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.

Use of the weibull distribution in bayesian decision theory

Abstract: : The Weibull distribution is useful in analyzing the probabilistic lifetimes of many electrical components and complex systems. It is attractive for Bayesian decision-making because its right-hand cumulative function is of an exponential form which allows all life-test data to be easily incorporated into the decision-making process. Unfortunately no natural conjugate prior distribution exists if both the shape and scale parameters of the Weibull distribution are assumed to be unknown. If the shape parameter is assumed known, however, Bayesian analysis becomes little more difficult than for the exponential distribution, a special case of the Weibull. Prior, posterior, and preposterior analyses are given for the case of known shape parameter. In connection with preposterior analysis several sampling plans are discussed. The paper concludes with an analysis of a problem in optical sampling.

On two families of bivariate distributions with exponential marginals: Aggregation and capital allocation

Abstract: In this paper, we consider two main families of bivariate distributions with exponential marginals for a couple of random variables ( X 1 , X 2 ) . More specifically, we derive closed-form expressions for the distribution of the sum S = X 1 + X 2 , the TVaR of S and the contributions of each risk under the TVaR-based allocation rule. The first family considered is a subset of the class of bivariate combinations of exponentials, more precisely, bivariate combinations of exponentials with exponential marginals. We show that several well-known bivariate exponential distributions are special cases of this family. The second family we investigate is a subset of the class of bivariate mixed Erlang distributions, namely bivariate mixed Erlang distributions with exponential marginals. For this second class of distributions, we propose a method based on the compound geometric representation of the exponential distribution to construct bivariate mixed Erlang distributions with exponential marginals. Notably, we show that this method not only leads to Moran–Downton’s bivariate exponential distribution, but also to a generalization of this bivariate distribution. Moreover, we also propose a method to construct bivariate mixed Erlang distributions with exponential marginals from any absolutely continuous bivariate distributions with exponential marginals. Inspired from Lee and Lin (2012), we show that the resulting bivariate distribution approximates the initial bivariate distribution and we highlight the advantages of such an approximation.

Distribution of LRT for Testing the Equality of Several 2-Parameter Exponential Distributions

Abstract: Exact distribution of the likelihood-ratio-test (LRT) criterion for testing the equality of several 2-parameter exponential distributions is obtained for the first time in a computational closed form. This is then used to obtain the s-significance points of the LRT.

Polynomial Approximations for Bivariate Aggregate Claims Amount Probability Distributions

Abstract: A numerical method to compute bivariate probability distributions from their Laplace transforms is presented. The method consists in an orthogonal projection of the probability density function with respect to a probability measure that belongs to a Natural Exponential Family with Quadratic Variance Function (NEF-QVF). A particular link to Lancaster probabilities is highlighted. The procedure allows a quick and accurate calculation of probabilities of interest and does not require strong coding skills. Numerical illustrations and comparisons with other methods are provided. This work is motivated by actuarial applications. We aim at recovering the joint distribution of two aggregate claims amounts associated with two insurance policy portfolios that are closely related, and at computing survival functions for reinsurance losses in presence of two non-proportional reinsurance treaties.