Showing papers on "Poisson distribution published in 1991"

Pairwise comparisons of mitochondrial DNA sequences in stable and exponentially growing populations.

Abstract: We consider the distribution of pairwise sequence differences of mitochondrial DNA or of other nonrecombining portions of the genome in a population that has been of constant size and in a population that has been growing in size exponentially for a long time. We show that, in a population of constant size, the sample distribution of pairwise differences will typically deviate substantially from the geometric distribution expected, because the history of coalescent events in a single sample of genes imposes a substantial correlation on pairwise differences. Consequently, a goodness-of-fit test of observed pairwise differences to the geometric distribution, which assumes that each pairwise comparison is independent, is not a valid test of the hypothesis that the genes were sampled from a panmictic population of constant size. In an exponentially growing population in which the product of the current population size and the growth rate is substantially larger than one, our analytical and simulation results show that most coalescent events occur relatively early and in a restricted range of times. Hence, the "gene tree" will be nearly a "star phylogeny" and the distribution of pairwise differences will be nearly a Poisson distribution. In that case, it is possible to estimate r, the population growth rate, if the mutation rate, mu, and current population size, N0, are assumed known. The estimate of r is the solution to ri/mu = ln(N0r) - gamma, where i is the average pairwise difference and gamma approximately 0.577 is Euler's constant.

Deformation mechanisms in negative Poisson's ratio materials : structural aspects

Abstract: Poisson's ratio in materials is governed by the following aspects of the microstructure: the presence of rotational degrees of freedom, non-affine deformation kinematics, or anisotropic structure. Several structural models are examined. The non-affine kinematics are seen to be essential for the production of negative Poisson's ratios for isotropic materials containing central force linkages of positive stiffness. Non-central forces combined with pre-load can also give rise to a negative Poisson's ratio in isotropic materials. A chiral microstructure with noncentral force interaction or non-affine deformation can also exhibit a negative Poisson's ratio. Toughness and damage resistance in these materials may be affected by the Poisson's ratio itself, as well as by generalized continuum aspects associated with the microstructure.

Confidence intervals for weighted sums of poisson parameters

Abstract: Directly standardized mortality rates are examples of weighted sums of Poisson rate parameters. If the numbers of events are large then normal approximations can be used to calculate confidence intervals, but these are inadequate if the numbers are small. We present a method for obtaining approximate confidence limits for the weighted sum of Poisson parameters as linear functions of the confidence limits for a single Poisson parameter, the unweighted sum. The location and length of the proposed interval depend on the method used to obtain confidence limits for the single parameter. Therefore several methods for obtaining confidence intervals for a single Poisson parameter are compared. For single parameters and for weighted sums of parameters, simulation suggests that the coverage of the proposed intervals is close to the nominal confidence levels. The method is illustrated using data on rates of myocardial infarction obtained as part of the WHO MONICA Project in Augsburg, Federal Republic of Germany.

The Pointwise Stationary Approximation for Queues with Nonstationary Arrivals

Abstract: We empirically explore the accuracy of an easily computed approximation for long run, average performance measures such as expected delay and probability of delay in multiserver queueing systems with exponential service times and periodic sinusoidal Poisson arrival processes. The pointwise stationary approximation is computed by integrating over time that is taking the expectation of the formula for the stationary performance measure with the arrival rate that applies at each point in time. This approximation, which has been empirically confirmed as a tight upper bound of the true value, is shown to be very accurate for a range of parameter values corresponding to a reasonably broad spectrum of real systems.

