Showing papers on "Poisson distribution published in 1992"
TL;DR: Zero-inflated Poisson (ZIP) regression as discussed by the authors is a model for counting data with excess zeros, which assumes that with probability p the only possible observation is 0, and with probability 1 − p, a Poisson(λ) random variable is observed.
Abstract: Zero-inflated Poisson (ZIP) regression is a model for count data with excess zeros. It assumes that with probability p the only possible observation is 0, and with probability 1 – p, a Poisson(λ) random variable is observed. For example, when manufacturing equipment is properly aligned, defects may be nearly impossible. But when it is misaligned, defects may occur according to a Poisson(λ) distribution. Both the probability p of the perfect, zero defect state and the mean number of defects λ in the imperfect state may depend on covariates. Sometimes p and λ are unrelated; other times p is a simple function of λ such as p = l/(1 + λ T ) for an unknown constant T . In either case, ZIP regression models are easy to fit. The maximum likelihood estimates (MLE's) are approximately normal in large samples, and confidence intervals can be constructed by inverting likelihood ratio tests or using the approximate normality of the MLE's. Simulations suggest that the confidence intervals based on likelihood ratio test...
3,440 citations
Book•
01 Jan 1992
TL;DR: In this paper, the authors propose a family of Discrete Distributions, which includes Hypergeometric, Mixture, and Stopped-Sum Distributions (see Section 2.1).
Abstract: Preface. 1. Preliminary Information. 2. Families of Discrete Distributions. 3. Binomial Distributions. 4. Poisson Distributions. 5. Neggative Binomial Distributions. 6. Hypergeometric Distributions. 7. Logarithmic and Lagrangian Distributions. 8. Mixture Distributions. 9. Stopped-Sum Distributions. 10. Matching, Occupancy, Runs, and q-Series Distributions. 11. Parametric Regression Models and Miscellanea. Bibliography. Abbreviations. Index.
2,106 citations
TL;DR: In this article, a family of two-dimensional, two-phase, composite materials with hexagonal symmetry was found with Poisson's ratios arbitrarily close to 1 as r → 0 and in this limit it was conjectured that the material deforms conlbrmally on a macroscopic scale.
Abstract: A family of two-dimensional, two-phase, composite materials with hexagonal symmetry is found with Poisson's ratios arbitrarily close to — 1. Letting k∗, k1,k2 and μ∗,μ1,μ2 denote the bulk and shear moduli of one such composite, stiff inclusion phase and compliant matrix phase, respectively, it is rigorously established that when k1 = K2r and μ1 = μ2r there exists a constant c depending only on k2, μ2 and the geometry such that k∗/μ∗
456 citations
TL;DR: In this paper, a maximum-penalized likelihood method is proposed for estimating a mixing distribution and it is shown that this method produces a consistent estimator, in the sense of weak convergence.
Abstract: A maximum-penalized-likelihood method is proposed for estimating a mixing distribution and it is shown that this method produces a consistent estimator, in the sense of weak convergence. In particular, a new proof of the consistency of maximum-likelihood estimators is given. The estimated number of components is shown to be at least as large as the true number, for large samples. Also, the large-sample limits of estimators which are constrained to have a fixed finite number of components are identified as distributions minimizing Kullback-Leibler divergence from the true mixing distribution. Estimation of a Poisson mixture distribution is illustrated using the distribution of traffic accidents presented by Simar.
442 citations
TL;DR: In this paper, general formulae are obtained for size-biased sampling from a Poisson point process in an abstract space where the size of a point is defined by an arbitrary strictly positive function.
Abstract: Some general formulae are obtained for size-biased sampling from a Poisson point process in an abstract space where the size of a point is defined by an arbitrary strictly positive function. These formulae explain why in certain cases (gamma and stable) the size-biased permutation of the normalized jumps of a subordinator can be represented by a stickbreaking (residual allocation) scheme defined by independent beta random variables. An application is made to length biased sampling of excursions of a Markov process away from a recurrent point of its statespace, with emphasis on the Brownian and Bessel cases when the associated inverse local time is a stable subordinator. Results in this case are linked to generalizations of the arcsine law for the fraction of time spent positive by Brownian motion.
392 citations
01 Jan 1992
TL;DR: A method to compare the Bayesian bootstrap with a parametric analysis is derived and the method is applied to the mean of a Poisson distribution.
