Showing papers on "Poisson distribution published in 1988"

Statistical Models for Earthquake Occurrences and Residual Analysis for Point Processes

Abstract: This article discusses several classes of stochastic models for the origin times and magnitudes of earthquakes. The models are compared for a Japanese data set for the years 1885–1980 using likelihood methods. For the best model, a change of time scale is made to investigate the deviation of the data from the model. Conventional graphical methods associated with stationary Poisson processes can be used with the transformed time scale. For point processes, effective use of such residual analysis makes it possible to find features of the data set that are not captured in the model. Based on such analyses, the utility of seismic quiescence for the prediction of a major earthquake is investigated.

Some arma models for dependent sequences of Poisson counts

Abstract: A family of models for discrete-time processes with Poisson marginal distributions is developed and investigated. They have the same correlation structure as the linear ARMA processes. The joint distribution of n consecutive observations in such a process is derived and its properties discussed. In particular, time-reversibility and asymptotic behaviour are considered in detail. A vector autoregressive process is constructed and the behaviour of its components, which are Poisson ARMA processes, is considered. In particular, the two-dimensional case is discussed in detail.

On the exceedance point process for a stationary sequence

Abstract: It is known that the exceedance points of a high level by a stationary sequence are asymptotically Poisson as the level increases, under appropriate long range and local dependence conditions. When the local dependence conditions are relaxed, clustering of exceedances may occur, based on Poisson positions for the clusters. In this paper a detailed analysis of the exceedance point process is given, and shows that, under wide conditions, any limiting point process for exceedances is necessarily compound Poisson. More generally the possible random measure limits for normalized exceedance point processes are characterized. Sufficient conditions are also given for the existence of a point process limit. The limiting distributions of extreme order statistics are derived as corollaries.

Computing Poisson probabilities

Abstract: We propose an algorithm to compute the set of individual (nonnegligible) Poisson probabilities, rigorously bound truncation error, and guarantee no overflow or underflow. Work and space requirements are modest, both proportional to the square root of the Poisson parameter. Our algorithm appears numerically stable. We know no other algorithm with all these (good) features. Our algorithm speeds generation of truncated Poisson variates and the computation of expected terminal reward in continuous-time, uniformizable Markov chains. More generally, our algorithm can be used to evaluate formulas involving Poisson probabilities.

Stein's method and poisson process convergence

Abstract: Stein's method of obtaining rates of convergence, well known in normal and Poisson approximation, is considered here in the context of approximation by Poisson point processes, rather than their one-dimensional distributions. A general technique is sketched, whereby the basic ingredients necessary for the application of Stein's method may be derived, and this is applied to a simple problem in Poisson point process approximation.

Approximate interval estimation of the ratio of binomial parameters: a review and corrections for skewness.

Abstract: Various methods for finding confidence intervals for the ratio of binomial parameters are reviewed and evaluated numerically. It is found that the method based on likelihood scores (Koopman, 1984, Biometrics 40, 513-517; Miettinen and Nurminen, 1985, Statistics in Medicine 4, 213-226) performs best in achieving the nominal confidence coefficient, but it may distribute the tail probabilities quite disparately. Using general theory of Bartlett (1953, Biometrika 40, 306-317; 1955, Biometrika 42, 201-203), we correct this method for asymptotic skewness. Following Gart (1985, Biometrika 72, 673-677), we extend this correction to the case of estimating the common ratio in a series of two-by-two tables. Computing algorithms are given and applied to numerical examples. Parallel methods for the odds ratio and the ratio of Poisson parameters are noted.

Mixed Poisson likelihood regression models for longitudinal interval count data.

Abstract: SUMMARY In many longitudinal studies it is desired to estimate and test the rate over time of a particular recurrent event Often only the event counts corresponding to the elapsed time intervals between each subject's successive observation times, and baseline covariate data, are available The intervals may vary substantially in length and number between subjects, so that the corresponding vectors of counts are not directly comparable A family of Poisson likelihood regression models incorporating a mixed random multiplicative component in the rate function of each subject is proposed for this longitudinal data structure A related empirical Bayes estimate of random-effect parameters is also described These methods are illustrated by an analysis of dyspepsia data from the National Cooperative Gallstone Study

Generalized Poisson Functionals

Abstract: With the aim of treating nonlinear systems with inputs being discrete and outputs being generalized functions, generalized Poisson functional are defined and analysed, where the -transforms and the renormalizational play essential roles. For Poisson functionals, the differential operators with respect to a Poisson white noise $$\dot P$$ (t), their adjoint operators and the multiplication operators by $$\dot P$$ (t) are defined. Since these operators involve the time parameter explicitly, they can be used to obtain information concerning the Poisson functional at each point in time. As an example, a new method for measuring the Wiener kernels of such functionals is outlined.

