Showing papers on "Natural exponential family published in 1996"
TL;DR: In this article, the Wishart distribution on a symmetric cone C is characterized by extending the Olkin-Rubin proof by using three modern ideas: (i) avoid artificial coordinates in differential geometry; (ii) the variance function of a natural exponential family F characterizes F; and (iii) symmetric matrices are a particular example of a Euclidean simple Jordan algebra.
Abstract: We characterize the Wishart distributions on a symmetric cone C. If $C = (0, +\infty)$, this has been done by Lukacs in 1955. If C is the cone of positive definite symmetric matrices, this has been done by Olkin and Rubin in 1962. We both shorten and extend the Olkin-Rubin proof (sometimes obscure) by using three modern ideas: (i) try to avoid artificial coordinates in differential geometry; (ii) the variance function of a natural exponential family F characterizes F; (iii) symmetric matrices are a particular example of a Euclidean simple Jordan algebra.
89 citations
01 Jan 1996
TL;DR: In this paper, a brief review of the Bennett and Hoeίfding inequalities is presented, as they apply to independent random variables, for the purpose of identifying the point where independence is actually utilized.
Abstract: A brief review of the Bennett and Hoeίfding inequalities is presented, as they apply to independent random variables, for the purpose of identifying the point where independence is actually utilized. On the basis of an observation, it follows that the inequalities remain in force whenever the expectation of a certain product is bounded by the product of the expectations of the factors involved. This requirement is satisfied, for example, when the underlying random variables are negatively associated. By a counterexample, it is demonstrated that the inequalities need not hold for positively associated random variables. Next, a Hoeffding-type inequality is established for a strong mixing sequence of random variables. The paper is concluded with the utilization of the Hoeffding inequality in order to construct a minimum distance estimate of the probability measure governing a sequence of negatively associated random variables.
59 citations
TL;DR: In this paper, the authors established several recurrence relations satisfied by the single and product moments of progressive Type-II right censored order statistics from an exponential distribution, which can then be used to compute all the means, variances and covariances of exponential progressive type-II, right-censored order statistics for all sample sizes n and all censoring schemes (R1, R2,..., Rm).
Abstract: In this paper, we establish several recurrence relations satisfied by the single and product moments of progressive Type-II right censored order statistics from an exponential distribution. These relations may then be used, for example, to compute all the means, variances and covariances of exponential progressive Type-II right censored order statistics for all sample sizes n and all censoring schemes (R1, R2, ..., Rm), m≤n. The results presented in the paper generalize the results given by Joshi (1978, Sankhyā Ser. B, 39, 362–371; 1982, J. Statist. Plann. Inference, 6, 13–16) for the single moments and product moments of order statistics from the exponential distribution.
44 citations
TL;DR: In this paper, the authors explore the reliability properties of the minimum and maximum lifetimes of two-component systems whose components are dependent and have the bi-variate exponential (BVE) distributions of Gumbel, Marshall and Olkin, Block and Basu, Freund, Friday and Patil, and Raftery.
Abstract: We explore the reliability properties of the minimum and maximum lifetimes of two–component systems whose components are dependent and have the bi–variate exponential (BVE) distributions of Gumbel, Marshall and Olkin, Block and Basu, Freund, Friday and Patil, and Raftery. In most cases, these random variables are either exponential or generalized hyperexponential (GH) distributions with three or fewer components. We determine the properties of the failure rates of such GH distributions in terms of the weights and the parameters of the constituent exponential random variables. We apply these general results and show that the failure rates of the order statistics from the above BVE distributions exhibit a variety of patterns. Their failure rate properties can differ substantially from those of order statistics of two independent exponential random variables
32 citations
29 citations
TL;DR: A simplified and extended theory of cuts in natural exponential families is established in this article, and an open question in this subject area is solved, in the negative, by a counter-example.
Abstract: A simplified and extended theory of cuts in natural exponential families is established. Further, an open question in this subject area is solved, in the negative, by a counter-example, and a link to the recent theory of variance functions for natural exponential families is pointed out.
28 citations
TL;DR: In this paper, the authors consider multidimensional extensions of their results which still yield quadratic or simple Quadratic Natural Exponential families on R. In this paper, we extend the results of the present paper.
Abstract: There exist several different characterizations of the class of quadratic natural exponential families onR, two of which use orthogonal polynomials. In Feinsilver (1986), the polynomials result from the derivation of the probability densities while Meixner (1934) adopts an exponential generating function. In this paper, we consider multidimensional extensions of their results which still yield quadratic or simple quadratic natural exponential families.
28 citations
TL;DR: In this paper, the existence problem for the optimal parameters for the exponential function approximating these data in the sense of total least squares was considered and sufficient conditions were given to guarantee the existence of such optimal parameters.
Abstract: Given the data , i = 1,...,m, we consider the existence problem for the optimal parameters for the exponential function approximating these data in the sense of total least squares. We give sufficient conditions which guarantee the existence of such optimal parameters.
22 citations
TL;DR: In this paper, a simple method for approximate conditional inference is described, which is applied to natural exponential family models where it is shown to provide accurate approximations to fully conditional estimates.
Abstract: SUMMARY A simple method for approximate conditional inference is described The methodology is applied to natural exponential family models where it is shown to provide accurate approximations to fully conditional estimates The approximation technique can be applied much more generally than in this particular class of models The technique depends only on the construction of certain 'projected scores' derived from higher order likelihood derivatives and their covariances and so can be used in many problems where it is relevant to control for the estimation of nuisance parameters
18 citations
TL;DR: In this paper, the authors select a family of prior distributions to combine with a given likelihood, the prime consideration is that the resulting posteriors should be members of tractable families of distributions.