Using Count Data Models in Travel Cost Analysis with Aggregate Data

Abstract: Estimators of recreational demand models frequently use continuous functional forms, such as ordinary least squares (OLS) on log transformed variables (e.g., Ziemer, Musser, and Hill). However, the nature of trip demand introduces complicating factors. First, trips occur in nonnegative quantities. Failure to control for this censoring will lead to biased estimation. Second, because trips are available only in integer quantities, the usual demand models, which correlate marginal quantity with marginal price, may be inapplicable. In light of these factors, a natural alternative is to use statistical models that explicitly recognize the "count" nature of trip demand. Several recent papers (e.g., Shaw, Smith, Grogger and Carson, Creel and Loomis) have applied count models to the travel cost model. These works largely have focused on truncated data sets based on choice-based samples. In this study the focus is on the older problem where zero-demanders are included. In particular, the application of several robust estimators of count models to aggregated data will be considered. The Poisson distribution forms the foundation for the count models examined in this study. Although the Poisson is a convenient distribution to work with, it imposes some stringent constraints on the demand distribution. In particular, the Poisson distribution assumes the variance of trip demand is equal to the expected value of trip demand. To loosen these constraints, a generalization of the Poisson, the negative binomial, is discussed. Robust estimation procedures, that permit further loosening of a priori assumptions are then reviewed. Permit data from the Boundary Waters Canoe Area are used to examine the effects of these count models on consumer surplus estimates and on coefficient variability.

A Universal Reduction Procedure for Hamiltonian Group Actions

Abstract: We give a universal method of inducing a Poisson structure on a singular reduced space from the Poisson structure on the orbit space for the group action. For proper actions we show that this reduced Poisson structure is nondegenerate. Furthermore, in cases where the Marsden-Weinstein reduction is well-defined, the action is proper, and the preimage of a coadjoint orbit under the momentum mapping is closed, we show that universal reduction and Marsden-Weinstein reduction coincide. As an example, we explicitly construct the reduced spaces and their Poisson algebras for the spherical pendulum.

Convergence of finite volume schemes for Poisson's equation on nonuniform meshes

Abstract: An error analysis of a family of cell-center finite volume schemes for Poisson’s equation on Cartesian product nonuniform meshes in two dimensions is presented. Optimal-order error estimates are derived in the discrete $H^1$ norm under minimum smoothness requirements on the exact solution and without any additional assumption on the regularity of the mesh. On quasi-uniform meshes analogous estimates are obtained in the maximum norm.

On a continuum percolation model

Abstract: Consider particles placed in space by a Poisson process. Pairs of particles are bonded together, independently of other pairs, with a probability that depends on their separation, leading to the formation of clusters of particles. We prove the existence of a non-trivial critical intensity at which percolation occurs (that is, an infinite cluster forms). We then prove the continuity of the cluster density, or free energy. Also, we derive a formula for the probability that an arbitrary Poisson particle lies in a cluster consisting of k particles (or equivalently, a formula for the density of such clusters), and show that at high Poisson intensity, the probability that an arbitrary Poisson particle is isolated, given that it lies in a finite cluster, approaches 1.

Momentum Mappings And Reduction of Poisson Actions

Abstract: An action σ: G × P→P of a Poisson Lie group G on a Poisson manifold P is called a Poisson action if σ is a Poisson map. It is believed that Poisson actions should be used to understand the “hidden symmetries” of certain integrable systems [STS2]. If the Poisson Lie group G has the zero Poisson structure, then σ being a Poisson action is equivalent to each transformation σ g : P→ P for g ∈ G preserving the Poisson structure on P. In this case, if the orbit space G \ P is a smooth manifold, it has a reduced Poisson structure such that the projection map P→G \ P is a Poisson map. If P is symplectic and if the action σ is generated by an equivariant momentum mapping J: P→ g*, the reduction procedure of Meyer [Me] and Marsden and Weinstein [Ms-We] gives a way of describing the symplectic leaves of G \ P as the quotients P µ := G µ \J −1 (µ), where µ∈ g* and G µ ⊂ G is the coadjoint isotropy subgroup of µ.

Quantization of the Poisson SU(2) and its Poisson homogeneous space — The 2-sphere

Abstract: We show that deformation quantizations of the Poisson structures on the Poisson Lie groupSU(2) and its homogeneous space, the 2-sphere, are compatible with Woronowicz's deformation quantization ofSU(2)'s group structure and Podles' deformation quantization of 2-sphere's homogeneous structure, respectively. So in a certain sense the multiplicativity of the Lie Poisson structure onSU(2) at the classical level is preserved under quantization.

