Showing papers on "Poisson distribution published in 1979"

Software Reliability Model for Modular Program Structure

Abstract: The paper treats a modular program in which transfers of control between modules follow a semi-Markov process. Each module is failure-prone, and the different failure processes are assumed to be Poisson. The transfers of control between modules (interfaces) are themselves subject to failure. The overall failure process of the program is described, and an asymptotic Poisson process approximation is given for the case when the individual modules and interfaces are very reliable. A simple formula gives the failure rate of the overall program (and hence mean time between failures) under this limiting condition. The remainder of the paper treats the consequences of failures. Each failure results in a cost, represented by a random variable with a distribution typical of the type of failure. The quantity of interest is the total cost of running the program for a time t, and a simple approximating distribution is given for large t. The parameters of this limiting distribution are functions only of the means and variances of the underlying distributions, and are thus readily estimable. A calculation of program availability is given as an example of the cost process. There follows a brief discussion of methods of estimating the parameters of the model, with suggestions of areas in which it might be used.

A note on the Poisson assumption in partial duration series models

Abstract: The Poisson process is frequently encountered in probability models of the partial duration series. This paper looks at the necessity for specifying the model in detail by such a process and also examines the validity of the Poisson distribution, in this context, as judged on data from 26 gaging stations on 20 catchments in Great Britain. When all the data are considered jointly the Poisson assumption has to be rejected although it is acceptable in some cases. It is also suggested that if dependence exists in the partial duration series it should be looked for in the point process from which the series arises rather than among successive peak magnitudes.

Existence of a negative Poisson ratio in fcc crystals

Abstract: The Poisson ratios along two principal crystal directions normal to a [110] uniaxial load are generally of opposite algebraic sign for fcc crystals. A theoretical basis for this behavior is revealed.

Parameter estimation in X-ray astronomy using maximum likelihood

Abstract: Methods of estimation of parameter values and confidence regions by maximum likelihood and Fisher efficient scores starting from Poisson probabilities are developed for the nonlinear spectral functions commonly encountered in X-ray astronomy. It is argued that these methods offer significant advantages over the commonly used alternatives called minimum chi-squared because they rely on less pervasive statistical approximations and so may be expected to remain valid for data of poorer quality. Extensive numerical simulations of the maximum likelihood method are reported which verify that the best-fit parameter value and confidence region calculations are correct over a wide range of input spectra.

Car-Following Headways on Freeways Interpreted by the Semi-Poisson Headway Distribution Model

Abstract: The goal of this work is to investigate driver car-following patterns on freeways, particularly as a function of traffic flow level, using a headway distribution model. A number of authors have developed “two-component” vehicular headway distribution models that assume vehicles on a road can be divided into two groups according to whether or not they are interacting with the vehicle ahead. A model of this type, the “semi-Poisson” model proposed by Buckley, is applied to a data base consisting of 42,000 observed headways from a single lane of an urban freeway over a range of flow from 900 to 2,000 vehicles per lane per hour. A previously developed computational method allows the distribution of followers headways to be calculated directly from the observed total headway distribution by numerically solving an integral equation without introducing a parametric form for the followers distribution. The resulting followers headway distribution is found to be independent of the flow with a mean of 1.32 s and a standard deviation of 0.52 s. No statistically significant discrepancies are found between the model results and the observed data. The theoretical basis for the semi-Poisson model is discussed and compared with those of other models in order to assess the plausibility of the interpretation with respect to car following.

Some considerations in the formulation of IC yield statistics

Abstract: Some published modifications of the Poisson distribution for describing IC yield are critiqued. It is shown that it is incorrect to obtain an average yield for a non-uniform defect population by integrating, either in the geometrical space or in the density space, the Poisson distribution with some assumed density distribution functions. The correct way, and happily also the simplest way, is to average the yields of regionally partitioned subpopulations in a discrete manner. The simple Poisson distribution would become rigorously correct when the size of an imaginary IC increases to one quarter of a wafer, regardless of the non-uniformity in defect density. It is also shown that both cases of clustering of defects, one due to interaction among defects themselves, and the other due to wafer regional preference, result in increased yield for a given defect density in a wafer. On the other hand when there are interactions between defects and IC active area elements, or when defects themselves have physical dimensions, there would be a decreased yield for a given defect density, and a non-zero intercept in the plot of the logarithm of yield vs the active device area.

The Computer Generation of Poisson Random Variables

Abstract: SUMMARY A comparison is made of methods of generating samples on a computer from the Poisson distribution. The well-known methods of counting the number of occurrences in a Poisson process and of sequentially searching through a table of cumulative probabilities have the disadvantage that the time required increases with thePoisson parameter ft. For fixed ft two modified search procedures are described which remain fast as ft increases. If ft varies from sample to sample the modified search procedures are not directly applicable. But fast methods can be found which use combinations of either modified search method and are appreciably faster than rejection methods.