Abstract: A method to compare the Bayesian bootstrap with a parametric analysis is derived. The method is applied to the mean of a Poisson distribution.
365 citations
TL;DR: In this article, a method for obtaining tests for overdispersion with respect to a natural exponential family is derived, which is designed to be powerful against arbitrary alternative mixture models where only the first two moments of the mixed distribution are specified.
Abstract: In this article a method for obtaining tests for overdispersion with respect to a natural exponential family is derived. The tests are designed to be powerful against arbitrary alternative mixture models where only the first two moments of the mixed distribution are specified. Various tests for extra-Poisson and extra-binomial variation are obtained as special cases; the use of a particular test may be motivated by a consideration of the mechanism through which the overdispersion may arise. The common occurrence of extra-Poisson and extra-binomial variation has been noted by several authors. However, the Poisson and binomial models remain valid in many instances and, because of their simplicity and appeal, it is of real interest to ascertain when they apply. This paper develops a unifying theory for testing for overdispersion and generalizes tests previously derived, including those by Fisher (1950), Collings and Margolin (1985), and Prentice (1986). It also shows the Pearson statistic to be a sc...
350 citations
TL;DR: Negative Poisson's ratio copper foam was prepared and characterized experimentally as mentioned in this paper, and the transformation into reentrant foam was accomplished by applying sequential permanent compressions above the yield point to achieve a triaxial compression.
Abstract: Negative Poisson's ratio copper foam was prepared and characterized experimentally. The transformation into re-entrant foam was accomplished by applying sequential permanent compressions above the yield point to achieve a triaxial compression. The Poisson's ratio of the re-entrant foam depended on strain and attained a relative minimum at strains near zero. Poisson's ratio as small as -0.8 was achieved. The strain dependence of properties occurred over a narrower range of strain than in the polymer foams studied earlier. Annealing of the foam resulted in a slightly greater magnitude of negative Poisson's ratio and greater toughness at the expense of a decrease in the Young's modulus.
299 citations
TL;DR: In this article, the authors show how approximate maximum likelihood estimation for fairly general point processes on the line can be performed with GLIM, based on a weighted sum approximation to an integral in the likelihood.
Abstract: This paper shows how approximate maximum likelihood estimation for fairly general point processes on the line can be performed with GLIM. The approximation is based on a weighted sum approximation to an integral in the likelihood. Various weighting schemes are briefly examined. The methodology is illustrated with an example, and its extension to Poisson processes in higher dimensions is briefly described.
260 citations
TL;DR: This paper concerns the use and implementation of maximum-penalized-likelihood procedures for choosing the number of mixing components and estimating the parameters in independent and Markov-dependent mixture models.
Abstract: SUMMARY This paper concerns the use and implementation of maximum-penalized-likelihood procedures for choosing the number of mixing components and estimating the parameters in independent and Markov-dependent mixture models. Computation of the estimates is achieved via algorithms for the automatic generation of starting values for the EM algorithm. Computation of the information matrix is also discussed. Poisson mixture models are applied to a sequence of counts of movements by a fetal lamb in utero obtained by ultrasound. The resulting estimates are seen to provide plausible mechanisms for the physiological process. The analysis of count data that are overdispersed relative to the Poisson distribution (i.e., variance > mean) has received considerable recent attention. Such data might arise in a clinical study in which overdispersion is caused by unexplained or random subject effects. Alternatively, we might observe a time series of counts in which temporal patterns in the data suggest that a Poisson model and its implied randomness are inappropriate. This paper is motivated by analysis of a time series of overdispersed count data generated in a study of central nervous system development in fetal lambs. Our data set consists of observed movement counts in 240 consecutive 5-second intervals obtained from a single animal. In analysing these data, we focus on the use of Poisson mixture models assuming independent observations and also Markov-dependent mixture models (or hidden Markov models). These models assume that the counts follow independent Poisson distributions conditional on the rates, which are generated from a mixing distribution either independently or with Markov dependence. We believe finite mixture models are particularly attractive because they provide plausible explanations for variation in the data. This paper will emphasize the following issues concerning estimation, inference, and application of mixture models: (i) choosing the number of model components; (ii) applying the EM algorithm to obtain parameter estimates; (iii) generating sufficiently many starting values to identify a global maximum of the likelihood; (iv) avoiding numerical instability
TL;DR: In this paper, the authors describe several methods that can be used to analyze count data and investigate the small-sample properties of these estimators in the presence of overdispersion and autocorrelation.
