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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, a useful subfamily of the exponential family has been considered and the ML estimation based on upper record values has been calculated for the parameter, Cumulative Density Function, and Probability Density Functions of the family.
Abstract: In this paper a useful subfamily of the exponential family has been considered. The ML estimation based on upper record values has been calculated for the parameter, Cumulative Density Function, and Probability Density Function of the family. Also, the relations between MLE based on record values and a random sample has been discussed. Additionally, some properties of these estimators have been investigated. Finally, it has been proven that these estimators have some useful properties for samples with large size
TL;DR: In this paper , the authors proposed a new class of probability distributions as an extended version of the exponential hyper-Poisson distribution and Weibull Poisson distribution, which they termed as exponential hyper poisson distribution.
Abstract: Here we propose a new class of probability distributions as an extended version of the exponential hyper-Poisson distribution and Weibull Poisson distribution. We investigate several important aspects of the distribution through deriving expressions for its probability density function (pdf), cumulative distribution function, survival function, failure rate function, pdf of the order statistics, r-th raw moments, etc. The method of maximum likelihood estimation procedures along with EM algorithm is discussed for estimating the parameters of the distribution and a test procedure is suggested for testing the significance of the additional parameters of the proposed model. The use of the proposed distribution is illustrated through real-life data sets. Further, a brief simulation study is carried out for evaluating the performance of the estimators obtained for the parameters of the distribution.
01 Mar 2023
TL;DR: In this article , the authors develop and compare e-variables for testing whether $k$ samples of data are drawn from the same distribution, the alternative being that they come from different elements of an exponential family.
Abstract: We develop and compare e-variables for testing whether $k$ samples of data are drawn from the same distribution, the alternative being that they come from different elements of an exponential family. We consider the GRO (growth-rate optimal) e-variables for (1) a 'small' null inside the same exponential family, and (2) a 'large' nonparametric null, as well as (3) an e-variable arrived at by conditioning on the sum of the sufficient statistics. (2) and (3) are efficiently computable, and extend ideas from Turner et al. [2021] and Wald [1947] respectively from Bernoulli to general exponential families. We provide theoretical and simulation-based comparisons of these e-variables in terms of their logarithmic growth rate, and find that for small effects all four e-variables behave surprisingly similarly; for the Gaussian location and Poisson families, e-variables (1) and (3) coincide; for Bernoulli, (1) and (2) coincide; but in general, whether (2) or (3) grows faster against the small null is family-dependent. We furthermore discuss algorithms for numerically approximating (1).
16 Apr 2023