Showing papers on "Natural exponential family published in 1994"
Book•
01 Jan 1994
TL;DR: Continuous Distributions (General) Normal Distributions Lognormal Distributions Inverse Gaussian (Wald) Distributions Cauchy Distribution Gamma Distributions Chi-Square Distributions Including Chi and Rayleigh Exponential Distributions Pareto Distributions Weibull Distributions Abbreviations Indexes
Abstract: Continuous Distributions (General) Normal Distributions Lognormal Distributions Inverse Gaussian (Wald) Distributions Cauchy Distribution Gamma Distributions Chi-Square Distributions Including Chi and Rayleigh Exponential Distributions Pareto Distributions Weibull Distributions Abbreviations Indexes
7,270 citations
Book•
31 Mar 1994
TL;DR: Inverse natural exponential families on R as discussed by the authors have been studied extensively in the literature, including a historical survey of the inverse Gaussian distribution and characterizations of combinations of these families.
Abstract: 1. A historical survey 2. Properties of the inverse Gaussian distribution 3. Characterizations 4. Combinations 5. Inverse natural exponential families on R 6. Statistical properties References Author index Subject index
236 citations
TL;DR: In this paper, a two shape parameter generalization of the well known family of the Weibull distributions is presented and its properties are studied, including skewness and kurtosis, density shapes and tail character, and relation of the members of the family to those of the Pearsonian system.
Abstract: A two shape parameter generalization of the well known family of the Weibull distributions is presented and its properties are studied. The properties examined include the skewness and kurtosis, density shapes and tail character, and relation of the members of the family to those of the Pear-sonian system. The members of the family are grouped in four classes in terms of these properties. Also studied are the extreme value distributions and the limiting distributions of the extreme spacings for the members of the family. It is seen that the generalized Weibull family contains distributions with a variety of density and tail shapes, and distributions which in terms of skewness and kurtosis approximate the main types of curves of the Pearson system. Furthermore, as shown by the extreme value and extreme spacings distributions the family contains short, medium and long tailed distributions. The quantile and density quantile functions are the principle tools used for the structural analysis of the family.
93 citations
TL;DR: In this article, it was shown that there are only six types of irreducible diagonal NEFs in ℝn, that are normal, Poisson, multinomial, negative multinomials, gamma, and hybrid.
Abstract: A natural exponential family (NEF)F in ℝn,n>1, is said to be diagonal if there existn functions,a1,...,an, on some intervals of ℝ, such that the covariance matrixVF(m) ofF has diagonal (a1(m1),...,an(mn)), for allm=(m1,...,mn) in the mean domain ofF. The familyF is also said to be irreducible if it is not the product of two independent NEFs in ℝk and ℝn-k, for somek=1,...,n−1. This paper shows that there are only six types of irreducible diagonal NEFs in ℝn, that we call normal, Poisson, multinomial, negative multinomial, gamma, and hybrid. These types, with the exception of the latter two, correspond to distributions well established in the literature. This study is motivated by the following question: IfF is an NEF in ℝn, under what conditions is its projectionp(F) in ℝk, underp(x1,...,xn)∶=(x1,...,xk),k=1,...,n−1, still an NEF in ℝk? The answer turns out to be rather predictable. It is the case if, and only if, the principalk×k submatrix ofVF(m1,...,mn) does not depend on (mk+1,...,mn).
68 citations
TL;DR: In this paper, the maximum likelihood estimation problem for the singly truncated normal family of distributions was studied and necessary and sufficient conditions in terms of the coefficient of variation were provided to obtain a solution to the likelihood equations.
Abstract: This paper is concerned with the maximum likelihood estimation problem for the singly truncated normal family of distributions. Necessary and suficient conditions, in terms of the coefficient of variation, are provided in order to obtain a solution to the likelihood equations. Furthermore, the maximum likelihood estimator is obtained as a limit case when the likelihood equation has no solution.
56 citations
TL;DR: It is shown that queueing models in which failure and repair times are represented by GE distributions can be analyzed with the same complexity as if these distributions were exponential.