Abstract: I begin this paper by describing several methods that can be used to analyze count data. Starting with relatively familiar maximum likelihood methods-Poisson and negative binomial regression-I then introduce the less well known (and less well understood) quasi-likelihood approach. This method (like negative binomial regression) allows one to model overdispersion, but it can also be generalized to deal with autocorrelation. I then investigate the small-sample properties of these estimators in the presence of overdispersion and autocorrelation by means of Monte Carlo simulations. Finally, I apply these methods to the analysis of data on the foundings of labor unions in the U.S. Quasi-likelihood methods are found to have some advantages over Poisson and negative binomial regression, especially in the presence of autocorrelation.
TL;DR: The negative Poisson ratio of the transverse strain to the corresponding axial strain was observed in α-cristobalite and other forms of silica with first-principles calculations and classical interatomic potentials as mentioned in this paper.
Abstract: THE Poisson ratio of a solid characterizes its response to uniaxial stress. It is defined as the negative ratio of the transverse strain to the corresponding axial strain. Normally, this ratio is positive, as most solids expand in the transverse direction when subjected to a uniaxial compression. Although a negative Poisson ratio is not forbidden by thermodynamics, it is rare in crystalline solids: the results of recent experiments1 which observed a negative Poisson ratio in α-cristobalite were therefore unexpected. We have investigated the elastic behaviour of α-cristobalite and other forms of silica with first-principles calculations and classical interatomic potentials. Our calculations reproduce the negative Poisson ratio in α-cristobalite, and predict that α-quartz, the most common form of crystalline silica, will also exhibit a negative Poisson ratio under large uniaxial tension. We attribute the occurrence of a negative Poisson ratio in low-density silica polymorphs to the high rigidity of the SiO4 tetrahedra.
TL;DR: A simple, completely general, and computationally efficient procedure for calculating probability distributions arising from fluctuation analysis and the formula for this procedure when cells in a colony have only grown for a finite number of generations after initial seeding are reported.
Abstract: Fluctuation analysis, which is often used to demonstrate random mutagenesis in cell lines (and to estimate mutation rates), is based on the properties of a probability distribution known as the Luria-Delbruck distribution (and its generalizations). The two main new results reported in this paper are (i) a simple, completely general, and computationally efficient procedure for calculating probability distributions arising from fluctuation analysis and (ii) the formula for this procedure when cells in a colony have only grown for a finite number of generations after initial seeding. It is also shown that the procedure reduces to one that was developed earlier when an infinite number of generations is assumed. The derivation of the generating function of the distribution is also clarified. The results obtained should also be useful to experimentalists when only a relatively short time elapses between seeding and harvestint cultures for fluctuation analysis.
TL;DR: In some production processes and administrative processes, the occurrence of certain events is best described by a geometric distribution as discussed by the authors, and control charts are developed for the total number of events and for the average of events in a fixed number.
Abstract: In some production processes and administrative processes, the occurrence of certain events is best described by a geometric distribution. Control charts are developed for the total number of events and for the average number of events in a fixed number..
TL;DR: This paper shows that the exact binomial and Poisson confidence limits can be expressed very simply in terms of the inverse beta and inverse gamma distributions, and describes two macros in the SAS programming language to perform the computations.
TL;DR: In this paper, the authors extend Stein's method to a compound Poisson distribution setting, where the distribution of random variables is a finite positive measure on $(0, ∞).
Abstract: The aim of this paper is to extend Stein's method to a compound Poisson distribution setting. The compound Poisson distributions of concern here are those of the form POIS$(
u)$, where $
u$ is a finite positive measure on $(0, \infty)$. A number of results related to these distributions are established. These in turn are used in a number of examples to give bounds for the error in the compound Poisson approximation to the distribution of a sum of random variables.
TL;DR: In this article, a complete statistical description of the properties of a cellular microstructure generated by a three-dimensional Poisson-Voronoi tesselation has been obtained by a rigorous computer simulation involving several hundred thousand cells.