On Exceedance Point Processes for Stationary Sequences under Mild Oscillation Restrictions

Abstract: It is known ([1]) that any point process limit for the (time normalized) exceedances of high levels by a stationary sequence is necessarily Compound Poisson, under general dependence restrictions. This results from the clustering of exceedances where the underlying Poisson points represent cluster positions, and the multiplicities correspond to cluster sizes.

Point Process Models with Applications to Safety and Reliability

Abstract: 1 Introduction.- 1.1 Arrivals in time.- 1.2 Reliability.- 1.3 Safety assessment.- 1.4 Random stress and strength.- Notes on the literature.- Problems.- 2 Point processes.- 2.1 The probabilistic context.- 2.2 Two methods of representation.- 2.3 Parameters of point processes.- 2.4 Transformation to a process with constant arrival rate.- 2.5 Time between arrivals.- Notes on the literature.- Problems.- 3 Homogeneous Poisson processes.- 3.1 Definition.- 3.2 Characterization.- 3.3 Time between arrivals for the hP process.- 3.4 Relations to the uniform distribution.- 3.5 A process with simultaneous arrivals.- Notes on the literature.- Problems.- 4 Application of point processes to a theory of safety assessment.- 4.1 The Reactor Safety Study.- 4.2 The annual probability of a reactor accident.- 4.3 A stochastic consequence model.- 4.4 A concept of rare events.- 4.5 Common mode failures.- 4.6 Conclusion.- Notes on the literature.- Problems.- 5 Renewal processes.- 5.1 Probabilistic theory.- 5.2 The renewal process cannot model equipment wearout.- Notes on the literature.- Problems.- 6 Poisson processes.- 6.1 The Poisson model.- 6.2 Characterization of regular Poisson processes.- 6.3 Time between arrivals for Poisson processes.- 6.4 Further observations on software error detection.- Notes on the literature.- Problems.- 7 Superimposed processes.- Notes on the literature.- Problems.- 8 Markov point processes.- 8.1 Theory.- 8.2 The Poisson process.- 8.3 Facilitation and hindrance.- Notes on the literature.- Problems.- 9 Applications of Markov point processes.- 9.1 Egg-laying dispersal of the bean weevil.- 9.2 Application of facilitation - hindrance to the spatial distribution of benthic invertebrates.- 9.3 The Luria-Delbruck model.- 9.4 Chance placement of balls in cells.- 9.5 A model for multiple vehicle automobile accidents.- 9.6 Engels' model.- Notes on the literature.- Problems.- 10 The order statistics process.- 10.1 The sampling of lifetimes.- 10.2 Derivation from the Poisson process.- 10.3 A Poisson model of equipment wearout.- Notes on the literature.- Problems.- 11 Competing risk theory.- 11.1 Markov chain model.- 11.2 Classical competing risks.- 11.3 Competing risk presentation of reactor safety studies.- 11.4 Delayed fatalities.- 11.5 Proportional hazard rates.- Notes on the literature.- Problems.- Further reading.- Appendix 1 Probability background.- A1.1 Probability distributions.- A1.2 Expectation.- A1.3 Transformation of variables.- A1.4 The distribution of order statistics.- A1.5 Conditional probability.- A1.6 Operational methods in probability.- A1.7 Convergence concepts and results in the theory of probability.- Notes on the literature.- Appendix 2 Technical topics.- A2.1 Existence of point process parameters.- A2.2 No simultaneous arrivals.- Solutions to a few of the problems.- References.- Author index.

Temporal and Magnitude Dependence in Earthquake Recurrence Models

Abstract: The effects of temporal and magnitude dependence among seismic recurrences, which are ignored in the conventional Poisson earthquake model, are studied. The potential impact of non-Poissonian assumptions on practical hazard estimates are considered. A broad set of recurrence models with memory are analyzed using convenient second-moment time-magnitude statistics to parameterize a general class of semi-Markov models. The conventional time- and slip-predictable models are included and studied as special cases. Conditions are identified under which the Poisson model provides a sufficient engineering hazard estimate. i.e., either conservative or unconservative by a factor of no more than three. The Poisson approximation is found to be sufficient for all but what is expected to be a small subset of the cases encountered in practice.

