia, 1962) and power transforms leading to Weibull distributions mentioned above (Moeschberger, 1974)). We are aware of a number of as yet unpublished works investigating and characterizing multivariate Weibull distributions (with not necessarily Weibull marginals) based—for example—on the property of possessing Weibull minima after arbitrary scaling.