Biological Applications of the Lagrangian Poisson Distribution

Abstract: Biology has frequently benefited from the introduction of simple mathematical models. Conspicuous examples are the Hardy-Weinberg law, logistic growth equation, the fundamental theorem of natural selection, and Cole's demonstration of the power of the simple (Cole 1946a) and compound (Cole 1946b) Poisson distributions. This paper describes the utility of the Lagrangian Poisson distribution as a tool for understanding biological processes. The Poisson distribution has been

Simulation of Nonhomogeneous Poisson Processes with Degree-Two Exponential Polynomial Rate Function

Abstract: A new method for simulating a nonhomogeneous Poisson process with rate function λ(t) = exp{α0 + α1t + a2t2} in a fixed time interval (0, t0] is given. The method is based on a decomposition of the process, it employs a rejection technique, in conjunction with a method for simulating the nonhomogeneous Poisson process with rate function exp {γ0 + γ1t} by generation of gap statistics from a random number of exponential random variables with suitably chosen parameters. The proposed method for the degree-two exponential polynomial model is more efficient than time-scale transformation of a homogeneous Poisson process, and should be applicable to other rate function models.

Asymptotic Coverage Distributions on the Circle

Abstract: Place $n$ arcs, each of length $a_n$, uniformly at random on the circumference of a circle, choosing the arc length sequence $a_n$ so that the probability of completely covering the circle remains constant. We obtain the limiting distribution of the uncovered proportion of the circle. We show that this distribution has a natural interpretation as a noncentral chi-square distribution with zero degrees of freedom by expressing it as a Poisson mixture of mass at zero with central chi-square deviates having even degrees of freedom. We also treat the case of proportionately smaller arcs and obtain a limiting normal distribution. Potential applications include immunology, genetics, and time series analysis.

Aspects of correlation in bivariate poisson distributions and processes

Abstract: Summary This paper reviews some interesting but scattered results that are known about correlation in bivariate Poisson distributions and processes and presents some new results. A particular concern in both contexts is with results and examples relating to negative correlation.

Upcrossing Rate Solution for Load Combinations

Abstract: An exact expression for the mean upcrossing rate of a sum of two poisson square-wave process representations of stochastic loading functions is used to explore the accuracy of an earlier approximation (due to Wen) of the extreme value distribution of such processes. The approximation is found to be accurate over a broad parameter range despite the fact that it ignores the randomness of load durations and implicitly assumes very infrequent load events. The limits of the mean upcrossing rate as an approximation to the extreme value distribution are also studied.

Poisson mixtures and quasi-infinite divisibility of distributions

Abstract: Any probability distribution on [0,∞) can function as the mixing distribution for a Poisson mixture, i.e. a mixture of Poisson distributions. The mixing distribution is called quasi-infinitely divisible (q.i.d.) if it renders the Poisson mixture infinitely divisible, or λ-q.i.d. if it does so after scaling by a factor λ> 0, or ∗-q.i.d. if it is λ-q.i.d. for some λ. These classes of distributions include the infinitely divisible distributions, and each exhibits many of the properties of the latter class but in weakened form. The paper presents the main properties of the classes and the class of Poisson mixtures, including characterisations of membership, relation with cumulants, and closure properties. Examples are given that establish among other things strict inclusions between the classes of mixing distributions.

On the KOLMOGOROV‐SMIRNOV Test for the POISSON Distribution with Unknown Mean

Abstract: The standard tables for the KOLMOGOROV-SMIRNOV test are valid only in the case of testing whether a set of observations is from a completely specified cumulative distribution, F0(X), with all parameters known. If the parameters are unknown and must be estimated from the sample, then the tables are not valid. A table is given in this paper for use with the KOLMOGOROV-SMIRNOV statistic in the case of testing whether a set of observations is from the POISSON distribution with an unknown mean that must be estimated from the sample. The table is obtained from a Monte Carlo calculation.

Rates of poisson convergence for u-statistics

Abstract: Poisson limit theorems for U-statistics are studied. A general rate of convergence is obtained; this rate is improved for the special case where the U-statistic arises from the consideration of distances between uniformly

Time domain synthesis via Poisson moment functionals

Abstract: This paper presents a method of time-domain synthesis of dynamical systems treating the process signals as distributions or generalized functions, in the manner originally established by L. Schwartz. The technique of synthesizing the transfer function or the state-space model of a system employs exponentially weighted series of the generalized time derivatives of the impulse distribution, known also as the Poisson moment functional expansion. General algorithms are developed and their application is illustrated in typical eases.