Sample size for Poisson regression

Abstract: SUMMARY For the Poisson regression model, an exact expression for Fisher's information matrix, based upon the moment generating function of the distribution of covariates, is calculated. This parallels a similar, approximate, calculation by Whittemore (1981) for logistic regression. The resulting asymptotic variance of the maximum likelihood estimate of the parameters is used to calculate the sample size required to test hypotheses about the parameters at a specified significance and power. Methods for calculating sample size are derived for various distributions of a single covariate, and for a family of multivariate exponential-type distributions of multiple covariates. The procedures are illustrated with two examples.

A two-state Markov mixture model for a time series of epileptic seizure counts.

Abstract: This paper discusses a model for a time series of epileptic seizure counts in which the mean of a Poisson distribution changes according to an underlying two-state Markov chain. The EM algorithm (Dempster, Laird, and Rubin, 1977, Journal of the Royal Statistical Society, Series B 39, 1-38) is used to compute maximum likelihood estimators for the parameters of this two-state mixture model and extensions are made allowing for nonstationarity. The model is illustrated using daily seizure counts for patients with intractable epilepsy and results are compared with a simple Poisson distribution and Poisson regressions. Some simulation results are also presented to demonstrate the feasibility of this model.

Large-deviation approximations to the distribution of scan statistics

Abstract: Suppose a Poisson process is observed on the unit interval. The scan statistic is defined as the maximum number of events observed as a window of fixed width is moved across the interval, and the distribution under homogeneity has been widely studied. Frequently, we may not wish to specify the window width in advance but to consider scan statistics with varying window widths. We propose a modification of the scan statistic based on a likelihood ratio criterion. This leads to a boundarycrossing problem for a two-dimensional random field, which we approximate using a large-deviation scaling under homogeneity. Similar results are obtained for Poisson processes observed in two dimensions. Numerical computations and simulations are used to illustrate the accuracy of the approximations. BOUNDARY CROSSING; CHANGE POINTS; POISSON PROCESS; RANDOM FIELD; SPATIAL STATISTICS

Tests for Detecting Overdispersion in the Positive Poisson Regression Model

Abstract: This article derives score tests for extra-Poisson variation in the positive or truncated-at-zero Poisson regression model against truncated-at-zero negative binomial family alternatives. It also develops size-corrected tests of overdispersion that are expected to improve their small-sample properties. Further, small-sample performance of the tests is investigated by means of Monte Carlo experiments. As an illustration, the proposed tests are applied to a model of strikes in U.S. manufacturing. The proposed tests have an interpretation as conditional moment tests and require only the positive Poisson model to be estimated. It is shown that most of the tests for overdispersion in the regular Poisson model given in the econometric and statistical literature can be obtained as special cases of the tests developed in this article. Monte Carlo experiments indicate that the size correction, based on the asymptotic expansions of the score function, is effective in improving the accuracy of the size and power of ...

On the geometric quantization of Poisson manifolds

Abstract: In a paper by Huebschmann [J Reine Angew Math 408, 57 (1990)], the geometric quantization of Poisson manifolds appears as a particular case of the quantization of Poisson algebras Here, this quantization is presented straightforwardly The results include a geometric prequantization integrality condition and its discussion in particular cases such as Dirac brackets, an adaptation of the notion of a polarization and a construction of a quantum Hilbert space, and a computational example In the last section of the paper the general prequantization representations in the sense of Urwin [Adv Math 50, 126 (1983)] are described for the Poisson and Jacobi manifolds

Information capacity of the Poisson channel

Abstract: The information capacity of the Poisson channel with random or time-varying noise intensity is obtained for time-varying peak and average constraints on the encoder intensity. The channel model is specified, and some definitions from information theory are given. Causal feedback is shown not to increase the channel capacity for the case of nonrandom noise intensity. For random noise intensity, use of causal feedback does increase capacity; the extent of that increase is quantified. Jamming is considered, and the optimal jamming signal is given. Poisson channels with thinning are introduced. >

Robust estimation of the variance in moment methods for extra-binomial and extra-Poisson variation.

