Showing papers on "Natural exponential family published in 1979"

Group reaction time distributions and an analysis of distribution statistics.

Abstract: A method of obtaining an average reaction time distribution for a group of subjects is described. The method is particularly useful for cases in which data from many subjects are available but there are only 10-20 reaction time observations per subject cell. Essentially, reaction times for each subject are organized in ascending order, and quantiles are calculated. The quantiles are then averaged over subjects to give group quantiles (cf. Vincent learning curves). From the group quantiles, a group reaction time distribution can be constructed. It is shown that this method of averaging is exact for certain distributions (i.e., the resulting distribution belongs to the same family as the individual distributions). Furthermore, Monte Carlo studies and application of the method to the combined data from three large experiments provide evidence that properties derived from the group reaction time distribution are much the same as average properties derived from the data of individual subjects. This article also examines how to quantitatively describe the shape of reaction time distributions. The use of moments and cumulants as sources of information about distribution shape is evaluated and rejected because of extreme dependence on long, outlier reaction times. As an alternative, the use of explicit distribution functions as approximations to reaction time distributions is considered.

Conjugate Priors for Exponential Families

Abstract: Let $X$ be a random vector distributed according to an exponential family with natural parameter $\theta \in \Theta$. We characterize conjugate prior measures on $\Theta$ through the property of linear posterior expectation of the mean parameter of $X : E\{E(X|\theta)|X = x\} = ax + b$. We also delineate which hyperparameters permit such conjugate priors to be proper.

Generalized hypergeometric, digamma and trigamma distributions

Abstract: In this paper we introduce and study new probability distributions named “digamma” and “trigamma” defined on the set of all positive integers. They are obtained as limits of the zero-truncated Type B3 generalized hypergeometric distributions (inverse Polya-Eggenberger or negative binomial beta distributions), and also by compounding the logarithmic series distributions.

A note on the moments of order statistics from doubly truncated exponential distribution

Abstract: In a recent paper [2], the author has obtained some recurrence relations between the moments of order statitics from the exponential and right truncated exponential distributions. In this paper, similar relations are derived for a doubly truncated exponential distribution. It is shown that one can obtain all the moments by using these recurrence relations.

Some aspects of the kemp families of distributions

Abstract: Two families of discrete distributions, one with the probability generating function (p.g.f) and the other with the p.g.f. have been examined. Under certain restrictions on the parameters, some members of the former class can be interpreted as compound distributions. These resulting distributions have the property of being over-, under-, or equi-dispersed. These properties are also examined for the latter class of distributions. Relationships among different members of the latter class are considered graphically by means of a criterion based on successive probabilities.

Bayesian approach to prediction and the spacings in the exponential distribution

Abstract: A series of independent samples are drawn from a general population with positive variationf(x,ϕ), x>0. Based on the Bayesian approach, a general predictive distribution is given, to predict a statistic in the future sample based on the statistics in the earlier samples (or stages). Few general classes of distributions of this type like Koopman-Pitman family, power function family and Burr's class of distributions are considered to show how this procedure works in predicting order statistics in the future sample. Also, the sum of the spacings in the future samples from an exponential population is predicted in terms of similar sum of spacings in the earlier samples. Discussion on the variance of this predictive distribution is dealt with. Finally, an illustrative example with simulated samples from an exponential population gives actual prediction of an order statistic as well as the sum of spacings in the future sample.

An extremal quotient test for exponential distributions

Abstract: A procedure based on the extremal quotient is proposed for testing the null hypothesis H 0: the population of the sample has an exponential distribution against H a : it does not have an exponential distribution. The proposed procedure is a nonparametric test which could lead to an early decision for the rejection ofH 0

On an exponential family

Abstract: This paper considers a generalization of the exponential type distributions in the class of exponential families. A characterization and a method of generating an exponential family from a given family are given. In particular the generalized gamma, the generalized Poisson, the inverse Gaussian distributions belonging to this family are discussed. The approximations of the cumulative sums for the generalized gamma and the generalized Poisson by the Chi-square are considered. Some of the results are extended to the bivariate case.

