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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01 Jan 2017
TL;DR: The role of exponential family of distributions has become increasingly important for developing generalized linear models that can be extended to repeated measures data as well as discussed by the authors, and several important properties are also shown.
Abstract: The role of exponential family of distributions has become increasingly important for developing generalized linear models that can be extended to repeated measures data as well. Chapter 3 provides an introduction to the exponential family of distribution with several examples. Some important properties are also shown.
2 citations
TL;DR: In this article, the compensator process of the indicator function corresponding to the failure times of two s-dependent components, having a bivariate exponential distribution, is calculated, and the compensators are shown to be independent of the failure time.
Abstract: The compensator process of the indicator function corresponding to the failure times of two s-dependent components, having a bivariate exponential distribution, is explicitly calculated.
2 citations
TL;DR: In this article, it was shown that the relation E(Q | X 1 + X + Xn) = 0 holds if and only if q(α 1(θ), θ, θ) is the jth moment of the natural exponential family generated by F.
Abstract: A polynomial Q = Q(X1, …, Xn) of degree m in independent identically distributed random variables with distribution function F is an unbiased estimator of a functional q(α1(F), …, αm(F)), where q(u1, …, um) is a polynomial in u1, …, um and αj(F) is the jth moment of F (assuming the necessary moment of F exists). It is shown that the relation E(Q | X1 + … + Xn) = 0 holds if and only if q(α1(θ), …, αm(θ)) ≡ 0, where αj(θ) is the jth moment of the natural exponential family generated by F. This result, based on the fact that X1 + … + Xn is a complete sufficient statistic for a parameter θ in a sample from a natural exponential family of distributions Fθ(x) = ∫−∞xeθu−k(θ)dF(u), explains why the distributions appearing as solutions of regression problems are the same as solutions of problems for natural exponential families though, at the first glance, the latter seem unrelated to the former.
2 citations
TL;DR: In this article, the authors describe complete minimal families of exponential functions and Riesz bases of functions which admit orthogonalizers of a special type, and formulate a condition for the uniqueness of the orthogonality.
Abstract: We describe complete minimal families of exponential functions and Riesz bases of exponential functions which admit orthogonalizers of a special type. We obtain a complete description of all orthogonalizers of the class of complete minimal family of exponential functions under consideration and formulate a simple condition which guarantees the uniqueness of the orthogonalizer.
2 citations
TL;DR: In this paper, a semi-diagonality for a Bhattacharyya matrix is introduced to give a characterization of the Letac-Mora class of real natural exponential families having cubic variance function.
2 citations