Showing papers on "Natural exponential family published in 2008"
TL;DR: In this paper, it was shown that a multivariate exponential power distribution is a scale mixture of normal distributions, with respect to a probability distribution function, when its kurtosis parameter belongs to the interval (0, 1).
Abstract: This paper shows that a multivariate exponential power distribution is a scale mixture of normal distributions, with respect to a probability distribution function, when its kurtosis parameter belongs to the interval (0, 1]. The corresponding mixing probability distribution function is presented. This result is used to design, through a Bayesian hierarchical model, an algorithm to generate samples of the posterior distribution; this is applied to a problem of quantitative genetics.
66 citations
TL;DR: In this paper, the problem of estimating the stress-strength reliability when the available data is in the form of record values was considered and the maximum likelihood estimators and the associated confidence intervals were derived.
Abstract: We consider the problem of estimating the stress-strength reliability when the available data is in the form of record values. The one parameter and two parameters exponential distribution are considered. In the case of two parameters exponential distributions we considered the case where the location parameter is common and the case where the scale parameter is common. The maximum likelihood estimators and the associated confidence intervals are derived.
62 citations
TL;DR: In this article, the authors present two types of symmetric scale mixture probability distributions which include the normal, Student t, Pearson Type VII, variance gamma, exponential power, uniform power and generalized t distributions.
Abstract: Summary
This paper presents two types of symmetric scale mixture probability distributions which include the normal, Student t, Pearson Type VII, variance gamma, exponential power, uniform power and generalized t (GT) distributions. Expressing a symmetric distribution into a scale mixture form enables efficient Bayesian Markov chain Monte Carlo (MCMC) algorithms in the implementation of complicated statistical models. Moreover, the mixing parameters, a by-product of the scale mixture representation, can be used to identify possible outliers. This paper also proposes a uniform scale mixture representation for the GT density, and demonstrates how this density representation alleviates the computational burden of the Gibbs sampler.
58 citations
TL;DR: In this paper, the generalized order statistics from heterogeneous distributions are introduced and their properties are investigated, and the case of proportional hazards leads to an interesting connection to the model of generalized ordering statistics.
Abstract: Progressively censored order statistics from heterogeneous distributions are introduced and their properties are investigated. After deriving the joint density function, some properties are established. In particular, the case of proportional hazards leads to an interesting connection to the model of generalized order statistics. Finally, the special case of exponential distribution is considered and some known results are generalized to this heterogeneous case, and their implications to robustness are highlighted.
37 citations
TL;DR: A sampling plan with a polynomial loss function for the exponential distribution based on Type-I and Type-II hybrid censored samples is considered, and a robustness study reveals that the proposed optimal sampling plans are quite robust.
Abstract: A sampling plan with a polynomial loss function for the exponential distribution is considered. From the distribution of the maximum likelihood estimator of the mean of an exponential distribution based on Type-I and Type-II hybrid censored samples, we obtain an explicit expression for the Bayes risk of a sampling plan with a quadratic loss function. Some numerical examples and comparisons are given to illustrate the effectiveness of the proposed method, and a robustness study reveals that the proposed optimal sampling plans are quite robust.
35 citations
TL;DR: In this paper, the linear exponential distribution (exponential and Rayleigh distributions) is considered and explicit expressions for single and product moments of ordinary order statistics and upper record values have been obtained.
Abstract: This article is concerned with the linear exponential distribution (exponential and Rayleigh distributions). Recurrence relations for single and product moments of generalized order statistics have been derived. Single and product moments of ordinary order statistics and upper k-records cases have been discussed as special cases from generalized order statistics. Explicit expressions for single and product moments of ordinary order statistics and upper record values have been obtained.
28 citations
TL;DR: In this article, the maximum likelihood estimation procedure of the parameters of the generalized exponential distribution when the data are left censored is considered, and the estimators of the unknown para-meters and the Fisher information matrix are obtained.
Abstract: The generalized exponential distribution proposed by Gupta and Kundu [Gupta, R.D and Kundu, D., 1999, Generalized exponential distributions. Australian and New Zealand Journal of Statistics, 41(2), 173–188.] is an important lifetime distribution in survival analysis. In this paper, we consider the maximum likelihood estimation procedure of the parameters of the generalized exponential distribution when the data are left censored. We obtain the maximum likelihood estimators of the unknown para-meters and the Fisher information matrix. Simulation studies are carried out to observe the performance of the estimators in small sample.