Abstract: When selecting a family of prior distributions to combine with a given likelihood, the prime consideration is that the resulting posteriors should be members of tractable families of distributions. Assuming a likelihood belonging to an exponential family we identify the family of priors which will lead to convenient posteriors in the sense that they belong to a specified exponential family.
TL;DR: In this article, the exact distribution of the maximum likelihood estimators in an exponential regression model is derived by inverting the characteristic function by a straightforward application of residue theory, and the distribution is a weighted sum of independent exponential random variables.
TL;DR: In this paper, the authors considered the three-parameter exponential power distribution with location parameter μ, scale parameter σ2 and shape parameter β and obtained simultaneous estimates of μ, σ 2 and β by method of moments and method of maximum likelihood.
Abstract: Consider the three-parameter exponential power distribution with location parameter μ, scale parameter σ2 and shape (power) parameter β. This is a general symmetric family of distributions with normal, double exponential and rectangular as special cases. Such distributions are used in Bayesian statistics as a wider choice of symmetric parent distribution and in classical statistics in determination of lack of normality. This note obtains simultaneous estimates of μ, σ2 and β by method of moments and method of maximum likelihood. It also studies the behavior of estimates of β through Monte Carlo simulation when values of μ and σ2 are set equal to zero and unity respectively.
TL;DR: In this article, three characteristic properties of a certain class of probability distributions are given based on recurrence relations between conditional moments, conditional variance and relationships between two moments of order statistics, which shed new light on some special cases obtained by the Weibull, power, Pareto, beta and Burr type XII distributions.
Abstract: Three characteristic properties of a certain class of probability distributions are given. They are based on recurrence relations between conditional moments, conditional variance and relationships between two moments of order statistics. These results shed new light on some special cases obtained by the Weibull, power, Pareto, beta and Burr type XII distributions.
TL;DR: In this article, some characterization results for the exponential distribution are given using mixing distributions, and the achieved results generalize some known results in this connection and have its relevance to some practical applications.
TL;DR: In this article, a general identity for the product moments of successive order statistics is given, which is valid in a class of probability distributions including Weibull, Pareto, exponential and Burr distributions.
Abstract: A general identity for the product moments of successive order statistics is given, which is valid in a class of probability distributions including Weibull, Pareto, exponential and Burr distributions.
01 Sep 1996
TL;DR: The projection filter as mentioned in this paper is an approximate finite-dimensional filter based on the differential geometric approach to statistics, and it has been shown to be useful in the case of exponential families.
Abstract: We present the projection filter, an approximate finite-dimensional filter based on the differential geometric approach to statistics. We recall the definition of the projection filter in the case of exponential families, and we give some hints about the selection of the coefficients in the exponential family.
TL;DR: In this article, a sequential procedure for estimating with prescribed proportional accuracy one parameter in a class of two-parameter exponential family of distributions is proposed, and the properties of the resulting stopping time and second-order analysis of the coverage probability associated with it are provided.
Abstract: We propose a sequential procedure for estimating with prescribed proportional accuracy one parameter in a class of two-parameter exponential family of distributions. We study the properties of the resulting stopping time and provide second-order analysis of the coverage probability associated with it by using Edgeworth expansion.
TL;DR: In this article, the conditional distribution of the maximum given the sample total for a random sample from the truncated exponential distribution was derived, and tests or associated confidence intervals for the truncation parameter with another parameter θ assumed unknown.
Abstract: This paper derives the conditional distribution of the maximum given the sample total for a random sample from the truncated exponential distribution. Based on that result, the paper develops tests or associated confidence intervals for the truncation parameter θ with another parameter θ assumed unknown.
TL;DR: In this paper, the existence of the maximum likelihood estimate (MLE) of the parameter for 1-dimensional exponential families was investigated and it was established that the MLE exists in most cases.
01 Mar 1996
TL;DR: In this paper, the authors considered a special case of the general location-scale family of censored samples from exponential distributions, and showed that the problem can be treated as a special problem for the normal distribution.
Abstract: The estimation δ = μ1 − μ2 and of the common μ when ϕ=0 is exhibited separately for ρ = σ2/σ1 specified, and unspecified, for two Type II censored samples from exponential distributions (μ i , σ i ). The problem is treated as a special case of the general location-scale family. As such, it presents a useful parallel to the well known results for the normal distribution.
TL;DR: In this article, the authors studied the properties of monotone systems and cold standby systems with exponential life distributions and dependent components, and proved that all but one of the components are degenerate at zero while the remaining one is exponential.
Abstract: Structures of monotone systems and cold standby systems with exponential life distributions and dependent components are studied. It is shown that a monotone system composed of components with multivariate HNBUE life distributions is essentially a series system composed of components with multivariate exponential life distributions. Also, it is proved that for cold standby systems composed of components with multivariate NBU life distributions, all but one of the components are degenerate at zero while the remaining one is exponential. In addition, several equivalent characterizations of multivariate exponential distribution are provided in the multivariate HNBUE life distribution class which include many existing results as special cases.
TL;DR: In this paper, an accelerated sequential procedure for estimating the mean p in the class of the natural exponential family of distributions having power variance function (NEF-PVF) is proposed.
Abstract: An accelerated sequential procedure for estimating the mean p in the class of the natural exponential family of distributions having power variance function (NEF-PVF), is proposed. The accelerated procedure is conducted under a combined loss of weighted squared estimation error and sarnpling cost. A class of bias-corrected estimators, which are natural for this suggested accelerated sampling scheme is proposed. The asymptotic properties of the suggested estimators are provided. In particular, one realizes the impact of acceleration on the regret and the tradeoff between bias reduction and regret reduction.