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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 article, the likelihood equations for the estimation of the parameters of an exponential family from a Type I censored sample are derived and an interpretation which suggests a particular iterative method of solution is given.
Abstract: SUMMARY The likelihood equations are derived for the estimation of the parameters of an exponential family from a Type I censored sample and are shown to have an interpretation which suggests a particular iterative method of solution. Some results relevant to the convergence properties of this method are given and the asymptotic variance-covariance matrix is derived. The method is illustrated by an example in which it is compared with an alternative method in the case of estimation for a doubly censored normal distribution.
30 citations
TL;DR: Yadav et al. as discussed by the authors analyzed the earthquake interoccurrence times of northeast India and its vicinity from eleven probability distributions, namely exponential, Frechet, gamma, generalized exponential, inverse Gaussian, Levy, lognormal, Maxwell, Pareto, Rayleigh, and Weibull distributions.
Abstract: This study analyzes earthquake interoccurrence times of northeast India and its vicinity from eleven probability distributions, namely exponential, Frechet, gamma, generalized exponential, inverse Gaussian, Levy, lognormal, Maxwell, Pareto, Rayleigh, and Weibull distributions. Parameters of these distributions are estimated from the method of maximum likelihood estimation, and their respective asymptotic variances as well as confidence bounds are calculated using Fisher information matrices. Three model selection criteria namely the Chi-square criterion, the maximum likelihood criterion, and the Kolmogorov–Smirnov minimum distance criterion are used to compare model suitability for the present earthquake catalog (Yadav et al. in Pure Appl Geophys 167:1331–1342, 2010). It is observed that gamma, generalized exponential, and Weibull distributions provide the best fitting, while exponential, Frechet, inverse Gaussian, and lognormal distributions provide intermediate fitting, and the rest, namely Levy, Maxwell Pareto, and Rayleigh distributions fit poorly to the present data. The conditional probabilities for a future earthquake and related conditional probability curves are presented towards the end of this article.
30 citations
TL;DR: In this article, the Bhattacharyya bounds are considered for the unbiased estimation of a parametric function when the sampling distribution is a member of an exponential family of distributions.
Abstract: SUMMARY Bhattacharyya bounds are considered for the unbiased estimation of a parametric function when the sampling distribution is a member of an exponential family of distributions. It is shown that the Bhattacharyya bounds converge to the variance of the best unbiased estimator. The application of this result to variance determination is demonstrated with examples from the negative binomial distribution and from the exponential distribution in a reliability theory context.
30 citations
TL;DR: In this paper, conditions under which quadratic and polynomial exponential models can be generated as mixtures of linear exponential models are derived for continuous sample spaces, but less restrictive in the discrete case.
Abstract: SUMMARY Conditions are derived under which quadratic and polynomial exponential models can be generated as mixtures of linear exponential models. The conditions are highly restrictive for continuous sample spaces, but less restrictive in the discrete case. Some properties of binary quadratic exponential models are explored with a view towards finding models that have properties suitable for epidemiological applications.
29 citations
TL;DR: In this paper, the decay of the maximum of the successive partial sums of independent and identically distributed random variables is asymptotically equal to c. Under conditions typical for complete exponential convergence, G(x)-G(x) is the limiting distribution function.
Abstract: Let Gn(x) be the distribution function of the maximum of the successive partial sums of independent and identically distributed random variables and G(x) its limiting distribution function. Under conditions, typical for complete exponential convergence, the decay of Gn(x) - G(x) is asymptotically equal to c.H(x)n- Qrn as n-* oo where c and v are known constants and H(x) is a function solely depending on x. COMPLETE EXPONENTIAL CONVERGENCE; MAXIMUM OF SUMS OF INDEPENDENT AND IDENTICALLY DISTRIBUTED RANDOM VARIABLES; SUPREMUM OF PROCESSES WITH STATIONARY INDEPENDENT INCREMENTS; WAITING TIME DISTRIBUTION; EXPONENTIAL RATE OF DECAY
29 citations