Showing papers on "Poisson distribution published in 2003"

Alternatives for logistic regression in cross-sectional studies: an empirical comparison of models that directly estimate the prevalence ratio

Abstract: Cross-sectional studies with binary outcomes analyzed by logistic regression are frequent in the epidemiological literature. However, the odds ratio can importantly overestimate the prevalence ratio, the measure of choice in these studies. Also, controlling for confounding is not equivalent for the two measures. In this paper we explore alternatives for modeling data of such studies with techniques that directly estimate the prevalence ratio. We compared Cox regression with constant time at risk, Poisson regression and log-binomial regression against the standard Mantel-Haenszel estimators. Models with robust variance estimators in Cox and Poisson regressions and variance corrected by the scale parameter in Poisson regression were also evaluated. Three outcomes, from a cross-sectional study carried out in Pelotas, Brazil, with different levels of prevalence were explored: weight-for-age deficit (4%), asthma (31%) and mother in a paid job (52%). Unadjusted Cox/Poisson regression and Poisson regression with scale parameter adjusted by deviance performed worst in terms of interval estimates. Poisson regression with scale parameter adjusted by χ2 showed variable performance depending on the outcome prevalence. Cox/Poisson regression with robust variance, and log-binomial regression performed equally well when the model was correctly specified. Cox or Poisson regression with robust variance and log-binomial regression provide correct estimates and are a better alternative for the analysis of cross-sectional studies with binary outcomes than logistic regression, since the prevalence ratio is more interpretable and easier to communicate to non-specialists than the odds ratio. However, precautions are needed to avoid estimation problems in specific situations.

Analysis of sports data by using bivariate Poisson models

Abstract: Summary. Models based on the bivariate Poisson distribution are used for modelling sports data. Independent Poisson distributions are usually adopted to model the number of goals of two competing teams. We replace the independence assumption by considering a bivariate Poisson model and its extensions. The models proposed allow for correlation between the two scores, which is a plausible assumption in sports with two opposing teams competing against each other. The effect of introducing even slight correlation is discussed. Using just a bivariate Poisson distribution can improve model fit and prediction of the number of draws in football games. The model is extended by considering an inflation factor for diagonal terms in the bivariate joint distribution. This inflation improves in precision the estimation of draws and, at the same time, allows for overdispersed, relative to the simple Poisson distribution, marginal distributions. The properties of the models proposed as well as interpretation and estimation procedures are provided. An illustration of the models is presented by using data sets from football and water-polo.

Zero-Inflated Negative Binomial Mixed Regression Modeling of Over-Dispersed Count Data with Extra Zeros

Abstract: In many biometrical applications, the count data encountered often contain extra zeros relative to the Poisson distribution. Zero-inflated Poisson regression models are useful for analyzing such data, but parameter estimates may be seriously biased if the nonzero observations are over-dispersed and simultaneously correlated due to the sampling design or the data collection procedure. In this paper, a zero-inflated negative binomial mixed regression model is presented to analyze a set of pancreas disorder length of stay (LOS) data that comprised mainly same-day separations. Random effects are introduced to account for inter-hospital variations and the dependency of clustered LOS observations. Parameter estimation is achieved by maximizing an appropriate log-likelihood function using an EM algorithm. Alternative modeling strategies, namely the finite mixture of Poisson distributions and the non-parametric maximum likelihood approach, are also considered. The determination of pertinent covariates would assist hospital administrators and clinicians to manage LOS and expenditures efficiently.

Observation-driven models for Poisson counts

Abstract: SUMMARY This paper is concerned with a general class of observation-driven models for time series of counts whose conditional distributions given past observations and explanatory variables follow a Poisson distribution. These models provide a flexible framework for modelling a wide range of dependence structures. Conditions for stationarity and ergodicity of these processes are established from which the large-sample properties of the maximum likelihood estimators can be derived. Simulations are provided to give additional insight into the finite-sample behaviour of the estimators. Finally an application to a regression model for daily counts of asthma presentations at a Sydney hospital is described.

