Showing papers on "Poisson distribution published in 1983"
TL;DR: In models for vital rates which include effects due to age, period and cohort, there is aliasing due to a linear dependence among these three factors both when age and period intervals are equal and when they are not.
Abstract: In models for vital rates which include effects due to age, period and cohort, there is aliasing due to a linear dependence among these three factors This dependence arises both when age and period intervals are equal and when they are not One solution to the dependence is to set an arbitrary constraint on the parameters Estimable functions of the parameters are invariant to the particular constraint applied For evenly spaced intervals, deviations from linearity are estimable but only a linear function of the three slopes is estimable When age and period intervals have different widths, further aliasing occurs It is assumed that the number of deaths in the numerator of the rate equation has a Poisson distribution The calculations are illustrated with data on mortality from prostate cancer among nonwhites in the US
549 citations
TL;DR: The method is illustrated by using a nonlinear model, derived from the multistage theory of carcinogenesis, to analyze lung cancer death rates among British physicians who were regular cigarette smokers.
Abstract: Models are considered in which the underlying rate at which events occur can be represented by a regression function that describes the relation between the predictor variables and the unknown parameters. Estimates of the parameters can be obtained by means of iteratively reweighted least squares (IRLS). When the events of interest follow the Poisson distribution, the IRLS algorithm is equivalent to using the method of scoring to obtain maximum likelihood (ML) estimates. The general Poisson regression models include log-linear, quasilinear and intrinsically nonlinear models. The approach considered enables one to concentrate on describing the relation between the dependent variable and the predictor variables through the regression model. Standard statistical packages that support IRLS can then be used to obtain ML estimates, their asymptotic covariance matrix, and diagnostic measures that can be used to aid the analyst in detecting outlying responses and extreme points in the model space. Applications of these methods to epidemiologic follow-up studies with the data organized into a life-table type of format are discussed. The method is illustrated by using a nonlinear model, derived from the multistage theory of carcinogenesis, to analyze lung cancer death rates among British physicians who were regular cigarette smokers.
423 citations
TL;DR: Since the probability of developing or dying from most neurologic disorders is relatively small, the Poisson distribution is often utilized to establish confidence intervals around rates or ratios for neurological disorders.
Abstract: Since the probability of developing or dying from most neurologic disorders is relatively small, the Poisson distribution is often utilized to establish confidence intervals around rates or ratios for diseases of the nervous system. This report describes a simplified method and provides a table of factors based on the Poisson distribution for calculating confidence intervals around estimates of rates and ratios derived from neuroepidemiologic studies.
289 citations
TL;DR: The normal, Poisson, gamma, binomial, negative binomial and NEFGHS distributions are the six univariate natural exponential families with quadratic variance functions (QVF) as mentioned in this paper.
Abstract: The normal, Poisson, gamma, binomial, negative binomial, and NEFGHS distributions are the six univariate natural exponential families (NEF) with quadratic variance functions (QVF). This sequel to Morris (1982) treats certain statistical topics that can be handled within this unified NEF-QVF formulation, including unbiased estimation, Bhattacharyya and Cramer-Rao lower bounds, conditional distributions and moments, quadratic regression, conjugate prior distributions, moments of conjugate priors and posterior distributions, empirical Bayes and $G_2$ minimax, marginal distributions and their moments, parametric empirical Bayes, and characterizations.
257 citations
TL;DR: In this paper, the asymptotic behavior of symmetric statistics of arbitrary order is studied. But the authors use as a tool a randomization of the sample size, which they use as an application to describe all limit distributions of square integrable $U$-statistics.
Abstract: The asymptotic behaviour of symmetric statistics of arbitrary order is studied. As an application we describe all limit distributions of square integrable $U$-statistics. We use as a tool a randomization of the sample size. A sample of Poisson size $N_\lambda$ with $EN_\lambda = \lambda$ can be interpreted as a Poisson point process with intensity $\lambda$, and randomized symmetric statistics are its functionals. As $\lambda \rightarrow \infty$, the probability distribution of these functionals tend to the distribution of multiple Wiener integrals. This can be considered as a stronger form of the following well-known fact: properly normalized, a Poisson point process with intensity $\lambda$ approaches a Gaussian random measure, as $\lambda \rightarrow \infty$.
145 citations
01 Jan 1983
TL;DR: For a stochastic epidemic of the type considered by Bailey and Kendall as mentioned in this paper, the distribution of the number remaining uninfected in a large epidemic has approximately the Poisson form.
