Showing papers on "Natural exponential family published in 2011"

A new family of generalized distributions

Abstract: Kumaraswamy [Generalized probability density-function for double-bounded random-processes, J. Hydrol. 462 (1980), pp. 79–88] introduced a distribution for double-bounded random processes with hydrological applications. For the first time, based on this distribution, we describe a new family of generalized distributions (denoted with the prefix ‘Kw’) to extend the normal, Weibull, gamma, Gumbel, inverse Gaussian distributions, among several well-known distributions. Some special distributions in the new family such as the Kw-normal, Kw-Weibull, Kw-gamma, Kw-Gumbel and Kw-inverse Gaussian distribution are discussed. We express the ordinary moments of any Kw generalized distribution as linear functions of probability weighted moments (PWMs) of the parent distribution. We also obtain the ordinary moments of order statistics as functions of PWMs of the baseline distribution. We use the method of maximum likelihood to fit the distributions in the new class and illustrate the potentiality of the new model with a...

An extension of the exponential distribution

Abstract: A generalization of the exponential distribution is presented. The generalization always has its mode at zero and yet allows for increasing, decreasing and constant hazard rates. It can be used as an alternative to the gamma, Weibull and exponentiated exponential distributions. A comprehensive account of the mathematical properties of the generalization is presented. A real data example is discussed to illustrate its applicability.

Instability, Sensitivity, and Degeneracy of Discrete Exponential Families.

Abstract: A number of discrete exponential family models for dependent data, first and foremost relational data, have turned out to be near-degenerate and problematic in terms of Markov chain Monte Carlo (MCMC) simulation and statistical inference. I introduce the notion of instability with an eye to characterize, detect, and penalize discrete exponential family models that are near-degenerate and problematic in terms of MCMC simulation and statistical inference. I show that unstable discrete exponential family models are characterized by excessive sensitivity and near-degeneracy. In special cases, the subset of the natural parameter space corresponding to nondegenerate distributions and mean-value parameters far from the boundary of the mean-value parameter space turns out to be a lower-dimensional subspace of the natural parameter space. These characteristics of unstable discrete exponential family models tend to obstruct MCMC simulation and statistical inference. In applications to relational data, I show that d...

Geometry of q-Exponential Family of Probability Distributions

Abstract: The Gibbs distribution of statistical physics is an exponential family of probability distributions, which has a mathematical basis of duality in the form of the Legendre transformation. Recent studies of complex systems have found lots of distributions obeying the power law rather than the standard Gibbs type distributions. The Tsallis q-entropy is a typical example capturing such phenomena. We treat the q-Gibbs distribution or the q-exponential family by generalizing the exponential function to the q-family of power functions, which is useful for studying various complex or non-standard physical phenomena. We give a new mathematical structure to the q-exponential family different from those previously given. It has a dually flat geometrical structure derived from the Legendre transformation and the conformal geometry is useful for understanding it. The q-version of the maximum entropy theorem is naturally induced from the q-Pythagorean theorem. We also show that the maximizer of the q-escort distribution is a Bayesian MAP (Maximum A posteriori Probability) estimator.

The exponentiated exponential distribution: a survey

Abstract: The exponentiated exponential distribution, a most attractive generalization of the exponential distribution, introduced by Gupta and Kundu (Aust. N. Z. J. Stat. 41:173–188, 1999) has received widespread attention. It appears, however, that many mathematical properties of this distribution have not been known or have not been known in simpler/general forms. In this paper, we provide a comprehensive survey of the mathematical properties. We derive expressions for the moment generating function, characteristic function, cumulant generating function, the nth moment, the first four moments, variance, skewness, kurtosis, the nth conditional moment, the first four cumulants, mean deviation about the mean, mean deviation about the median, Bonferroni curve, Lorenz curve, Bonferroni concentration index, Gini concentration index, Renyi entropy, Shannon entropy, cumulative residual entropy, Song’s measure, moments of order statistics, L moments, asymptotic distribution of the extreme order statistics, reliability, distribution of the sum of exponentiated exponential random variables, distribution of the product of exponentiated exponential random variables and the distribution of the ratio of exponentiated exponential random variables. We also discuss estimation by the method of maximum likelihood, including the case of censoring, and provide simpler expressions for the Fisher information matrix than those given by Gupta and Kundu. It is expected that this paper could serve as a source of reference for the exponentiated exponential distribution and encourage further research.