Calculus on Gaussian and Poisson white noises

Abstract: Recently one of the authors has introduced the concept of generalized Poisson functionals and discussed the differentiation, renormalization, stochastic integrals etc ([8], [9]), analogously to the works of T Hida ([3], [4], [5]) Here we introduce a transformation for Poisson fnnctionals with the idea as in the case of Gaussian white noise (cf [10], [11], [12], [13]) Then we can discuss the differentiation, renormalization, multiple Wiener integrals etc in a way completely parallel with the Gaussian case The only one exceptional point, which is most significant, is that the multiplications are described by for the Gaussian case, for the Poisson case, as will be stated in Section 5 Conversely, those formulae characterize the types of white noises

Review of the stirling numbers, their generalizations and Statistical Applications

Abstract: The basic properties of the Stirling numbers and their generalizations are reviewed. Statistical applications of these numbers in expressing (i) the distrib u t l o n ot successes In Poisson (generalized Bernoulli) trials, (ii) the occupancy distributions, (iii) the convolutions of truncated power series distributions and related minimun variance unbiased estimators and (iv) the generalized discrete distributions and factorial moments are presented.

Reexamination of dynamic problems of elasticity for negative Poisson’s ratio

Abstract: Recently, isotropic elastic materials with a negative Poisson’s ratio have been manufactured. Since most of the theoretical results of linear elasticity focus on a positive Poisson’s ratio, the need arises for their extension and reexamination. The above materials may have a variety of technological applications so the motivation for this study is not purely academic. The article deals first with some of the limit cases arising when Poisson’s ratio takes on an extreme value. For models represented by these limit cases, the material and structure responses may not be treated independently from each other. Then such basic dynamic elasticity problems as reflection from a free surface, propagation of Rayleigh waves, and lateral vibrations of beams and plates are reconsidered for the case of a negative Poisson’s ratio. It is shown, in particular, that the static definition of the shear factor in Timoshenko beam theory may not be satisfactory in all cases. Extensive numerical results are also given.

Closure and Monotonicity Properties of Nonhomogeneous Poisson Processes and Record Values

Abstract: Interconnections between occurrence times of nonhomogeneous Poisson processes, record values, minimal repair times, and the relevation transform are explained. A number of properties of the distributions of occurrence times and interoccurrence times of a nonhomogeneous Poisson process are proved when the mean-value function of the process is convex, starshaped, or superadditive. The same results hold for upper record values of independently identically distributed random variables from IFR, IFRA, and NBU distributions.

Cracked media, Poisson's ratio and the structure of the upper oceanic crust

Abstract: SUMMARY Theoretical models of water-filled cracks within a solid are in general agreement, and indicate that the Poisson’s ratio of the composite material will typically increase for thin cracks with aspect ratios less than A, but will decrease for thick cracks with aspect ratios between &, and i. Near-spherical pores cause little change in Poisson’s ratio. This is true both for randomly oriented, isotropic crack models, and, at certain directions, for aligned crack models which predict seismic anisotropy . Observed anomalous values of Poisson’s ratio in oceanic Layer 2 can often be explained using these models as an effect of cracks, and do not necessarily require the presence of rocks of unusual composition. In particular, the low Poisson’s ratios reported by Spudich & Orcutt (1980) and Au & Clowes (1984) can approximately, but not always exactly, be fitted with a model consisting of thick cracks within higher Poisson’s ratio rock more typical of the upper oceanic crust. However, this interpretation is near the limit of existing theories of cracked material, so other explanations for the anomalous observations must also be considered.

Parameter Estimation for Probabilistic Document-Retrieval Models.

Abstract: A probabilistic document-retrieval system may be seen as a sequential learning process, in which the system learns the characteristics of relevant documents, or more formally, it learns the parameters of probability distributions describing the frequencies of feature occurrences in relevant and nonrelevant documents. Probability distributions that may be used to describe the distribution of features include binary and Poisson distributions. Techniques for estimating the parameters of distributions are suggested. We have tested a proposal that parameters of distributions describing the distribution of features in nonrelevant documents be estimated from the parameters of the corresponding distributions of the entire database; the confidence parameter of such an estimate resulting in the highest average precision is given. Tests of several methods for estimating the parameters of distributions describing the distribution of features in relevant documents suggest that small values of the confidence parameter be used in our initial estimates of parameters for relevant documents. © 1988 John Wiley & Sons, Inc.

