Showing papers on "Poisson distribution published in 1994"

Accounting for Excess Zeros and Sample Selection in Poisson and Negative Binomial Regression Models

Abstract: We present several modifications of the Poisson and negative binomial models for count data to accommodate cases in which the number of zeros in the data exceed what would typically be predicted by either model. The excess zeros can masquerade as overdispersion. We present a new test procedure for distinguishing between zero inflation and overdispersion. We also develop a model for sample selection which is analogous to the Heckman style specification for continuous choice models. An application is presented to a data set on consumer loan behavior in which both of these phenomena are clearly present.

Accounting for Excess Zeros and Sample Selection in Poisson and Negative Binomial Regression Models

Abstract: We present several modifications of the Poisson and negative binomial models for count data to accommodate cases in which the number of zeros in the data exceed what would typically be predicted by either model. The excess zeros can masquerade as overdispersion. We present a new test procedure for distinguishing between zero inflation and overdispersion. We also develop a model for sample selection which is analogous to the Heckman style specification for continuous choice models. An application is presented to a data set on consumer loan behavior in which both of these phenomena are clearly present.

Wide-area traffic: the failure of Poisson modeling

Abstract: Network arrivals are often modeled as Poisson processes for analytic simplicity, even though a number of traffic studies have shown that packet interarrivals are not exponentially distributed. We evaluate 21 wide-area traces, investigating a number of wide-area TCP arrival processes (session and connection arrivals, FTPDATA connection arrivals within FTP sessions, and TELNET packet arrivals) to determine the error introduced by modeling them using Poisson processes. We find that user-initiated TCP session arrivals, such as remote-login and file-transfer, are well-modeled as Poisson processes with fixed hourly rates, but that other connection arrivals deviate considerably from Poisson; that modeling TELNET packet interarrivals as exponential grievously underestimates the burstiness of TELNET traffic, but using the empirical Tcplib[DJCME92] interarrivals preserves burstiness over many time scales; and that FTPDATA connection arrivals within FTP sessions come bunched into “connection burst”, the largest of which are so large that they completely dominate FTPDATA traffic. Finally, we offer some preliminary results regarding how our findings relate to the possible self-similarity of wide-area traffic.

Poisson structure induced (topological) field theories

Abstract: A class of two-dimensional field theories, based on (generically degenerate) Poisson structures and generalizing gravity-Yang–Mills systems, is presented. Locally, the solutions of the classical equations of motions are given. A general scheme for the quantization of the models in a Hamiltonian formulation is found. A BRS-formulation is outlined briefly.

The distribution of the likelihood ratio for mixtures of densities from the one-parameter exponential family

Abstract: We here consider testing the hypothesis ofhomogeneity against the alternative of a two-component mixture of densities. The paper focuses on the asymptotic null distribution of 2 log λ n , where λ n is the likelihood ratio statistic. The main result, obtained by simulation, is that its limiting distribution appears pivotal (in the sense of constant percentiles over the unknown parameter), but model specific (differs if the model is changed from Poisson to normal, say), and is not at all well approximated by the conventional χ (2) 2 -distribution obtained by counting parameters. In Section 3, the binomial with sample size parameter 2 is considered. Via a simple geometric characterization the case for which the likelihood ratio is 1 can easily be identified and the corresponding probability is found. Closed form expressions for the likelihood ratio λ n are possible and the asymptotic distribution of 2 log λ n is shown to be the mixture giving equal weights to the one point distribution with all its mass equal to zero and the χ2-distribution with 1 degree of freedom. A similar result is reached in Section 4 for the Poisson with a small parameter value (θ≤0.1), although the geometric characterization is different. In Section 5 we consider the Poisson case in full generality. There is still a positive asymptotic probability that the likelihood ratio is 1. The upper precentiles of the null distribution of 2 log λ n are found by simulation for various populations and shown to be nearly independent of the population parameter, and approximately equal to the (1–2α)100 percentiles of χ (1) 2 . In Sections 6 and 7, we close with a study of two continuous densities, theexponential and thenormal with known variance. In these models the asymptotic distribution of 2 log λ n is pivotal. Selected (1−α) 100 percentiles are presented and shown to differ between the two models.

