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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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01 Jan 1998TL;DR: In this paper, the background is modelled as independent observations from an exponential family of distributions with a known 'null' value of the natural parameter, while the signal is given by independent observations with a different value on a particular subregion of the spatial domain.
Abstract: This paper is concerned with statistics that scan a multidimensional spatial region to detect a signal against a noisy background. The background is modelled as independent observations from an exponential family of distributions with a known 'null' value of the natural parameter, while the signal is given by independent observations from the same exponential family, but with a different value of the parameter on a particular subregion of the spatial domain. The main result is an extension to multidimensional time of the method of Pollak and Yakir, which relies on a change of measure motivated by change-point analysis, to evaluate approximately the null distribution of the likelihood ratio statistic. Both large-deviation and Poisson approximations are obtained.
82 citations
TL;DR: In this paper, the authors study probability distributions of convergent random series of a special structure, called perpetuities, and prove that such distributions are of pure type: degenerate, absolutely continuous, or continuously singular.
Abstract: We study probability distributions of convergent random series of a special structure, called perpetuities. By giving a new argument, we prove that such distributions are of pure type: degenerate, absolutely continuous, or continuously singular. We further provide necessary and sufficient criteria for the finiteness of p-moments, p>0, as well as exponential moments. In particular, a formula for the abscissa of convergence of the moment generating function is provided. The results are illustrated with a number of examples at the end of the article.
81 citations
TL;DR: It is shown that additional information about the distributions is more important for the interarrival-time than for the service time, and one kind of shape constraint — assuming that the distribution is a mixture of exponential distributions — can be examined analytically.
Abstract: To evaluate queueing approximations based on a few parameters (e.g., the first two moments) of the interarrival-time and service-time distributions, we examine the set of all possible values of the mean queue length given this partial information. In general, the range of possible values given such partial information can be large, but if in addition shape constraints are imposed on the distributions, then the range can be significantly reduced. The effect of shape constraints on the interarrival-time distribution in a GI/M/1 queue was investigated in Part II (see "On Approximations for Queues, II: Shape Constraints," this issue) by restricting attention to discrete probability distributions with probability on a fixed finite set of points and then solving nonlinear programs. In this paper we show how one kind of shape constraint — assuming that the distribution is a mixture of exponential distributions — can be examined analytically. By considering GI/G/1 queues in which both the interarrival-time and service-time distributions are mixtures of exponential distributions with specified first two moments, we show that additional information about the distributions is more important for the interarrivai time than for the service time.
81 citations
TL;DR: In this paper, the Topp-Leone family of distributions is proposed and general expression for density and distribution function of the new family is given. But the proposed family is not suitable for the case of large numbers of nodes.
Abstract: In this paper we have proposed a new family of distributions; the Topp–Leone family of distributions. We have given general expression for density and distribution function of the new family. Expression for moments and hazard rate has also been given. We have also given an example of the proposed family.
81 citations
Abstract: The exponentiated exponential distribution, a most attractive generalization of the exponential distribution, introduced by Gupta and Kundu (Aust. N. Z. J. Stat. 41:173–188, 1999) has received widespread attention. It appears, however, that many mathematical properties of this distribution have not been known or have not been known in simpler/general forms. In this paper, we provide a comprehensive survey of the mathematical properties. We derive expressions for the moment generating function, characteristic function, cumulant generating function, the nth moment, the first four moments, variance, skewness, kurtosis, the nth conditional moment, the first four cumulants, mean deviation about the mean, mean deviation about the median, Bonferroni curve, Lorenz curve, Bonferroni concentration index, Gini concentration index, Renyi entropy, Shannon entropy, cumulative residual entropy, Song’s measure, moments of order statistics, L moments, asymptotic distribution of the extreme order statistics, reliability, distribution of the sum of exponentiated exponential random variables, distribution of the product of exponentiated exponential random variables and the distribution of the ratio of exponentiated exponential random variables. We also discuss estimation by the method of maximum likelihood, including the case of censoring, and provide simpler expressions for the Fisher information matrix than those given by Gupta and Kundu. It is expected that this paper could serve as a source of reference for the exponentiated exponential distribution and encourage further research.
81 citations