Showing papers on "Natural exponential family published in 2002"

The /spl alpha/-/spl mu/ distribution: a general fading distribution

Abstract: This paper presents a general fading distribution - the /spl alpha/-/spl mu/ distribution - that includes the Nakagami-m and the Weibull as special cases. One-Sided Gaussian, Rayleigh, and Negative Exponential distributions are also special cases of the /spl alpha/-/spl mu/ Distribution.

Improved exponential bounds and approximation for the Q-function with application to average error probability computation

Abstract: We present new exponential bounds for the Gaussian Q-function or, equivalently, of the complementary error function er f c(.). More precisely, the new bound is in the form of the sum of exponential functions that, in the limit, approaches the exact value. Then, a quite accurate and simple approximated expression given by the sum of two exponential functions is reported. Moreover, some new simple bounds for the inverse er f c(.) are derived. The results are applied to the general problem of evaluating the average error probability in fading channels. An example of application to the computation of the pairwise error probability of space-time codes is also presented.

Single observation unbiased priors

Abstract: This paper studies a class of default priors, which we call single observation unbiased priors (SOUP). A prior for a parameter is a SOUP if the corresponding posterior mean of the parameter based on a single observation is an unbiased estimator of the parameter. We prove that, under mild regularity conditions, a default prior for a convolution parameter is "noninformative" in the sense of yielding a posterior inference invariant under amalgamation only if it is a SOUP. Therefore, when amalgamation invariance is desirable, as in our motivating example of performing imputation for census undercount, SOUP is the only possible coherent "noninformative" prior for Bayesian predictions that will be utilized under aggregation. The use of SOUP also mutually calibrates Bayesian and frequentist inferences for aggregates of convolution parameters across many small areas. We describe approaches that identify SOUPs in many common models, in particular a constructive duality method that identifies SOUPs in pairs of distribution families. We introduce O-completeness, a necessary and sufficient condition for a prior distribution to be uniquely characterized by the corresponding posterior mean. Uniqueness of the SOUP is determined by the O-completeness of the dual family. O-completeness of a natural exponential family is implied by its completeness. Hence, the Diaconis-Ylvisaker characterization of the conjugate prior for natural exponential families via linear posterior expectation is a direct consequence of the completeness of such families. For most of the examples we have examined, the inverse of the variance function is the SOUP for the mean parameter of the corresponding family, suggesting that Hartigan's results on asymptotic unbiasedness can be generalized to some families with discrete parameters. We also discuss a possible extension of Berger's result on the inadmissibility of unbiased estimators, as the nonexistence of SOUP can be a first step in establishing such inadmissibility.

Spatial mixture models based on exponential family conditional distributions

Abstract: Spatial statistical models are applied in many problems for which dependence in observed random variables is not easily explained by a direct scientific mechanism. In such situations there may be a latent spatial process that acts to produce the observed spatial pattern. Scientific interest often centers on the latent process and the degree of spatial dependence that characterizes it. Such latent processes may be thought of as spatial mixing distributions. We present methods for the specification of flexible joint distributions to model spatial processes through multi-parameter exponential family conditional distributions. One approach to the analysis of these models is Monte Carlo maximum likelihood, and an approach based on independence pseudomodels is presented for formulating importance sampling distributions that allow such an analysis. The methods developed are applied to a problem of forest-health monitoring, where the numbers of affected trees in spatial field plots are modeled using a spatial beta-binomial mixture.

A generalization of the classical discrete distributions

Abstract: Some results related to the family of Inflated-parameter generalized power series distributions (IGPSD) are presented. The probability mass functions, useful properties, and recursion formulas are given. A characterization of the Inflated-parameter geometric distribution in terms of the ρ-type lack of memory property is given. The distributions are discussed as counting distributions in risk theory.

General Fading Distributions

Abstract: This paper presents two general fading distributions - the kappa-mu Distribution and the eta-mu Distribution. The kappa-mu Distribution includes the Rice and the Nakagami-m distributions as special cases. The eta-mu Distribution includes the Hoyt and the Nakagami-m distributions as special cases. Therefore, in both fading distributions, the One-Sided Gaussian and the Rayleigh distributions also constitute special cases and the Lognormal distribution may be well-approximated. Preliminary results show that these new distributions provide a very good fitting to experimental data.

Non parametric mixture priors based on an exponential random scheme

Abstract: We propose a general procedure for constructing nonparametric priors for Bayesian inference. Under very general assumptions, the proposed prior selects absolutely continuous distribution functions, hence it can be useful with continuous data. We use the notion ofFeller-type approximation, with a random scheme based on the natural exponential family, in order to construct a large class of distribution functions. We show how one can assign a probability to such a class and discuss the main properties of the proposed prior, namedFeller prior. Feller priors are related to mixture models with unknown number of components or, more generally, to mixtures with unknown weight distribution. Two illustrations relative to the estimation of a density and of a mixing distribution are carried out with respect to well known data-set in order to evaluate the performance of our procedure. Computations are performed using a modified version of an MCMC algorithm which is briefly described.

Reliability for beta models

Abstract: In the area of stress-strength models there has been a large amount of work as regards estimation of the reliability R = Pr(X2 < X1) when X1 and X2 are independent random variables belonging to the same univariate family of distributions. The algebraic form for R = Pr(X2 < X1) has been worked out for the majority of the well-known distributions in- cluding Normal, uniform, exponential, gamma, weibull and pareto. How- ever, there are still many other distributions for which the form of R is not known. We have identified at least some 30 distributions with no known form for R. In this paper we consider some of these distributions and derive the corresponding forms for the reliability R. The calculations involve the use of various special functions.

