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, the estimation of the probability that a single observation will lie in a certain Borel set, when this probability is regarded as a function of the parameters 6 and/or p, is studied.
Abstract: 1. Summary. Unbiased estimators for functions of a location parameter 0 and a scale parameter p are expressed as uniknown functions in integral equations of convolution type, and are then obtained by integral transform methods. An outline of the paper is contained in Section 3. The main results consist in the application of various derived expressions to the exponential distribution with parameters 6 and p, the gamma and Weibull distributions with parameter p, and to general distributions with truncation parameter 0. In the latter case, a simple formula is given for a minimum variance unbiased estimator of any absolutely continuous function of 0; this extends slightly a result of Davis [3j concerning distributions of exponential type. Throughout the paper particular attention is paid to the estimation of the probability that a single observation will lie in a certain Borel set, when this probability is regarded as a function of the parameters 6 and/or p. Extensions to sample points of m observations and Borel sets in m-space are in most cases immediate.
116 citations
TL;DR: The generalized hyperexponential (GH) probability distribution as discussed by the authors is a generalization of the exponential probability distribution with the objective of preserving the computationally attractive feature of "memorylessness" possessed by the exponential distribution.
Abstract: This paper examines in detail the class of generalized hyperexponential (GH) probability distribution functions. The family is compared to and contrasted with similar popular classes of distributions used in stochastic modeling. Each of these families arises from a desire to preserve the computationally attractive feature of “memorylessness” possessed by the exponential probability distribution while extending the representations to a broader class in order to approximate an arbitrary probability distribution function. Thus the simple structure and attractive properties of the GH probability distribution functions are presented with a view toward facilitating the mathematical operations which frequently occur in practice.
114 citations
TL;DR: In this paper, the authors proposed the use of quantile functions to define the W function of the T-X family of distributions, which provides a new method of generating univariate distributions.
Abstract: The cumulative distribution function (CDF) of the T-X family is given by R{W(F(x))}, where R is the CDF of a random variable T, F is the CDF of X and W is an increasing function defined on [0, 1] having the support of T as its range. This family provides a new method of generating univariate distributions. Different choices of the R, F and W functions naturally lead to different families of distributions. This paper proposes the use of quantile functions to define the W function. Some general properties of this T-X system of distributions are studied. It is shown that several existing methods of generating univariate continuous distributions can be derived using this T-X system. Three new distributions of the T-X family are derived, namely, the normal-Weibull based on the quantile of Cauchy distribution, normal-Weibull based on the quantile of logistic distribution, and Weibull-uniform based on the quantile of log-logistic distribution. Two real data sets are applied to illustrate the flexibility of the distributions.
113 citations
IBM1
TL;DR: In this article, the authors introduced and studied new probability distributions named "digamma" and "trigamma", which are obtained as limits of the zero-truncated Type B3 generalized hypergeometric distributions (inverse Polya-Eggenberger or negative binomial beta distributions) by compounding the logarithmic series distributions.
Abstract: In this paper we introduce and study new probability distributions named “digamma” and “trigamma” defined on the set of all positive integers. They are obtained as limits of the zero-truncated Type B3 generalized hypergeometric distributions (inverse Polya-Eggenberger or negative binomial beta distributions), and also by compounding the logarithmic series distributions.
111 citations
01 Aug 2006
TL;DR: In this article, two new fading distributions, α-η-μ and α-κ-μ distributions, are presented, which include the α-μ, Nakagami-m, Weibull, Rice, Rayleigh, Exponential, and the One-Sided Gaussian distributions as special cases.
Abstract: In this paper two new fading distributions, the α-η-μ Distribution and α-κ-μ Distribution, are presented. The α-η-μ distribution includes the α-μ, Nakagami-m, Nakagami-q, Weibull, Hoyt, Rayleigh, Exponential, and the One-Sided Gaussian distributions as special cases. The α-κ-μdistribution includes the α-μ, Nakagami-m, Weibull, Rice, Rayleigh, Exponential, and the One-Sided Gaussian distributions as special cases. Furthermore, it proposes estimators for the involved parameters and uses field measurements to validate the distributions.
111 citations