Showing papers on "Natural exponential family published in 2004"

Families of distributions arising from distributions of order statistics

Abstract: Consider starting from a symmetric distributionF on ℜ and generating a family of distributions from it by employing two parameters whose role is to introduce skewness and to vary tail weight. The proposal in this paper is a simple generalisation of the use of the collection of order statistic distributions associated withF for this purpose; an alternative derivation of this family of distributions is as the result of applying the inverse probability integral transformation to the beta distribution. General properties of the proposed family of distributions are explored. It is argued that two particular special cases are especially attractive because they appear to provide the most tractable instances of families with power and exponential tails; these are the skewt distribution and the logF distribution, respectively. Limited experience with fitting the distributions to data in their four-parameter form, with location and scale parameters added, is described, and hopes for their incorporation into complex modelling situations expressed. Extensions to the multivariate case and to ℜ+ are discussed, and links are forged between the distributions underlying the skewt and logF distributions and Tadikamalla and Johnson'sLU family.

Record Values Theory and Applications

Abstract: Chapter 1 Record Statistics Exponential Distribution Generalized Extreme Value Distributions Generalized Pareto Distribution Power Function Distribution Geometric Distribution Some Selected Distributions Additional Topics Complements and Problems Reference

Discriminating between gamma and generalized exponential distributions

Abstract: Recently the two-parameter generalized exponential distribution was introduced by the authors. It is observed that a generalized exponential distribution has several properties which are quite similar to a gamma distribution. It is also observed that a generalized exponential distribution can be used quite effectively in many situations where a skewed distribution is needed. In this paper, we use the ratio of the maximized likelihoods in choosing between a generalized exponential distribution and a gamma distribution. We obtain asymptotic distributions of the logarithm of the ratio of the maximized likelihoods under null hypotheses and use them to determine the sample size needed to discriminate between two overlapping families of distributions for a user specified probability of correct selection and a tolerance limit.

Poissonian Exponential Functionals, q-Series, q-Integrals, and the Moment Problem for log-Normal Distributions

Abstract: Moments formulae for the exponential functionals associated with a Poisson process provide a simple probabilistic access to the so-called q-calculus, as well as to some recent works about the moment problem for the log-normal distributions.

Adaptive model selection and assessment for exponential family distributions

Abstract: In many scientific and engineering problems, selecting the optimal model from a large pool of candidate models is important, particularly in data mining. In the literature, model assessment in the context of nonnormal distributions has not yet received a lot of attention. Indeed, many existing model selection criteria such as the Bayes information criterion and Cp may not be suitable for a situation in which the conditional mean and variance of the response are dependent, such as in generalized linear model regression. In this article we propose a new adaptive model selection criterion and construct an approximately unbiased Kullback–Leibler loss estimator for model assessment in the context of exponential family distributions. This permits comparing any arbitrary complex modeling procedures. Our proposal uses a concept called generalized degrees of freedom that generalizes the concept originally proposed for the normal distribution. The proposed procedure is implemented for the binomial and Poisson distr...

Small-area estimation based on natural exponential family quadratic variance function models and survey weights

Abstract: SUMMARY We propose pseudo empirical best linear unbiased estimators of small-area means based on natural exponential family quadratic variance function models when the basic data consist of survey-weighted estimators of these means, area-specific covariates and certain summary measures involving the weights. We also provide explicit approximate mean squared errors of these estimators in the spirit of Prasad & Rao (1990), and these estimators can be readily evaluated. A simulation study is undertaken to evaluate the performance of the proposed inferential procedure. We estimate also the proportion of poor children in the 5-17 years age-group for the different counties in one of the states in the United States.

Identification of models using failure rate and mean residual life of doubly truncated random variables

Abstract: In this paper, we study the relationship between the failure rate and the mean residual life of doubly truncated random variables. Accordingly, we develop characterizations for exponential, Pareto II and beta distributions. Further, we generalize the identities for the Pearson and the exponential family of distributions given respectively in Nair and Sankaran (1991) and Consul (1995). Applications of these measures in the context of lengthbiased models are also explored.

Generalized exponential distribution: Moments of order statistics

Abstract: In this paper, we consider the generalized exponential distribution (GED) with shape parameter α. We establish several recurrence relations satisfied by the single and the product moments for order statistics from the GED. The relationships can be written in terms of polygamma and hypergeometric functions and used in a simple recursive manner in order to compute the single and the product moments of all order statistics for all sample sizes.

