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 article, it was shown that the distribution of random variables with common distribution is an element of a linear one-parameter exponential family of Poisson distributions, for all θ∈Θ and n=1, 2,... if and only ifF is a family of scale transformed Poisson distribution.
Abstract: LetX
1, ...,X
n
be i.i.d. random variables with common distribution an element of a linear one-parameter exponential family
indexed by a natural parameter θ. It is proved that the distribution of
$$\sum\limits_{i = 1}^n {X_i } $$
is an element ofF, for all θ∈Θ andn=1, 2, ... if and only ifF is a family of scale transformed Poisson distributions.
3 citations
3 citations
TL;DR: In this article, the authors derived closed-form expressions for the related variance functions in terms of the exponential tilting invariants and particular special functions for specific subclasses of Poisson-Tweedie mixtures and identified the Levy measure for all the members of this threeparameter family.
Abstract: Poisson-Tweedie mixtures are the Poisson mixtures for which the mixing measure is generated by those members of the family of Tweedie distributions whose support is non-negative. This class of non-negative integer-valued distributions is comprised of Neyman type A, back-shifted negative binomial, compound Poisson-negative binomial, discrete stable and exponentially tilted discrete stable laws. For a specific value of the “power” parameter associated with the corresponding Tweedie distributions, such mixtures comprise an additive exponential dispersion model. We derive closed-form expressions for the related variance functions in terms of the exponential tilting invariants and particular special functions. We compare specific Poisson-Tweedie models with the corresponding Hinde-Demetrio exponential dispersion models which possess a comparable unit variance function. We construct numerous local approximations for specific subclasses of Poisson-Tweedie mixtures and identify Levy measure for all the members of this three-parameter family.
3 citations
TL;DR: Some new characterizations of probability distributions based on properties of record values shifted by independent exponential variables are obtained and explicit expressions for the inverse functions of the corresponding variables are presented.
Abstract: Some new characterizations of probability distributions based on properties of record values shifted by independent exponential variables are obtained. Explicit expressions for the inverse functions of the corresponding variables are presented. Among others some characterizations of exponential distributions and limiting distributions of maximal and minimal extreme values are given.
3 citations
14 Sep 2011
TL;DR: In this article, the authors proposed several near-exact distributions for the likelihood ratio test statistic, which provide very accurate but yet very manageable approximations to the exact distribution, much adequate for practical purposes.
Abstract: The distribution of the likelihood ratio test statistic to test the equality of several one or two‐parameter Exponential distributions either for censored or non‐censored samples has been studied by several authors. These statistics are of interest in many areas, namely in reliability and lifetime studies. We propose several near‐exact distributions for these statistics, which provide very accurate but yet very manageable approximations to the exact distribution, much adequate for practical purposes.
3 citations