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.

Asymptotically most informative procedure in the case of exponential families

Abstract: Recently we showed the following fact in our paper [2]. We considered in [2] two binomial trials Ely E2 having unknown means plf p2 respectively. And we have introduced the notion of costs such that we must pay costs clf c2 to the observation of a result given by the trials Eu E2 respectively. In each step we are admitted to select one of the two trials Ely E2. Be continued the selections by some way we denoted the sequence of trials till n-th step as E\\ •••, E and the sequence of costs till n-th step as C c υ , •••, C ( W . Of course we may select at i-th step E from the two trials Elf E2 depending previous i—1 data Xlf •••, Xz-i given by E\\ •••, E~. A procedure <£ was given in [2] such that the sum of information given by two dimensional likelihood ratio relative to the sum of costs till n-th step to discriminate p{>p2 or pi ••, k) respectively. And we introduced the boundary π: μ-θ=p(θ=(θ1, •••, θk)) as a hyperplane in k dimensional euclidean space where μ=(μi, •••, μk) is any fixed k dimensional unit vector having all non-zero components and p is any fixed nonnegative number and μ-θ is the inner product of two vectors μ and θ. Moreover we use the notion of costs introduced by Kunisawa [6], as we used the notion in [2], [3], then we can get some information of θ3 by paying of cost cj(j=l, •••, k) respectively. Then we shall show analogously that under the generalized procedure 3£* given in the following Section 3 the sum of information relative to the sum of costs payed till n-th step to discriminate μ-θ larger than p or not is asymptotically

Bayesian inference and prediction for generalized inverted exponential distribution for Type-II censored data

Abstract: The manuscript introduces Bayesian estimation and prediction of a generalized version of the inverted exponential distribution for Type-II censored data. It further reflects on the Bayesian estimation of the unknown parameters under the squared error loss function, assuming that both the scale and the shape parameters of the distribution have a gamma prior and are independently distributed. Under these priors, the importance sampling technique is used to calculate Bayes estimates and the corresponding highest posterior density intervals. Bayes estimates are also computed using Lindley’s approximation and the Metropolis-Hastings algorithm. Monte Carlo simulations are performed to compare the performance of the proposed Bayes estimates. This article seeks to extend the posterior predictive density of future observations, as well as construct a predictive interval with a given coverage probability. A data analysis is performed for illustrative purposes.

Simple, Efficient Closed-Form Approximations for Beta Percentiles, Exponential Prediction Intervals and Confidence Bounds on Exponential and Binomial Parameters.

Abstract: : The use of Patnaik's (1949) Chi-square approximation to the non-central Chi-square distribution and the Wilson-Hilferty (1931) transformation of Chi-square to approximate normality are explored herein as a simple, efficient means of finding one-or-two-order statistic confidence bounds on parameters of the one- and two-parameter negative exponential distributions. Such methods can be used when it is known that r of n sample items, r < or = n, have failed during a life test, but the times of some early failures are not known exactly. An important implication of the result applying to approximate confidence bounds on the mean of an exponential distribution from a single order statistic is that of obtaining simple-closed-form approximations to percentiles of the Beta distribution (with integer parameters). It is shown that a generalization of the approximation applies to Beta variates with noninteger parameters. (Modified author abstract)

D-distribution and its applications

Abstract: We derive the distribution of the sum ofn independent doubly truncated Poisson variables not necessarily identically distributed. This distribution is called a D-distribution. Its p.d.f. can be expressed in terms of a D-number and an incomplete exponential function which are both defined in this paper. We investigate the relationship between these numbers and distributions, and use these relationships to derive recurrence relations and other properties of the D-distribution. A minimum variance unbiased estimates of the p.d.f. of this D-distribution is also obtained. Some example are included at the end to illustrate the use of this D-distribution.