Showing papers on "Poisson distribution published in 1976"

Doubly stochastic Poisson processes

Abstract: Definitions and basic properties.- Some miscellaneous results.- Characterization and convergence of non-atomic random measures.- Limit theorems.- Estimation of random variables.- Linear estimation of random variables in stationary doubly stochastic Poisson sequences.- Estimation of second order properties of stationary doubly stochastic Poisson sequences.

Maximum Likelihood Estimation of a Compound Poisson Process

Abstract: The problem of estimating the compounding distribution of a compound Poisson process from independent observations of the compound process has been analyzed by Tucker (1963). A maximum likelihood method is proposed. The existence, uniqueness and convergence of the resulting estimator are derived. One obtains practical solutions by means of a very simple algorithm which is briefly described. A numerical example is presented in the risk business framework.

Correlations in stochastic theories of chemical reactions

Abstract: A comprehensive study of correlations in linear and nonlinear chemical reactions is presented using coupled chemical and diffusion master equations. As a consequence of including correlations in linear reactions the approach to the steady-state Poisson distribution from an initial non-Poissonian distribution is given by a power law rather than the exponential predicted by neglecting correlations. In nonlinear reactions we show that a steadystate Poisson distribution is achieved in small volumes, whereas in large volumes a non-Poissonian distribution is built up via the correlation. The spatial correlation function is calculated for two examples, one which exhibits an instability, the other which exhibits a second-order phase transition, and correlation length and correlation time are calculated and shown to become infinite as the critical point is approached. The critical exponents are found to be classical.

On the Application of Symmetric Dirichlet Distributions and their Mixtures to Contingency Tables

Abstract: This paper is a continuation of a paper in the Annals of Statistics (1976), 4 1159-1189 where, among other things, a Bayesian approach to testing independence in contingency tables was developed. Our first purpose now, after allowing for an improvement in the previous theory (which also has repercussions on earlier work on the multinomial), is to give extensive numerical results for two-dimensional tables, both sparse and nonsparse. We deal with the statistics $X^2, \Lambda$ (the likelihood-ratio statistic), a slight transformation $G$ of the Type II likelihood ratio, and the number of repeats within cells. The latter has approximately a Poisson distribution for sparse tables. Some of the "asymptotic" distributions are surprisingly good down to exceedingly small tail-area probabilities, as in the previous "mixed Dirichlet" approach to multinomial distributions (J. Roy. Statist. Soc. B. 1967, 29 399-431; J. Amer. Statist. Assoc. 1974, 69 711-720). The approach leads to a quantitative measure of the amount of evidence concerning independence provided by the marginal totals, and this amount is found to be small when neither the row totals nor the column totals are very "rough" and the two sets of totals are not both very flat. For Model 3 (all margins fixed), the relationship is examined between the Bayes factor against independence and its tail-area probability.

Optimum Design for Infrequent Disturbances

Abstract: Sporadic disturbances such as earthquakes and tornadoes are idealized as renewal processes. Uncertainties are classified according to their time correlation into disturbance, structure, and analyst random variables. Only the latter admit Bayesian updating. In this light a result due to Hasofer is revised. Structures are idealized as having a single degree-of-freedom and as having potential limit states in cascade, i.e., limit states can be entered only in a fixed order. Equivalent second-moment (beta method) criteria are developed. Treatment is then specialized to earthquake resistant design, in which disturbances are idealized as generalized Poisson processes. Explicit optimal-design formulas are given for this case. Appendices include novel second-moment approximate treatment of uncertainties not requiring calculation of derivatives. Those variables whose distributions are evidently close to Gaussian or lognormal are given a treatment that is exact for these types of distribution.

Introductory medical statistics

Abstract: Data presentation Describing cuves and distributions The normal distribution curve Introduction to sampling, errors and accuracy Introduction to probability Binomial probabilities Poisson probabilities Introduction to statistical significance The chi-squared test The Fisher exact probability test The t-test Difference between proportions for independent and for non-independent (McNemar's test for paired proportions) samples Wilcoxon, Mann-Whitney and sign tests Survival rate calculations The logrank and Mantel-Haenszel tests Regression and correlation Analysis of variance Multivariate analysis: the Cox proportional hazards model Sensitivity and specificity Clinical trials Cancer treatment success, cure and quality of life Risk specification with emphasis on ionising radiation Types of epidemiological study: case-control, cohort and cross-sectional Glossary of rates and ratios: terminology in vital statistics References Index

