Showing papers on "Poisson distribution published in 1993"

Bias and variance of angular correlation functions

Abstract: We present a general method for calculating the bias and variance of estimators for w(θ) based on galaxy-galaxy (DD), random-random (RR), and galaxy-random (DR) pair counts and describe a procedure for quickly estimating these quantities given an arbitrary two-point correlation function and sampling geometry. These results, based conditionally upon the number counts, are accurate for both high and low number counts. We show explicit analytical results for the variances in the estimators DD/RR, DD/DR, which turn out to be considerably larger than the common wisdom Poisson estimate and report a small bias in DD/DR in addition to that due to the integral constraint. Further, we introduce and recommend an improved estimator (DD−2DR+RR)/RR, whose variance is nearly Poisson

Nonlinear Poisson Brackets: Geometry and Quantization

Abstract: Poisson manifolds Analog of the group operation for nonlinear Poisson brackets Poisson brackets in $\mathbb R^2n$ and semiclassical approximation Asymptotic quantization Appendix I: Formulas of noncommutative analysis II: Calculus of symbols and commutation relations.

A comparison of approximate interval estimators for the bernoulli parameter

Abstract: The goal of this paper is to compare the accuracy of two approximate confidence interval estimators for the Bernoulli parameter p. The approximate confidence intervals are based on the normal and Poisson approximations to the binomial distribution. Charts are given to indicate which approximation is appropriate for certain sample sizes and point estimators.

Effective Elastic Moduli of Porous Ceramic Materials

Abstract: This paper compares the applicability of a few theoretical models for determining effective elastic moduli, using published experimental data on ceramic materials in a porosity range of 0–40% and on a cellular material with a porosity of about 90%. As the experimental data for the effective Poisson's ratio involve a large scatter, a set of numerical experiments using the finite element method was carried out to obtain the variation of the effective Poisson's ratio with porosity. These variations show that the effective Poisson's ratio approaches 0.25 with increasing porosity, irrespective of the material Poisson's ratio. The effect of pore shapes on the effective elastic moduli and the Poisson's ratio has also been analyzed using FEM.

Restricted generalized poisson regression model

Abstract: The family of generalized Poisson distribution has been found useful in describing over-dispersed and under-dispersed count data We propose the use of restricted generalized Poisson regression model to predict a response variable affected by one or more explanatory variables Approximate tests for the adequacy of the model and the estimation of the parameters are considered Restricted generalized Poisson regression model has been applied to an observed data set

M t /G/∞ queues with sinusoidal arrival rates

Abstract: In this paper we describe the mean number of busy servers as a function of time in an Mt/G/∞ queue having a nonhomogeneous Poisson arrival process with a sinusoidal arrival rate function. For an Mt/G/∞ model with appropriate initial conditions, it is known that the number of busy servers at time t has a Poisson distribution for each t, so that the full distribution is characterized by its mean. Our formulas show how the peak congestion lags behind the peak arrival rate and how much less is the range of congestion than the range of offered load. The simple formulas can also be regarded as consequences of linear system theory, because the mean function can be regarded as the image of a linear operator applied to the arrival rate function. We also investigate the quality of various approximations for the mean number of busy servers such as the pointwise stationary approximation and several polynomial approximations. Finally, we apply the results for sinusoidal arrival rate functions to treat general periodic arrival rate functions using Fourier series. These results are intended to provide a better understanding of the behavior of the Mt/G/∞ model and related Mt/G/s/r models where some customers are lost or delayed.

Sampling to Detect Rare Species

Abstract: Often a sampling program has the objective of detecting the presence of one or more species. One night wish to obtain a species list for the habitat, or to detect the presence of a rare and possibly endangered species. How can the sampling effort necessary for the detection of a rare species can be determined? The Poisson and the negative binomial are two possible spatial distributions that could be assumed. The Poisson assumption leads to the simple relationship n = -(1/m)log @b, where n is the number of quadrats needed to detect the presence of a species having density m, with a chance @b (the Type 2 error probability) that the species will not be collected in any of the n quadrats. Even if the animals are not randomly distributed the Poisson distribution will be adequate if the mean density is very low (i.e., the species is rare, which we arbitrarily define as a true mean density of 0.95. Only 8 of the 273 cases represented rare species that failed this requirement. Thus we conclude that a Poisson-based estimate of necessary sample size will generally be adequate and appropriate.

Poisson law for Axiom A diffeomorphisms

Abstract: Let f be an Axiom A diffeomorphism, Ω its non wandering set, µ the Gibbs measure for the Lipschitz continuous potential. We consider the (suitably normalized) return times of the orbit to the e-neighborhood of a point z ∈ Ω and prove that for µ-a.e. z the sequence of the normalized return times converges to the Poisson point process in finite dimensional distribution as e → 0.