Including systematic uncertainties in confidence interval construction for Poisson statistics

Abstract: One way to incorporate systematic uncertainties into the calculation of confidence intervals is by integrating over probability density functions parametrizing the uncertainties. In this paper we present a development of this method which takes into account uncertainties in the prediction of background processes and uncertainties in the signal detection efficiency and background efficiency, and allows for a correlation between the signal and background detection efficiencies. We implement this method with the likelihood ratio (usually denoted as the Feldman-Cousins) approach with and without conditioning. We present studies of coverage for the likelihood ratio and Neyman ordering schemes. In particular, we present two different types of coverage tests for the case where systematic uncertainties are included. To illustrate the method we show the relative effect of including systematic uncertainties in the case of the dark matter search as performed by modern neutrino telescopes.

Shot noise Cox processes

Abstract: Shot noise Cox processes constitute a large class of Cox and Poisson cluster processes in ℝ d , including Neyman-Scott, Poisson-gamma and shot noise G Cox processes It is demonstrated that, due to the structure of such models, a number of useful and general results can easily be established The focus is on the probabilistic aspects with a view to statistical applications, particularly results for summary statistics, reduced Palm distributions, simulation with or without edge effects, conditional simulation of the intensity function and local and spatial Markov properties

Modelling time series count data: an Autoregressive Conditional Poisson model

Abstract: This paper introduces and evaluates new models for time series count data. The Autoregressive Conditional Poisson model (ACP) makes it possible to deal with issues of discreteness, overdispersion (variance greater than the mean) and serial correlation. A fully parametric approach is taken and a marginal distribution for the counts is specified, where conditional on past observations the mean is autoregressive. This enables to attain improved inference on coefficients of exogenous regressors relative to static Poisson regression, which is the main concern of the existing literature, while modelling the serial correlation in a flexible way. A variety of models, based on the double Poisson distribution of Efron (1986) is introduced, which in a first step introduce an additional dispersion parameter and in a second step make this dispersion parameter time-varying. All models are estimated using maximum likelihood which makes the usual tests available. In this framework autocorrelation can be tested with a straightforward likelihood ratio test, whose simplicity is in sharp contrast with test procedures in the latent variable time series count model of Zeger (1988). The models are applied to the time series of monthly polio cases in the U.S between 1970 and 1983 as well as to the daily number of price change durations of .75$ on the IBM stock. A .75$ price-change duration is defined as the time it takes the stock price to move by at least .75$. The variable of interest is the daily number of such durations, which is a measure of intradaily volatility, since the more volatile the stock price is within a day, the larger the counts will be. The ACP models provide good density forecasts of this measure of volatility.

Multiscale Fitting Procedure Using Markov Modulated Poisson Processes

Abstract: This paper proposes a parameter fitting procedure using Markov Modulated Poisson Processes (MMPPs) that leads to accurate estimates of queuing behavior for network traffic exhibiting long-range dependence behavior. The procedure matches both the autocovariance and marginal distribution of the counting process. A major feature is that the number of states is not fixed a priori, and can be adapted to the particular trace being modeled. The MMPP is constructed as a superposition of L 2-MMPPs and one M-MMPP. The 2-MMPPs are designed to match the autocovariance and the M-MMPP to match the marginal distribution. Each 2-MMPP models a specific time-scale of the data. The procedure starts by approximating the autocovariance by a weighted sum of exponential functions that model the autocovariance of the 2-MMPPs. The autocovariance tail can be adjusted to capture the long-range dependence characteristics of the traffic, up to the time-scales of interest to the system under study. The procedure then fits the M-MMPP parameters in order to match the marginal distribution, within the constraints imposed by the autocovariance matching. The number of states is also determined as part of this step. The final MMPP with M2 L states is obtained by superposing the L 2-MMPPs and the M-MMPP. We apply the inference procedure to traffic traces exhibiting long-range dependence and evaluate its queuing behavior through simulation. Very good results are obtained, both in terms of queuing behavior and number of states, for the traces used, which include the well-known Bellcore traces.

Order Fill Rate, Leadtime Variability, and Advance Demand Information in an Assemble-to-Order System

Abstract: We study an assemble-to-order system with stochastic leadtimes for component replenishment. There are multiple product types, of which orders arrive at the system following batch Poisson processes. Base-stock policies are used to control component inventories. We analyze the system as a set of queues driven by a common, multiclass batch Poisson input, and derive the joint queue-length distribution. The result leads to simple, closed-form expressions of the first two moments, in particular the covariances, which capture the dependence structure of the system. Based on the joint distribution and the moments, we derive easy-to-compute approximations and bounds for the order fulfillment performance measures. We also examine the impact of demand and leadtime variability, and investigate the value of advance demand information.

