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.

Can data recognize its parent distribution

Abstract: This study is concerned with model selection of lifetime and survival distributions arising in engineering reliability or in the medical sciences. We compare various distributions—including the gamma, Weibull, and lognormal—with a new distribution called geometric extreme exponential. Except for the lognormal distribution, the other three distributions all have the exponential distribution as special cases. A Monte Carlo simulation was performed to determine sample sizes for which survival distributions can distinguish data generated by their own families. Two methods for decision are by maximum likelihood and by Kolmogorov distance. Neither method is uniformly best. The probability of correct selection with more than one alternative shows some surprising results when the choices are close to the exponential distribution.

The generalized Cauchy family of distributions with applications

Abstract: A family of generalized Cauchy distributions, T-Cauchy{Y} family, is proposed using the T-R{Y} framework. The family of distributions is generated using the quantile functions of uniform, exponential, log-logistic, logistic, extreme value, and Frechet distributions. Several general properties of the T-Cauchy{Y} family are studied in detail including moments, mean deviations and Shannon’s entropy. Some members of the T-Cauchy{Y} family are developed and one member, gamma-Cauchy{exponential} distribution, is studied in detail. The distributions in the T-Cauchy{Y} family are very flexible due to their various shapes. The distributions can be symmetric, skewed to the right or skewed to the left.

Some Inferences about Gamma Parameters with an Application to a Reliability Problem

Abstract: Lehmann and Scheffe have shown how to construct uniformly most powerful unbiased tests of certain hypotheses when the assumed distributions belong to an “exponential family.” The present paper is concerned with particular cases which arise from independent gamma variates with scale parameters θ1 and θ2. Conditional distributions are given which are appropriate for testing γ = γ0, where γ = c 1/θ1+c 2/θ2. As an application, suppose that a series system has two dissimilar components with expected lives θ1 and θ2. When component failures are exponentially distributed, so are system failures, the mean being 1/(1/θ1+1/θ2). From independent estimates of θ1 and θ2 confidence limits can be found for this mean, or for the probability of successful system operation up to any fixed “mission time.” With appropriate restrictions, more general distributions including the Weibull can also be treated.

Some Exponentiated Distributions

Abstract: In this paper we study a number of new exponentiated distributions. The survival function, failure rate and moments of the distributions have been derived using certain special functions. The behavior of the failure rate has also been studied.