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, it was shown that under mild regularity assumptions, ϕ and ϕ−1 are continuous with respect to the Levy metric in P and Euclidean metric in θ.
Abstract: Let P={P
θ: θ∈Θ} be an exponential family of probability distributions with θ the canonical parameter and consider the one to one mapping ϕ: P
θ→θ. It is shown that, under mild regularity assumptions, ϕ and ϕ
−1 are continuous with respect to the Levy metric in P and Euclidean metric in θ.
2 citations
TL;DR: In this article, a new class of α-modified binomial distribution has been proposed, and its distributional properties like probability generating function (pgf), moments, and their interrelations have been studied.
Abstract: A new class of α-modified binomial distribution has been proposed, and its distributional properties like probability generating function (pgf), moments, and their interrelations have been studied. Two new α-modified Poisson distributions and Poisson distribution have been obtained as limiting distributions. Modified binomial and Poisson distributions introduced by Berg and Jaworski (1988) have been seen as particular cases. Mixture distributions of α-modified binomial distributions have been derived. A new distributions called α-modified binomial distributions of type j, their moment properties, limiting distributions as α-modified Poisson distribution of type j, their different convolution properties, pgf, parameter estimators have been studied. Two more new distributions namely Doubly α-modified binomial distributions of type (i, j) and α-modified weighted generalized Poisson distributions of type (j − 1) have also been studied. Various α-modified binomial and Poisson distributions of Berg and Mutafchi...
2 citations
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TL;DR: In this article, the authors investigated the coverage probability of the parametric bootstrap percentile confidence intervals under and over the nominal level in one-sample and two-sample cases, respectively, and proposed a local asymptotic framework to study this subtle coverage behavior.
Abstract: The asymptotic behaviour of the commonly used bootstrap percentile confidence interval is investigated when the parameters are subject to linear inequality constraints. We concentrate on the important one- and two-sample problems with data generated from general parametric distributions in the natural exponential family. The focus of this paper is on quantifying the coverage probabilities of the parametric bootstrap percentile confidence intervals, in particular their limiting behaviour near boundaries. We propose a local asymptotic framework to study this subtle coverage behaviour. Under this framework, we discover that when the true parameters are on, or close to, the restriction boundary, the asymptotic coverage probabilities can always exceed the nominal level in the one-sample case; however, they can be, remarkably, both under and over the nominal level in the two-sample case. Using illustrative examples, we show that the results provide theoretical justification and guidance on applying the bootstrap percentile method to constrained inference problems.
2 citations
12 Oct 2009
TL;DR: In this paper, the authors extend some methods for obtaining the Wald score and likelihood ratio confidence intervals for the mean of the Takacs (generalized negative binomial) distribution, and also show that the likelihood ratio intervals are better than Wald interval, and the Wald interval has the poorest performance.
Abstract: In this paper, we extend some methods for obtaining the Wald score and likelihood ratio confidence intervals for the mean of the Takacs (generalized negative binomial) distribution. Also, we present these confidence intervals for the mean of the binomial and negative binomial distributions as the special cases of the Takacs distribution. The Takacs distribution is a member of the natural exponential family with cubic variance function (NEF-CVF). The coverage probabilities for these confidence intervals are estimated by means of a simulation study. The results show that the score and likelihood ratio intervals are better than Wald interval, and the Wald interval has the poorest performance.
2 citations
TL;DR: For the general multivariate exponential family of distributions, it was shown in this paper that Rao's test criterion based on efficient scores is algebraically identical to the general chi-squared criterion, based on maximum likelihood estimates.
2 citations