Showing papers on "Poisson distribution published in 1985"

Estimation of a common effect parameter from sparse follow-up data.

Abstract: Breslow (1981, Biometrika 68, 73-84) has shown that the Mantel-Haenszel odds ratio is a consistent estimator of a common odds ratio in sparse stratifications. For cohort studies, however, estimation of a common risk ratio or risk difference can be of greater interest. Under a binomial sparse-data model, the Mantel-Haenszel risk ratio and risk difference estimators are consistent in sparse stratifications, while the maximum likelihood and weighted least squares estimators are biased. Under Poisson sparse-data models, the Mantel-Haenszel and maximum likelihood rate ratio estimators have equal asymptotic variances under the null hypothesis and are consistent, while the weighted least squares estimators are again biased; similarly, of the common rate difference estimators the weighted least squares estimators are biased, while the estimator employing "Mantel-Haenszel" weights is consistent in sparse data. Variance estimators that are consistent in both sparse data and large strata can be derived for all the Mantel-Haenszel estimators.

Dressing transformations and Poisson group actions

Abstract: On explique les proprietes de Poisson des transformations d'habillage en theorie des solitons

Some simple models for discrete variate time series

Abstract: Simple models are presented for use in the modeling and generation of sequences of dependent discrete random variables. The models are essentially Markov Chains, but are structurally autoregressions, and so depend on only a few parameters. The marginal distribution is an intrinsic component in the specification of each model, and the Poisson, Geometric, Negative Binomial and Binomial distributions are considered. Details are also given for the introduction of time-dependence into the means of the sequences so that seaonality can be treated simply.

Use of poisson regression models in estimating incidence rates and ratios

Abstract: Summarizing relative risk estimates across strata of a covariate is commonly done in comparative epidemiologic studies of incidence or mortality. Conventional Mantel-Haenszel and rate standardization techniques used for this purpose are strictly suitable only when there is no interaction between relative risk and the covariate, and tests for interaction typically are limited to examination for departures from linearity. Poisson regression modeling offers an alternative technique which can be used for summarizing relative risk and for evaluating complex interactions with covariates. A more general application of Poisson regression is its utility in modeling disease rates according to postulated etiologic mechanisms of exposures or according to disease expression characteristics in the population. The applications of Poisson regression analysis to problems of summarizing relative risk and disease rate modeling are illustrated with examples of cancer incidence and mortality data, including an example of a nonlinear model predicted by the multistage theory of carcinogenesis.

Energy-Level Statistics of Integrable Quantum Systems

Abstract: Using a simple example we show that the distribution for the energy levels for integrable systems is not the uncorrelated Poisson distribution as is commonly believed. In particular, the spectrum was found to be rather rigid. We conjecture that these are typical properties of the integrable quantum systems.

New approach to the statistical properties of energy levels.

Abstract: The joint distribution of energy eigenvalues of a Hamiltonian is derived by means of the usual statistical laws of classical many-body systems. It makes a transition from the Poisson type to the Gaussian type depending on the value of a single parameter characteristic of the Hamiltonian.

Recurrence rates and probability estimates for the New Madrid Seismic Zone

Abstract: An updated frequency-magnitude relation for the New Madrid seismic zone is used to derive conditional probabilities for future, large New Madrid earthquakes. We estimate that there is a 40–63% probability of an mb ≥ 6.0(Ms ≥ 6.3) event occurring by the year 2000 and an 86–97% probability by the year 2035. The estimates for a great 1812-type event (Ms ≥ 8.3) are less than 1% probability by 2000 A.D. and less than 4% by 2035 A.D. These probabilities are contingent on many factors, a number of which remain assumptions because of the lack of a geological or paleoseismological chronology of past New Madrid activity. A conditional probability requires knowledge of a mean recurrence time, the type of distribution, and the standard deviation of actual repeat times about this mean. Four assumed distribution functions (Gaussian, lognormal, Weibull, and Poisson) were fit to recurrence estimates based on a combination of historical and instrumental seismicity data. Standard deviation was allowed to vary between one third and one half of the mean recurrence time, and a range of conditional probabilities was generated for time intervals of 15 and 50 years from the year 1985. The largest uncertainty in this procedure was the size of the seismic source area to use for recurrence estimation. Calculations were done for both a large and a small source zone which led to variation in estimated recurrence intervals by a factor of 2. The large source zone was favored for the final probability estimates because of the large crustal volume required to store elastically the strain energy for great New Madrid earthquakes.

Normal forms for smooth Poisson structures

Abstract: On classe, localement, les tenseurs de Poisson C ∞ de rang variable, soumis a une condition de non-degenerescence

Testing Goodness of Fit for the Poisson Assumption When Observations are Not Identically Distributed

Abstract: Tests for detecting negative binomial departures from a Poisson model are studied for the one-way-layout and regression-through-the-origin cases. Approximations to the null and alternative distributions of these test statistics are presented. Locally optimal tests and tests suggested in the literature are compared in terms of asymptotic relative efficiency. Small sample comparisons are included.