On Rényi and Tsallis entropies and divergences for exponential families

Abstract: Many common probability distributions in statistics like the Gaussian, multinomial, Beta or Gamma distributions can be studied under the unified framework of exponential families. In this paper, we prove that both R\'enyi and Tsallis divergences of distributions belonging to the same exponential family admit a generic closed form expression. Furthermore, we show that R\'enyi and Tsallis entropies can also be calculated in closed-form for sub-families including the Gaussian or exponential distributions, among others.

Towards the geometry of estimation of distribution algorithms based on the exponential family

Abstract: In this paper we present a geometrical framework for the analysis of Estimation of Distribution Algorithms (EDAs) based on the exponential family. From a theoretical point of view, an EDA can be modeled as a sequence of densities in a statistical model that converges towards distributions with reduced support. Under this framework, at each iteration the empirical mean of the fitness function decreases in probability, until convergence of the population. This is the context of stochastic relaxation, i.e., the idea of looking for the minima of a function by minimizing its expected value over a set of probability densities. Our main interest is in the study of the gradient of the expected value of the function to be minimized, and in particular on how its landscape changes according to the fitness function and the statistical model used in the relaxation. After introducing some properties of the exponential family, such as the description of its topological closure and of its tangent space, we provide a characterization of the stationary points of the relaxed problem, together with a study of the minimizing sequences with reduced support. The analysis developed in the paper aims to provide a theoretical understanding of the behavior of EDAs, and in particular their ability to converge to the global minimum of the fitness function. The theoretical results of this paper, beside providing a formal framework for the analysis of EDAs, lead to the definition of a new class algorithms for binary functions optimization based on Stochastic Natural Gradient Descent (SNGD), where the estimation of the parameters of the distribution is replaced by the direct update of the model parameters by estimating the natural gradient of the expected value of the fitness function.

General results for the beta-modified Weibull distribution

Abstract: We study in detail the so-called beta-modified Weibull distribution, motivated by the wide use of the Weibull distribution in practice, and also for the fact that the generalization provides a continuous crossover towards cases with different shapes. The new distribution is important since it contains as special sub-models some widely-known distributions, such as the generalized modified Weibull, beta Weibull, exponentiated Weibull, beta exponential, modified Weibull and Weibull distributions, among several others. It also provides more flexibility to analyse complex real data. Various mathematical properties of this distribution are derived, including its moments and moment generating function. We examine the asymptotic distributions of the extreme values. Explicit expressions are also derived for the chf, mean deviations, Bonferroni and Lorenz curves, reliability and entropies. The estimation of parameters is approached by two methods: moments and maximum likelihood. We compare by simulation the perform...

Absolute continuous bivariate generalized exponential distribution

Abstract: Generalized exponential distribution has been used quite effectively to model positively skewed lifetime data as an alternative to the well known Weibull or gamma distributions. In this paper we introduce an absolute continuous bivariate generalized exponential distribution by using a simple transformation from a well known bivariate exchangeable distribution. The marginal distributions of the proposed bivariate generalized exponential distributions are generalized exponential distributions. The joint probability density function and the joint cumulative distribution function can be expressed in closed forms. It is observed that the proposed bivariate distribution can be obtained using Clayton copula with generalized exponential distribution as marginals. We derive different properties of this new distribution. It is a five-parameter distribution, and the maximum likelihood estimators of the unknown parameters cannot be obtained in closed forms. We propose some alternative estimators, which can be obtained quite easily, and they can be used as initial guesses to compute the maximum likelihood estimates. One data set has been analyzed for illustrative purposes. Finally we propose some generalization of the proposed model.

Expectation identities of lower generalized order statistics from generalized exponential distribution and a characterization

Abstract: In this paper, we study the explicit expressions and some recurrence relations for single and product moments of lower generalized order statistics from generalized exponential distribution. The results include as particular cases the above relations for moments of order statistics and lower records. Further, using a recurrence relation for single moments we obtain a characterization of generalized exponential distribution.