New Methods for Estimating Tail Probabilities and Extreme Value Disributions.

Abstract: : This research has focused on the problem of estimating probabilities in the upper tail of an underlying distribution and the corresponding quantiles based on a random sample from the distribution. Two estimation procedures, exponential tail and transformed exponential tail, were defined and their bias and variance properties were thoroughly studied both analytically and by means of an extensive Monte Carlo experiment. The experiment involved several forms of each of the two procedures; twenty underlying distributions were simulated, including a variety of Weibull and lognormal distributions; four sample sizes were considered--100, 200, 400 and 800. Careful study of the analytic and Monte Carlo results showed that exponential tail and transformed exponential tail procedures worked quite well, but indicated a potential for substantial further improvement by properly combining them. (Author)

Differential Relations, in the Original Parameters, which Determine the First Two Moments of the Multiparameter Exponential Family

Abstract: We study general multiparameter exponential families of distribution and obtain differential equations relating the first two moments of the sufficient statistics to the normalization constant. Another result illuminates the structure of both the second order partial derivatives of the likelihood and their expected values.

The order statistics of exponential, power function and pareto distributions and some applications

Abstract: Some relations between the exponential, the Pareto and the Power function distributions and their order statistics are given. These relations are employed to obtain some characterization theorems of Pareto and Power distributions.

Sample Sizes for Testing Equality of Location Parameters of Two Exponential Distributions

Abstract: A table for determining minimal sample sizes n1 = n2 = n for testing the hypothesis of equality of location parameters a4 and a2 of two two-parametric exponential distributions for a first kind risk α=0,01 and 0,05 are given in such a way that the second kind risk β≦β0 as long as |a1–a>2|>d.

Charaterizations of the exponential distribution by conditional moments

Abstract: In this note, two characterizations of the exponential distribution are established. Theorem 1: Let a > -1, a ~ O. A positive random variable X has an exponential distribution if and only if E[CX-y)aI X > y] is a finite constant for all y ~ O. Theorem 2: Let Xl ,X2, ..• ,Xn be i.i.d. random variables having a continuous distribution FCx), and XCl)'XC2)' ... 'XCn) their order statistics. Let a > 0 and m = 1,2, ..• ,n-l be given. Then FCx) is an exponential distribution up to a location parameter if and only if E[CX Cm l ) XCm))aIXCm) = y] is a finite constant for almost all y with respect to FCy). *This research was supported by the Air Force Office of Scientific Research under contract AFOSR-75-2796 and the Office of Naval Research under contract NOOOI4-75-C-0809. CHARACTERIZATIONS OF THE EXPONENTIAL DISTRIBUTION BY CONDITIONAL MOMENTS

Probability Inequalities of Exponential Type for Continuous Parameter Processes and their Applications

Abstract: ' ' (1.1) IE(Xel.ce-s)-rE(Xe)IS-q(s) (a.s.) or -(1.2) EIE("XolS-s)rE(Xo)[ lll a(s)・ In [4], the author has given an exponential bound of the probability for the sum of random variables satisfying (1.1) or (1.2) with the discrete parameter. Our results in this paper are obtained by the same method as that of [4]. Theorem 1 and Theorem 3 are some extension of those in [1]. Examples satisfying (1.1) are presented in [1]. If {Xt} is a strictly stationary strong mixing process of uniforrnly bounded random variables, then (1.2) is satisfied. (cf. [2])

Transient solution of a certain type of heterogeneous queues

Abstract: This paper considers a limited space, single channel, first come first served heterogeneous queue model. There are two operating statesE andF. For either state, the time in transition from one state to the other, follows an exponential distribution. In stateE, there are Poisson arrivals and exponentially distributed service time. In stateF, the arrival rate is zero and service time has a different exponential distribution. Free use of probability generating function and Laplace transform has been made. The steady state has been discussed and a particular case has been derived in the end.