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, the authors provide the application of such tools suitable for deriving the exact cumulative distribution functions and density functions of extreme values, products, and ratios in symmetric populations.
Abstract: The exact distributions of many functions of random vectors are derived in the literature mainly for the case of a Gaussian vector distribution or under the assumption that the vector follows a spherical or an elliptically contoured distribution. Numerous standard statistical applications are given for these cases. Deriving analogous results, if the sample distribution comes from a large family of probability laws, needs to make use of new analytical tools from the area of exact distribution theory. The present paper provides the application of such tools suitable for deriving the exact cumulative distribution functions and density functions of extreme values, products, and ratios in $$l_{2,p}$$
-symmetrically distributed populations. Accompanying simulation studies are presented in cases of power-exponentially distributed populations and for different sample sizes. As an application, well-known results on the increasing failure rate properties of extremes from Gaussian samples are extended to $$p$$
-power exponential sample distributions.
7 citations
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TL;DR: In this paper, it was shown that it is possible to separate off, from the start, those factors which do not agree with the exponential, and to use the remaining factors as coefficients of a rigorously defined generalized hypergeometric series.
Abstract: The direct expansion of the individual terms in (2) and (3) involves repeated use of combinatorial algebra with restrictions on the indices; the resulting coefficients, apart from one factor, are of the form corresponding to the factorization of the exponential of a series of derivatives. This was first found by Sylvester [18] for the inversion of power series; the same feature is shown by the corresponding expressions obtained by Mellin [13], Birkeland [3] and Frame [8] in terms of generalized hypergeometric series, and by the expansions for the simultaneous inversion of several power series [5], [19]. This similarity to the exponential series suggests that it may be possible to separate off, from the start, those factors which do not agree with the exponential, and to use the remaining factors as coefficients of a rigorously
7 citations
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30 Jun 2002TL;DR: In this paper, generalized MLE and MLE in an information closure were proposed to solve the problem of non-existence of a maximum likelihood estimate (MLE) in an exponential family or subfamily.
Abstract: Remedies to non-existence of a maximum likelihood estimate (MLE) in an exponential family or subfamily are offered: generalized MLE, and MLE in an information closure.
7 citations
01 Jan 2009
TL;DR: In this article, the distribution of concomitant of order statistics for Bivariate Pseudo-Exponential distribution has been obtained, and the expression for moments has also been obtained.
Abstract: In this paper we have obtained the distribution of concomitant of order statistics for Bivariate Pseudo-Exponential distribution. The expression for moments has also been obtained. We have found that only the fractional moments of the distribution exist. The tables for survival function and hazard function has also been obtained.
7 citations
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TL;DR: The gamma integral distribution as mentioned in this paper is a subclass of the exponential integral distribution and is closely related to both the gamma and the beta functions, and the physical relevance of this new distribution is discussed.
7 citations