Discrete software reliability growth models

Abstract: A general description of a discrete software reliability growth model, which adopts the number of test runs or the number of executed test cases as the unit of error detection period, is presented. Two classes of discrete software reliability growth models are proposed and discussed. These models can be described by non-homogeneous Poisson processes, in which the random variable is denned as the number of errors detected by n test runs (n = 0, 1, 2,…). The application and comparison of these models to actual software error data are shown.

An asymptotic expansion for the null distribution of the efficient score statistic

Abstract: SUMMARY An asymptotic expansion for the null distribution of the efficient score statistic for testing a composite hypothesis in the presence of nuisance parameters is derived, and an interpretation of the terms occurring in the expansion is given. The use of the expansion to modify the percentage points of the large-sample reference X2 distribution is discussed. The first three moments of the null distribution are obtained and are used as a check on the accuracy of the algebra via a comparison with the first three moments of the index of dispersion test for homogeneity of Poisson parameters.

Spatial analysis of density dependent pattern in coniferous forest stands

Abstract: An investigation of spatial pattern in relatively sparse Pinus ponderosa-P. Jeffreyi stands showed that a simple Poisson model of random distribution described the pattern at 5 to 50 m scales in the denser stands examined when allowance is made for inhibition between nearest neighbors. There is evidence for a clumped distribution in large quadrats for the sparsest stands, which concurs with prior work where a mixed Poisson model was fit to the data. The technique used was innovative in that it involved digitally recording tree locations from high resolution aerial photos, which allowed for the automatic application of several statistical techniques in order to determine how pattern varies with plot density and scale. Point locations were recorded for six 11.3 ha plots in three density regions of a 340 ha study area in northeastern California, USA. The inter-event distance distribution, and one- and two-dimensional power spectra were calculated, and variable quadrat analysis was performed for the data sets. The second order and spectral analyses showed no evidence of a distinctive clumped pattern at any scale, and all analyses showed that the pattern was regular at the scale of the average inter-plant distance in the denser stands. For the sparser stands, the counts in large quadrats did not fit a Poisson distribution, but were better fit by a mixed Poisson model describing aggregated pattern.

Asymptotic Number of Roots of Cauchy Location Likelihood Equations

Abstract: The number of local maxima of the Cauchy location likelihood function which are not global maxima is asymptotically Poisson distributed with mean parameter $1/\pi.$

Estimation of Interaction Potentials of Marked Spatial Point Patterns Through the Maximum Likelihood Method

Abstract: The likelihood procedure in estimating interaction potentials of spatial point patterns is developed for a set of point locations with attached marks, where the potential functions depend on the marks. Approximate log likelihood functions are derived under the assumption that the point patterns do not deviate much from the Poisson pattern in some sense. Some methods examining this assumption are provided. Analyses of ecological data sets are discussed through a model selection procedure.

The generalized poisson distribution when the sample mean is larger than the sample variance

Abstract: The generalized Poisson distribution (GPD), containing two parameters and studied by many researchers, was found to fit data arising in various situations and in many fields. It is generally assumed that both parameters (θ,λ) are non-negative, and hence the distribution will have a variance larger than the mean. However, it appears that the distribution, as a descriptive model, fits many data for negative values of λ, which implies that the mean must be greater than the variance. Thetruncation of the GPD, proposed to remedy the deficiency in the model in this case, is investigated on the basis of the error analysis on the computer and it is suggested that the truncation error for most of the cases is negligible and that the model can be used without any correction for truncation.

Approximations for compound Poisson and Pólya processes

Abstract: Suppose Xi _ 0 are i.i.d., i = 1, 2, ? ? ?. We derive a saddlepoint approximation for PIE"l Xk > y} as y--*and t is fixed, where N(t), t 0, is either a Poisson or a P61lya process. These results are then compared and contrasted with the well-known Esscher approximation. ASYMPTOTIC EXPANSIONS; COMPOUND PROCESSES; ESSCHER APPROXIMATION; RUIN THEORY; SADDLEPOINT METHOD

Estimation of cancer mortality rates: a Bayesian analysis of small frequencies

Abstract: A Bayesian method is presented for estimating mortality rates of specific diseases when the frequency of deaths over a specified time period is assumed to have a Poisson distribution with mean proportional to the population size. The estimators use information from related populations each having its own rate which is assumed distributed according to a common prior distribution about which some information is available. The method is illustrated using data from an epidemiological study on the geographic variation of cancer mortality in Missouri. Comparisons are also made with a simpler empirical Bayes method. (EXCERPT)

Poisson boundaries of discrete groups of matrices

Abstract: If μ is a probability measure on a countable group there is defined a notion of the Poisson boundary for μ which enables one to represent all bounded μ-harmonic functions on the group. It is shown that for discrete groups of matrices this boundary can be identified with the boundary of the corresponding Lie group.