Abstract: For a stochastic epidemic of the type considered by Bailey [1] and Kendall [3], Daniels [2] showed that 'when the threshold is large but the population size is much larger, the distribution of the number remaining uninfected in a large epidemic has approximately the Poisson form.' A simple, intuitive proof is given for this result without use of Daniels's assumption that the original number of infectives is 'small'. The proof is based on a construction of the epidemic process which is more explicit than the usual description. LIMITING POISSON DISTRIBUTION; STOCHASTIC EPIDEMIC
126 citations
TL;DR: In this paper, Stein's method of proving limit theorems for sums of dependent random variables is used to derive Poisson approximations for a class of statistics, constructed from finitely exchangeable random variables.
Abstract: Stein's (1970) method of proving limit theorems for sums of dependent random variables is used to derive Poisson approximations for a class of statistics, constructed from finitely exchangeable random variables. Let be exchangeable random elements of a space and, for I a k -subset of , let X I be a 0–1 function. The statistics studied here are of the form
where N is some collection of k -subsets of . An estimate of the total variation distance between the distributions of W and an appropriate Poisson random variable is derived and is used to give conditions sufficient for W to be asymptotically Poisson. Two applications of these results are presented.
101 citations
TL;DR: In this article, an empirical investigation of the assumptions of the NBD model in the context of purchasing at individual stores is presented, and a formal testing procedure against the assumption that individuals' inter-purchase times follow an exponential distribution is presented.
Abstract: This paper presents an empirical investigation of the assumptions of the NBD model in the context of purchasing at individual stores. A formal testing procedure against the assumption that individuals' inter‐purchase times follow an exponential distribution shows that the Poisson assumption of the NBD model holds for the majority of consumers. The theoretical negative binomial also fits closely for centrally located stores. For suburban stores the relevant population hypothesis is of some importance, and refitting the model to a local subset of the sample removes certain small but consistent discrepancies. The NBD model may thus play an important role in studies of urban consumer behaviour.
81 citations
TL;DR: If the lognormal model is representative of the coliform distribution; the arithmetic mean sample count is a poor estimator of the true mean coliform density, and the probability of water in a distribution system containing small patches with large coliform densities without detection by routine monitoring is finite.
Abstract: Nine small water distribution systems were sampled intensively to determine the patterns of dispersion of coliforms. The frequency distributions of confirmed coliform counts were compatible with either the negative-binomial or the lognormal distribution. They were not compatible with either the Poisson or Poisson-plus-added zeroes distribution. The implications of the use of the lognormal distributional model were further evaluated because of its previous use in water quality studies. The geometric means from 14 data sets ranged from 10(-6) to 0.2 coliforms per 100 ml, and the geometric standard deviations were between 10 and 100, with one exception. If the lognormal model is representative of the coliform distribution; the arithmetic mean sample count is a poor estimator of the true mean coliform density, and the probability of water in a distribution system containing small patches with large coliform densities without detection by routine monitoring is finite. These conclusions have direct bearing on the interpretation of microbiological quality standards for drinking water.
63 citations
TL;DR: In this paper, the authors extended and unified the theory of simultaneous estimation for the discrete exponential family and showed that new simultaneous Poisson means estimators perform more favorably than those previously proposed.
Abstract: This paper extends and unifies the theory of simultaneous estimation for the discrete exponential family. We discuss construction of estimators which theoretically dominate the uniformly minimum variance unbiased estimator (UMVUE) under a weighted squared error loss function, and show by means of computer simulation results that new simultaneous Poisson means estimators perform more favorably than those previously proposed. Our improved estimators shift the UMVUE towards a possibly nonzero point or a data-based point.
59 citations
TL;DR: Examining extensive amino acid sequence data now available for five protein families, it is evident that most estimates of total base substitutions between genes are badly in need of revision.
Abstract: An examination has been conducted of the extensive amino acid sequence data now available for five protein families - the alpha crystallin A chain, myoglobin, alpha and beta hemoglobin, and the cytochromes c - with the goal of estimating the true spatial distribution of base substitutions within genes that code for proteins. In every case the commonly used Poisson density failed to even approximate the experimental pattern of base substitution. For the 87 species of beta hemoglobin examined, for example, the probability that the observed results were from a Poisson process was the minuscule 10 to the -44th. Analogous results were obtained for the other functional families. All the data were reasonably, but not perfectly, described by the negative binomial density. In particular, most of the data were described by one of the very simple limiting forms of this density, the geometric density. The implications of this for evolutionary inference are discussed. It is evident that most estimates of total base substitutions between genes are badly in need of revision.
TL;DR: Tests based upon combinatorial methods are presented to examine the significance of the number of contiguous counties among those with high rates and suggest that the practice of dividing the counties into high- and low-risk categories on the basis of the ordered rates alone should be questioned.