26 citations
TL;DR: Conditions on the probability density are derived which determine whether or not the distribution of the digits of all the unnormalized differences converges to Benford's law, shifted exponential behavior, or oscillates between the two, and it is shown that the Pareto distribution leads to oscillating behavior.
Abstract: Fix a base and let have the standard exponential distribution; the distribution of digits of base is known to be very close to Benford's law. If there exists a such that the distribution of digits of times the elements of some set is the same as that of , we say that set exhibits shifted exponential behavior base Let be i.i.d.r.v. If the 's are Unif, then as the distribution of the digits of the differences between adjacent order statistics converges to shifted exponential behavior. If instead 's come from a compactly supported distribution with uniformly bounded first and second derivatives and a second-order Taylor series expansion at each point, then the distribution of digits of any consecutive differences and all normalized differences of the order statistics exhibit shifted exponential behavior. We derive conditions on the probability density which determine whether or not the distribution of the digits of all the unnormalized differences converges to Benford's law, shifted exponential behavior, or oscillates between the two, and show that the Pareto distribution leads to oscillating behavior.
24 citations
01 Jan 2008
TL;DR: In this article, a new class of bivariate Gompertz distributions is presented, which is based on a latent random variable with exponential distribution, and a mixture of the suggested bivariate distributions is also derived.
Abstract: A new class of bivariate Gompertz distributions is presented in this paper. The model introduced here is of Marshall-Olkin type. The used procedure is based on a latent random variable with exponential distribution. A mixture of the suggested bivariate distributions is also derived. The obtained results in this paper generalize those of MarshallOlkin bivariate exponential distribution and other present in the literature.
21 citations
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TL;DR: This article proposed a generalized linear model with a linear predictor and a link function for the mean of response Y as a function of predictors X. The model has a similar level of flexibility as the proportional odds model.
Abstract: I propose a new class of generalized linear models. As with the existing models, these new models are specified via a linear predictor and a link function for the mean of response Y as a function of predictors X. However, here, the “baseline” distribution of Y when the linear predictor is zero is left unspecified and is estimated from the data. The response distribution when the linear predictor differs from zero is then generated via exponential tilting of the baseline distribution, yielding a response model that is a member of the natural exponential family, with corresponding canonical link and variance functions. The resulting model has a similar level of flexibility as the proportional odds model. Maximum likelihood estimators are developed for response distribution with finite support, and the new model is studied and illustrated through simulations and example analyses from aging and psychiatry research.
19 citations
TL;DR: In this paper, the Weibull distribution manifold and the generalized exponential distribution manifold were investigated and their geometric structures were obtained, and the geometric structures of the non-exponential distribution manifold was analyzed.
Abstract: Investigating the geometric structures of the distribution manifolds is a basic task in information geometry. However, by so far, most works are on the distribution manifolds of exponential family. In this paper, we investigate two non-exponential distribution manifolds —the Weibull distribution manifold and the generalized exponential distribution manifold. Then we obtain their geometric structures.
TL;DR: In this paper, a generalization of the Gindikin theorem for natural exponential families with Laplace transforms with variance function of type 1 pm ⊗ m-φ(m)M v, where m v is the symmetric matrix associated to the quadratic form v and m → φ(n) is a real function.
Abstract: We find the distributions in R n for the independent random variables X and Y such that E(X|X + Y) = a(X + Y) and E(g(X)|X + Y) = bq(X + Y) where q runs through the set of all quadratic forms on R n orthogonal to a given quadratic form v. The essential part of this class is provided by distributions with Laplace transforms (1 - 2(c, s) + v(s)) -p that we describe completely, obtaining a generalization of a Gindikin theorem. This leads to the classification of natural exponential families with the variance function of type 1 pm ⊗ m-φ(m)M v , where M v is the symmetric matrix associated to the quadratic form v and m → φ(m) is a real function. These natural exponential families extend the classical Wishart distributions on Lorentz cones already considered by Jensen, and later on by Faraut and Koranyi.