An intervened Poisson distribution and its medical application

Abstract: Among probability distributions that are used to describe a chance mechanism whose observational apparatus becomes active only when at least one event occurs is the zero-truncated Poisson distribution (ZTPD). A modified version of the ZTPD, which we call an intervened Poisson distribution (IPD), is discussed in this paper. We give a genesis of IPD and obtain its statistical properties. A numerical example is included to illustrate the results.

Approximation of aggregate claims distributions by compound poisson distributions

Abstract: New error bounds are derived for the approximation of aggregate claims distributions by compound Poisson distributions. These approximations can be recommended in most cases in which the normal approximation fails.

Some goodness-of-fit methods for the Poisson plus added zeros distribution.

Abstract: Methods for making inferences about the Poisson plus added zeros distribution and the truncated Poisson distribution are presented and illustrated with bacteriological data. Some of the methods are designed for testing the compatibility of the zero frequency with the Poisson distribution, whereas others are given for testing the goodness of fit for the truncated Poisson. In particular, a modified form of the Fisher index of dispersion is presented which is suitable for the truncated case. It is shown that the use of the usual expression of the index of dispersion for testing the adequacy of the truncated Poisson is not correct and leads to accepting inadequate fits more frequently than expected on the basis of test of significance. Furthermore, three test statistics are presented for testing the compatability of the zero frequency with the Poisson distribution. The results of the simulation show that two test statistics, one due to Cochran (W. G. Cochran, Biometrics 10:417-451, 1954) and the other to Rao and Chakravarti (C. R. Rao and I. M. Chakravarti, Biometrics 12:264-282, 1956), are preferable to those from the likelihood ratio test.

Compound Poisson Approximations for Sums of Random Variables

Abstract: : This document shows that a sum of dependent random variables is approximately compound Poisson when the variables are rarely nonzero and, given they are nonzero, their conditional distributions are nearly identical. It give several upper bounds on the total-variation distance between the distribution of such a sum and a compund Poisson distribution. Included is an example for Markovian occurrences of a rare event. The bounds are consistent with those that are known for Poisson approximations for sums of uniformly small random variables. (Author)

Poisson boundaries of discrete groups of matrices

Abstract: if g is a probability measure on a countable group there is defined a notion of the Poisson boundary for g which enables one to represent all bounded g-harmonic functions on the group. It is shown that for discrete groups of matrices this boundary can be identified with the boundary of the corresponding Lie group. The behavior at infinity of a countable group G is partly described by boundaries. We consider here Poisson boundaries of random walks on T: Let/~ be a probability measure on G, and call a function h on G/z-harmonic when for any g in G, h(g)= '~ h(gg')l~(g'). g'EG

Maximum-likelihood estimation applied to electron microscopic autoradiography

Abstract: A new method for analysis of electron microscope autoradiographs is described which is based on the maximum-likelihood method of statistics for estimating the intensities of radioactivity in organelle structures. We adopted a Poisson statistical model to describe the autoradiographic grain distributions that we prove results from the underlying Poisson nature of the radioactive decays as well as the additive errors introduced during the formation of grains. Within the model, an interative procedure derived from the expectation-maximization algorithm of mathematical statistics is used to generate the maximum-likelihood estimates. The algorithm has the properties that at every stage of the iteration process the likelihood of the data increases; and for all initial nonzero starting points the algorithm converges to the maximum-likelihood estimates of the organelle intensities. The maximum-likelihood approach differs from the mask-analysis method, and other published quantitative algorithms in the following ways: (1) In deriving estimates of the radioactivity intensities the maximum-likelihood algorithm requires that we obtain the actual locations of the grains as well as the micrograph geometries; each micrograph is digitized so that both the grain locations as well as the geometries of the organelle structures can be used. (2) The maximum-likelihood algorithm iteratively computes the minimum-meansquared-error estimate of the underlying emission locations that resulted in the observed grain distributions, from which intensity estimates are generated; this algorithm does not minimize a chi-squared error statistic. (3) The maximum-likelihood approach is based on a Poisson model and is therefore valid for low-count experiments; there are no minimum constraints on data collection for any single organelle compartment. (4) The maximum-likelihood algorithm requires the form of the point-spread function describing the emission spread; a probability matrix based on the use of overlay masks is not required. (5) The maximum-likelihood algorithm does not change for different organelle geometries; arbitrary geometries are incorporated by maximizing the likelihood-function subject to the geometry constraints. We have performed a preliminary evaluation of the quantitative accuracy of the maximum-likelihood and mask-analysis algorithms. Based on two different phantoms in which we compared the squared error resulting from the two algorithms, we find that the new maximum-likelihood approach provides substantially improved estimates of the radioactivity intensities of the phantoms.