Showing papers on "Natural exponential family published in 1972"

Computer methods for sampling from the exponential and normal distributions

Abstract: distributed raw, dora m~m}~.rs into expo~e~ttaRy a=d normally dis~rib~ed q~mntilies~ W.~e most ef~kien~ ones are compared, i~ terms of memory reqairemenN a~'~d sNeed, wi#~ some ne~' a~gori~bms, A rmmber of pro

Consistency and Asymptotic Normality of MLE's for Exponential Models

Abstract: Conditions are given for the strong consistency and asymptotic normality of the MLE (maximum likelihood estimator) for multiparameter exponential models. Because of the special structure assumed, the conditions are less restrictive than required by general theorems in this area. The technique involves certain convex functions on Euclidean spaces that arise naturally in the present context. Some examples are considered; among them, the multinomial distribution. Some convexity and continuity properties of multivariate cumulant generating functions are also discussed.

Goodness of Fit Tests for the Gamma and Exponential Distributions

Abstract: Goodness of fit tests based on generalized minimum x 2 techniques are developed for the gamma and exponential distributions. The power of these tests has been found for several alternative families of distributions by utilizing the asymptotic non-null distribution of the test statistic. The tests behave very well for the types of alternatives considered here. Applications to some failure data of Proschan (1963) are included for illustrative purposes.

On a Characterization by Conditional Expectations

Abstract: In a recent paper, Shanbhag [3] gave characterizations for the exponential and geometric distributions in terms of conditional expectations. The present note gives a general theorem on characterization by conditional expectations. A special form of the theorem characterizes the Weibull distribution (and hence Shanbhag's result for the exponential distribution). Another interesting special form of the theorem leads to a characterization of the uniform distribution. Applications of these characterizations are also indicated.

Maximum likelihood theory and applications for distributions generated when observing a function of an exponential family variable

Abstract: Maximum likelihood theory and applications for distributions generated when observing a function of an exponential family variable

On the distribution of a symmetric statistic from a mixed population

Abstract: The density of a symmetric statistic T = g(X 1, X 2, …, Xn ), for a random sample from a mixed population with density f(x) = pf 1(x) + pf 2), is a binomial mixture of the densities of the statist.ics Tk = g(Xk1 , Xk2 , Xkn ), k = 0, 1, … n. where Xki 's are independent with density f 1(x) if i ≤ k and density f 2(x) if i > k. It is shown how to find the distributions of some important symmetric statistics like sample mean, sample variance, and order statistics by using Tk 's. The results are applied to normal and exponential mixtures.

Sufficient Statistics and Discrete Exponential Families

Abstract: $\{P_\theta\}$ is a set of probabilities on a countable set $_\chi$ such that $P_\theta(x) > 0$ for each $x$ and $\theta$. We prove that if $\{P_\theta\}$ is not an exponential family, then each sufficient statistic for $n$ independent observations must be one-to-one, modulo permutations, on an infinite product set (which does not depend on the sufficient statistic).

Bayesian Prior Distributions for Systems with Exponential Failure-time Data

Abstract: In this paper, confidence bounds on the reliability of a serial system composed of exponential subsystems are considered. Both the classical and the Bayesian analyses are discussed. The main result is that for the case in which there are no previous data, then there are no prior distributions on the subsystem reliabilities that are independent of current data and that yield the uniformly most accurate unbiased confidence bounds available through classical techniques.

Empirical Bayes Point Estimation in a Family of Probability Distributions

Abstract: The empirical Bayes approach has been described in detail in the literature [1], [2], [3], [6], thus the problem will only be briefly summarized here. Let X be a random variable whose probability distribution depends in a known way on an unknown real parameter 0, with 0 itself being a random variable with unknown a priori distribution function G (0). The aim is to estimate 0 on the basis of the observed value x, which may be vector valued, so that the estimator has small squared error. The problem presents itself repeatedly and independently with the same unknown probability distribution function G (0) and a known family of distribution functions {F (x I 0): 0 e 0), 0 being the set of all possible values of 0. It is well known that for a squared-error loss function the Bayes estimator is the mean of the posterior probability distribution, that is, 0 = E (0 I x). It has been demonstrated by Rutherford and Krutchkoff [4], that if the prior has a known bound for any moment higher than the second, then one can obtain e-asymptotic optimality by a method of truncating a consistent sequence of estimators for the Bayes estimator.

On-Line Analysis of Exponential Time Functions

Abstract: A hybrid computer configuration has been devised for the fast on-line estimation of the parameters of a second order exponential time function. Experimental results are given for a range of time constant ratios and the effect of signal noise is considered.