A new family of distributions based on probability generating functions

Abstract: This paper examines a method for generating new classes of distributions which arise naturally in practice. The generated classes of distributions include the well known Marshall and Olkin class of distributions and can be thought of as mixing two discrete distributions or a discrete distribution with an absolutely continuous distribution. Properties of these classes of distributions are derived and a number of existing results in the literature are recovered as special cases. Finally, failure rates for a special class of distributions which are obtained when the discrete distribution is assumed to have a Harris form are given.

Exponentiated Modified Weibull Distribution

Abstract: In this paper we consider the exponentiated modified Weibull distribution. The modified Weibull distribution, Weibull distribution and the exponentiated exponential distribution are found to be particular cases of this family. We derive the analytical shape of the corresponding density functions and hazard rate functions. The rth moment and the moment generating function are determined. Finally the distribution of order statistics and the least squares estimators of the parameters are discussed.

A new bivariate distribution with weighted exponential marginals and its multivariate generalization

Abstract: Gupta and Kundu (Statistics 43:621–643, 2009) recently introduced a new class of weighted exponential distribution. It is observed that the proposed weighted exponential distribution is very flexible and can be used quite effectively to analyze skewed data. In this paper we propose a new bivariate distribution with the weighted exponential marginals. Different properties of this new bivariate distribution have been investigated. This new family has three unknown parameters, and it is observed that the maximum likelihood estimators of the unknown parameters can be obtained by solving a one-dimensional optimization procedure. We obtain the asymptotic distribution of the maximum likelihood estimators. Small simulation experiments have been performed to see the behavior of the maximum likelihood estimators, and one data analysis has been presented for illustrative purposes. Finally we discuss the multivariate generalization of the proposed model.

Estimation in Functional Regression for General Exponential Families

Abstract: This paper studies a class of exponential family models whose canonical parameters are specified as linear functionals of an unknown infinite-dimensional slope function. The optimal minimax rates of convergence for slope function estimation are established. The estimators that achieve the optimal rates are constructed by constrained maximum likelihood estimation with parameters whose dimension grows with sample size. A change-of-measure argument, inspired by Le Cam's theory of asymptotic equivalence, is used to eliminate the bias caused by the non-linearity of exponential family models.

The Exponential Family and Statistical Applications

Abstract: The exponential family is a practically convenient and widely used unified family of distributions on finite-dimensional Euclidean spaces parametrized by a finite-dimensional parameter vector. Specialized to the case of the real line, the exponential family contains as special cases most of the standard discrete and continuous distributions that we use for practical modeling, such as the normal, Poisson, binomial, exponential, Gamma, multivariate normal, and so on. The reason for the special status of the exponential family is that a number of important and useful calculations in statistics can be done all at one stroke within the framework of the exponential family. This generality contributes to both convenience and larger-scale understanding. The exponential family is the usual testing ground for the large spectrum of results in parametric statistical theory that require notions of regularity or Cramer–Rao regularity. In addition, the unified calculations in the exponential family have an element of mathematical neatness. Distributions in the exponential family have been used in classical statistics for decades. However, it has recently obtained additional importance due to its use and appeal to the machine learning community. A fundamental treatment of the general exponential family is provided in this chapter. Classic expositions are available in Barndorff-Nielsen (1978), Brown (1986), and Lehmann and Casella (1998). An excellent recent treatment is available in Bickel and Doksum (2006).

Moment Distributions of Phase Type

Abstract: Moment distributions of phase-type and matrix-exponential distributions are shown to remain within their respective classes. We provide a probabilistic phase-type representation for the former case and an alternative representation, with an analytically appealing form, for the latter. First order moment distributions are of special interest in areas like demography and economics, and we calculate explicit formulas for the Lorenz curve and Gini index used in these disciplines.

The Effect of Non-normality in the Power Exponential Distributions

Abstract: As an alternative to the multivariate normal distribution we have dealt with a wider class of distributions, including the normal, that considers slightly different tail behavior than the normal tail. This is the multivariate exponential power family of distributions with a kurtosis parameter to give the possible forms of the distributions. To measure distribution deviations the Kullback-Leibler divergence will be used as an asymmetric dissimilarity measure from an information-theoretic basis. Thus, a local quantitative description of the non-normality could be established for joint distributions in this family as well as the impact this perturbation causes in the marginal and conditional distributions.