TL;DR: The generalized exponential function is proposed to approximate the underlying Weibull or Gamma distributions, and then solves for the RF using Laplace transform.
Abstract: When the inter-renewal time follows the Weibull or the Gamma distribution, the analytical renewal function (RF) usually is not tractable, and approximation method has been used. Instead of approximating RF directly, this article proposes the generalized exponential function to approximate the underlying Weibull or Gamma distributions, and then solves for the RF using Laplace transform. Parameters for generalized exponential function can be obtained by solving a simple optimization problem. The method can obtain accurate RF approximations where the inter-renewals follow Weibull or Gamma distributions, yet analytical RF is still desirable. Comprehensive analysis shows that the new model is mathematically accurate and computationally convenient to approximate the Weibull RF given its shape parameter β ∈ [1, 5]. For the Gamma distribution, the proposed model can achieve good approximations when the Gamma shape parameter k ∈ [1, 10]. These are the typical ranges of shape parameters when modeling the product re...
TL;DR: In this article, three different definitions of the concept of mean square exponential stability in mean square are introduced and it is shown that they are not always equivalent and that they may not always be equivalent.
Abstract: The problem of the mean square exponential stability for a class of discrete-time linear stochastic systems subject to independent random perturbations and Markovian switching is investigated. The case of the linear systems whose coefficients depend both to present state and the previous state of the Markov chain is considered. Three different definitions of the concept of exponential stability in mean square are introduced and it is shown that they are not always equivalent. One definition of the concept of mean square exponential stability is done in terms of the exponential stability of the evolution defined by a sequence of linear positive operators on an ordered Hilbert space. The other two definitions are given in terms of different types of exponential behavior of the trajectories of the considered system. In our approach the Markov chain is not prefixed. The only available information about the Markov chain is the sequence of probability transition matrices and the set of its states. In t...
01 Jan 2008
TL;DR: In this article, a two-parameter family of distri- butions is introduced, which includes the ordinary exponential distribution as a special case, and several statistical and reliability aspects of this model are explored.
Abstract: This paper introduces a two-parameter family of distri- butions which includes the ordinary exponential distribution as a special case. This distribution exhibits monotone hazard rate and may be a competitor to the families of two parameter gamma and Weibull distributions. Various statistical and reliability aspects of this model is explored. Several numerical examples based on real data show the ∞exibility of the new distribution for modeling proposes.
05 Jul 2008
TL;DR: This work presents a new, computationally efficient, learning algorithm based on an approximate likelihood function that can be interpreted as attempting to induce stationary distributions of observations, features and states which match their empirically observed counterparts.
Abstract: Exponential Family PSR (EFPSR) models capture stochastic dynamical systems by representing state as the parameters of an exponential family distribution over a shortterm window of future observations. They are appealing from a learning perspective because they are fully observed (meaning expressions for maximum likelihood do not involve hidden quantities), but are still expressive enough to both capture existing models and predict new models. While maximum-likelihood learning algorithms for EFPSRs exist, they are not computationally feasible. We present a new, computationally efficient, learning algorithm based on an approximate likelihood function. The algorithm can be interpreted as attempting to induce stationary distributions of observations, features and states which match their empirically observed counterparts. The approximate likelihood, and the idea of matching stationary distributions, may apply to other models.
TL;DR: In this article, a test statistic is constructed based on CDF-transformed observations and the corresponding moments of arbitrary positive order for generalized exponential distributions, and the proposed test compares well with standard methods based on the empirical distribution function.
Abstract: Recently, many standard families of distributions have been generalized by exponentiating their cumulative distribution function (CDF). In this paper, test statistics are constructed based on CDF–transformed observations and the corresponding moments of arbitrary positive order. Simulation results for generalized exponential distributions show that the proposed test compares well with standard methods based on the empirical distribution function.