Abstract: When faced with data in the form of overdispersed counts or proportions, moment methods allow consistent parameter estimation when only the form of the mean and variance is specified. If the variance form is misspecified, these methods still yield consistent parameter estimates, though with lower efficiency, and the variances of the estimates will be inconsistent. A variance correction is available that yields consistent variance estimates in these circumstances. The asymptotic and small-sample efficiencies of this correction are calculated, and its performance under variance misspecification is studied. A group-randomized breast self-examination prevention study that is now underway serves as a focal point for the study of these properties. The use of the variance correction in modelling is illustrated on a teratology data set.

Hemacytometer Cell Count Distributions: Implications of Non‐Poisson Behavior

Abstract: The variance of observed cell counts using a hemacytometer was analyzed The variance was found to be greater than expected under the normally assumed Poisson distribution because of variations in the volume of the counting area Furthermore, counts from the two sides of the hemacytometer were found to be correlated Total cell count on a hemacytometer was better described using a normal distribution with quadratic variance, N(lambda, lambda + sigma-lambda-2), with sigma = 0012 Because hemacytometer cell counts are not Poisson distributed, modified counting protocols are recommended

The Structure of Genealogies and the Distribution of Fixed Differences between DNA Sequence Samples from Natural Populations

Abstract: When two samples of DNA sequences are compared, one way in which they may differ is in the presence of fixed differences, which are defined as sites at which all of the sequences in one sample are different from all of the sequences in a second sample. The probability distribution of the number of fixed differences is developed. The theory employs Wright-Fisher genealogies and the infinite sites mutation model. For the case when both samples are drawn randomly from the same population it is found that genealogies permitting fixed differences are very unlikely. Thus the mere presence of fixed differences between samples is statistically significant, even for small samples. The theory is extended to samples from populations that have been separated for some time. The relationship between a simple Poisson distribution of mutations and the distribution of fixed differences is described as a function of the time since populations have been isolated. It is shown how these results may contribute to improved tests of recent balancing or directional selection.

Poisson approximations for runs and patterns of rare events

Abstract: Consider a sequence of Bernoulli trials with success probability p, and let N,,k denote the number of success runs of length k ?_i 2 among the first n trials. The Stein-Chen method is employed to obtain a total variation upper bound for the rate of convergence of N,,k to a Poisson random variable under the standard condition

Scaling properties of the eigenvalue spacing distribution for band random matrices

Abstract: The authors show that the spacing distribution for eigenvalues of band random matrices is described by a single parameter b2/N, where b is the band half-width and N is the size of the matrices. It is also shown that the eigenvalue's density obeys the semicircle law. The found scaling behaviour suggests that the fluctuation properties in the intermediate regime, between Wigner-Dyson and Poisson, are universal.

Confidence Intervals for Welfare Measures with Application to a Problem of Truncated Counts

Abstract: Demand for deer hunting trips was estimated using statistical models based on the normal, Poisson, and negative binomial probability laws. Some of the models accounted for existing sampling truncation. Estimates of Marshallian and Hicksian welfare measures are presented, accompanied by 90 percent confidence intervals based on Krinsky and Robb's procedure. For each of the statistical models, the Hicksian measures are found to be very close to the Marshallian measures, with similar confidence intervals. Accounting for the truncation of the dependent variable has a statistically significant effect on the resulting estimates of welfare measures. Copyright 1991 by MIT Press.