Abstract: A complete statistical description of the properties of a cellular microstructure generated by a three-dimensional Poisson-Voronoi tesselation has been obtained by a rigorous computer simulation involving several hundred thousand cells. A two-parameter gamma distribution is found to be a good fit to the cell's face, volume, and surface area distributions. For a sample size of several thousand cells or less, a lognormal distribution can also be used to approximate these distributions. The individual face, area, and edge length distributions are also obtained.
TL;DR: In this paper, the authors give a simple upper bound on the total variation distance between a random permutation and an independent Poisson process and show that this distance decays to zero superexponentially fast as a function of n/b \rightarrow \infty.
Abstract: The total variation distance between the process which counts cycles of size $1,2,\ldots, b$ of a random permutation of $n$ objects and a process $(Z_1,Z_2,\ldots, Z_b)$ of independent Poisson random variables with $\mathbb{E}Z_i = 1/i$ converges to 0 if and only if $b/n \rightarrow 0$. This Poisson approximation can be used to give simple proofs of limit theorems and bounds for a wide variety of functionals of random permutations. These limit theorems include the Erdos-Turan theorem for the asymptotic normality of the log of the order of a random permutation, and the DeLaurentis-Pittel functional central limit theorem for the cycle sizes. We give a simple explicit upper bound on the total variation distance to show that this distance decays to zero superexponentially fast as a function of $n/b \rightarrow \infty$. A similar result holds for derangements and, more generally, for permutations conditioned to have given numbers of cycles of various sizes. Comparison results are included to show that in approximating the cycle structure by an independent Poisson process the main discrepancy arises from independence rather than from Poisson marginals.
Journal Article•
TL;DR: In this paper, a Poisson regression model is proposed to establish empirical relationships between truck accidents and key highway geometric design variables, such as horizontal curvature, vertical grade, and shoulder width.
Abstract: A Poisson regression model is proposed to establish empirical relationships between truck accidents and key highway geometric design variables. For a particular road section, the number of trucks involved in accidents over 1 year was assumed to be Poisson-distributed. The Poisson rate was related to the road section's geometric, traffic, and other explanatory variables (or covariates) by a loglinear function, which ensures that the rate is always nonnegative. The primary data source used was the Highway Safety Information System (HSIS), administered by FHWA. Highway geometric and traffic data for rural Interstate highways and the associated truck accidents in one HSIS state from 1985 to 1987 were used to illustrate the proposed model. The maximum likelihood method was used to estimate the model coefficients. The final model suggested that annual average daily traffic per lane, horizontal curvature, and vertical grade were significantly correlated with truck accident involvement rate but that shoulder width had comparably less correlation. Goodness-of-fit test statistics indicated that extra variation (or overdispersion) existed in the developed Poisson model, which was most likely due to the uncertainties in truck exposure data and omitted variables in the model. This suggests that better quality in truck exposure data and additional covariates could probably improve the current model. Subsequent analyses suggested, however, that this overdispersion did not change the conclusions about the relationships between truck accidents and the examined geometric and traffic variables.
TL;DR: In this paper, a Poisson pacemaker/accumulator system for perceiving and remembering time is studied and it is shown that were the Poisson process the only source of variance, the distribution of estimates of a remembered time would follow a compound gamma law.
TL;DR: In this article, the Stein-Chen method for Poisson approximation is adapted into a form suitable for obtaining error estimates for the approximation of the whole distribution of a point process on a suitable topological space by that of a Poisson process.
Book•
01 May 1992
TL;DR: Inverse spectral results for generic bounded domains: planar domains interpolating Hamiltonians approximations of closed geodesics by periodic reflecting rays Poisson relation for generic strictly convex domains.
Abstract: Part 1 Preliminaries from differential topology and microlocal analysis: jets and transversality theorems generalized bicharacteristics wave front sets of distributions. Part 2 Reflecting rays: billiard ball map periodic rays for several convex bodies Poincare map scattering rays examples. Part 3 Generic properties of reflecting rays: generic properties and smooth embeddings elementary generic properties absence of tangent segments non-degeneracy of reflecting rays. Part 4 Bumpy metrics: Poincare map for closed geodesics local perturbations of smooth surfaces non-degeneracy and transversality global perturbations of smooth surfaces. Part 5 Poisson relation for manifolds with boundary: Poisson relation for convex domains Poisson relation for arbitrary domains. Part 6 Poisson summation formula for manifolds with boundary: global parametrix for mixed problem Poisson summation formula. Part 7 Inverse spectral results for generic bounded domains: planar domains interpolating Hamiltonians approximations of closed geodesics by periodic reflecting rays Poisson relation for generic strictly convex domains. Part 8 Poisson relation for the scattering kernel: representation of the scattering kernel Poisson relation for the scattering kernel. Part 9 Singularities of the scattering kernel for generic domains. Part 10 Scattering invariants for several strictly convex domains: hyperbolicity of scattering trajectories existence of scattering rays and asymptotic of their sojourn times asymptotic of the coefficients of the main singularity.