Specifying interdependence in networked systems

Abstract: Realistic assessments of the reliability of networked systems, series and parallel systems being special cases, require that we account for interdependence between the component life-lengths. The key to doing this is the specification and use of a suitable probability model in two or more dimensions. Consequently, several multivariate probabilistic models have been proposed in the literature. Many of these models have marginal distributions that are exponential; the ones by Gumbel, and by Marshall and Olkin being some of the earliest and the best known. The purpose of this paper is two fold: The first purpose is to articulate the nature of dependence encapsulated by such models, using a perspective which is best appreciated by a user. Specifically, we anchor on the bivariate case, and focus attention on the conditional mean as a measure of dependence. The second purpose, motivated by the first, is to introduce a new family of multivariate distributions with exponential marginals, whose conditional mean fills a void in the general forms of the conditional means of the available models. The method of "copulas" is used to generate this new family of distributions. Attention is focused on the case of exponential marginals, because the notion of "hazard potentials" enables us to use multivariate distributions with exponential marginals as a seed for generating multivariate distributions with marginals other than the exponential.

The Möbius distribution on the disc

Abstract: A simple new family of distributions is proposed which has support the unit disc in two dimensions The density functions of the family are unimodal, monotonic or uniantimodal The bivariate symmetric beta distributions, which include the uniform distribution, are special cases, but many members of the family are skew The distributions have three parameters, one controlling orientation, one controlling degree of concentration and the third controlling skewness, or more precisely off-centredness Importantly, these parameters are globally orthogonal An illustrative example of fitting the model to data is given Conditional and marginal distributions are considered The new distributions are compared favourably with an earlier suggestion of the same author

Estimation of the Scale Parameter of Generalized Exponential Distribution Using Order Statistics

Abstract: In this paper we have developed an estimator for the scale parameter of generalized exponential distribution using an appropriate Ustatistic defined by the Best Linear Unbiased Estimator (BLUE) based on order statistics of a random sample of size 2 as the kernel. We have compared our estimator with the maximum likelihood estimator and an unbiased estimator based on sample mean.

A Non-Linear Exponential(NLINEX) Loss Function in Bayesian Analysis

Abstract: In this paper we have proposed a new loss function, namely, non-linear exponential(NLINEX) loss function, which is quite asymmetric in nature. We obtained the Bayes estimator under exponential(LINEX) and squared error(SE) loss functions. Moreover, a numerical comparison among the Bayes estimators of power function distribution under SE, LINEX, and NLINEX loss function have been made.

On the role of mixtures in Bayesian nonparametrics

Abstract: Riassunto: Combinazioni lineari convesse di distribuzioni (“modelli mistura”) sono ampiamente utilizzate, in diversi contesti applicativi, per modellare l’eterogeneit à d dati, o come strumento per ottenere modelli flessibili. In queste note si discute il ruolo dei modelli mistura nell’inferenza Bayesiana nonparametrica. Basandoci su risultati di Feller, presentiamo una procedura costruttiva di approssimazione di una funzione di ripartizione mediante misture. Ci ò fornisce un quadro di riferimento generale per studiare iniziali basate su misture per l’inferenza bayesiana nonparametrica. Si riprendono alcuni risultati nel caso unidimensionale, in particolare relativi alla propriet à di consistenza della distribuzione finale, e si suggeriscono estensioni al caso di dati multidimensionali.

Development of Markov random field models based on exponential family conditional distributions

Abstract: This article is concerned with construction of multivariate models through specifying con­ ditional distributions. Restricting attention to conditionals specified in exponential families, a class of exponential family conditional models may be formulated. Although this formula­ tion allows one to immediately construct and identify a valid joint distribution from a given set of exponential family conditionals, the form of the joint is too complicated. As a way to make an identified model more applicable, we compare it with an expanded form of the joint density called the negpotential function (Besag, 1974; Kaiser and Cressie, 2000) and presents a necessary form of each term in the negpotential function. Specifying a neighborhood sys­ tem, the obtained model results in a Markov random field (MRF). Our approach provides us with a flexible way of MRF construction that can deal with the entire class of exponential family distributions, and we use this result in development of spatial models for bounded-sum random variables. In an attempt to obtain a desirable dependence structure we propose a new exponential family conditional model by choosing appropriate sufficient statistics for the conditionals.

Prediction Intervals of an Ordered Observation from One-Parameter Exponential Distribution Based on Multiple Type II Censored Samples

Abstract: This paper provides a suitable pivotal quantity to present the prediction interval of the j(superscript th) future ordered-observation in a sample of size n from one-parameter exponential distribution in case of a multiple type II censored sample. In addition, we also discuss the approximate prediction interval, the best linear unbiased estimate and approximate maximum likelihood estimate of X(subscript j) based on the censored sample. As in application, the total duration time in a life test and the failure time of a j-out-of-n system may be predicated. Finally, a simulated study and three illustrative examples are included.