Branching diffusion processes in population genetics

Abstract: A branching random field is considered as a model of either of two situations in genetics in which migration or dispersion plays a role. Specifically we consider the expected number of individuals NA in a (geographical) set A at time t, the covariance of NA and NB for two sets A, B, and the probability I(x, y, u) that two individuals found at locations x, y at time t are of the same genetic type if the population is subject to a selectively neutral mutation rate u. The last also leads to limit laws for the average degree of relationship of individuals in various types of branching random fields. We also find, the equations that the mean and bivariate densities satisfy, and explicit formulas when the underlying migration process is Brownian motion. BRANCHING PROCESS; BRANCHING RANDOM FIELD; IDENTITY BY DESCENT; GENETICS; MIGRATION; POISSON RANDOM FIELD; POINT PROCESS; STEPPING-STONE MODEL

On Confidence Sequences

Abstract: This paper is concerned with confidence sequences, i.e., sequences of confidence regions which contain the true parameter for every sample size simultaneously at a prescribed level of confidence. By making use of generalized likelihood ratio martingales, confidence sequences are constructed for the unknown parameters of the binomial, Poisson, uniform, gamma and other distributions. It is proved that for the exponential family of distributions, the method of using generalized likelihood ratio martingales leads to a sequence of intervals which have the desirable property of eventually shrinking to the population parameter. The problem of nuisance parameters is considered, and in this connection, boundary crossing probabilities are obtained for the sequence of Student's $t$-statistics, and a limit theorem relating to the boundary crossing probabilities for the Wiener process is proved.

Simulation of nonhomogeneous Poisson processes with log linear rate function

Abstract: An efficient method for simulating a nonhomogeneous Poisson process with rate function A(t) = exp (co +ac,t) is given. The method is based on an identity relating the nonhomogeneous Poisson process to the gap statistics from a random number of exponential random variables with suitably chosen parameters; it avoids costly ordering and taking of logarithms required by direct simulation methods and is more efficient than time scale transformations of a homogeneous Poisson process.

Buffer Behavior with Poisson Arrivals and Bulk Geometric Service

Abstract: This correspondence considers a buffered system with Poisson arrivals and m output channels subjected to geometrically distributed interruptions. The generating function of the buffet occupancy is derived and is a transcendental function. An example demonstrates the results.

Parallel Poisson and Biharmonic solvers

Abstract: In this paper we develop direct and iterative algorithms for the solution of finite difference approximations of the Poisson and Biharmonic equations on a square, using a number of arithmetic units in parallel. Assuming ann×n grid of mesh points, we show that direct algorithms for the Poisson and Biharmonic equations require 0(logn) and 0(n) time steps, respectively. The corresponding speedup over the sequential algorithms are 0(n 2) and 0(n 2logn). We also compare the efficiency of these direct algorithms with parallel SOR and ADI algorithms for the Poisson equation, and a parallel semi-direct method for the Biharmonic equation treated as a coupled pair of Poisson equations.

Some properties of line segment processes

Abstract: This paper formulates the random process of line-segments in the Euclidean plane. Under conditions more general than Poisson, expressions are obtained, for Borel A ⊂ R 2 , for the first moments of M ( A ), the number of segment mid-points in A ; N ( A ), the number of segments which intersect with convex A ; S ( A ), the total length within A of segments crossing A ; and C ( A ) the number of segment-segment crossings within A . In the case of Poisson mid-points, the distribution of the r th nearest line-segment to a given point is found.

A continuous-time population model with poisson recruitment

Abstract: A continuous-time model of a multigrade system is developed, which includes Poisson arrivals, interaction between grades and a leaving process. It therefore constitutes a continuous-time analogue of Pollard's hierarchical population model with Poisson recruitment. An expression is found for the first and second moments of grade size at any time. A general formulation of the joint probability generating function of the numbers in each grade is given, and the limiting distribution of grade size is shown to be Poisson. MULTIGRADE SYSTEM; CONTINUOUS-TIME STOCHASTIC MODEL; MANPOWER PLANNING MODEL; POPULATION MODEL; MARKOV PROCESSES

Random record models

Abstract: We study record times, mainly, and sizes in the following context. Let X, denote the size of the nth event occurring in a point stochastic pacing process A. The X, is i.i.d. and * is, variously, Poisson, negative binomial, renewal and Furry. Explicit distributions of first-record times are found, domains of attraction studied and the asymptotic lognormality of the nth-record time is shown for Poisson 1P.

Some alternative approaches to multiparameter estimation

Abstract: SUMMARY Bayesian and classical alternatives are considered, interpreted, and compared with simultaneous estimation methods in the literature. The main topics explored are the estimation of several normal means, and of several binomial or Poisson parameters. An empirical comparison and three real examples are included.

On the Multivariate Poisson Normal Distribution

Abstract: The factorial moment generating function (FMGF) of the multivariate Poisson normal (or Hermite) distribution is derived as the limiting form of a multinomial process and as the FMGF of a mixture of independent Poisson distributions when the parameters have a multivariate normal distribution. The FMGF is then used as starting point to derive a limiting form, a series expansion, marginal and conditional distributions as well as for estimating the parameters by using jointly the method of moments and least squaresf or correlated variables.

