Showing papers on "Natural exponential family published in 1997"

A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families

Abstract: SUMMARY A new way of introducing a parameter to expand a family of distributions is introduced and applied to yield a new two-parameter extension of the exponential distribution which may serve as a competitor to such commonly-used two-parameter families of life distributions as the Weibull, gamma and lognormal distributions. In addition, the general method is applied to yield a new three-parameter Weibull distribution. Families expanded using the method introduced here have the property that the minimum of a geometric number of independent random variables with common distribution in the family has a distribution again in the family. Bivariate versions are also considered.

Fitting mixtures of exponentials to long-tail distributions to analyze network performance models

Abstract: Traffic measurements from communication networks have shown that many quantities characterizing network performance have long-tail probability distributions, i.e., with tails that decay more slowly than exponentially. Long-tail distributions can have a dramatic effect upon performance, but it is often difficult to describe this effect in detail, because performance models with component long-tail distributions tend to be difficult to analyze. We address this problem by developing an algorithm for approximating a long-tail distribution by a finite mixture of exponentials. The fitting algorithm is recursive over time scales. At each stage, an exponential component is fit in the largest remaining time scale and then the fitted exponential component is subtracted from the distribution. Even though a mixture of exponentials has an exponential tail, it can match a long-tail distribution in the regions of primary interest when there are enough exponential components.

On the asymptotics of M-hypothesis Bayesian detection

Abstract: In two-hypothesis detection problems with i.i.d. observations, the minimum error probability decays exponentially with the amount of data, with the constant in the exponent equal to the Chernoff distance between the probability distributions characterizing the hypotheses. We extend this result to the general M-hypothesis Bayesian detection problem where zero cost is assigned to correct decisions, and find that the Bayesian cost function's exponential decay constant equals the minimum Chernoff distance among all distinct pairs of hypothesized probability distributions.

On exponential sums over smooth numbers

Abstract: This paper is concerned with the theory and applications of exponential sums over smooth numbers. Despite the numerous applications stemming from suitable estimates for these exponential sums, thus far little attention has been paid to any but the simplest cases (see, for example, [19], [20], [23]). Our primary objective is the development of a method for estimating mean values of exponential sums over smooth numbers, such mean values being fundamental to subsequent applications. Having established such a method, we develop estimates of use in applications of such exponential sums inside the fabric of the Hardy-Littlewood method. For the purposes of illustrating the power of our new estimates, we draw corollaries concerning the distribution of the fractional parts of polynomials, and for Waring's problem with polynomial summands. There are also consequences of our methods for problems involving the global solubility of simultaneous additive equations, but we defer an account of such developments to a later occasion (see [30]). These applications by no means exhaust the available supply. Our estimates will also be useful in considering problems concerning simultaneous small values of additive forms (see, for example, [18], Chapter 11), and the simultaneous distribution modulo l of additive forms (see [6]).

On characterizing distributions via linearity of regression for order statistics

Abstract: summary Let X1Xn be a random sample from an absolutely continuous distribution with the corresponding order statistics X1:n≤X2:n≤Xn:n. A complete solution of the problem, posed in 1967 by T. Ferguson, of determining the distribution by linearity of regression of Xk+2:n with respect to Xk:n is given. The only possible distributions are of the exponential, power and Pareto type. A linear regression relation for exponents of order statistics is also considered.

On the theory of entire matrix-functions of exponential type

Abstract: It is shown that if an entire matrix function U(λ) is J inner, then it is of exponential type and (1+λ2)-1ln +||U(λ)|| is summable. These results are applied to the monodromy matrix of a canonical system. In particular, the rate of growth is calculated. [Abstract added by the editors.]

Divergence measures between populations: applications in the exponential family

Abstract: Toussaint (1974) introduced a divergence measure between s populations as a generalization of the J-divergence. The asymptotic distribution of Toussaint's measure is obtained when the parameters are substituted by their maximum likelihood estimators and the s probability density functions belong to the exponential farnily. A procedure to test statistical hypotheses about s populations is given These results can also be applied to multinomial populations.

