Showing papers on "Natural exponential family published in 2000"

A new test for random events of an exponential distribution

Abstract: A new statistical test procedure is described to evaluate whether a set of radioactive-decay data is compatible with the assumption that these data originate from the decay of a single radioactive species. Criteria to detect contributions from other radioactive species and from different event sources are given. The test is applicable to samples of exponential distributions with two or more events.

Computer Generation of Statistical Distributions

Abstract: : This report presents a collection of computer-generated statistical distributions which are useful for performing Monte Carlo simulations. The distributions are encapsulated into a C++ class, called "Random", so that they can be used with any C++ program. The class currently contains 27 continuous distributions, 9 discrete distributions, data-driven distributions, bivariate distributions, and number-theoretic distributions. The class is designed to be flexible and extensible, and this is supported in two ways: (1) a function pointer is provided so that the user-programmer can specify an arbitrary probability density function, and (2) new distributions can be easily added by coding them directly into the class. The format of the report is designed to provide the practitioner of Monte Carlo simulations with a handy reference for generating statistical distributions. However, to be self-contained, various techniques for generating distributions are also discussed, as well as procedures for estimating distribution parameters from data. Since most of these distributions rely upon a good underlying uniform distribution of random numbers, several candidate generators are presented along with selection criteria and test results. Indeed, it is noted that one of the more popular generators is probably overused and under what conditions it should be avoided.

Microcanonical foundation for systems with power-law distributions

Abstract: Starting from a microcanonical basis with the principle of equal a priori probability, it is shown using the method of steepest descents that besides ordinary Boltzmann-Gibbs theory with the exponential distribution a theory describing systems with power-law distributions can also be derived.

Inference based on the empirical probability generating function for mixtures of poisson distributions

Abstract: Tail asymptotics of certain mixtures of Poisson distributions show that they are Paretian, their tail index being one of the parameters deening these laws. Estimators similar to those proposed by Press 15] for continuous stable laws are then used for the estimation of the parameters of such laws and asymptotic properties are proved. Inference is based on the empirical probability generating function. As special cases the discrete stable distribution, the discrete Linnik distribution, and the Sibuya distribution are examined.

On the distribution of exponential sums.

Abstract: We discuss three problems of the following kind: given a set A ⊆ Fp of n := |A| residues modulo a prime p, how are the absolute values |SA(z)| of the corresponding exponential sums SA(z) := ∑ a∈A e 2πi p ; z ∈ Fp distributed in the interval [0, n]?

Exponential approximation for maintained Weibull distributed component

Abstract: Exponential distribution is widely used in reliability and maintainability studies although it is well known that the constant failure rate assumption may not be valid. The purpose of this paper is to investigate the use of exponential distribution as an approximation. In fact, for components undergoing regular maintenance or replacement, the exponential assumption can be acceptable. In this paper, the exponential approximation for regularly maintained Weibull component is studied. The approximated exponential distribution using the average failure rate is compared with the exact reliability. The asymptotic relative error is derived, which can be used to adjust the exponential approximation when needed. Based on the framework of exponential approximation for Weibull distributed components, the problems of decision‐making regarding the optimal maintenance time and spare allocation are also addressed.

Diffusion approximations for a single node accessed by congestion-controlled sources

Abstract: We consider simple models of congestion control in high-speed networks, and develop diffusion approximations which could be useful for resource allocation. We first show that, if the sources are ON-OFF type with exponential ON and OFF times, then, under a certain scaling, the steady-state distribution of the number of active sources can be described by a combination of two appropriately truncated and renormalized normal distributions. For the case where the source arrival process is Poisson and the service times are exponential, the steady-state distribution consists of appropriately normalized and truncated Gaussian and exponential distributions. We then consider the case where the arrival process is a general renewal process with finite coefficient of variation and service-time distributions that are phase type, and show the impact of these distributions on the steady-state distribution of the number of sources in the system. We also establish an insensitivity to service-time distribution when the arrival process is Poisson. We use these results to relate the capacity of a bottleneck node to performance measures of interest for best effort traffic.

Goodness-of-fit tests based on characterizations of continuous distributions

Abstract: We construct goodness-of-fit tests for continuous distributions using their characterizations in terms of moments of order statistics and moments of record values. Our approach is based on characterizations presented in [2]–[4], [5], [9].

