Showing papers on "Natural exponential family published in 1967"

A Multivariate Exponential Distribution

Abstract: A number of multivariate exponential distributions are known, but they have not been obtained by methods that shed light on their applicability. This paper presents some meaningful derivations of a multivariate exponential distribution that serves to indicate conditions under which the distribution is appropriate. Two of these derivations are based on “shock models,” and one is based on the requirement that residual life is independent of age. It is significant that the derivations all lead to the same distribution. For this distribution, the moment generating function is obtained, comparison is made with the case of independence, the distribution of the minimum is discussed, and various other properties are investigated. A multivariate Weibull distribution is obtained through a change of variables.

A generalized bivariate exponential distribution

Abstract: In a previous paper (Marshall and Olkin (1966)) the authors have derived a multivariate exponential distribution from points of view designed to indicate the applicability of the distribution. Two of these derivations are based on “shock models” and one is based on the requirement that residual life is independent of age. The practical importance of the univariate exponential distribution is partially due to the fact that it governs waiting times in a Poisson process. In this paper, the distribution of joint waiting times in a bivariate Poisson process is investigated. There are several ways to define “joint waiting time”. Some of these lead to the bivariate exponential distribution previously obtained by the authors, but others lead to a generalization of it. This generalized bivariate exponential distribution is also derived from shock models. The moment generating function and other properties of the distribution are investigated.

Finite mixtures of distributions from the exponential family

Abstract: Abstract General “successive substitutions” iteration equations are developed for obtaining estimates for finite mixtures of distributions from the exponential family. These, in general, correspond to relative maximums of the likelihood function. It is assumed that the number of distributions is known, and that the mixtures are from distributions of the same type, but with different parameter values. The particular equations for the Poisson, binomial, and exponential distributions are given, as well as examples of the results of the procedure for each distribution. From the examples tried, it was observed that the likelihood function increased at each iteration. Graphs of the asymptotic variances of the estimates are given, and two sampling experiments comparing estimates obtained by this scheme with moment estimates are also given.

Tests of Composite Hypotheses for the Multivariate Exponential Family

Abstract: This paper is concerned with testing the hypothesis that the parameter in a multivariate exponential distribution lies in a linear subspace of the natural parameter space. Our main result characterizes a complete class of tests which is independent of the particular exponential distribution. This class is, in fact, complete relative to the stronger ordering among tests which compares conditional power, given a certain statistic, pointwise. The conclusion holds without any restriction on the exponential distribution. Many of the tests are admissible, but examples show that although the class is essentially the smallest class complete relative to all exponential distributions, it is not in general minimally complete. Some special cases where the class is minimally complete are discussed.

Sample Sequences of Maxima

Abstract: Let $X_1, X_2, \cdots, X_n, \cdots$ be a sequence of independent, identically distributed random variables with common distribution function $F$. Let $Z_n = \max \{X_1, X_2, \cdots X_n\}$. Conditions for the stability and relative stability of such sequences with the various modes of convergence have been given by Geffroy [3], and Barndorff-Nielsen [1]. The principal result of this paper is Theorem 2.1, which is an analogue for maxima of the law of the iterated logarithm for sums (Loeve [6] pages 260-1). In Section 3, it is indicated that the theorem is satisfied by a wide class of distributions, and specific forms are given for the normal and exponential distributions.

Multivariate logarithmic and exponential regression models

Abstract: : This study analyzes a multivariate exponential regression function. Two basic types of error assumptions are examined: multiplicative (logarithmic model) and additive (exponential model). The usual method of taking natural logarithms of the regression relationship and then using linear least-squares estimators for the parameter estimates assumes that the error is multiplicative. Analysis shows that only the estimate of the parameter in the coefficient term and its distribution are affected by whether the hypothetical regression function is equal to the expected value (the mean) of the dependent variable Y, or to the median. The other parameter estimates, their distribution, and the prediction interval of Y are not affected. The study also examines the exponential, or additive, model in which the error is assumed to be normally distributed and added to the function. This leads to the more difficult problem of least-squares estimation of a nonlinear form. Such a solution is not exact and may not be unique. Methods of comparing the two models are given. The bulk of the Memorandum describes, lists, and gives instructions for use of the Multivariate Logarithmic and Exponential Computer Program.

On probability distributions for filtered white noise

Abstract: When white noise generated by an underlying Poisson process is filtered by any member of a large class of stable (not necessarily linear nor stationary) filters, the first-order probability distributions of the filter output is infinitely divisible. The Kolmogorov canonical form for the characteristic function is displayed and related to the parameters of the white noise and of the filter. In certain linear stationary cases, cubclasses of the infinitely divisible distributions are identified. An invariance property of the bilateral exponential distribution is demonstrated.

On the distribution of products of independent beta variables

Abstract: : The problem of finding the probability distribution of a product of a number of identically distributed, independent random variables had been solved by an application of the Mellin transform for normal and Cauchy distributions (Springer and Thompson) and for exponential, Gamma and Weibull distributions (Lomnicki). The present paper shows that it can be solved by similar methods for Beta distributions. This is of practical importance, since most physical quantities with which an engineer is dealing have finite ranges, while all the distributions previously studied had infinite ranges.

On the Application of the Double Exponential and Weibull Distributions to Failure Data

Abstract: ened, which leads to a loss of stability in the protein. Also, by opening these hydrophobic areas, polar bonds which have been masked so far are now accessible to the water and can be broken. The reversibility of the work reduction by rinsing the fibers in distilled water, after treatment and extension in tenside solutions, supports the theory of hydrophobic interactions. A similar rever’ sibility was found for alcohol-water mixtures [3, 12].

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

Tolerance Regions for a Joint Exponential Distribution

Abstract: The evaluation of the reliability of a system of components, when the components are assumed to follow a joint exponential distribution, is considered The approach used is to develop tolerance regions for the joint exponential distribution or to estimate the probability content of the appropriate specification region