Confidence Intervals for the Mean of a Poisson Distribution: A Review

Abstract: The paper provides a comprehensive review of methodology for setting confidence intervals for the parameter of a Poisson distribution. The results are illustrated by a numerical example.

Poisson vs. GOE Statistics in Integrable and Non-Integrable Quantum Hamiltonians

Abstract: We calculate the level statistics by finding the eigenvalue spectrum for a variety of one-dimensional many-body models, namely the Heisenberg chain, the t-J model and the Hubbard model. In each case the generic behaviour is GOE, however at points corresponding to models known to be exactly integrable Poisson statistics are found, in agreement with an argument we outline.

Poisson structures: towards a classification

Abstract: In the present note we give an explicit description of certain class of Poisson structures. The methods lead to a classification of Poisson structures in low dimensions and suggest a possible approach for higher dimensions.

Moments of the Counts Distribution in the 1.2 Jansky IRAS Galaxy Redshift Survey

Abstract: We have measured the count probability distribution function (CPDF) in a series of 10 volume-limited subsamples of a deep redshift survey of IRAS galaxies. The CPDF deviates significantly from both the Poisson and Gaussian limits in all but the largest volumes. We derive the volume-averaged 2-, 3-, 4-, and 5 point correlation functions from the moments of the CPDF and find them all to be reasonably well described by power laws. Weak systematic effects with the sample size provide evidence for stronger clustering of galaxies of higher luminosity on small scales. Nevertheless, remarkably tight relationships hold between the correlation functions of different order

Poisson versus GOE statistics in integrable and non-integrable quantum hamiltonian

Abstract: We calculate the level statistics by finding the eigenvalue spectrum for a variety of one-dimensional many-body models, namely the Heisenberg chain, the t-J model and the Hubbard model. In each case the generic behaviour is GOE, however at points corresponding to models known to be exactly integrable Poisson statistics are found, in agreement with an argument we outline.

Large deviations and the maximum entropy principle for marked point random fields

Abstract: We establish large deviation principles for the stationary and the individual empirical fields of Poisson, and certain interacting, random fields of marked point particles in ℝd. The underlying topologies are induced by a class of not necessarily bounded local functions, and thus finer than the usual weak topologies. Our methods yield further that the limiting behaviour of conditional Poisson distributions, as well as certain distributions of Gibbsian type, is governed by the maximum entropy principle. We also discuss various applications and examples.

Software reliability model with optimal selection of failure data

Abstract: The possibility of obtaining more accurate predictions of future failures by excluding or giving lower weight to the earlier failure counts is suggested. Although data aging techniques such as moving average and exponential smoothing are frequently used in other fields, such as inventory control, the author did not find use of data aging in the various models surveyed. A model that includes the concept of selecting a subset of the failure data is the Schneidewind nonhomogeneous Poisson process (NHPP) software reliability model. In order to use the concept of data aging, there must be a criterion for determining the optimal value of the starting failure count interval. Four criteria for identifying the optimal starting interval for estimating model parameters are evaluated The first two criteria treat the failure count interval index as a parameter by substituting model functions for data vectors and optimizing on functions obtained from maximum likelihood estimation techniques. The third uses weighted least squares to maintain constant variance in the presence of the decreasing failure rate assumed by the model. The fourth criterion is the familiar mean square error. It is shown that significantly improved reliability predictions can be obtained by using a subset of the failure data. The US Space Shuttle on-board software is used as an example. >

Some autoregressive moving average processes with generalized Poisson marginal distributions

Abstract: Some simple models are introduced which may be used for modelling or generating sequences of dependent discrete random variables with generalized Poisson marginal distribution. Our approach for building these models is similar to that of the Poisson ARMA processes considered by Al-Osh and Alzaid (1987,J. Time Ser. Anal.,8, 261–275; 1988,Statist. Hefte,29, 281–300) and McKenzie (1988,Adv. in Appl. Probab.,20, 822–835). The models have the same autocorrelation structure as their counterparts of standard ARMA models. Various properties, such as joint distribution, time reversibility and regression behavior, for each model are investigated.