Moments for the canonical parameter of an exponential family under a conjugate distribution

Abstract: SUMMARY For exponential families, the moment generating function of a conjugate exponential family distribution for the canonical parameter can be conveniently written in terms of the corresponding normalising constant for the conjugate densities. This result can be used to find the moments of the canonical parameter and certain functions thereof. This is illustrated for the class of natural exponential families having a simple quadratic variance function. Two applications are outlined. The first is concerned with the analysis of dynamic generalised linear models, as put forward by West, Harrison & Migon (1985). The second refers to the calculations of the expected logarithmic divergence for exponential families under a conjugate distribution.

A finite dimensional filter with exponential conditional density

Abstract: In this paper we consider the continuous-time nonlinear filtering problem, which has an infinite-dimensional solution in general, as proved by Chaleyat-Maurel and Michel (1984). There are few examples of nonlinear systems for which the optimal filter is finite dimensional, in particular the Kalman, Benes, and Daum filters. In the present paper, we construct new classes of scalar nonlinear systems admitting finite-dimensional filters. We consider a given (nonlinear) diffusion coefficient for the state equation, a given (nonlinear) observation function, and a given finite-dimensional exponential family of probability densities. We construct a drift for the state equation such that the resulting nonlinear system admits a finite-dimensional filter evolving in the prescribed exponential family, provided the coefficients of the exponential family include the observation function and its square.

Family of Pearson discrete distributions generated by the univariate hypergeometric function 3F2(α1, α2, α3; γ1, γ2; λ)

Abstract: In this work we present a study of the Pearson discrete distributions generated by the hypergeometric function 3F2(α1, α2, α3;γ1, γ2; λ), a univariate extension of the Gaussian hypergeometric function, through a constructive methodology. We start from the polynomial coefficients of the difference equation that lead to such a function as a solution. Immediately after, we obtain the generating probability function and the differential equation that it satisfies, valid for any admissible values of the parameters. We also obtain the differential equations that satisfy the cumulants generating function, moments generating function and characteristic function, From this point on, we obtain a relation in recurrences between the moments about the origin, allowing us to create an equation system for estimating the parameters by the moment method. We also establish a classification of all possible distributions of such type and conclude with a summation theorem that allows us study some distributions belonging to this family. © 1997 by John Wiley & Sons, Ltd.

9 Some extensions in the robust estimation of parameters of exponential and double exponential distributions in the presence of multiple outliers

Abstract: Publisher Summary This chapter discusses the robust estimation of parameters of exponential and double exponential distributions in the presence of multiple outliers. It provides an alternative, more direct proof of the above recursion algorithm and to extend the numerical results to include larger sample sizes and also to accomodate a larger number of outliers. It considers generalizations of the Chikkagoudar–Kunchur estimator discussed by Balakrishnan and Barnett, provides a further generalization, and extends these results to include larger sample sizes and more outliers. The moments of order statistics from the multiple-outlier exponential model and some equations relating them are used in the moments of order statistics from the multiple-outlier double exponential model to examine the robustness properties of various linear estimators of the location and scale parameters of the double exponential (Laplace) distribution in the presence of one or more outliers. These results generalize those obtained by Balakrishnan and Ambagaspitiya, who examined the case of a single outlier.

Tauber theory for infinitely divisible variance functions

Abstract: We study a notion of Tauber theory for infinitely divisible natural exponential families, showing that the variance function of the family is (bounded) regularly varying if and only if the canonical measure of the Levy-Khinchine representation of the family is (bounded) regularly varying. Here a variance function V is called bounded regularly varying if V(μ)\sim cμp either at zero or infinity, with a similar definition for measures. The main tool of the proof is classical Tauber theory.

Characterizations of uniform and exponential distributions via moments of the kth record values randomly indexed

Abstract: We characterize uniform and exponential distributions via moments of the kth record statistics. Too and Lin’s (1989) results are contained in our approach.

