Topic
Natural exponential family
About: Natural exponential family is a research topic. Over the lifetime, 1973 publications have been published within this topic receiving 60189 citations. The topic is also known as: NEF.
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90 citations
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01 Jun 1978
TL;DR: In this article, the poisson process was used for characterisation of truncated distributions based on properties of order statistics, and multivariate exponential distributions for poisson processes were used.
Abstract: Preliminaries and basic results.- Characterizations based on truncated distributions.- Characterizations by properties of order statistics.- Characterizations of the poisson process.- Characterizations of multivariate exponential distributions.- Miscellaneous results.
90 citations
TL;DR: In this article, the Wishart distribution on a symmetric cone C is characterized by extending the Olkin-Rubin proof by using three modern ideas: (i) avoid artificial coordinates in differential geometry; (ii) the variance function of a natural exponential family F characterizes F; and (iii) symmetric matrices are a particular example of a Euclidean simple Jordan algebra.
Abstract: We characterize the Wishart distributions on a symmetric cone C. If $C = (0, +\infty)$, this has been done by Lukacs in 1955. If C is the cone of positive definite symmetric matrices, this has been done by Olkin and Rubin in 1962. We both shorten and extend the Olkin-Rubin proof (sometimes obscure) by using three modern ideas: (i) try to avoid artificial coordinates in differential geometry; (ii) the variance function of a natural exponential family F characterizes F; (iii) symmetric matrices are a particular example of a Euclidean simple Jordan algebra.
89 citations
TL;DR: In this paper, the authors examined predictive distributions, concentrating on measuring their fit to the true distribution by average Kullback-Leibler divergence, and the notion of an "averaged bootstrap" predictive distribution was introduced.
Abstract: SUMMARY The paper examines predictive distributions, concentrating on measuring their fit to the true distribution by average Kullback-Leibler divergence. The notion of an 'averaged bootstrap' predictive distribution is introduced. This predictive distribution is shown to be asymptotically superior to the estimative distribution, in terms of average KullbackLeibler divergence, when the true distribution is in a natural exponential family. Smallsample results are presented for the Poisson and binomial distributions which suggest that the bootstrap distribution performs well in these cases.
89 citations
TL;DR: In this article, the generalized beta of the second kind (GB2) family of distributions is used for modeling insurance loss processes and the results suggest that seemingly slight differences in modeling the tails can result in large differences in premiums and quantiles for the distribution of total insurance losses.
Abstract: This paper investigates the use of a four parameter family of probability distributions, the generalized beta of the second kind (GB2), for modeling insurance loss processes. The GB2 family includes many commonly used distributions such as the lognormal, gamma and Weibull. The GB2 also includes the Burr and generalized gamma distributions. Members of this family and their inverse distributions have significant potential for improving the distributional fit in many applications involving thin or heavy-tailed distributions. Members of the GB2 family can be generated as mixtures of well-known distributions and provide a model for heterogeneity in claims distributions. Examples are presented which consider models of the distribution of individual and of aggregate losses. The results suggest that seemingly slight differences in modeling the tails can result in large differences in reinsurance premiums and quantiles for the distribution of total insurance losses.
88 citations