Uniform saddlepoint approximations

Abstract: The validity of the saddlepoint expansion evaluated at the point y is considered in the limit y tending to oo. This is done for the expansions of the density and of the tail probability of the mean En Xi/n of n i.i.d. random variables and also for the expansion of the tail probability of a compound Poisson sum EXi, where N is a Poisson random variable. We consider both general conditions that ensure the validity of the expansions and study the four classes of densities for X1 introduced in

Seasonality comparisons among groups using incidence data.

Abstract: A new test using incidence data is developed for testing whether two or more groups have the same seasonal pattern. The method fits sine waves to the data with a fundamental period of one cycle per year, and has the possibility of using higher harmonics, when necessary, to adequately model the data. The seasonal pattern can, therefore, have an arbitrary shape. The method allows for different length time intervals and different size populations at risk in the time intervals. Maximum likelihood estimation, based on the Poisson distribution, is used to determine the parameters of the model. Likelihood ratio tests and Akaike's information criterion (AIC) are used to determine the number of harmonics, and to test hypotheses. This method has been used to test for seasonal patterns in the incidence of insulin-dependent diabetes mellitus (IDDM) in Colorado among persons aged 0-17 years. Comparisons of seasonal patterns are made between males and females, and three age groups, each controlling for the other effect as in analysis of variance. Other potential applications of this approach are also discussed. A basic program is available for an IBM-PC to carry out these analyses.

Conditional PASTA

Abstract: Let Y be a stochastic process representing the state of a system and N a doubly stochastic Poisson process whose intensity varies with the state of a random environment represented by a stochastic process X. In this context a generalization of ''Pasta'' (Poisson Arrivals See Time Averages) is shown to be valid. Various applications of the result are given.

Comparison of frequency distributions in flow cytometry.

Abstract: A number of methods have previously been considered for the statistical comparison of flow cytometric frequency distributions. For two distributions, the foremost of these is the Kolmogorov-Smirnov (K-S) test, which has been criticized as “too sensitive.” We discuss some alternative methods based on the Poisson distribution. The assumption of Poisson variation within channels allows the use of channel-by-channel confidence intervals and chi-square tests. These are simple and more appropriate for discrete data than the K-S test. Graphical displays of these and other techniques are presented. We also attempt to set the problem in an appropriate context. We argue that any statistical procedure must rest on a reasonable understanding of the nature of the variability in the system. This understanding takes the form of an appropriate probability model, which may be approximate but must provide a reasonably accurate description of the data. Incomplete understanding of the data can lead to inappropriate analysis. We discuss the assumptions that underlie our techniques and consider extensions to more complex situations.

An Introduction to Probabilistic Modeling

Abstract: Basic concepts and elementary models discrete probability probability densities Gaus and Poisson convergences additional exercises solutions to additional exercises.

Asymptotic gaussianity of some estimators for reduced factorial moment measures and product densities of stationary poisson cluster processes

Abstract: The main purpose of this paper is to present cer~trai limit tileoreins including functional limit theorems for empirical factorial moment measures and kernel-type product density estimators when the underlying point process is a regular infinitely divisible one.The requied moment conditions are minimal, they are necessary to ensure finite variances of the estimators under consideration. In the special case of a stationary poisson process the obtained results are used to construct a goodness-of-fit test for the function λK(t), 0≤t≤T, denoting the mean number of points within a sphere with radius t around a typical point of the process.

Inference about the mean of a poisson distribution in the presence of a nuisance parameter

Abstract: Summary In this note we examine the problem of estimating the mean of a Poisson distribution when a nuisance parameter is present. Using a condition of Cox (1958) about ancillarity in the presence of a nuisance parameter, we justify that inference about the parameter should be carried out using the conditional distribution given the appropriate ancillary statistics. A small simulation study has been done to compare the performance of the conditional likelihood approach and the standard likelihood approach.

On a relationship between Uspensky's theorem and Poisson approximations

Abstract: In this paper we show that Uspensky's expansion theorem for the Poisson approximation of the distribution of sums of independent Bernoulli random variables can be rewritten in terms of the Poisson convolution semigroup. This gives rise to exact evaluations and simple remainder term estimations for the deviations of the distributions in study with respect to various probability metrics, generalizing results of Shorgin (1977, Theory Probab. Appl., 22, 846–850). Finally, we compare the sharpness of Poisson versus normal approximations.