Umvu estimation for certain family of exponential distributions

Abstract: In this paper we study certain properties of estimable and UMUV-estimable functions in a subfamily of the one-parameter exponential family of distributions for which there exists a sufficient and complete statistic following a Gamma distribution. These results are applied to the problem of estimation in the transformed chi-square family.

Identification methods for exponential possibility distributions

Abstract: This paper proposes three kinds of models for obtaining exponential possibility distributions, namely, the upper approximation model, the lower approximation model and the integrated model to reflect the different viewpoints. An example of the possibility distribution in a portfolio selection problem is shown.

Evaluation of exponential product kernel for quadratic time-frequency distributions applied to ultrasonic signals

Abstract: The display of energy of ultrasonic backscattered echoes simultaneously on a joint time-frequency (t-f) plane reveals critical information pertaining to time of arrival and frequency of echoes. The quadratic t-f distributions play important role in displaying the energy of the signal on a joint t-f plane. The t-f energy distribution of the signal is dependent on a weighting function, kernel, of generalized quadratic t-f distribution. This kernel, a function of product of time lag and frequency lag variables, controls the t-f concentration of the signal and the suppression of artifacts generated by the quadratic t-f distribution. A generalized exponential product (GEP) kernel function is explored in this paper, Exponential (i.e., Choi-Williams) distribution is a special case of this generalized exponential distribution. A whole family of Quadratic exponential distributions can be generated by varying the parameters of the generalized exponential product kernel. We evaluate these parameters on the basis of optimum concentration of the ultrasonic backscattered echoes, resolution of defect echoes, suppression of the cross-terms artifacts, and performance in the presence of noise. These parameters are evaluated by reducing the cross-terms and keeping auto-terms on the ambiguity plane close to the ideal. It is shown that by controlling the parameters of the generalized exponential product kernel we can achieve better performance in the form of time-frequency concentration, and resolution for multiple echoes as compared to exponential distribution. The application of GEP kernel to ultrasonic experimental data, with properly chosen parameters, not only discern the defect echo embedded in grain echoes but diminish the cross-terms generated by the bilinear structure of the t-f distribution.

Randomized confidence intervals of a parameter for a family of discrete exponential type distributions

Abstract: For a family of one-parameter discrete exponential type distributions, the higher order approximation of randomized confidence intervals derived from the optimum test is discussed. Indeed, it is shown that they can be asymptotically constructed by means of the Edgeworth expansion. The usefulness is seen from the numerical results in the case of Poisson and binomial distributions.

Estimation of an Exponential Distribution

Abstract: Consider a sample from an exponential distribution with unknown parameter θ From the sample, we wish to estimate the entire distribution, (Borel sets). If an estimator is used for , we are concerned with proximity of to , under total variation distance, the maximum likelihood estimate of θ, define , where , the 1 – (a/2) percentile from the standard normal. Then, . The preceding approximation to the 1 — α percentile of is very accurate even for small n We also consider the problem of obtaining a confidence band for the survival function, possessing minimal maximum width. Finally, a class of estimators of is compared to the maximum likelihood estimator from the viewpoint of total variation distance loss function.

Overdispersed Exponential Regression Models

Abstract: This paper introduces two-parameter exponential family distributions into the stochastic components of generalized linear models (GLMs), in order to model patterns of dispersion unexpected under ordinary GLMs, including overdispersion and underdispersion. Exponential regression models can be fitted using the iteratively re-weighted least squares algorithm and allow full likelihood analyses.

Asymptotic properties of mle in exponential family nonlinear models

Abstract: In this paper, a necessary condition for the existence of any weakly consistentestimator is presented in exponential family nonlinear models. Then we give one morenecessary condition for a consistent estimator in generalized linear models. Under mildregularity conditions, the existence, the strong consistency and the asymptotic normalityof MLE are proved in exponential family nonlinear models. Our results may be regardedas a further work of [1] in generalized linear models.