Adaptive estimation of the intensity of inhomogeneous Poisson processes via concentration inequalities

Abstract: In this paper, we establish oracle inequalities for penalized projection estimators of the intensity of an inhomogeneous Poisson process. We study consequently the adaptive properties of penalized projection estimators. At first we provide lower bounds for the minimax risk over various sets of smoothness for the intensity and then we prove that our estimators achieve these lower bounds up to some constants. The crucial tools to obtain the oracle inequalities are new concentration inequalities for suprema of integral functionals of Poisson processes which are analogous to Talagrand's inequalities for empirical processes.

Bounds on the Capacity of the Discrete-Time Poisson Channel

Abstract: The large-inputs asymptotic capacity of a peak and average power limited discrete-time Poisson channel is derived using a new rm (non-asymptotic) lower bound and an asymptotic upper bound. The upper bound is based on the dual expression for channel capacity and the recently introduced notion of capacityachieving input distributions that escape to innity. The lower bound is based on a lemma that lower bounds the entropy of a conditionally Poisson random variable in terms of the dierential entropy of the conditional mean.

Poisson kernel characterization of Reifenberg flat chord arc domains

Abstract: In this paper we prove the conjecture stated by the authors in Free boundary regularity for harmonic measures and Poisson kernels (Ann. of Math. 150 (1999) 369–454) concerning the free boundary regularity problem for the Poisson kernel below the continuous threshold. We show that if Ω is a Reifenberg flat chord arc domain, and the logarithm of the Poisson kernel has vanishing mean oscillation then the unit normal vector to the boundary also has vanishing mean oscillation.

Random effects Cox models: A Poisson modelling approach

Abstract: SUMMARY We propose a Poisson modelling approach to nested random effects Cox proportional hazards models. An important feature of this approach is that the principal results depend only on the first and second moments of the unobserved random effects. The orthodox best linear unbiased predictor approach to random effects Poisson modelling techniques enables us to justify appropriate consistency and optimality. The explicit expressions for the random effects given by our approach facilitate incorporation of a relatively large number of random effects. The use of the proposed methods is illustrated through the reanalysis of data from a large-scale cohort study of particulate air pollution and mortality previously reported by Pope et al. (1995).

Efficient and accurate likelihood for iterative image reconstruction in x-ray computed tomography

Abstract: We report a novel approach for statistical image reconstruction in X-ray CT. Statistical image reconstruction depends on maximizing a likelihood derived from a statistical model for the measurements. Traditionally, the measurements are assumed to be statistically Poisson, but more recent work has argued that CT measurements actually follow a compound Poisson distribution due to the polyenergetic nature of the X-ray source. Unlike the Poisson distribution, compound Poisson statistics have a complicated likelihood that impedes direct use of statistical reconstruction. Using a generalization of the saddle-point integration method, we derive an approximate likelihood for use with iterative algorithms. In its most realistic form, the approximate likelihood we derive accounts for polyenergetic X-rays and poisson light statistics in the detector scintillator, and can be extended to account for electronic additive noise. The approximate likelihood is closer to the exact likelihood than is the conventional Poisson likelihood, and carries the promise of more accurate reconstruction, especially in low X-ray dose situations.

The poisson bracket for poisson forms in multisymplectic field theory

Abstract: We present a general definition of the Poisson bracket between differential forms on the extended multiphase space appearing in the geometric formulation of first order classical field theories and, more generally, on exact multisymplectic manifolds. It is well defined for a certain class of differential forms that we propose to call Poisson forms and turns the space of Poisson forms into a Lie superalgebra.

Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model

Abstract: General boundary conditions ("branes") for the Poisson sigma model are studied. They turn out to be labeled by coisotropic submanifolds of the given Poisson manifold. The role played by these boundary conditions both at the classical and at the perturbative quantum level is discussed. It turns out to be related at the classical level to the category of Poisson manifolds with dual pairs as morphisms and at the perturbative quantum level to the category of associative algebras (deforming algebras of functions on Poisson manifolds) with bimodules as morphisms. Possibly singular Poisson manifolds arising from reduction enter naturally into the picture and, in particular, the construction yields (under certain assumptions) their deformation quantization.

