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: It is shown that queueing models in which failure and repair times are represented by GE distributions can be analyzed with the same complexity as if these distributions were exponential.
Abstract: Failures of machines have a significant effect on the behavior of manufacturing systems. As a result it is important to model this phenomenon. Many queueing models of manufacturing systems do incorporate the unreliability of the machines. Most models assume that the times to failure and the times to repair of each machine are exponentially distributed (or geometrically distributed in the case of discrete-time models). However, exponential distributions do not always accurately represent actual distributions encountered in real manufacturing systems. In this paper, we propose to model failure and repair time distributions bygeneralized exponential (GE) distributions (orgeneralized geometric distributions in the case of a discretetime model). The GE distribution can be used to approximate distributions with any coefficient of variation greater than one. The main contribution of the paper is to show that queueing models in which failure and repair times are represented by GE distributions can be analyzed with the same complexity as if these distributions were exponential. Indeed, we show that failures and repair times represented by GE distributions can (under certain assumptions) be equivalently represented by exponential distributions.
41 citations
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41 citations
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21 Nov 2005TL;DR: In this article, the authors present a model for estimating the good fit of a test set for exponential distribution with respect to a set of parameters using Monte-Carlo simulation and Bayes rules.
Abstract: Acronyms Notations List of Figures List of Tables GENERAL STATISTICAL THEORY Basic Concepts The System of Real Numbers Some Useful Algebraic Results Set Theory Introduction to Probability Theory Random Variables Joint Probability Distributions Moment Generating Function Order Statistics Characteristic Function Some Common Probability Distributions Discrete Distributions Continuous Distributions Some Limit Theorems Concepts of Statistical Inference Introduction Sufficiency and Completeness Methods of Estimation Methods of Evaluating Estimators Elements of Hypothesis Testing Set (or Interval) Estimation Elements of Decision Theory Introduction Optimality Criteria Loss Functions Admissible, Minimax and Bayes Rules Computational Aspects Preliminaries Numerical Integration Monte-Carlo Simulation Bootstrap Method of Resampling Testing and Interval Estimation Based on Computations EXPONENTIAL AND OTHER POSITIVELY SKEWED DISTRIBUTIONS WITH APPLICATIONS Exponential Distribution Preliminaries Characterization of Exponential Distribution Estimation of Parameter(s) Goodness of Fit Tests for Exponential Distributions Gamma Distribution Preliminaries Characterization of Gamma Distribution Estimation of Parameters Goodness of fit Tests for Gamma Distribution Weibull Distribution Preliminaries Characterization of Weibull Distribution Estimation of Parameters Goodness of Fit Tests for Weibull Distribution Extreme Value Distributions Preliminaries Characterizations of Extreme Value Distributions Estimation of Parameters Goodness of Fit Tests for Extreme Value Distributions System Reliability Preliminaries Single Component Systems Reliability of a Series System with Componentwise Data Reliability of a Parallel System with Componentwise Data Bibliography Selected Statistical Tables Index
41 citations
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TL;DR: It is shown that these distributions can be expressed as mixtures of unified skew-elliptical distributions, and then use these mixture representations to derive their moment generating functions and moments, when they exist.
41 citations
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TL;DR: A new adaptive model selection criterion is proposed and an approximately unbiased Kullback–Leibler loss estimator is constructed for model assessment in the context of exponential family distributions using a concept called generalized degrees of freedom that generalizes the concept originally proposed for the normal distribution.
Abstract: In many scientific and engineering problems, selecting the optimal model from a large pool of candidate models is important, particularly in data mining. In the literature, model assessment in the context of nonnormal distributions has not yet received a lot of attention. Indeed, many existing model selection criteria such as the Bayes information criterion and Cp may not be suitable for a situation in which the conditional mean and variance of the response are dependent, such as in generalized linear model regression. In this article we propose a new adaptive model selection criterion and construct an approximately unbiased Kullback–Leibler loss estimator for model assessment in the context of exponential family distributions. This permits comparing any arbitrary complex modeling procedures. Our proposal uses a concept called generalized degrees of freedom that generalizes the concept originally proposed for the normal distribution. The proposed procedure is implemented for the binomial and Poisson distr...
41 citations