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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TL;DR: In this paper, the authors derived third-order asymptotic expansions for the non null distribution functions of four classic statistics under a sequence of local alternatives in one-parameter exponential family models.
Abstract: In this article we derive third-order asymptotic expansions for the non null distribution functions of four classic statistics under a sequence of local alternatives in one-parameter exponential family models. Our results are quite general and cover a wide range of important distributions.
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03 Jul 2021
TL;DR: In this paper, it was shown that a natural exponential family F is reproducible iff it possesses a variance function which is a power function of its mean, i.i.d.
Abstract: Let F=Fθ:θ∈Θ⊂R be a family of probability distributions indexed by a parameter θ and let X1,⋯,Xn be i.i.d. r.v.’s with L(X1)=Fθ∈F. Then, F is said to be reproducible if for all θ∈Θ and n∈N, there exists a sequence (αn)n≥1 and a mapping gn:Θ→Θ,θ⟼gn(θ) such that L(αn∑i=1nXi)=Fgn(θ)∈F. In this paper, we prove that a natural exponential family F is reproducible iff it possesses a variance function which is a power function of its mean. Such a result generalizes that of Bar-Lev and Enis (1986, The Annals of Statistics) who proved a similar but partial statement under the assumption that F is steep as and under rather restricted constraints on the forms of αn and gn(θ). We show that such restrictions are not required. In addition, we examine various aspects of reproducibility, both theoretically and practically, and discuss the relationship between reproducibility, convolution and infinite divisibility. We suggest new avenues for characterizing other classes of families of distributions with respect to their reproducibility and convolution properties .
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TL;DR: In this article, Bar-Lev, Bshouty and Enis showed that a polynomial with a simple root at 0 and a complex root with positive imaginary part is the variance function of some one-parameter natural exponential family (NEF) if and only if the real part of the complex root is not positive.
01 Jan 2016
TL;DR: A uniformly-sampled-autoregressive-moving-average (USAM) model for a second-order linear stochastic system is derived, it is shown that exponential smoothing is a limiting case of the USAM model, and the optimal value of the exponential-smoothing parameter is discussed.