Decompounding: an estimation problem for Poisson random sums

Abstract: Given a sample from a compound Poisson distribution, we consider estimation of the corresponding rate parameter and base distribution. This has applications in insurance mathematics and queueing theory. We propose a plug-in type estimator that is based on a suitable inversion of the compounding operation. Asymptotic results for this estimator are obtained via a local analysis of the decompounding functional.

Point and interval estimation of the population size using the truncated Poisson regression model

Abstract: A method is presented to derive point and interval estimates of the total number of individuals in a heterogenous Poisson population. The method is based on the Horvitz-Thompson approach. The zero-truncated Poisson regression model is fitted and results are used to obtain point and interval estimates for the total number of individuals in the population. The method is assessed by performing a simulation experiment computing coverage probabilities of Horvitz-Thompson confidence intervals for cases with different sample sizes and Poisson parameters. We illustrate our method using capture-recapture data from the police registration system providing information on illegal immigrants in four large cities in the Netherlands.

Estimating Frailty Models via Poisson Hierarchical Generalized Linear Models

Abstract: Frailty models extend proportional hazards models to multivariate survival data. Hierarchical-likelihood provides a simple unified framework for various random effect models such as hierarchical generalized linear models, frailty models, and mixed linear models with censoring. Wereview the hierarchical-likelihood estimation methods for frailty models. Hierarchical-likelihood for frailty models can be expressed as that for Poisson hierarchical generalized linear models. Frailty models can thus be fitted using Poisson hierarchical generalized linear models. Properties of the new methodology are demonstrated by simulation. The new method reduces the bias of maximum likelihood and penalized likelihood estimates.

Estimating the Size of a Criminal Population from Police Records Using the Truncated Poisson Regression Model

Abstract: The truncated Poisson regression model is used to arrive at point and interval estimates of the size of two offender populations, i.e. drunk drivers and persons who illegally possess firearms. The dependent capture‐recapture variables are constructed from Dutch police records and are counts of individual arrests for both violations. The population size estimates are derived assuming that each count is a realization of a Poisson distribution, and that the Poisson parameters are related to covariates through the truncated Poisson regression model. These assumptions are discussed in detail, and the tenability of the second assumption is assessed by evaluating the marginal residuals and performing tests on overdispersion. For the firearms example, the second assumption seems to hold well, but for the drunk drivers example there is some overdispersion. It is concluded that the method is useful, provided it is used with care.

Computing with the COM-Poisson distribution

Abstract: The Conway-Maxwell-Poisson (COM-Poisson) is a generalization of the Poisson distribution which can model both under-dispersed and over-dispersed data. However, the distribution, moments, and MLE cannot be computed in closed form. This paper describes computational schemes and handy approximations for the COM-Poisson.

Regular variation in the mean and stable limits for Poisson shot noise

Abstract: Poisson shot noise is a natural generalization of a compound Poisson process when the summands are stochastic processes starting at the points of the underlying Poisson process. We study the limiting behaviour of Poisson shot noise when the limits are infinite-variance stable processes. In this context a sufficient condition for this convergence turns up which is closely related to multivariate regular variation -- we call it regular variation in the mean. We also show that the latter condition is necessary and sufficient for the weak convergence of the point processes constructed from the normalized noise sequence and also for the weak convergence of its extremes.

Dirac submanifolds and Poisson involutions

Abstract: Dirac submanifolds are a natural generalization in the Poisson category of the symplectic submanifolds of a symplectic manifold. They correspond to symplectic subgroupoids of the symplectic groupoid of the given Poisson manifold. In particular, Dirac submanifolds arise as the stable loci of Poisson involutions. In this paper, we make a general study of these submanifolds including both local and global aspects. In the second part of the paper, we study Poisson involutions and the induced Poisson structures on their stable loci. In particular, we discuss the Poisson involutions on a special class of Poisson groups, and more generally Poisson groupoids, called symmetric Poisson groups, and symmetric Poisson groupoids. Many well-known examples, including the standard Poisson group structures on semi-simple Lie groups, Bruhat Poisson structures on compact semi-simple Lie groups, and Poisson groupoid structures arising from dynamical r-matrices of semi-simple Lie algebras are symmetric, so they admit a Poisson involution. For symmetric Poisson groups, the relation between the stable locus Poisson structure and Poisson symmetric spaces is discussed. As a consequence, we prove that the Dubrovin Poisson structure on the space of Stokes matrices U+ (appearing in Dubrovin's theory of Frobenius manifolds) is a Poisson symmetric space.