TL;DR: A review of the continuous and discrete distributions introduced by the eminent Professors Marshall and Olkin is provided in this article, where the topics reviewed include: bivariate geometric distribution, extreme value behavior, bivariate negative binomial distribution, Bivariate exponential distribution, concomitants, reliability, distributions of sums and ratios, Ryu's bivariate exponential and Weibull distributions some hitherto unknown results about these distributions are also mentioned
Abstract: A review is provided of the continuous and discrete distributions introduced by the eminent Professors Marshall and Olkin The topics reviewed include: bivariate geometric distribution, extreme value behavior, bivariate negative binomial distribution, bivariate exponential distribution, concomitants, reliability, distributions of sums and ratios, Ryu’s bivariate exponential distribution, bivariate Pareto distribution and generalized exponential and Weibull distributions Some hitherto unknown results about these distributions are also mentioned
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10 Jul 2008TL;DR: In this paper, an extension of record models, well known as k-records, is considered, and Bayesian estimation as well as prediction based on krecords are presented when the underlying distribution is assumed to have a general form.
Abstract: In this paper, an extension of record models, well known as k-records, is considered. Bayesian estimation as well as prediction based on k-records are presented when the underlying distribution is assumed to have a general form. The proposed procedure is applied to the Exponential, Weibull and Pareto models in one parameter case. Also, the two-parameter Exponential distribution, when both parameters are unknown, is studied in more details.
Since the ordinary record values are contained in the k-records, by putting k = 1, the results for usual records can be obtained as special case.
01 Jan 2008
TL;DR: In this paper, the problem of pointwise estimation of a regression function under certain shape constraints, using a number of different statistics that can be viewed as measures of discrepancy from a postulated null hypothesis, is addressed.
Abstract: We address the problem of pointwise estimation of a regression function under certain shape constraints, using a number of different statistics that can be viewed as measures of discrepancy from a postulated null hypothesis. Pointwise confidence sets are obtained via the usual inversion technique that exploits the duality between construction of confidence sets for a parameter of interest and testing pointwise hypotheses about that parameter. Monotonicity, unimodality and U–shapes are considered. A major advantage of these proposed methods lies in the fact that the statistics of interest are approximately pivotal for large sample sizes and therefore enable inference to be carried out without the need to estimate difficult nuisance parameters. Multivariate generalizations are briefly discussed.
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TL;DR: In this article, the assumption of unimodality for variables with distributions belonging to the two-parameter exponential family with known or constant dispersion parameter was discussed and a non-parametric method based on monotonicity properties was proposed.
Abstract: In this paper we discuss statistical methods for curve-estimation under the assumption of unimodality for variables with distributions belonging to the two-parameter exponential family with known or constant dispersion parameter. We suggest a non-parametric method based on monotonicity properties. The method is applied to Swedish data on laboratory verified diagnoses of influenza and data on inflation from an episode of hyperinflation in Bulgaria.
TL;DR: It is proved that for every probability measure μ and every λ
Abstract: Abstract We consider the problem of the asymptotic size of the random maximum-weight matching of a sparse random graph, which we translate into dynamics of the operator in the space of distribution functions. A tight condition for the uniqueness of the globally attracting fixed point is provided, which extends the result of Karp and Sipser [Maximum matchings in sparse random graphs. 22nd Ann. Symp. on Foundations of Computer Science (Nashville, TN, 28–30 October, 1981). IEEE, New York, 1981, pp. 364–375] from deterministic weight distributions (Dirac measures μ) to general ones. Given a probability measure μ which corresponds to the weight distribution of a link of a random graph, we form a positive linear operator Φμ (convolution) on distribution functions and then analyze a family of its exponents, with parameter λ, which corresponds to the connectivity of a sparse random graph. The operator 𝕋 relates the distribution F on the subtrees to the distribution 𝕋F on the node of the tree by 𝕋F=exp (−λΦμF). We prove that for every probability measure μ and every λe, there is no fixed point attractor. We further establish that the uniqueness of the invariant measure of the underlying operator is not a monotone property of the average connectivity; this parallels similar non-monotonicity results in the statistical physics context.
03 Sep 2008
TL;DR: This paper proposes a method to apply the exponential family PCA to mixture models and finds a sub-optimal solution by linear programming formulation.
Abstract: Dimension reduction for a set of distribution parameters has been important in various applications of datamining. The exponential family PCA has been proposed for that purpose, but it cannot be directly applied to mixture models that do not belong to an exponential family. This paper proposes a method to apply the exponential family PCA to mixture models. A key idea is to embed mixtures into a space of an exponential family. The problem is that the embedding is not unique, and the dimensionality of parameter space is not constant when the numbers of mixture components are different. The proposed method finds a sub-optimal solution by linear programming formulation.