Abstract: Failures of machines have a significant effect on the behavior of manufacturing systems. As a result it is important to model this phenomenon. Many queueing models of manufacturing systems do incorporate the unreliability of the machines. Most models assume that the times to failure and the times to repair of each machine are exponentially distributed (or geometrically distributed in the case of discrete-time models). However, exponential distributions do not always accurately represent actual distributions encountered in real manufacturing systems. In this paper, we propose to model failure and repair time distributions bygeneralized exponential (GE) distributions (orgeneralized geometric distributions in the case of a discretetime model). The GE distribution can be used to approximate distributions with any coefficient of variation greater than one. The main contribution of the paper is to show that queueing models in which failure and repair times are represented by GE distributions can be analyzed with the same complexity as if these distributions were exponential. Indeed, we show that failures and repair times represented by GE distributions can (under certain assumptions) be equivalently represented by exponential distributions.
41 citations
TL;DR: This paper moves away from the parametric framework of much of statistical differential geometry and instead employs divergence in a covariate setting and obtains a differential geometric interpretation of standard ideas of association.
Abstract: SUMMARY In applications of differential geometry to problems of parametric statistical inference, the notion of divergence is often used to measure the separation between two parametric densities. In this paper we move away from the parametric framework of much of statistical differential geometry and instead employ divergence in a covariate setting. Many data-analyses involve investigation of how the conditional distribution f(y I X = x) of a random variable Y changes with covariate values x. We propose using local divergence between conditional distributions as a measure of this change and thus as a general measure of association between Y and the covariates X. Under certain regularity conditions we define a class of divergences which are locally the Rao distance (LR). The limiting LR divergence is bounded below by the signal-to-noise ratio, with equality holding if and only if the conditional density comes from the natural exponential family. The correlation curve and in particular the correlation coefficient are simply transformations of the signal-to-noise ratio and thus are transformations of local divergence. We therefore obtain a differential geometric interpretation of standard ideas of association. The class of LR divergences is broad and includes the Kullback-Leibler divergence and Renyi oc-information measures. We can therefore interpret local association as local utility or information gain of order cx. We obtain comparable results without regularity conditions.
30 citations
TL;DR: In this paper, conditions under which quadratic and polynomial exponential models can be generated as mixtures of linear exponential models are derived for continuous sample spaces, but less restrictive in the discrete case.
Abstract: SUMMARY Conditions are derived under which quadratic and polynomial exponential models can be generated as mixtures of linear exponential models. The conditions are highly restrictive for continuous sample spaces, but less restrictive in the discrete case. Some properties of binary quadratic exponential models are explored with a view towards finding models that have properties suitable for epidemiological applications.
29 citations
TL;DR: Shanbhag's clever method for finding the Jorgensen set of the family of Wishart distributions on symmetric matrices is extended here to Wishart distribution on asymmetric cones, such as Hermitian matrices on complex numbers or quaternions.
Abstract: Shanbhag’s clever method for finding the Jorgensen set of the family of Wishart distributions on symmetric matrices is extended here to Wishart distributions on symmetric cones, such as Hermitian matrices on complex numbers or quaternions. The idea is also extended to various other multivariate distributions, including the natural exponential family associated with the set of normal distributions onR with unknown mean and variance.
26 citations
TL;DR: In this article, the moments of a NBWUE family of life distributions were derived for weak and moderate convergence conditions, and the equivalence of weak convergence and moment convergence was established under mild conditions.
26 citations
TL;DR: In this paper, the authors derived recurrence relations for the single and product moments of order statistics arising from n independent non-identically distributed right-truncated exponential random variables.
Abstract: By considering order statistics arising from n independent non-identically distributed right-truncated exponential random variables, we derive in this paper several recurrence relations for the single and the product moments of order statistics These recurrence relations are simple in nature and could be used systematically in order to compute all the single and the product moments of order statistics for all sample sizes in a simple recursive manner The results for order statistics from a multiple-outlier model (with a slippage of p observations) from a right-truncated exponential population are deduced as special cases These results will be useful in assessing robustness properties of any linear estimator of the unknown parameter of the right-truncated exponential distribution, in the presence of one or more outliers in the sample These results generalize those for the order statistics arising from an iid sample from a right-truncated exponential population established by Joshi (1978, 1982)
TL;DR: In this paper, the authors compared parameterizations for natural exponential families with quadratic variance functions (QVF's) according to the nearness to normality of the likelihood and posterior distribution.