Simultaneously Modeling Joint and Marginal Distributions of Multivariate Categorical Responses

Abstract: We discuss model-fitting methods for analyzing simultaneously the joint and marginal distributions of multivariate categorical responses. The models are members of a broad class of generalized logit and loglinear models. We fit them by improving a maximum likelihood algorithm that uses Lagrange's method of undetermined multipliers and a Newton-Raphson iterative scheme. We also discuss goodness-of-fit tests and adjusted residuals, and give asymptotic distributions of model parameter estimators. For this class of models, inferences are equivalent for Poisson and multinomial sampling assumptions. Simultaneous models for joint and marginal distributions may be useful in a variety of applications, including studies dealing with longitudinal data, multiple indicators in opinion research, cross-over designs, social mobility, and inter-rater agreement. The models are illustrated for one such application, using data from a recent General Social Survey regarding opinions about various types of government s...

A Conditional Approach to Point Process Modelling of Elevated Risk

Abstract: SUMMARY We consider the problem of investigating the elevation in risk for a specified disease in relation to possible environmental factors. Our starting point is an inhomogeneous Poisson point process model for the spatial variation in the incidence of cases and controls in a designated geographic region, as proposed by Diggle. We develop a conditional approach to inference which converts the point process model to a non-linear binary regression model for the spatial variation in risk. Simulations suggest that the usual asymptotic approximations for likelihood-based inference are more reliable in this conditional setting than in the original point process setting. We present an application to some data on the spatial distribution of asthma in relation to three industrial locations.

Sequence Comparison Significance and Poisson Approximation

Abstract: The Chen-Stein method of Poisson approximation has been used to establish theorems about comparison of two DNA or protein sequences. The most useful result for sequence alignment applies to alignment scoring with no gaps. However, there has not been a valid method to assign statistical significance to alignment scores with gaps. In this paper we extend Poisson approximation techniques using the Aldous clumping heuristic to a practical method of estimating statistical significance.

Use of Poisson Regression and Time Series Analysis for Detecting Changes over Time in Rates of Child Injury following a Prevention Program

Abstract: The use of two statistical methods to quantify time trends (Poisson regression and time series analysis) is illustrated in analyses of changes in child injury incidence after implementation of a community-based injury prevention program in Central Harlem, New York City. The two analytical methods are used to quantify changes in the rate of injury following the program, while taking into account the underlying annual and seasonal trends. Rates of severe injury during the period from 1983 to 1991 among children under the age of 17 years living in Central Harlem and in the neighboring community of Washington Heights are analyzed. The two methods provide similar point estimates of the effect of the intervention and have a good fit to the data. Although time series analysis has been promoted as the method of choice in analysis of sequential observations over long periods of time, this illustration suggests that Poisson regression is an attractive and viable alternative. Poisson regression provides a versatile analytical method for quantifying the time trends of relatively rare discrete outcomes, such as severe injuries, and provides a useful tool for epidemiologists involved with program evaluation.

Global existence, uniqueness and asymptotic behaviour of solutions of the Wigner–Poisson and Schrödinger-Poisson systems

Abstract: We prove global existence and uniqueness of classical solutions of the Wigner–Poisson and Schrodinger–Poisson systems of equations for both repulsive and attractive potentials. In the repulsive case, we prove decay estimates for the particle density, the potential and the solutions.

Asymptotic Properties of Estimators for the Parameters of Spatial Inhomogeneous Poisson Point Processes

Abstract: Consider a spatial point pattern realized from an inhomogeneous Poisson process on a bounded Borel set , with intensity function λ (s; θ), where . In this article, we show that the maximum likelihood estimator and the Bayes estimator are consistent, asymptotically normal, and asymptotically efficient as the sample region . These results extend asymptotic results of Kutoyants (1984), proved for an inhomogeneous Poisson process on [0, T] , where T →∞. They also formalize (and extend to the multiparameter case) results announced by Krickeberg (1982), for the spatial domain . Furthermore, a Cramer–Rao lower bound is found for any estimator of θ. The asymptotic properties of and are considered for modulated (Cox (1972)), and linear Poisson processes.