TL;DR: In this paper, two measures to highlight the possible effect of an observation on the UMVU estimate are proposed: conditional bias and the asymptotic mean sensitivity curve (AMSC), which depend on parametric function under consideration at the true and unknown value of the parameter.
TL;DR: This paper developed empirical Bayes (EB) confidence intervals for population means with distributions belonging to the natural exponential family-quadratic variance function (NEF-QVF) family when the sample size for a particular population is moderate or large.
Abstract: The paper develops empirical Bayes (EB) confidence intervals for population means with distributions belonging to the natural exponential family-quadratic variance function (NEF-QVF) family when the sample size for a particular population is moderate or large. The basis for such development is to find an interval centred around the posterior mean which meets the target coverage probability asymptotically, and then show that the difference between the coverage probabilities of the Bayes and EB intervals is negligible up to a certain order. The approach taken is Edgeworth expansion so that the sample sizes from the different populations need not be significantly large. The proposed intervals meet the target coverage probabilities asymptotically, and are easy to construct. We illustrate use of these intervals in the context of small area estimation both through real and simulated data. The proposed intervals are different from the bootstrap intervals. The latter can be applied quite generally, but the order of accuracy of these intervals in meeting the desired coverage probability is unknown.
01 Jan 2008
Abstract: The paper is devoted to some functional inequalities related to the exponential mapping.
TL;DR: In this article, it was shown that the uniformly minimum variance unbiased (UMVU) estimator of the generalized variance always exists for any natural exponential family, however, this estimator is often difficult to obtain.
Abstract: It has been shown that the uniformly minimum variance unbiased (UMVU) estimator of the generalized variance always exists for any natural exponential family. In practice, however, this estimator is often di¢ cult to obtain. This paper explicitly identi…es the results in complete bivariate and symmetric multivariate gamma models, which are diagonal quadratic exponential families. For the non-independent multivariate gamma models, it is then pointed out that the UMVU and the maximum likelihood estimators are not proportional as conjectured for models belonging in certain quadratic exponential families. AMS 2000 subject classi…cation: 62F10; 62H12; 62H99
TL;DR: In this article, a matrix formula for second-order covariances of maximum likelihood estimators in exponential family nonlinear models was given, thus generalizing the result of Cordeiro (2004) valid for generalized linear models with known dispersion parameter.
Abstract: This article gives a matrix formula for second-order covariances of maximum likelihood estimators in exponential family nonlinear models, thus generalizing the result of Cordeiro (2004) valid for generalized linear models with known dispersion parameter. Some simulations show that the second-order covariances for exponential family nonlinear models can be quite pronounced in small to moderate sample sizes.
TL;DR: In this article, continuous exponential families are applied to linking forms via a single-group design, where a distribution from the continuous bivariate exponential family is used that has selected moments that match those of the bivariate distribution of scores on the forms to be linked.
Abstract: Continuous exponential families are applied to linking forms via a single-group design. In this application, a distribution from the continuous bivariate exponential family is used that has selected moments that match those of the bivariate distribution of scores on the forms to be linked. The selected continuous bivariate distribution then yields continuous univariate marginal distributions for the two forms. These marginal distributions then provide distribution functions and quantile functions that may be employed in equating. Normal approximations are obtained for the sample distributions of the conversion functions.
TL;DR: In this article, the authors consider multi-parameter auto-models whose local conditional distributions belong to a mixed state exponential family and present some experimental results for modelling motion measurements from video sequences.
Abstract: In several application fields like daily pluviometry data modelling, or motion analysis from image sequences, observations contain two components of different nature. A first part is made with discrete values accounting for some symbolic information and a second part records a continuous (real-valued) measurement. We call such type of observations "mixed-state observations". This paper introduces spatial models suited for the analysis of these kinds of data. We consider multi-parameter auto-models whose local conditional distributions belong to a mixed state exponential family. Specific examples with exponential distributions are detailed, and we present some experimental results for modelling motion measurements from video sequences.