Maximum likelihood estimates of the fractal dimension for random spatial patterns

Abstract: SUMMARY Two approximate maximum likelihood estimates of the fractal dimension are suggested for random point patterns and planar curves, based on the distributions of the Palm probability and the spectrum. For estimation of the fractal dimension of self-similar patterns in space, we usually measure the slope of a linear log-log relationship between pairs of points. The box- counting method plots the number of pixels which intersects the pattern under consider- ation versus length of the pixel unit, and the walking-dividers method plots the total length of polygons which approximate the considered curves versus length of the polygon's side, and so on; see Mandelbrot (1982, p. 33), for example. Using these methods, there has been a large number of papers reporting the dimensions of fractal sets, especially since Mandelbrot (1977). Second-order properties of a stochastic process such as the auto-covariance and the spectrum are also available for examining self-similarity and measuring its indices. This is especially easy for stochastic processes on a real line. For instance, Ogata (1988) and Ogata & Abe (1991) applied one-dimensional point processes to the occurrence times of major earthquakes in the world and Japan for a time span of about one century, investigating the self-similarity by the auto-covariance, spectrum, dispersion-time curves and so-called R/S statistic. In particular, fractal dimension and the Hurst number, or the self-similarity index, are estimated objectively by maximizing a spectral log likelihood. The maximum likelihood method is generally expected to provide an efficient estimate together with its standard error. Therefore, in this paper, we develop maximum likelihood methods for planar point and curve patterns. Two types of approximate likelihood functions are considered. One is equivalent to an isotropic Poisson likelihood by modelling an intensity function of points under the Palm probability, and the other is the so-called spectral likelihood based on the distribution of the periodogram, which is an estimate of the spectrum. These two independent estimation methods are compared by using a certain artificially generated clustering point pattern, then a map of epicentres of earthquakes and a topographic contour line are analyzed to compare the two methods.

Extensions of Palm's theorem: a review

Abstract: This note reviews the transient behavior the M/G/∞ queue with nonhomogeneous Poisson or compound Poisson input and nonstationary service distribution. In the case of nonhomogeneous Poisson input, the number of customers in the queueing system over time turns out to have a Poisson distribution. The generality of the nonhomogeneity/nonstationarity assumptions and the ease of use of the resulting Poisson distribution broaden the area of applications for Poisson models. These results have found use in modeling multi-echelon repair systems in situations where the number of arrivals or number in service has a variance-to-mean ratio of unity (the Poisson case) or greater than unity.

Nonhomogeneous Poisson model for volcanic eruptions

Abstract: A simple Poisson process is more specifically known as a homogeneous Poisson process since the rateλ was assumed independent of time t. The homogeneous Poisson model generally gives a good fit to many volcanoes for forecasting volcanic eruptions. If eruptions occur according to a homogeneous Poisson process, the repose times between consecutive eruptions are independent exponential variables with meanθ=1/λ. The exponential distribution is applicable when the eruptions occur “at random” and are not due to aging, etc. It is interesting to note that a general population of volcanoes can be related to a nonhomogeneous Poisson process with intensity factorλ(t). In this paper, specifically, we consider a more general Weibull distribution, WEI (θ, β), for volcanism. A Weibull process is appropriate for three types of volcanoes: increasing-eruption-rate (β>1), decreasing-eruption-rate (β<1), and constant-eruption-rate (β=1). Statistical methods (parameter estimation, hypothesis testing, and prediction intervals) are provided to analyze the following five volcanoes: Also, Etna, Kilauea, St. Helens, and Yake-Dake. We conclude that the generalized model can be considered a goodness-of-fit test for a simple exponential model (a homogeneous Poisson model), and is preferable for practical use for some nonhomogeneous Poisson volcanoes with monotonic eruptive rates.