A comparison of the generalized estimating equation approach with the maximum likelihood approach for repeated measurements

Abstract: Liang and Zeger proposed an extension of generalized linear models to the analysis of longitudinal data. Their approach is closely related to quasi-likelihood methods and can handle both normal and non-normal outcome variables such as Poisson or binary outcomes. Their approach, however, has been applied mainly to non-normal outcome variables. This is probably due to the fact that there is a large class of multivariate linear models available for normal outcomes such as growth models and random-effects models. Furthermore, there are many iterative algorithms that yield maximum likelihood estimators (MLEs) of the model parameters. The multivariate linear model approach, based on maximum likelihood (ML) estimation, specifies the joint multivariate normal distribution of outcome variables while the approach of Liang and Zeger, based on the quasi-likelihood, specifies only the marginal distributions. In this paper, I compare the approach of Liang and Zeger and the ML approach for the multivariate normal outcomes. I show that the generalized estimating equation (GEE) reduces to the score equation only when the data do not have missing observations and the correlation is unstructured. In more general cases, however, the GEE estimation yields consistent estimators that may differ from the MLEs. That is, the GEE does not always reduce to the score equation even when the outcome variables are multivariate normal. I compare the small sample properties of the GEE estimators and the MLEs by means of a Monte Carlo simulation study.

Lower Bounds for Contamination Bias: Globally Minimax Versus Locally Linear Estimation

Abstract: We study how robust estimators can be in parametric families, obtaining a lower bound on the contamination bias of an estimator that holds for a wide class of parametric families. This lower bound includes as a special case the bound used to establish that the median is bias minimax among location equivariant estimators, and it is tight or nearly tight in a variety of other settings such as scale estimation, discrete exponential families and multiple linear regression. The minimum variation distance estimator has contamination bias within a dimension-free factor of this bound. A second lower bound applies to locally linear estimates and implies that such estimates cannot be bias minimax among all Fisher-consistent estimates in higher dimensions. In linear regression this class of estimates includes the familiar M-estimates, GM-estimates and S-estimates. In discrete exponential families, yet another lower bound implies that the "proportion of zeros" estimate has minimax bias if the median of the distribution is zero, a common situation in some fields. This bound also implies that the information-standardized sensitivity of every Fisher consistent estimate of the Poisson mean and of the Binomial proportion is unbounded.

Optimal Control and Poisson Reduction

Abstract: : In this paper, the author has worked out explicitly the Poisson reduction of certain G-invariant optimal control problems on Lie groups. The approach presented yields an algorithm for constructing regular extremals.

Quantization of poisson superalgebras and speciality of jordan poisson superalgebras

Abstract: The problem of speciality and i-speciality is considered for Jordan superalgebras related with Poisson superalgebras. The quantizations of Poisson superalgebras play an important role in the consideration. The i-speciality of Jordan Poisson superalgebras is obtained by means of quantization of Poisson superalgebras. The criterium for speciality of Jordan Poisson superalgebras is given also.

On random polynomials over finite fields

Abstract: We consider random monic polynomials of degree n over a finite field of q elements, chosen with all qn possibilities equally likely, factored into monic irreducible factors. More generally, relaxing the restriction that q be a prime power, we consider that multiset construction in which the total number of possibilities of weight n is qn. We establish various approximations for the joint distribution of factors, by giving upper bounds on the total variation distance to simpler discrete distributions. For example, the counts for particular factors are approximately independent and geometrically distributed, and the counts for all factors of sizes 1, 2, …, b, where b = O(n/log n), are approximated by independent negative binomial random variables. As another example, the joint distribution of the large factors is close to the joint distribution of the large cycles in a random permutation. We show how these discrete approximations imply a Brownian motion functional central limit theorem and a Poisson-Dirichiet limit theorem, together with appropriate error estimates. We also give Poisson approximations, with error bounds, for the distribution of the total number of factors.

Heterogeneity in Mantel-Haenszel-type models

Abstract: SUMMARY The Mantel-Haenszel problem involves inferences about a common odds ratio in a set of 2 x 2 tables. Although it is a fairly standard practice to test whether the odds ratios are indeed constant, there is remarkably little methodology available for proceeding when there is evidence of some heterogeneity. Our interest is in models where the log odds ratios Ok, for tables k = 1, 2, .. ., K, are thought of as a sample from a population with mean 0 and standard deviation a, and inferences are desired regarding the parameters (0, a). By Mantel-Haenszel-type models we mean to include the generalization involving pairs of Poisson observations rather than 2 x 2 tables, where the Ok are logarithms of ratios of the Poisson means within pairs. Direct computation of the likelihood function for (0, a) in these settings involves numerical integration, and the main point here is a simple approximation to this likelihood. The approximation is based on Laplace's method, and is very accurate for practical applications. Inference regarding the parameter 0 of primary interest may be made from the profile likelihood function. An alternative approach to likelihood methods, based on approximations to the marginal means and variances, is also considered. The methods explored here can be readily generalized to settings where the parameters Ok depend on covariables as well.