Stochastic Order and MLE of the Mean of the Exponential Distribution

Abstract: The maximum likelihood estimator of the mean of the exponential distribution, based on various data structures has been studied extensively. However, the order preserving property of these estimators is not found in the literature. This article discusses this property. Suppose that two samples of the same size are drawn from two independent exponential populations that have different means. It is shown in this article that the regular stochastic ordering holds between the two MLEs corresponding to the two exponential means, based on various censored data. In particular, conditions are given on inspection times so that the result is also true for grouped data.

The log-eig distribution: a new probability model for lifetime data

Abstract: In this paper, we propose a new probability model called the log-EIG distribution for lifetime data analysis. Some important properties of the proposed model and maximum likelihood estimation of its parameters are discussed. Its relationship with the exponential inverse Gaussian distribution is similar to that of the lognormal and the normal distributions. Through applications to well-known datasets, we show that the log-EIG distribution competes well, and in some instances even provides a better fit than the commonly used lifetime models such as the gamma, lognormal, Weibull and inverse Gaussian distributions. It can accommodate situations where an increasing failure rate model is required as well as those with a decreasing failure rate at larger times.

Information closure of exponential families and generalized maximum likelihood estimates

Abstract: Remedies to non-existence of a maximum likelihood estimate (MLE) in an exponential family or subfamily are offered: generalized MLE, and MLE in an information closure.

Exponential Inequalities for U-Statistics of Order Two with Constants

Abstract: We wish in these notes to further advance our knowledge of exponential inequalities for U–statistics of order two. These types of inequalities are already present in Hoeffding seminal papers [6], [7] and have seen further development since then. For example, exponential bounds were obtained by Hanson andWright [5] (and the references therein), Bretagnolle [1], and most recently by Gine, Latala, and Zinn [4]. As indicated in [4], the exponential bound there is optimal since it involves a mixture of exponents corresponding to a Gaussian chaos of order two behavior, and (up to logarithmic factors) to the product of a normal and of a Poisson random variable and to the product of two independent Poisson random variables. These various behaviors can be obtained as limits in law of triangular arrays of canonical U -statistics of degree two (with possibly non varying kernels). The methods of proof of [4] rely on precise moment inequalities of Rosenthal type which are of independent interest (and which are valid for U– statistics of arbitrary order). In case of order two, these moment inequalities together with Talagrand inequality for empirical processes provided the exponential bound. Here, we present a different proof of their result which also provide information about the constants which is often needed in statistical applications. Our approach still rely on Talagrand inequality

Estimation of the Parameters of a Mixture of Exponential and Weibull Distributions with Progressive Censoring

Abstract: A method is proposed for estimating the parameters of a mixture of exponential and Weibull distributions using censored samples. Preliminary estimates obtained by graphical analysis are refined by the method of maximum likelihood. The efficiency of the method is confirmed by the results of a statistical modeling.

Characterizations of generalized mixtures of geometric and exponential distributions based on order statistics

Abstract: Let X1 and X2 be independent and indentically distributed (i.i.d.) nondegenerate random variables with probability mass (or density) function (p.m.f. (or p.d.f.)) f (x), and let X(1) 0 if and only if X1 and X2 have the geometric distribution. In addition, Ahmed and Yehia (1993) have also provided an extension of this result when X1 and X2 have been taken from a mixture of two geometric distributions. As for, similar results characterize the geometric distribution or mixture of two geometric distributions based on E[X(2)|X(1) = x], reference may be made to Ferguson (1967), Kirmani and Alam (1980), Nagaraja (1988), Ahmed and Yehia (1993) and Lopez-Blazquez and Mino (1998). In this paper, we first characterize the generalized mixtures of geometric distribution with p. m. f. as

A Universal Distribution for Component Failure Times

Abstract: It is proposed to use a mixture of Weibull and exponential distributions to describe component reliability on the basis of a realistic model for resource consumption.

Factorization of discrete probability distributions

Abstract: We formulate necessary and sufficient conditions for an arbitrary discrete probability distribution to factor according to an undirected graphical model, or a log-linear model, or other more general exponential models. This result generalizes the well known Hammersley-Clifford Theorem.

New Dispersion Function in the Rank Regression

Abstract: In this paper we introduce a new score generating (unction for the rank regression in the linear regression model. The score function compares the 'th and s\`th power of the tail probabilities of the underlying probability distribution. We show that the rank estimate asymptotically converges to a multivariate normal. further we derive the asymptotic Pitman relative efficiencies and the most efficient values of and s under the symmetric distribution such as uniform, normal, cauchy and double exponential distributions and the asymmetric distribution such as exponential and lognormal distributions respectively

Estimation in an Exponential Distribution with Common Location and Scale Parameters

Abstract: We consider several point estimators including the jackknife estimator for the estimation of the common location and scale parameters and the right tail probability in an exponential distribution . Several interval estimators of the common location and scale parameters arc also obtained.

Exponential stability of nonlinear stochastic large-scale systems with delays

Abstract: We establish a new criteria of p-order moment exponential stability and the almost sure exponential stability for stochastic functional differential equation. Applying the derived criteria to nonlinear stochastic large-systems with variable multi-delay, we get the algebraic criteria of the delay-independent mean square exponential stability and the almost sure exponential stability for this type of stochastic large-scale systems.

On exponential stability of Wonham filter

Abstract: We give elementary proof of a stability result concerning an exponential asymptotic ($t\to\infty$) for filtering estimates generated by wrongly initialized Wonham filter. This proof is based on new exponential bound having independent interest.

Limit distributions of unbiased estimators in natural exponential families

Abstract: We obtain the possible limit distributions of unbiased estimators of functions of the parameter of a natural exponential family. The limit distribution depends on j , the order of the first non-zero derivative at the true (but usually unknown) value of the parameter. We show that if j \geq 2 then the umvu and the maximum likelihood estimators are not asymptotically equivalent.