Statistics and Data with R: An applied approach through examples

Abstract: Preface. Part I: Data in statistics and R. 1. Basic R. 1.1 Preliminaries. 1.2 Modes. 1.3 Vectors. 1.4 Arithmetic operators and special values. 1.5 Objects. 1.6 Programming. 1.7 Packages. 1.8 Graphics. 1.9 Customizing the workspace. 1.10 Projects. 1.12 Assignments. 2. Data in statistics and in R. 2.1 Types of data. 2.2 Objects that hold data. 2.3 Data organization. 2.4 Data import, export and connections. 2.5 Data manipulation. 2.6 Manipulating strings. 2.7 Assignments. 3. Presenting data . 3.1 Tables and the flavors of apply () 3.2 Bar plots. 3.3 Histograms. 3.4 Dot charts. 3.5 Scatter plots. 3.6 Lattice plots. 3.7 Three-dimensional plots and contours. 3.8 Assignments. Part II: Probability, densities and distributions . 4. Probability and random variables . 4.1 Set theory. 4.2 Trials, events and experiments. 4.3 Definitions and properties of probability. 4.4 Conditional probability and independence. 4.5 Algebra with probabilities. 4.6 Random variables. 4.7 Assignments. 5. Discrete densities and distributions . 5.1 Densities. 5.2 Distribution. 5.3 Properties. 5.4 Expected values. 5.5 Variance and standard deviation. 5.6 The binomial. 5.7 The Poisson. 5.8 Estimating parameters. 5.9 Some useful discrete densities. 5.10 Assignments. 6. Continuous distributions and densities . 6.1 Distributions. 6.2 Densities. 6.3 Properties. 6.4 Expected values. 6.5 Variance and standard deviation. 6.6 Areas under density curves. 6.7 Inverse distributions and simulations. 6.8 Some useful continuous densities. 6.9 Assignments. 7. The normal and sampling densities . 7.1 The normal density. 7.2 Applications of the normal. 7.3 Data transformations. 7.4 Random samples and sampling densities. 7.5 A detour: using R efficiently. 7.6 The sampling density of the mean. 7.7 The sampling density of proportion. 7.8 The sampling density of intensity. 7.9 The sampling density of variance. 7.10 Bootstrap: arbitrary parameters of arbitrary densities. 7.11 Assignments. Part III: Statistics . 8. Exploratory data analysis . 8.1 Graphical methods. 8.2 Numerical summaries. 8.3 Visual summaries. 8.4 Assignments. 9. Point and interval estimation . 9.1 Point estimation. 9.2 Interval estimation. 9.3 Point and interval estimation for arbitrary densities. 9.4 Assignments. 10. Single sample hypotheses testing . 10.1 Null and alternative hypotheses. 10.2 Large sample hypothesis testing. 10.3 Small sample hypotheses testing. 10.4 Arbitrary parameters of arbitrary densities. 10.5 p -values. 10.6 Assignments. 11. Power and sample size for single samples . 11.1 Large sample. 11.2 Small samples. 11.3 Power and sample size for arbitrary densities. 11.4 Assignments. 12. Two samples . 12.1 Large samples. 12.2 Small samples. 12.3 Unknown densities. 12.4 Assignments. 13. Power and sample size for two samples . 13.1 Two means from normal populations. 13.2 Two proportions. 13.3 Two rates. 13.4 Assignments. 14. Simple linear regression . 14.1 Simple linear models. 14.2 Estimating regression coefficients. 14.3 The model goodness of fit. 14.4 Hypothesis testing and confidence intervals. 14.5 Model assumptions. 14.6 Model diagnostics. 14.7 Power and sample size for the correlation coefficient. 14.8 Assignments. 15. Analysis of variance . 15.1 One-way, fixed-effects ANOVA. 15.2 Non-parametric one-way ANOVA. 15.3 One-way, random-effects ANOVA. 15.4 Two-way ANOVA. 15.5 Two-way linear mixed effects models. 15.6 Assignments. 16. Simple logistic regression . 16.1 Simple binomial logistic regression. 16.2 Fitting and selecting models. 16.3 Assessing goodness of fit. 16.4 Diagnostics. 16.5 Assignments. 17. Application: the shape of wars to come . 17.1 A statistical profile of the war in Iraq. 17.2 A statistical profile of the second Intifada. References . R Index. General Index.

On hazard rate processes

Abstract: Hazard rate processes are discussed in the context of doubly stochastic Poisson processes. We derive an explicit expression for the reliability function corresponding to an increasing hazard rate processes with independent increments. Also, bounds are obtained for the reliability function of a system with a general hazard rate process.