Showing papers on "Poisson distribution published in 1986"

Confidence limits for small numbers of events in astrophysical data

Abstract: The calculation of limits for small numbers of astronomical counts is based on standard equations derived from Poisson and binomial statistics; although the equations are straightforward, their direct use is cumbersome and involves both table-interpolations and several mathematical operations Convenient tables and approximate formulae are here presented for confidence limits which are based on such Poisson and binomial statistics The limits in the tables are given for all confidence levels commonly used in astrophysics

Econometric models based on count data. Comparisons and applications of some estimators and tests

Abstract: This paper deals with specification, estimation and tests of single equation reduced form type equations in which the dependent variable takes only non-negative integer values. Beginning with Poisson and compound Poisson models, which involve strong assumptions, a variety of possible stochastic models and their implications are discussed. A number of estimators and their properties are considered in the light of uncertainty about the data generation process. The paper also considers the role of tests in sequential revision of the model specification beginr ing with the Poisson case and provides a detailed application of the estimators and tests to a model of the number of doctor consultations.

Double Exponential Families and Their Use in Generalized Linear Regression

Abstract: In one-parameter exponential families such as the binomial and Poisson, the variance is a function of the mean. Double exponential families allow the introduction of a second parameter that controls variance independently of the mean. Double families are used as constituent distributions in generalized linear regressions, in which both means and variances are allowed to depend on observed covariates. The theory is applied to two examples—a logistic regression and a large two-way contingency table. In such cases the binomial model of variance is often untrustworthy. For example, because genuine random sampling was infeasible, the subjects may have been obtained in clumps so that the statistician should really be using smaller sample sizes. Clumped sampling is just one of many possible causes of overdispersion, a habitual source of concern to users of binomial and Poisson models. This article concerns a class of regression families that allow the statistician to model overdispersion while carrying ...

The natural variability of vital rates and associated statistics.

Abstract: The first concern of this work is the development of approximations to the distributions of crude mortality rates age-specific mortality rates age-standardized rates standardized mortality ratios and the like for the case of a closed population or period study. It is found that assuming Poisson birthtimes and independent lifetimes implies that the number of deaths and the corresponding midyear population have a bivariate Poisson distribution....It is...suggested that situations in which explanatory variables are present may be modelled via a doubly stochastic Poisson distribution for the number of deaths with mean proportional to the population size and an exponential function of a linear combination of the explanatories. Such a model is fit to mortality data for Canadian females classified by age and year. A dynamic variant of the model is further fit to the time series of total female deaths alone by year. The models with extra-Poisson variation are found to lead to substantially improved fits. (EXCERPT)

On yield, fault distributions, and clustering of particles

Abstract: Increasing the levels of semiconductor integration to larger chips with more transistors causes the fault and defect distributions of VLSI memory chips to deviate increasingly further from simple random Poisson statistics. The spatial distributions of particles on semiconductor wafers have been analyzed to gain insight into the nature of integrated circuit defect statistics. The analysis was done using grids of squares known as quadrats. It was found that the cluster parameter, which until now has been treated as a constant, did vary with quadrat area. The results also show that the deviation from Poisson statistics continues to increase into the realm of wafer-scale integration or WSI. Computer simulations were used to verify this effect.

The Poisson Function of Finite Elasticity

Abstract: The Poisson function is introduced to study in a simple tension test the lateral contractive response of compressible and incompressible, isotropic elastic materials in finite strain. The relation of the Poisson function to the classical Poisson’s ratio and its behavior for certain constrained materials are discussed. Some experimental results for several elastomers, including two natural rubber compounds of the same kind studied in earlier basic experiments by Rivlin and Saunders, are compared with the derived relations. A special class of compressible materials is also considered. It is proved that the only class of compressible hyperelastic materials whose response functions depend on only the third principal invariant of the deformation tensor is the class first introduced in experiments by Blatz and Ko. Poisson functions for the Blatz-Ko polyurethane elastomers are derived; and our experimental data are reviewed in relation to a volume constraint equation used in their experiments.