Data Set Models and Exponential Families in Statistical Physics and Beyond

Abstract: The exponential family of models is defined in a general setting, not relying on probability theory. Some results of information geometry are shown to remain valid. Exponential families both of classical and of quantum mechanical statistical physics fit into the new formalism. Other less obvious applications are predicted. For instance, quantum states can be modeled as points in a classical phase space and the resulting model belongs to the exponential family.

Characterizations of distributions via order statistics with random exponential shifts

Abstract: A class of probability distributions is characterized via equalities in law between two order statistics shifted by independent exponential variables. An explicit formula for the quintile function of the identified family of distributions is obtained. The results extend some known characterizations of exponential and logistic distributions.

Generalized Beta Generated-II Distributions

Abstract: A family of univariate distributions, generated by beta random variables, has been proposed by Jones [9]. This broad family of univariate distributions has received considerable attention in the recent literature since it possesses great flexibility while fitting symmetric as well as skewed models with varying tail weights. This paper introduces and studies a new broad class of univariate distributions which is defined by means of a generalized beta distribution and includes Jones family as a particular case. Some properties of the proposed class of distributions are discussed. These properties include its moments, generalized moments, representation and relationship with other distributions, expressions for Shannon entropy. Two examples are given and the paper is completed with some conclusions.

Theoretical Analysis of Inverse Generalized Exponential Models

Abstract: This article presents the theoretical analysis of Inverse Generalized Exponential Models. The shapes of the properties of the Inverse Generalized Exponential distribution are discussed. Inverse moments of Inverse Generalized Exponential distribution are derived. In this article we have presented the relationship between shape parameter and other properties such probability distribution, distribution function, reliability function, hazard function and cumulative hazard function, mean, median, mode, variance, coefficient of variation, coefficient of skewness and coefficient of kurtosis models are presented graphically and mathematically. Here we compare these relevant parameters such as shape, scale parameters by using Monte carol simulation. The Inverse Generalized Exponential models are the reliability models can be used in the reliability engineering discipline. We have developed this new reliability model and are the generalization of the inverse exponential distribution. The Inverse Generalized Exponential distribution approaches to the inverse exponential distribution when 1   and 0 0  t . The Inverse Generalized Exponential models can be used as a standard in reliability for modeling time-dependent failure data. The Inverse Generalized Exponential distribution can be used to model a variety of failure characteristics such as infant mortality, random failures, wear-out, and failure-free periods. The Inverse Generalized Exponential distribution can also be used to determine the cost effectiveness and maintenance periods of reliability-centered maintenance activities. This paper focuses on the Theoretical analysis of the Inverse Generalized Exponential distribution's to model in which some operational time has already been accumulated for the equipment of interest. This paper presents the relationship between shape parameter and other properties such as ) (

Discrete Exponential Bayesian Networks: An Extension of Bayesian Networks to Discrete Natural Exponential Families

Abstract: In this paper, we develop the notion of discrete exponential Bayesian network, parametrization of Bayesian networks (BNs) using more general discrete quadratic exponential families instead of usual multinomial ones. We then introduce a family of prior distributions which generalizes the Dirichlet prior classically used with discrete Bayesian network. We develop the posterior distribution for our discrete exponential BNs leading to bayesian estimations of the parameters of our models and one new scoring function extending the Bayesian Dirichlet score used for structure learning. These theoretical results are finally illustrated for Poisson and Negative Binomial BNs.

Some characterizations of families of distributions, including logistic and exponential ones, by properties of order statistics

Abstract: New characterizations of distributions based on properties of the maximal order statistics are obtained. The families of distributions that are characterized by some properties of maxima include exponential and logistic distributions as partial cases. Bibliography: 4 titles.

Bayes sequential estimation for a particular exponential family of distributions under LINEX loss

Abstract: The problem of sequentially estimating an unknown distribution parameter of a particular exponential family of distributions is considered under LINEX loss function for estimation error and a cost c > 0 for each of an i.i.d. sequence of potential observations X1, X2, . . . A Bayesian approach is adopted and conjugate prior distributions are assumed. Asymptotically pointwise optimal and asymptotically optimal procedures are derived.