Abstract: The clustering of cases of a rare disease is considered. The number of events observed for each unit is assumed to have a Poisson distribution, the mean of which depends upon the population size and the cluster membership of that unit. Here a cluster consists of those units that are homogeneous in their rate of occurrence of the rare events under study. A sample of units is modeled by a mixture of Poisson distributions, one for each cluster, the mixing parameters being the proportions of all units represented by the components of the mixture. Maximum likelihood and Bayes approaches are employed to determine criteria for separating a sample into groups of units with homogeneous rates. A likelihood ratio test for the significance of a two-component mixture is presented as an example. The performance of the criteria is illustrated with data on the spatial occurrence of sudden infant deaths (SIDs) in North Carolina counties over a four-year period. The results suggest that the practice of dividing the counties into high- and low-risk categories on the basis of the ordered rates alone should be questioned. Tests based upon combinatorial methods are also presented to examine the significance of the number of contiguous counties among those with high rates.
TL;DR: In this paper, the authors investigate the maximum number of points that have a given point as their nearest neighbor in the Poisson process. But they only consider the case where the points are independently and uniformly distributed in a d-dimensional cube of volume n.
Abstract: We investigate, for several models of point processes, the (random) number N of points which have a given point as their nearest neighbor. The largedimensional limit of Poisson processes is treated by considering N = Nd for n points independently and uniformly distributed in a d-dimensional cube of volume n and showing that lim,_, lim ,Na d Poisson (A = 1). An asymptotic Poisson (A = 1) distribution also holds for many of the other models. On the other hand, we find that limn--lim ,,, N D 0. Related results concern the (random) volume, Vol%, of a Voronoi polytope (or Dirichlet cell) in the cube model; we find that limd.oolimn-.Vol D 1 while lim_,, lim__, Vold 9 0. POISSON PROCESSES; DIRICHLET CELLS
TL;DR: In this article, the asymptotic behavior of the maximum of a set of independent, identically distributed Poisson random variables is investigated, and it is shown that the convergence from the maximum to the extreme two-point distribution is very slow and of an oscillatory nature.
Abstract: The asymptotic behaviour of the maximum of a set of independent, identically distributed Poisson random variables is investigated; the convergence of the maximum to the extreme two-point distribution is shown to be very slow and of an oscillatory nature.
TL;DR: In this paper, the authors examine the mathematical assumptions underlying the Poisson distribution, and their correspondence to the theories, and they find that a negatively contagious poisson distribution is equally consistent with the observed distribution of grade of multiples, since its basic assumptions are more consistent with our understanding of the discovery process.
Abstract: After discussing the issues of validity and reliability in the measurement of multiple discoveries and their implications for testing alternative theories, we examine the mathematical assumptions underlying the Poisson distribution, and their correspondence to the theories. We find that a negatively contagious Poisson distribution is equally consistent with the observed distribution of grade of multiples, and since its basic assumptions are more consistent with our understanding of the discovery process, there are ample grounds for rejecting the alternative `chance' theory advanced by Simonton, among others. Lastly, we explore some elements of the Zeitgeist theory which appear to provide a more plausible interpretation of the phenomena, particularly regarding the role of communication in modern science, and report some findings in support of this approach.
TL;DR: In this paper, two stochastic models of the earthquake process are used to generate sequences featuring seismic gaps, foreshock and aftershock episodes, background seismicity, and other patterns which resemble those of observed sequences.
Abstract: Two stochastic models of the earthquake process are used to generate sequences featuring seismic gaps, foreshock and aftershock episodes, background seismicity, and other patterns which resemble those of observed sequences. The models are based on the following statistical assumptions. (a) The distribution of earthquake magnitude is exponential. (b) The distribution in time of earthquake occurrences (excluding aftershocks) is Poisson. (c) The distribution in space of earthquake occurrences (excluding aftershocks) is Poisson. (d) The probability of occurrence of aftershocks increases with main event magnitude and follows Omori9s law. (e) The distribution of distances between main event and aftershocks is exponential, with mean proportional to 10 0.57 M , where M is the magnitude of the main event. (f) The distribution of aftershock magnitudes is exponential and related to the main event magnitude by Bath9s law. One model generates sequences in time only; the other generates events in time and space. The similarity between realizations of the stochastic models and observed earthquake sequences suggests that there may be no information in seismic gaps about the time of occurrence or the magnitude of the next large event in the region.
TL;DR: In this article, the authors show analytically how such restrictions interfere with the underlying hypotheses of the Poisson process commonly used to model flood counts, and caution against imposing restrictions that may render this simple and appealing model inapplicable.