On Birthday, Collectors', Occupancy and Other Classical Urn Problems

Abstract: Summary An urn contains r different balls. Balls are drawn with replacement until any k balls have been obtained at least m times each. How many draws are necessary? How many balls have been drawn exactly v times? Special cases of such problems are often named as birthday, collectors', dixie cup or occupancy problems. This paper presents a unified approach to such problems by imbedding in Poisson processes. In this way we see that many classical urn problems are closely related to properties of order statistics and extreme values from the gamma distribution. Both exact and asymptotic results are derived. Finally, a brief discussion is given on other drawing schemes.

Testing for spatial association between a point process and another stochastic process

Abstract: Motivated by a problem in geology, this paper proposes some tests of the spatial association between a point process and some other stochastic process of geometric structures, G. All the tests are performed conditionally on the realization of G. Under the null hypothesis that the point process is a stationary Poisson process independent of G, some of these statistics have well‐known distributional properties, even in small samples. The Poisson assumption is relaxed using a conditional Monte Carlo test suggested by Lotwick and Silverman (1982). The tests are applied to a geological data set.

A Semigroup Approach to Poisson Approximation

Abstract: The aim of this paper is twofold: first, to show that Poisson approximation problems for independent summands can in a natural way be treated in a suitable operator semigroup framework, allowing at the same time for an asymptotically precise evaluation of the leading term with respect to the total variation distance; second, to determine asymptotically those Poisson distributions which minimize this distance for given Bernoulli summands. Besides semigroup methods, coupling techniques as well as direct computations are used.

Sensitivity analysis via likelihood ratios

Abstract: We present a new method of obtaining derivatives of expectations with respect to various parameters. For example, if l is the rate of a Poisson process, NT is the number of Poisson events in (0, T),and p is nearly any function of the sample path (e.g. a performance measure in a queuing network), then we show that d/dl El(p) = El ((NT/l - T)p), which yields an obvious algorithm. We have proven that the method works for a wide class of parameters and performance measures in regenerative simulation. We also report on the method's limitations and on some numerical experiments.

Limit Theorems for a Triangular Scheme of $U$-Statistics with Applications to Inter-Point Distances

Abstract: The Annals of Probability 1986, Vol. 14, No. 4, 1347-1358 LIMIT THEOREMS FOR A TRIANGULAR SCHEME OF U-STATISTICS WITH APPLICATIONS TO INTER-POINT DISTANCES BY S. RAO JAMMALAMADAKA1 AND SVANTE J ANSON University of California, Santa Barbara and Uppsala University The asymptotic distribution of a “triangular” scheme of U-statistics is studied. Two limit theorems, applicable in different situations, are given. One theorem yields convergence to a normal distribution; the other includes Poisson limits and other limit laws. Applications to statistics based on small interpoint distances in a sample are given. 1. Introduction. There has been a renewed interest in the limit theory relating to U-statistics (Hoefi‘ding, 1948) especially focussing on degenerate kernels and nonnormal limits. See, for instance, Rubin and Vitale (1980), Dynkin and Mandelbaum (1983), and Berman and Eagleson (1983). In Section 2 of this paper we study a “triangular” scheme of U-statistics and establish their asymptotic normality under suitable conditions. It is worth noting that one can obtain normal limit laws even with degenerate kernels. Other infinitely divisible limit laws for the triangular scheme, including the Poisson limits, are examined in the next section. Applications to limit distributions of statistics based on interpoint distances are discussed in Section 4. These applications, which were the source of motivation for the results derived here, were suggested by the studies one of us made about statistics based on spacings. [See, for instance, Holst and Rao (1981).] They were also sparked by the recent papers of Bickel and Breiman (1983) and Onoyama et al. (1983). Theorems yielding asymptotic normality in this setting have also been proved by Weber (1983) using a martingale approach. His theorems yield the same conclusion as Theorem 2.1, but under difi‘erent (nonequivalent) conditions. A few words about notation: We write “ —gd ” to denote convergence in distribution an “ —gc ” to denote complete convergence [cf. Loeve (1963), page 178]. We write Po(}\) to denote the Poisson distribution with mean A and Np(p., 2) to denote a p-dimensional normal distribution with mean vector p. and covariance matrix 2. 2. Asymptotic normality of a triangular scheme of U-statistics. Let X1, X2,... be a sequence of independently and identically distributed (i.i.d.) random variables. We will use X and Y to denote two independent random Received October 1984; revised April 1985. ‘Formerly J. S. Rao. AMS 1980 subject classifications. Primary 60F05, 60E07; secondary 62E20. Key words and phrases. U-statistics, triangular scheme, multigraph, limit laws, infinitely divisible distributions, inter-point distances. 1347 ‘T Institute of Mathematical Statistics is collaborating with J STOR to digitize, preserve, and extend access to ef‘%}73i The Annals ofProbabiIity. 5T0 R ® www.jstor.org