Abstract: Parameterizations for natural exponential families (NEF's) with quadratic variance functions (QVF's) are compared according to the nearness to normality of the likelihood and posterior distribution. Nonnormality of the likelihood (posterior) is measured using two criteria. The first is the magnitude of a standardized third derivative of the log-likelihood (logposterior density); the second is a comparison of the probability of particular tail regions under the normalized likelihood (posterior distribution) and under the corresponding normal approximation. A relationship is given that links these two criteria. Sample sizes are recommended for adequate normality in the likelihood for various parameterizations of the NEF-QVF models, and these results are extended to Bayesian models with a conjugate prior.
TL;DR: In this article, a family of exponential four-step methods for numerical integration of the one-dimensional Schrodinger equation was developed, which allows it to be fitted automatically to exponential functions.
Abstract: A family of exponential four-step methods is developed for the numerical integration of the one-dimensional Schrodinger equation. The formula developed contains certain free parameters which allows it to be fitted automatically to exponential functions. The new methods integrate more exponential functions, and are very simple compared with the well-known sixth algebraic order Runge-Kutta type methods. Numerical results indicate that the new methods are much more accurate than other exponentially fitted methods.
TL;DR: In this article, it was shown that a class of Levy processes (processes with independent stationary increments) is connected in a natural way to many exponential families of continuous-time stochastic processes.
TL;DR: In this paper, the authors use an economic approach of Mendel to derive new bivariate exponential lifetime distributions, which are based on the principle of indifference and make use of the fact that the parameter of interest is a measurable function of observable quantities.
Abstract: We use an economic approach of Mendel to derive new bivariate exponential lifetime distributions. Features distinguishing this approach from the existing ones are (1) it makes use of the principle of indifference; (2) our parameter of interest is a measurable function of observable quantities; (3) the assessment of the probability measure for random lifetimes is performed by assessing that for random lifetime costs with a change of variables; and (4) characterization properties other than the bivariate loss-of-memory property are used to construct distributions. For the infinite population case, our distributions correspond to mixtures of existing bivariate exponential distributions such as the Freund distribution, the Marshall-Olkin distribution, and the Friday-Patil distribution. Moreover, a family of natural conjugate priors for Bayesian Freund (-type) bivariate exponential distributions is discussed.
TL;DR: In this article, the authors modified the bootstrap predictive distribution by replacing the maximum likelihood estimator of the unknown parameter by its minimum Hellinger distance estimator, which is asymptotically superior to the corresponding estimative distribution in terms of the average Kullback-Leibler divergence.
Abstract: SUMMARY The bootstrap predictive distribution considered by Harris (1989) is modified by replacing the maximum likelihood estimator of the unknown parameter by its minimum Hellinger distance estimator. The predictive distribution thus obtained is asymptotically superior to the corresponding estimative distribution in terms of the average Kullback-Leibler divergence, when the true distribution is in the natural exponential family. Simulation results are provided for the binomial and normal distributions which suggest that the proposed predictive distributions are robust and can perform better than the likelihood based methods under data contamination.
TL;DR: In this paper, the authors derived asymptotic expansions of posterior distributions for a two-dimensional exponential family, which includes normal, gamma, inverse gamma and inverse Gaussian distributions, and applied a version of Stein's identity to assess the posterior distributions.
Abstract: Asymptotic expansions of posterior distributions are derived for a two-dimensional exponential family, which includes normal, gamma, inverse gamma and inverse Gaussian distributions. Reparameterization allows us to use a data-dependent transformation, convert the likelihood function to the two-dimensional standard normal density and apply a version of Stein's identity to assess the posterior distributions. Applications are given to characterize optimal noninformative priors in the sense of Stein, to suggest the form of a high-order correction to the distribution function of a sequential likelihood ratio statistic and to provide confidence intervals for one parameter in the presence of other nuisance parameters.