Abstract: In the application of partial duration series models of flood analysis it is occasionally observed that successive exceedances are correlated. To reduce this correlation, some investigators tend to impose certain restrictions on the interarrival times of flood events in order that these events will not occur close together in bunches. We show analytically how such restrictions interfere with the underlying hypotheses of the Poisson process commonly used to model flood counts, and we caution against imposing restrictions that may render this simple and appealing model inapplicable.
TL;DR: In this paper, the coherence properties and photon statistics of stationary light obtained by the superposition of nonstationary emissions occurring at random times, in accordance with a homogeneous Poisson point process, were examined.
Abstract: We examine the coherence properties and photon statistics of stationary light obtained by the superposition of nonstationary emissions occurring at random times, in accordance with a homogeneous Poisson point process. The individual emissions are assumed to be in a coherent, chaotic, or $n$ state. The statistical nature of the emission times results in fluctuations of the relative contributions of different emissions at a given time. This is manifested by an additional positive term, exhibiting particlelike properties, in the normalized second-order correlation function. Thus, the photon-counting variance is increased. For coherent emissions, interference between the randomly delayed emissions produces additional wavelike noise. In the limit when the emissions overlap strongly, the field exhibits the correlation properties of chaotic light, regardless of the statistics of the individual emissions. In the opposite limit, when emissions seldom overlap, the light intensity is describable by a shot-noise stochastic process, and the detected photocounts show an enhanced particlelike noise, which has its largest value when the counting time is long. In that limit, the photocounts obey the Neyman type-$A$ and generalized Polya-Aeppli distributions, when the individual emissions are coherent and chaotic, respectively. When the individual emissions correspond to the $n$ state, the Poisson emission times result in bunching which reduces or eliminates the inherent antibunching associated with the $n$ state.
TL;DR: This paper examines various methods of ‘arithmetizing’ the claim size distribution so that stop-loss premiums can be recursively calculated and develops a decision strategy for choosing a method when both error and computer costs are constrained.
Abstract: This paper examines various methods of ‘arithmetizing’ the claim size distribution so that stop-loss premiums can be recursively calculated. Claim frequencies are assumed to be Poisson. A decision strategy for choosing a method when both error and computer costs are constrained is developed.
TL;DR: A two-parameter description of any computer is given that characterizes the performance of serial, pipelined, and array-like architectures and a family of FACR direct methods for solving Poisson's equation is optimized on the basis of this characterization.
Abstract: A two-parameter description of any computer is given that characterizes the performance of serial, pipelined, and array-like architectures. The first parameter (r∞) is the traditional maximum performance in megaflops, and the new second parameter (n½) measures the apparent parallelism of the computer. For computers with a single instruction stream (unicomputers), the relative performance of two algorithms on the same computer depends only on n½ and the average vector length of the algorithm. The performance of a family of FACR direct methods for solving Poisson's equation is optimized on the basis of this characterization.
TL;DR: Several methods of estimating the ascertainment probability from the distribution of the number of ascertainments per proband have been presented in the literature and these methods are compared.
Abstract: Several methods of estimating the ascertainment probability from the distribution of the number of ascertainments per proband have been presented in the literature. Here these methods are compared. The Skellam often gives the best fit, but there are counter examples. The estimate from a Poisson distribution is generally close to the best solution.
TL;DR: In this paper, the authors presented a Poisson-like distribution called the Hermite, which is the exact distribution of demand during lead time when unit demand is Poisson, P(Λ), and lead time is normally distributed, N(μ, σ2), so long as (μ/σ2)≥Λ.
Abstract: Heretofore, the Poisson and the Laplace distributions have been used to model demand during lead time for slow-moving items. In this paper, we present a Poisson-like distribution called the Hermite. The advantage of the Hermite is that it is as simple to use as the Poisson and the Laplace are. Moreover, the Hermite is the exact distribution of demand during lead time when unit demand is Poisson, P(Λ), and lead time is normally distributed, N(μ, σ2), so long as (μ/σ2)≥Λ. Thus, the Hermite can enhance the accuracy of analysis as well as add to the tools available to the analyst.
TL;DR: Sampling models are investigated for counts of mosquitoes from a malaria field survey conducted by the World Health Organization in Nigeria and an algorithm, based on iterative proportional fitting, is devised for finding maximum likelihood estimates.