Mixed Compound Poisson Distributions

Abstract: The distribution of total claims payable by an insurer is considered when the frequency of claims is a mixed Poisson random variable. It is shown how in many cases the total claims density can be evaluated numerically using simple recursive formulae (discrete or continuous).Mixed Poisson distributions often have desirable properties for modelling claim frequencies. For example, they often have thick tails which make them useful for long-tailed data. Also, they may be interpreted as having arisen from a stochastic process. Mixing distributions considered include the inverse Gaussian, beta, uniform, non-central chi-squared, and the generalized inverse Gaussian as well as other more general distributions.It is also shown how these results may be used to derive computational formulae for the total claims density when the frequency distribution is either from the Neyman class of contagious distributions, or a class of negative binomial mixtures. Also, a computational formula is derived for the probability distribution of the number in the system for the M/G/1 queue with bulk arrivals.

A Multivariate Polya Model of Brand Choice and Purchase Incidence

Abstract: In this paper we develop an integrated stochastic model of purchase timing and brand selection which incorporates the influence of marketing mix variables, seasonality and trend, and also allows for various individual choice mechanisms. Our approach rests on the assumptions of a zero-order choice process, a Poisson timing process and purchase rates following a multivariate Gamma Distribution over the population, the scale parameters of which vary according to marketing activities and time. The resulting model is a Multivariate Polya Process, and the distribution of brand choice probabilities turns out to be a Generalized Dirichlet Distribution. Thus, most currently used zero-order models can be considered to be special cases of this approach. Furthermore, we derive a number of market diagnostics which provide insights into market structure and demonstrate the model's use for marketing strategy simulation. Based on extensive testing of the underlying hypotheses we finally validate the model using empirical data and show that it fits the market in question.

The question of the total gene number in drosophila melanogaster

Abstract: A statistical analysis has been carried out on the distribution and allelism of nearly 500 sex-linked, X-ray-induced, cytologically normal and rearranged lethal mutations in Drosophila melanogaster that were obtained by G. Lefevre. The mutations were induced in four different regions of the X chromosome: (1) 1A1-3E8, (2) 6D1-8A5, (3) 9E1-11A7 and (4) 19A1-20F4, which together comprise more than one-third of the entire chromosome.—The analysis shows that the number of alleles found at different loci does not fit a Poisson distribution, even when the proper procedures are taken to accomodate the truncated nature of the data. However, the allele distribution fits a truncated negative binomial distribution quite well, with cytologically normal mutations fitting better than rearrangement mutations. This indicates that genes are not equimutable, as required for the data to fit a Poisson distribution.—Using the negative binomial parameters to estimate the number of genes that did not produce a detectable lethal mutation in our experiment (n) gave a larger number than that derived from the use of the Poisson parameter. Unfortunately, we cannot estimate the total numbers of nonvital loci, loci with undetectable phenotypes and loci having extremely low mutabilities. In any event, our estimate of the total vital gene number was far short of the total number of bands in the analyzed regions; yet, in several short intervals, we have found more vital genes than bands; in other intervals, fewer. We conclude that the one-band, one-gene hypothesis, in its literal sense, is not true; furthermore, it is difficult to support, even approximately.—The question of the total gene number in Drosophila will, not doubt, eventually be solved by molecular analyses, not by statistical analysis of mutation data or saturation studies.

