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.

An extension of the exponential distribution

Abstract: A generalization of the exponential distribution is presented. The generalization always has its mode at zero and yet allows for increasing, decreasing and constant hazard rates. It can be used as an alternative to the gamma, Weibull and exponentiated exponential distributions. A comprehensive account of the mathematical properties of the generalization is presented. A real data example is discussed to illustrate its applicability.

A generalized bivariate exponential distribution

Abstract: In a previous paper (Marshall and Olkin (1966)) the authors have derived a multivariate exponential distribution from points of view designed to indicate the applicability of the distribution. Two of these derivations are based on “shock models” and one is based on the requirement that residual life is independent of age. The practical importance of the univariate exponential distribution is partially due to the fact that it governs waiting times in a Poisson process. In this paper, the distribution of joint waiting times in a bivariate Poisson process is investigated. There are several ways to define “joint waiting time”. Some of these lead to the bivariate exponential distribution previously obtained by the authors, but others lead to a generalization of it. This generalized bivariate exponential distribution is also derived from shock models. The moment generating function and other properties of the distribution are investigated.

Approximating probability distributions to reduce storage requirements

Abstract: The measurement and/or storage of high order probability distributions implies exponential increases in equipment complexity. This paper considers the possibility of storing several of the lower order component distributions and using this partial information to form an approximation to the actual high order distribution. The approximation method is based on an information measure for the “closeness” of two distributions and on the criterion of maximum entropy. Approximations consisting of products of appropriate lower order distributions are proved to be optimum under suitably restricted conditions. Two such product approximations can be compared and the better one selected without any knowledge of the actual high order distribution other than that implied by the lower order distributions.