Maximal functions: Poisson integrals on symmetric spaces

Abstract: Let u be a harmonic function on a symmetric space which is the Poisson integral of a function f in Lp, 1 ≤ p ≤ ∞. Then u converges restrictedly and admissibly to f almost everywhere. This result is proved by obtaining an appropriate maximal theorem which takes into account the structure of the Poisson kernel.

Inverse moments of discrete distributions

Abstract: In this paper an expression for the inverse moment of order r is given for the truncated binomial and Poisson distributions. This enables one to obtain inverse moments in a finite series. Some applications and multivariate generalizations are also given. The method also enables one to obtain relations between inverse moments and factorial moments and distributions of sums of variables.

Sampling from the poisson distribution on a computer

Abstract: This paper describes a method of sampling from the Poisson distribution on a computer that appears to be less costly than a recently suggested method in [1]. The proposed method relies on a conventional search using the inverse transform approach for nonintegral μ 7. For large μ the sampling cost is proportional to μ1/2. The paper also shows that for μ≥15 an incidental error occurs by using max (0, [μ+Yμ1/2+0.5]) whereY is fromN(0,1). Since the sampling cost ofY is a constant, this approach places an upper bound on the cost of generating a Poisson variate.

Power Sum Distributions: An Easier Approach Using the Wald Distribution

Abstract: Power sum distributions arise in communications noise and highway noise models. Inverse Gaussian or Wald distributions with the same index of dispersion are similar and infinitely divisible, and often closely approximate lognormal distributions, and, thus, are more suitable than lognormals for theoretical studies. The Wald distribution is an exact model for Poisson distributed point sources with Erlang (O) emissions of noise power.

An extension of erlang's loss formula

Abstract: : A two server loss system is considered with N classes of Poisson arrivals, where the service distribution function and server preferences are arrival class dependent. The stationary state probabilities are derived and found to be independent of the form of the service distributions.

Analysis of intercellular distributions of chromatid aberrations.

Abstract: 59 intercellular distributions of chemically induced and spontaneous chromatid aberrations were analyzed for goodness of fit in respect of the Poisson (PD), the geometrical (GD), and the negative binomial distributions (NBD). The data are excellently described by the NBD. This distribution can be deduced from a model out of the queueing theory and is based on the hypothesis of restitution. Estimators are obtained for the ratio of restitution and induction processes involved in the origin of chromosomal aberrations.

Modeling laser velocimeter signals as triply stochastic Poisson processes

Abstract: Previous models of laser Doppler velocimeter (LDV) systems have not adequately described dual-scatter signals in a manner useful for analysis and simulation of low-level photon-limited signals. At low photon rates, an LDV signal at the output of a photomultiplier tube is a compound nonhomogeneous filtered Poisson process, whose intensity function is another (slower) Poisson process with the nonstationary rate and frequency parameters controlled by a random flow (slowest) process. In the present paper, generalized Poisson shot noise models are developed for low-level LDV signals. Theoretical results useful in detection error analysis and simulation are presented, along with measurements of burst amplitude statistics. Computer generated simulations illustrate the difference between Gaussian and Poisson models of low-level signals.

A Characterization of Convoluted Poisson Distributions with Applications to Estimation

Abstract: Define X as the sum of a Poisson random variable Y with parameter θ and a nonnegative integer valued variable Z independent of Y. If the distribution of Z is known, X has a one-parameter convoluted Poisson distribution. Such distributions are here characterized as solutions to a system of differential equations specifying the rate of change of the cumulative distribution function relative to a parametric change. The characterization is used in point and interval estimation of θ. It is shown that for a certain subclass of convoluted Poisson distributions, the maximum likelihood estimate of θ is the unique solution of the likelihood equation and is easily approximated numerically.

Estimation of the Parameters of the Triple and Quadruple Stuttering-Poisson Distributions

Abstract: In this article we introduce multiple stuttering-Poisson distributions and discuss their genesis, properties and applications. We consider a special case of triple stuttering-Poisson distribution with probability generating function G(t) = Exp [a(t − 1) + b(t 2 − 1) + c(t 3 − l)] and investigate (1) maximum likelihood estimation, (2) moment estimation, and (3) mixed moment estimation of the parameters a, b, c. First order terms in the expressions for the biases and covariances of moment estimators are presented. The joint asymptotic efficiencies of moment estimators are tabulated for a few selected parameter points. The first order terms in the biases of maximum likelihood estimators and moment estimators are also given for a few selected parameter points. As judged from the contribution of the first order term, the possible bias of maximum likelihood estimators can be substantial over a considerable region of the parameter space. The moment estimators have low joint asymptotic efficiency over a large par...