TL;DR: A statistical method for selecting the Gibbs parameter inMAP image restoration from Poisson data using Gibbs priors is presented and a simple iterative feedback algorithm is presented to statistically select the parameter as the MAP image restoration is being performed.
Abstract: A statistical method for selecting the Gibbs parameter in MAP image restoration from Poisson data using Gibbs priors is presented. The Gibbs parameter determines the degree to which the prior influences the restoration. The presented method yields a MAP restored image, minimally influenced by the prior, for which a statistic falls within an appropriate confidence interval. The method assumes that a close approximation to the blurring function is known. A simple iterative feedback algorithm is presented to statistically select the parameter as the MAP image restoration is being performed. This algorithm is heuristically based on a model reference control formulation, but it requires only a minimal number of iterations for the parameter to settle to its statistically specified value. The performance of the statistical method for selecting the prior parameter and that of the iterative feedback algorithm are demonstrated using both 2-D and 3-D images. >
TL;DR: In this paper, the authors provide a general study on quadratic Poisson structures on a vector space and obtain a decomposition for any quadrastic Poisson structure. And they classify all the three-dimensional quad-ratic poisson structures up to a Poisson diffeomorphism.
Abstract: We provide a general study on quadratic Poisson structures on a vector space. In particular, we obtain a decomposition for any quadratic Poisson structures. As an application, we classify all the three-dimensional quadratic Poisson structures up to a Poisson diffeomorphism.
TL;DR: CP is found to be superior to Poisson where clumping of failures exists and its predictive validity is comparable to the Musa-Okumoto log-Poisson model in certain cases.
Abstract: The probability density estimation of the number of software failures in the event of clustering or clumping of the software failures is considered. A discrete compound Poisson (CP) prediction model is proposed for the random variable X/sub rem/, which is the remaining number of software failures. The compounding distributions, which are assumed to govern the failure sizes at Poisson arrivals, are respectively taken to be geometric when failures are forgetful and logarithmic-series when failures are contagious. The expected value ( mu ) of X/sub rem/ is calculated as a function of the time-dependent Poisson and compounding distribution based on the failures experienced. Also, the variance/mean parameter for the remaining number of failures, q/sub rem/, is best estimated by q/sub past/ from the failures already experienced. Then, one obtains the PDF of the remaining number of failures estimated by CP( mu ,q). CP is found to be superior to Poisson where clumping of failures exists. Its predictive validity is comparable to the Musa-Okumoto log-Poisson model in certain cases. >
TL;DR: This paper will cover the bonus-malus system in automobile insurance, which is based on the distribution of the number of car accidents, and the modelling and fitting of that distribution are considered.
Abstract: In this paper, we will cover the bonus-malus system m automobile insurance. Bonus-malus systems are based on the distribution of the number of car accidents Therefore, the modelling and fitting of that dlsmbuhon are considered. Fitting of data Js done using the Polsson mverse Gaussmn distribution, which shows a good fit Building the bonus system is done by minimizing the insurer's risk, according to LEMA~RE'S (1985) bonus system.
TL;DR: In this paper, the authors show that Pearson's chi-squared and related tests are not appropriate for all frequency-type data, such as the number of encounters between individuals or performances of a behaviour, and a superior approach for the analysis of these data is demonstrated using parametric and nonparametric analysis of variance (ANOVA).
TL;DR: In this paper, it was shown that for an i.i.d. sample, wild bootstrap works under the same conditions as bootstrap under the assumption that the normal approximation with estimated variance works.
Abstract: We show for an i.i.d. sample that bootstrap estimates consistently the distribution of a linear statistic if and only if the normal approximation with estimated variance works. An asymptotic approach is used where everything may depend onn. The result is extended to the case of independent, but not necessarily identically distributed random variables. Furthermore it is shown that wild bootstrap works under the same conditions as bootstrap.