TL;DR: A generalized linear empirical Bayes model is developed for empirical bayes analysis of several means in natural exponential families as discussed by the authors, including the Normal, Poisson, Binomial, Gamma, and two others, using the extended quasi-likelihood of Nelder and Pregibon.
Abstract: A generalized linear empirical Bayes model is developed for empirical Bayes analysis of several means in natural exponential families. A unified approach is presented for all natural exponential families with quadratic variance functions (the Normal, Poisson, Binomial, Gamma, and two others.) The hyperparameters are estimated using the extended quasi-likelihood of Nelder and Pregibon (1987), which is easily implemented via the GLIM package. The accuracy of these estimates is developed by asymptotic approximation of the variance. Two data examples are illustrated.
Journal Article•
TL;DR: In this article, the positive dependence of a subclass of multivariate exponential distributions is examined, characterized by an index vector k and a parameter vector $\lambda$, which are used as an ordering to yield degrees of positive dependence.
Abstract: The positive dependence of a subclass of multivariate exponential distributions is examined. This class is characterized by an index vector k and a parameter vector $\lambda$, which are used as an ordering to yield degrees of positive dependence. The results presented have a direct implication on the reliability function of a system and the survival probability function of a shock model, and consequently on the optimal assembly of systems.
TL;DR: In this article, it is shown that the power-quadratic distributions of exponential-type life distributions can be partially ordered by hazard rates whose precise expression and asymptotic behavior can also be obtained by using special mathematical functions.
01 Jan 1994
TL;DR: In this paper, a simple class of generalized sequential likelihood ratio tests is introduced for testing hypotheses in multivariate exponential families, which have asymptotically optimal frequentist properties and also provide approximate Bayes solutions with respect to a large class of prior distributions.
Abstract: A simple class of generalized sequential likelihood ratio tests is introduced for testing hypotheses in multivariate exponential families. These sequential tests have asymptotically optimal frequentist properties and also provide approximate Bayes solutions with respect to a large class of prior distributions. 1? Introduction. Let X\,X2,- ? ? be i.i.d. ? ? 1 random vectors whose common multivariate density (with respect to some nondegenerate dominating measure v) belongs to the exponential family
TL;DR: In this article, a set of simultaneous upper confidence intervals for all ratios to the largest scale parameter is derived for k independent exponential populations with different scale and location (possibly unknown) parameters.
Abstract: Consider k independent exponential populations with different scale and location (possibly unknown) parameters. A set of simultaneous upper confidence intervals for all ratios to the largest scale parameter is derived. The data are assumed to be: (1) complete; or (2) incomplete with type-II censoring. The cases of known and unknown location parameters are treated separately. >
TL;DR: In this paper, an integral function Iμ(α, x; a, b) satisfies a functional recurrence relation which is exploited to find a closed form evaluation of some incomplete integrals.
Abstract: We define an integral function Iμ(α, x; a, b) for non-negative integral values of μ byIt is proved that Iμ(α, x; a, b) satisfies a functional recurrence relation which is exploited to find a closed form evaluation of some incomplete integrals. New integral representations of the exponential integral and complementary error functions are found as special cases.
TL;DR: In this paper, an improved and simplified version of the unifying probability density function of [1] was presented, which is shown to be the parent of the Rayleigh distribution in addition to the Weibull-, gamma-, Erlang-, χ 2 -and exponential distributions.
01 Jan 1994
TL;DR: In this article, the class of bivariate distributions having the almost-lack-of-memory property (ALM-distribution) is introduced and the exact form of these distributions in a sub-class with almost independent components is derived.
Abstract: The class of bivariate distribution having the almost-lack-of-memory property (ALM-distribution) is introduced and the exact form of these distributions in a sub-class with almost independent components is derived. Some of the corresponding probability properties are discussed.
TL;DR: The Poisson probability method for detecting stochastic randomness in three-dimensional space has resulted in the formal calculation of a useful class of exponential integrals as discussed by the authors, which can be used to detect randomness.
Abstract: The authors' Poisson probability method for detecting stochastic randomness in three-dimensional space has resulted in the formal calculation of a useful class of exponential integrals.