Abstract: SUMMARY Sampling models are investigated for counts of mosquitoes from a malaria field survey conducted by the World Health Organization in Nigeria. The data can be described by a negative binomial model for two-way classified counted data, where the cell means are constrained to satisfy row-by-column independence and the parameter k is constant across rows. An algorithm, based on iterative proportional fitting, is devised for finding maximum likelihood estimates. Sampling properties of the estimates and likelihood-ratio statistics for the small sample sizes of the data are investigated by Monte Carlo experiments. The WHO reported an observation that the relative efficiencies of four trapping methods vary over time. Out of eight villages in the survey area, this observation is found to be true in only the one village that is near a swamp. Estimates of mosquito population density in malaria field studies depend fundamentally on the methods used to catch mosquitoes. Various trapping methods exploit different aspects of the mosquito vector's behavior. When several methods of counting mosquitoes are used simultaneously, a statistical model which describes all the methods together permits a rigorous comparison of the various density estimates. In this paper, we shall construct such a statistical model which describes four sampling methods. We have examined data on Anopheles gambiae from a study on the epidemiology and control of malaria conducted by the World Health Organization (WHO) and the Government of Nigeria in Garki, northern Nigeria (Molineaux and Gramiccia, 1980). We describe the mosquito sampling methods used in the Garki study in ?2, and introduce there the question which motivated this work: whether the relative efficiencies of these methods vary over time. In ?3, we demonstrate the inadequacy of a model of Poisson sampling to describe the data. From among several alternatives to the Poisson distribution, we select the negative binomial in ?4. In ?5 we develop the model of independence under negative binomial sampling which describes the data for all villages but one. The hypothesis-testing procedures which we use in ?5 are validated by Monte Carlo results in ?6. In ?7 we compare the methods across villages.
TL;DR: This work finds expressions for the expected delay of packets that are valid in light and heavy traffic and provides an example of a method that often gives fast approximate solutions for bursty traffic models that are not themselves tractable but become so when the offered traffic is assumed to be Poisson.
Abstract: Assuming a particular model for “bursty” traffic at a packet-switching node, we find expressions for the expected delay of packets that are valid in light and heavy traffic. Each expression consists of a “correction factor” multiplied by the expected delay experienced by packets when the arrivals are “smooth” (Poisson) and of the same average rate. Approximate values for the correction factor in arbitrary traffic can be obtained by interpolation. This provides an example of a method that often gives fast approximate solutions for bursty traffic models that are not themselves tractable but become so when the offered traffic is assumed to be Poisson.
TL;DR: Bounds for the probabilities and expected values of the waiting time for a detection are derived for independent identically distributed zero-one Bernoulli trials, binomial, and Poisson random variables.
Abstract: The moving window detection procedure for discrete data is studied. Bounds for the probabilities and expected values of the waiting time for a detection are derived. The bounds are evaluated for independent identically distributed zero-one Bernoulli trials, binomial, and Poisson random variables, and also for the two-state stationary Markov chain. The results are applicable to the theory of radar detection, time sharing systems, and quality control.
TL;DR: Permission to copy without fee all or part of this work is granted provided that the copies are not made or dlstmbuted for direct commercial advantage, the ACM copyright notice and the title of …
Abstract: A) r e t u r n s a s a m p l e S f r o m t h e s t a n d a r d g a m m a d i s t r i b u t i o n w i t h p r o b a b i l i t y d e n s i t y f u n c t i o n y(x) = xa-ie-X/F(a), w h e r e a-~ A > 0. delivers s a m p l e s T f r o m t h e s t a n d a r d n o r m a l d i s t r i b u t i o n. T h e integer v a r i a b l e I R initially defines t h e s e e d of t h e b a s i c r a n d o m n u m b e r sequence. I R is to be e q u a t e d to s o m e p o s i t i v e i n t e g e r of t h e f o r m 4 x K + 1 (our s t a n d a r d s e q u e n c e for t e s t r u n s was b a s e d on I R = 1). A caI1 of a n y of t h e five r o u t i n e s will cause the initialization to t a k e effect (inside S U N I F () , s e c o n d part) a n d to r e s e t I R to-1. F r o m t h e n on this I R < 0 signifies t h a t h e n c e f o r t h t h e p a r a m e t e r I R is a d u m m y variable. Permission to copy without fee all or part of this maternal is granted provided that the copies are not made or dlstmbuted for direct commercial advantage, the ACM copyright notice and the title of …
TL;DR: For several queueing systems, sufficient conditions are given ensuring that from the coincidence of some time-stationary and customerstationary characteristics of the number of customers in the system such as idle or loss probabilities, it follows that the arrival process is Poisson as discussed by the authors.
Abstract: For several queueing systems, sufficient conditions are given ensuring that from the coincidence of some time-stationary and customer-stationary characteristics of the number of customers in the system such as idle or loss probabilities it follows that the arrival process is Poisson.