Density Approximation by Summary Statistics: An Information-theoretic Approach

Abstract: In the case of exponential families, it is a straightforward matter to approximate a density function by use of summary statistics; however, an appropriate approach to such approximation is far less clear when an exponential family is not assumed. In this paper, a maximin argument based on information theory is used to derive a new approach to density approximation from summary statistics which is not restricted by the assumption of validity of an underlying exponential family. Information-theoretic criteria are developed to assess loss of predictive power of summary statistics under such minimal knowledge. Under these criteria, optimal density approximations in the maximin sense are obtained and shown to be related to exponential families. Conditions for existence of optimal density approximations are developed. Applications of the proposed approach are illustrated, and methods for estimation of densities are provided in the case of simple random sampling. Large-sample theory for estimates is developed.

Bootstrap relative errors and sub-exponential distributions

Abstract: For the purposes of this paper, a distribution is sub-exponential if it has finite variance but its moment generating function is infinite on at least one side of the origin. The principal aim here is to study the relative error properties of the bootstrap approximation to the true distribution function of the sample mean in the important sub-exponential cases. Our results provide a fairly general description of how the bootstrap approximation breaks down in the tail when the underlying distribution is sub-exponential and satisfies some very mild additional conditions. The broad conclusion we draw is that the accuracy of the bootstrap approximation in the tail depends, in a rather sensitive way, on the precise tail behaviour of the underlying distribution. Our results are applied to several sub-exponential distributions, including the lognormal. The lognormal case is of particular interest because, as the simulation studies of Lee and Young have shown, bootstrap confidence intervals can have very poor coverage accuracy when applied to data from the lognormal.

Negative exponential disparity based family of goodness-of-fit tests for multinomial models

Abstract: This paper investigates a new family of goodness-of-fit tests based on the negative exponential disparities. This family includes the popular Pearson's chi-square as a member and is a subclass of the general class of disparity tests (Basu and Sarkar, 1994) which also contains the family of power divergence statistics. Pitman efficiency and finite sample power comparisons between different members of this new family are made. Three asymptotic approximations of the exact null distributions of the negative exponential disparity famiiy of tests are discussed. Some numerical results on the small sample perfomance of this family of tests are presented for the symmetric null hypothesis. It is shown that the negative exponential disparity famiiy, Like the power divergence family, produces a new goodness-of-fit test statistic that can be a very attractive alternative to the Pearson's chi-square. Some numerical results suggest that, application of this test statistic, as an alternative to Pearson's chi-square, coul...

On the Characteristic Properties of Exponential Distribution

Abstract: New characterizations for the exponential distribution are given in terms of record values and the probabilities of finite sums of independent and identically distributed nonnegative random variables provided that the underlying distribution is either new better than used or new worse than used.

Abelian theorems for a class of probability distributions inR d and their application

Abstract: A class of multidimensional absolutely continuous distributions is considered. Each of them has a moment-generating function that is finite in a bounded set S and, therefore, generates a family of so-called conjugate or associated distributions. At the focus of our attention are the limiting distributions for this family that appear as the conjugating parameter tends to the boundary of S. As in the one-dimensional case, each such limiting distribution can be obtained as a consequence of an Abelian theorem.

Conditional estimation in exponential models

Abstract: A two-sided conditional confidence interval for the parameter of an exponential probability distribution is constructed. The construction relies on a decision following a preliminary test of significance for the equality of two exponential population means. The coverage probability, the expected length together with the coefficient of variation of this interval are studied. A shrinkage version of the interval is also proposed. Furthermore, a numerical study on the accuracy of the interval estimator is performed.

General location transform of the order statistics from the exponential, pareto and weibull, with application to maximum likelihood estimation

Abstract: Based one some common distribution properties of the order statistics and the transformation theory by Efron(1982), we determine unified explicit general location transformations, which map the distributions of the order statistics from the Exponential, Pareto and Weibull to a standard normal distribution. This result is used to derive analytical formulas for the maximum likelihood estimators of the shape parameter of these distributions of order statistics. The presented exact method is applied to catastrophe earthquake life reinsurance.

Statistical rules of group classification of bivariate exponential populations

Abstract: We consider maximum likelihood estimators and unbiased estimators for bivariate exponential distributions under various a priori hypotheses about the parameters.

Some aspects of multivariate Rayleigh and exponential distributions

Abstract: We obtain a general form of the multivariate Rayleigh and exponential probability density functions (p.d.f.s) when these are generated by correlated Gaussian random variables. A general expression for the exponential characteristic function (c.f.) is also derived.