The poisson-inverse gaussian disiribuiion as a model for species abundance

Abstract: An inverse Gaussian mixture of Poisson distributions(the P-IG distribution) is considered as a model for species abundance data,, Minimum chi-square and maximum likelihood methods of estimation for the zero-truncated P-IG distribution are developed, Ihe performance of the P-IG distribution is illustrated and discussed for several well-known sets of insect abundance data.

Upper Bounds for Single Server Queues with Doubly Stochastic Poisson Arrivals

Abstract: Consider the following type of single server queues. Arrivals are according to a Doubly Stochastic Poisson process with a stationary, ergodic random intensity {λt}. Service times are independent, identically distributed, also independent from arrivals. It is proven that the mean stationary work-load is not greater than E[ωλ0], where ωa denotes the mean stationary work-load in the M/GI/1 queue with arrival intensity a and the same service process. Similar results are given for the mean stationary queue size and the mean stationary delay.

A Characterization of the Spatial Poisson Process and Changing Time

Abstract: Watanabe proved that if $X_t$ is a point process such that $X_t - t$ is a martingale, then $X_t$ is a Poisson process and this result was generalized by Bremaud for doubly stochastic Poisson processes. Here we define two-parameter point processes and extend this property without needing the strong martingale condition. Using this characterization, we study the problem of transforming a two-parameter point process into a two-parameter Poisson process by means of a family of stopping lines as a time change. Nualart and Sanz gave conditions in order to transform a square integrable strong martingale into a Wiener process. Here, we do the same for the Poisson process by a similar method but under more general conditions.

Some Statistical Applications of Poisson's Work

Abstract: Statistical applications and repercussions of Poisson's work are reviewed in historical perspective with special reference to (i) the distinction he made between two kinds of probability; (ii) the law of large numbers; (iii) the Poisson distribution; (iv) the difference between two proportions; (v) legalistic statistics; (vi) Poisson's summation formula; (vii) the Cauchy distribution; and (viii) the Poisson bracket.

Probability Density Distribution of Large-Scale Atmospheric Flow

Abstract: Publisher Summary This chapter discusses the probability density distribution of large-scale atmospheric flow. There is evidence that the large-scale dynamics of the atmosphere are consistent with the presence of more than one regime. A barotropic atmosphere confined to a β-channel and flowing over topography is considered. It is found that for each day an average over a latitude band was calculated, obtaining a 64-dimensional array. It was then projected onto Fourier space by employing a standard fast Fourier transform. It is observed that for the same day, the mean zonal wind was evaluated geostrophically from the geopotential height gradient in the latitudinal band considered above. It is suggested that the measure of planetary-scale waves studied show a marked bimodality, which is independent of the known externally forced periodicities of the system. The distribution of the persistence in a basin of attraction of a fixed point follows a Poisson distribution. It is found that the bimodal nature of the density is a consequence of the interannual variability.

Disjunctive kriging revisited: Part I

Abstract: Difficulties in applying disjunctive kriging (D.K.) with an anamorphosis to a normal distribution have led to an interest in D.K. based on other distributions. After reviewing Gaussian D.K., this paper reviews other types of D.K. based on other infinitely divisible distributions (gamma, Poisson, and negative binomial).

On the Mean Time between Failures for Repairable Systems

Abstract: Much of the recent work on modeling repairable systems involves Poisson processes with nonconstant intensity functions, viz, nonhomogeneous Poisson processes. Since times between failures are not identically distributed when the process is nonhomogeneous, it is not clear what concept should take the place of the mean time between failures in assessing the reliability of a repairable system. A number of alternate concepts can be found in the literature. We investigate the relationship between two of the most frequently considered alternatives: the reciprocal of the intensity function, and the mean waiting time from t until the next failure. Theorem 1 states a necessary and sufficient condition for the mean time until the next failure to be asymptotically proportional to the reciprocal of the intensity function. Some examples, including the familiar log-linear and power-intensity processes satisfy this condition. A monotonicity property is also established between these two concepts which could be used to obtain conservative statistical confidence limits for the mean time until the next failure, based on results which are already available for the intensity function of the power-intensity process. However, further study of concepts such as the rate of convergence would be needed in order to determine the degree of approximation of the nominal confidence level to the actual level.