Conjugate priors represent strong pre-experimental assumptions

Abstract: . It is well known that Jeffreys’ prior is asymptotically least favorable under the entropy risk, i.e. it asymptotically maximizes the mutual information between the sample and the parameter. However, in this paper we show that the prior that minimizes (subject to certain constraints) the mutual information between the sample and the parameter is natural conjugate when the model belongs to a natural exponential family. A conjugate prior can thus be regarded as maximally informative in the sense that it minimizes the weight of the observations on inferences about the parameter; in other words, the expected relative entropy between prior and posterior is minimized when a conjugate prior is used.

Epidemic Change Model for the Exponential Family

Abstract: We assume that a sequence of independent observations are given from an exponential family. It is hypothesised that the sequence has the same natural parameter θ0. We would like to test if this natural parameter has been subjected to an epidemic change after an unknown point, for an unknown duration in the sequence. The likelihood ratio statistic for testing such an hypothesis is derived and then it's asymptotic null distribution is derived. We discuss the asymptotic behaviour of the maximum likelihood estimates of the change points. We prove that the asymptotic non-null distribution is of the likelihood ratio statistic is normal. Some special cases of the exponential family are considered in detail.

The Inference of Gini's Mean Difference

Abstract: In this paper, the probability density function, the men and variance of Gini’s mean difference are derived by using the characterization of exponential distribution. Some inferences for the distribution and its more related results will be studied and presented.

Martingales Defined by Reciprocals of Sums and Related Characterizations

Abstract: We prove that linearly transformed inverses of cumulative sums form backward martingales for gamma, inverse Gaussian, and Kendall and Borel–Tanner sequences of independent, identically distributed random variables. Conversely, a characterization of the family of these four distributions by linearity of regression of inverses of sums is obtained. The results in both directions are derived via the technique of variance functions of natural exponential families.

Selecting test of distribution in the Hinde-Demétrio family

Abstract: Consider the Hinde-Demetrio distributions concentrated on nonnegative integers. These distributions are exponential dispersion models characterized by the unit variance function Vp(�) = � + � p , � > 0 and where p 2 f2;3;���g is associated to one model of the family. It is known that their probability mass functions pk do not have, in general, an explicit expression. We show that the ratios rk = k pk=pk 1; k = (1)

Characterizations of distributions by linear forms of order statistics

Abstract: Exponential and normal distributions will be characterized by the independence of linear forms and in order statistics X 1, n ≤ … ≤ X n, n . As a by-product, we show that no L-statistic with a i ≥ 0 and can be a complete and sufficient maximum likelihood estimator of a location parameter except from the arithmetic mean X¯ in the normal and X 1, n (X n, n ) in the exponential (reflected exponential) case.

Information closures of exponential families

Abstract: The closure in variation distance of any subfamily with a convex set of canonical parameters of an exponential family has been described previously, in terms of the concept of extension of the full exponential family. This contribution describes the closure in reversed information divergence (rI-closure), via variation closures of auxiliary subfamilies. Also, variation convergence and rI-convergence in the extension are characterized.

A Test for Two-Sample Problem Based on Sample Spacings

Abstract: A criterion W for testing a difference in scale between two continuous populations based on sample spacings is proposed. Under the null hypothesis the exact probability distribution of the test statistic and its expectation and variance are derived and critical values are tabled. Based on Monte Carlo estimates the power of the test W and two other tests are compared for samples from uniform, normal and double exponential distributions, and W and two other tests are compared for samples from chi-square and exponential distributions. Power results, obtained by simulation, indicate that the test based on W is better than the other tests for testing a difference in scale when two samples are from light-tailed symmetric distributions or skewed distributions with left most end points at zero.

A local probability exponential inequality for the large deviation of an empirical process indexed by an unbounded class of functions and its application

Abstract: A local probability exponential inequality for the tail of large deviation of an empirical process over an unbounded class of functions is proposed and studied. A new method of truncating the original probability space and a new symmetrization method are given. Using these methods, the local probability exponential inequalities for the tails of large deviations of empirical processes with non-i.i.d. independent samples over unbounded class of functions are established. Some applications of the inequalities are discussed. As an additional result of this paper, under the conditions of Kolmogorov theorem, the strong convergence results of Kolmogorov on sums of non-i.i.d. independent random variables are extended to the cases of empirical processes indexed by unbounded classes of functions, the local probability exponential inequalities and the laws of the logarithm for the empirical processes are obtained.