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, a simple generalisation of the use of the collection of order statistic distributions associated with symmetric distributions is presented, and an alternative derivation of this family of distributions is as the result of applying the inverse probability integral transformation to the beta distribution.
Abstract: Consider starting from a symmetric distributionF on ℜ and generating a family of distributions from it by employing two parameters whose role is to introduce skewness and to vary tail weight. The proposal in this paper is a simple generalisation of the use of the collection of order statistic distributions associated withF for this purpose; an alternative derivation of this family of distributions is as the result of applying the inverse probability integral transformation to the beta distribution. General properties of the proposed family of distributions are explored. It is argued that two particular special cases are especially attractive because they appear to provide the most tractable instances of families with power and exponential tails; these are the skewt distribution and the logF distribution, respectively. Limited experience with fitting the distributions to data in their four-parameter form, with location and scale parameters added, is described, and hopes for their incorporation into complex modelling situations expressed. Extensions to the multivariate case and to ℜ+ are discussed, and links are forged between the distributions underlying the skewt and logF distributions and Tadikamalla and Johnson'sLU family.
440 citations
TL;DR: An algorithm for approximating a long-tail distribution by a hyperexponential distribution (a finite mixture of exponentials) is developed, proving that, in prinicple, it is possible to approximate distributions from a large class, including the Pareto and Weibull distributions, arbitrarily closely by hyperexPonential distributions.
434 citations
TL;DR: A comprehensive treatment of the mathematical properties of the beta exponential distribution generated from the logit of a beta random variable is provided and an expression for the Fisher information matrix is provided.
414 citations
TL;DR: The Weibull distribution is the most important distribution for problems in reliability as discussed by the authors, and it has been studied extensively in the literature, including in the context of the wider Weibbull-G family of distributions.
Abstract: The Weibull distribution is the most important distribution for problems in reliability. We study some mathematical properties of the new wider Weibull-G family of distributions. Some special models in the new family are discussed. The properties derived hold to any distribution in this family. We obtain general explicit expressions for the quantile function, ordinary and incomplete moments, generating function and order statistics. We discuss the estimation of the model parameters by maximum likelihood and illustrate the potentiality of the extended family with two applications to real data.
391 citations
TL;DR: The discrete Weibull distribution is defined to correspond with the continuous time Weibell distribution in continuous time as mentioned in this paper, and a few properties of the discrete Webull distribution are discussed.
Abstract: The discrete Weibull distribution is defined to correspond with the Weibull distribution in continuous time. A few properties of the discrete Weibull distribution are discussed.
389 citations