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.

Parameterizations for Natural Exponential Families with Quadratic Variance Functions

Abstract: Parameterizations for natural exponential families (NEF's) with quadratic variance functions (QVF's) are compared according to the nearness to normality of the likelihood and posterior distribution. Nonnormality of the likelihood (posterior) is measured using two criteria. The first is the magnitude of a standardized third derivative of the log-likelihood (logposterior density); the second is a comparison of the probability of particular tail regions under the normalized likelihood (posterior distribution) and under the corresponding normal approximation. A relationship is given that links these two criteria. Sample sizes are recommended for adequate normality in the likelihood for various parameterizations of the NEF-QVF models, and these results are extended to Bayesian models with a conjugate prior.

A family of four-step exponential fitted methods for the numerical integration of the radial Schrödinger equation

Abstract: A family of exponential four-step methods is developed for the numerical integration of the one-dimensional Schrodinger equation. The formula developed contains certain free parameters which allows it to be fitted automatically to exponential functions. The new methods integrate more exponential functions, and are very simple compared with the well-known sixth algebraic order Runge-Kutta type methods. Numerical results indicate that the new methods are much more accurate than other exponentially fitted methods.

General Exponential Models for Discrete Observations

Abstract: : A class of models generalizing exponential families is defined via the algebraic structure of the sufficient statistics. The maximum likelihood estimate for the unknown parameter is shown to exist and be unique. The sequence of sufficient statistics from successive repetitions of experiments corresponding to a general exponential model is shown to form an extreme family of Markov chains as defined by Lauritzen (1974).

Conditional Tail Moments of the Exponential Family and Its Related Distributions

Abstract: The risk measure is a central theme in the risk management literature. For good reasons, the conditional tail expectation (CTE) has received much interest in both insurance and finance applications. It provides for a measure of the expected riskiness in the tail of the loss distribution. In this article we derive explicit formulas of the CTE and higher moments for the univariate exponential family class, which extends the natural exponential family, using the canonical representation. In addition we show how to compute the conditional tail expectations of other related distributions using transformation and conditioning. Selected examples are presented for illustration, including the generalized Pareto and generalized hyperbolic distributions. We conclude that the conditional tail expectations of a wide range of loss distributions can be analytically obtained using the methods shown in this article.

On the estimation of stress strength reliability parameter of inverted exponential distribution

Abstract: This paper aims to estimate the stress-strength reliability parameter R = P(Y < X), when X and Y are independent inverted exponential random variable We have also discussed some fundamental properties of the considered distribution The maximum likelihood estimator (MLE) of R and its asymptotic distribution are obtained The Bayesian estimation of the reliability parameter has been also discussed under the assumption of independent gamma prior Numerical integration technique is used for Bayesian computation The proposed estimators are compared in terms of their mean squared errors through the simulation study Two real data sets representing survival of head and neck cancer patients are fitted using the inverted exponential distribution and used to estimate the stress-strength parameters and reliability