Asymptotic Optimal Sequential Estimation: The Poisson Case

Abstract: The problems of estimating sequentially the intensity parameter of a homogeneous Poisson process and of estimating sequentially the mean of a sequence of i.i.d. Poisson rv's, are considered. The procedures suggested are shown to perform well for large values of the parameter and/or for small sampling cost. Having bounded regret, the procedure for estimating the mean of the Poisson sequence is asymptotically Bayes w.r.t. any sequence of a priori densities, which spread mass in a suitably smooth manner.

Breakdown of a stationary solution to the Nernst–Planck–Poisson equations

Abstract: This report shows an oscillatory instability, at sufficiently high voltage, of a representative steady solution to equations describing the passage of current through an unstirred layer of electrolyte adjacent to an ideal cation exchange membrane. This instability may be the harbinger of further “turbulent-like” transitions that correspond to observed unsteadiness at high voltages.

On assessing the potential severity of an outbreak of a rare infectious disease: a bayesian approach

Abstract: Summary In this paper an infective population, during the early stages of the outbreak of a disease, is approximated by a Galton-Watson process. Attention is focused on the threshold theorem which assesses an epidemic as major if the associated Galton-Watson process is supercritical (i.e. the mean μ of the offspring distribution is greater than one). A Bayesian formulation is adopted together with the assumption of a power series offspring distribution and an approximate form is found for P(μ >1) computed from the posterior distribution of μ. Exact results are given for the case of a Poisson offspring distribution. The results are illustrated with applications to three sets of data on smallpox outbreaks. The Bayesian approach has a number of advantages over classical methods and in particular allows the cases μ 1 to be treated without distinction.

Photocounting distributions for exponentially decaying sources.

Abstract: Exact photocounting distributions are obtained for a pulse of light whose intensity is exponentially decaying in time, when the underlying photon statistics are Poisson. It is assumed that the starting time for the sampling interval (which is of arbitrary duration) is uniformly distributed. The probability of registering n counts in the fixed time T is given in terms of the incomplete gamma function for n >/= 1 and in terms of the exponential integral for n = 0. Simple closed-form expressions are obtained for the count mean and variance. The results are expected to be of interest in certain studies involving spontaneous emission, radiation damage in solids, and nuclear counting. They will also be useful in neurobiology and psychophysics, since habituation and sensitization processes may sometimes be characterized by the same stochastic model.

Negative moments of a modified power series distribution and bias of the maximum likelihood estimator

Abstract: The class of Modified Power Series distributions (MPSD) containing Lagrangian Poisson (LPD) (Consul and Jain, 1973) and Lagrangian binomial distributions (LBD) (Jain and Consul, 1971) was studied by Gupta (1974). We investigate the problem of finding the negative momentsE[X-r ], of displaced and decapitated Modified Power Series Distributions. We derive the relationship between rand (r-1) negative moments. The negative moments of the decapitated and displaced LPD are obtained. These results are, then, used to find the exact amount of bias in the ML estimators of the parameters in the LPD and the LBD. We have also given the variances of the ML estimator and the minimum variance unbiased estimator of the parameter in the LPD.

Filtering of a diffusion process with poisson-type observation

Abstract: We consider a filtering problem, where the signal Xt is a Markov diffusion process, and the observation is a marked point process (for instance a Poisson process), whose predictable projection (the stochastic intensity in the case of a point process) is a given function of the signal Xt.

Recent Developments in the Computer Generation of Poisson Random Variables

Abstract: Two recent methods of generating samples on a computer from the Poisson distribution are compared with those in an earlier survey. Recommendations are made for algorithms which are either compact or fast, but unfortunately not both. Two cases are distinguished: that in which the Poisson parameter is fixed and that in which it changes from sample to sample.

Rates of convergence in a modified empirical bayes estimation problem involving poisson distributions

Abstract: Let {tn}be a known sequence of real numbers and let {(ρn ,Xn )} be a sequence of independent pairs where the ρn are iid G on [0,β] and unobservable and, given ρn , Xn has a Poisson distribution with parameter ρn Xn. Asymptotically optimal empirical Bayes rules in the squared error loss estimation problem are exhibited where rate of risk convergence is near n-1 . Conditions are set forth where similar results hold in general discrete exponential families.

Limiting results for arrays of binary random variables on rectangular lattices under sparseness conditions

Abstract: In this article we give limiting results for arrays {Xij (m, n) (i, j) Dmn } of binary random variables distributed as particular types of Markov random fields over m x n rectangular lattices Dmn. Under some sparseness conditions which restrict the number of X ij (m, n)'s which are equal to one we show that the random variables (l = 1, ···, r) converge to independent Poisson random variables for 0 < d1 < d2 < · ·· < dr when m→∞ nd∞. The particular types of Markov random fields considered here provide clustering (or repulsion) alternatives to randomness and involve several parameters. The limiting results are used to consider statistical inference for these parameters. Finally, a simulation study is presented which examines the adequacy of the Poisson approximation and the inference techniques when the lattice dimensions are only moderately large.