Inference for stationary random fields given poisson samples

Abstract: : This paper examines some questions of statistical inference -- specifically, estimation of the mean and covariance function, as well as linear state estimation -- for stationary random fields observable only at the points of a (likewise) Poisson process. Given a d-dimensional random field and a Poisson process independent of it, suppose that it is possible to observe only the location of each point of the Poisson process and the value of the random field at that (randomly located) point. Nonparametric estimators of the mean and covariance function of the random field - based on observation over compact sets of single realizations of the Poisson samples - are constructed. Under fairly mild conditions these estimators are consistent (in various senses) as the set of observation becomes unbounded in a suitable manner. The state estimation problem of minimum mean squares reconstruction of unobserved values of the random field is also examined.

Iterative Algorithms for Bulk Arrival, Bulk Service Queues with Poisson and Non-Poisson Arrivals

Abstract: Iterative numerical procedures are investigated for the study of queue lengths and waiting times in bulk arrival, bulk service queues For queues with compound Poisson arrivals, the procedure uses the imbedded Markov chain to study a variety of vehicle dispatching strategies The technique is easy to implement, numerically stable and, for most problems that arise in transportation, computationally faster than other approaches that have been proposed For queues with non-Poisson arrivals, an iterative scheme is devised for calculating a discretized form of the waiting time distribution using the concept of unfinished work Numerical experiments with different discretization strategies indicate that a high level of accuracy can be attained at a very moderate cost

Continuous‐Time Versus Discrete‐Time Point Process Models for Rainfall Occurrence Series

Abstract: Several authors have had apparent success in applying continuous-time point process models to rainfall occurrence sequences. In this paper, it is shown that if rainfall occurrences are interpreted as the events of a point process (and not as a censored sample), the continuous-time point process methodology and estimation procedures are not directly applicable since they fail to account for the time discreteness of the sample process. This is demonstrated analytically by studying the effects of discretization on selected statistical properties of a Poisson process, a Neyman-Scott process, and a renewal Cox process with Markovian intensity. In general, the study of rainfall occurrences under the continuous-time point process framework may result in misleading inferences regarding clustering (dispersion), and consequently incorrect interpretations of the underlying rainfall generating mechanisms. For example, daily rainfall occurrence structures underdispersed relative to the Poisson process are usually overdispersed relative to the Bernoulli process (the discrete-time analogue of the Poisson). These findings are confirmed by the statistical analysis of six daily rainfall records representative of a range of U.S. climates, two of which are described in detail.

Robust Density Estimation Through Distance Measurements

Abstract: A minor modification to Morisita's method permits the definition of a robust density estimator based on nearest neighbor by sectors. In a finite population frame, these distances are used to evaluate the probability of selecting the nearest individual to a point chosen at random. In simulations of regular, Poisson, and aggregated patterns, the estimator was shown to be robust not only with regard to unbiasedness, but also to efficiency. This is of particular interest in aggregated cases. See full-text article at JSTOR

Computation of the chi square and Poisson distribution

Abstract: The paper deals with the computation of upper and lower tail probabilities of the chi-square and Poisson distribution with a specified relative accuracy on both tails for virtually all possible parameter values. With some supplement the proposed algorithms will also work for the general gamma distribution. If the parameters are small, open forward and backward recursion is used for the summation with an adaptive number of steps depending on the specified accuracy. For large parameters asymptotic expansions related to the central limit theorem are applied for the approximation. The basic ideas of the proposed methods will also be applicable to other elementary statistical distributions such as the binomial, beta, and F-distribution as well as the hypergeometric distribution.