Towards a theory of word length distribution

Abstract: A method for modeling word length distributions and different models are presented. The compound Poisson and Ord family of distributions seems to be adequate. The relationship of word length to other language phenomena is discussed.

Parameter Estimation for Markov Modulated Poisson Processes

Abstract: A Markov modulated Poisson process (MMPP) is a doubly stochastic Poisson process whose intensity is controlled by a finite state continuous-time Markov chain. MMPPs have during the last decade been used to model traffic flows in communication networks as well as environmental data. We give a brief survey of methods, most of which are based on moment matching, that have earlier been proposed for estimating the parameters of MMPPs. Then we turn to likelihood based methods, prove a strong consistency property of the maximum likelihood estimator, and discuss some practical methods for calculating MLEs for two-state MMPPs

Evaluation of cluster-based rectangular pulses point process models for rainfall

Abstract: This technical note presents a comparison of cluster-based point rainfall models using the historical hourly rainfall data observed between 1949 and 1976 at Denver, Colorado. The Denver data are used to analyze the performance of three classes of models, namely, the Bartlett-Lewis model, the geometric Neyman-Scott model and the Poisson Neyman-Scott model. The original formulation of the structure of each model, as well as the modified description developed in order to improve the zero depth probability, is considered in this study. Rodriguez-Iturbe et al.(1987a) concluded that it is unlikely that empirical analysis of rainfall data can be used to choose between the Bartlett-Lewis model and the Neyman-Scott model. In a subsequent paper, Rodriguez-Iturbe et al. (1987b) argued that the choice of the distribution of the number of cells per storm for the Neyman-Scott model, either geometric or Poisson, has no general bias effect on the stochastic structure. Some investigators (e.g., Burlando and Rosso, 1991), however, reported results contradictory to those of the previous authors. In light of these observations this note investigates the performance of the cluster-based models. For the Denver data the geometric Neyman-Scott model yields better results compared to the Poisson Neyman-Scott model. Moreover, the Bartlett-Lewis model is shown to be very sensitive to the sets of moment equations used in the parameter estimation. This sensitivity is not observed in the Neyman-Scott scheme and is believed to be a drawback for applying the Bartlett-Lewis model in hydrologic simulation studies.

Precision of incidence predictions based on poisson distributed observations

Abstract: Disease incidence predictions are useful for a number of administrative and scientific purposes. The simplest ones are made using trend extrapolation, on either an arithmetic or a logarithmic scale. This paper shows how approximate confidence prediction intervals can be calculated for such predictions, both for the total number of cases and for the age-adjusted incidence rates, by assuming Poisson distribution of the age and period specific numbers of incident cases. Generalizations for prediction models, for example, using power families and extra-Poisson variation, are also presented. Cancer incidence predictions for the Stockholm-Gotland Oncological Region in Sweden are used as an example.

A comparison of normal and nonnormal mixed models for number of lambs born in Norwegian sheep.

Abstract: A comparison of linear (LM), threshold (TM), and Poisson (PM) mixed models for genetic analysis of number of lambs born (NLB) from 1-yr-old ewes was conducted using 37,718 and 18,633 records of two Norwegian breeds, Dala and Spaelsau, respectively. Models fitted included flock-year as a fixed effect and the random effect of sire. In the Poisson model, the residual variation was assumed to be Poisson, whereas it was normal in LM and multinomial in TM. The models were compared with respect to goodness of fit, predictive ability, and ranking of sires. Goodness of fit and predictive ability were assessed via the mean squared error and the correlation between observed NLB and fitted (predicted) values. Predictive ability was evaluated by estimating effects of sire and flock-year using a random half of the data and then using these estimates to predict records on the other half of the data. The heritability of NLB for Dala was estimated to be .20, .39, and .08 with LM, TM, and PM, respectively. For Spaelsau, corresponding estimates were .12, .26, and .00, respectively. In the PM, problems of low or zero estimates of sire variances were encountered. Hence, an alternative sire variance (PM-L) was approximated from the heritability estimated on the outward scale by REML. All models performed similarly with respect to goodness of fit, predictive ability, and ranking of sires. The TM was very slightly better for both breeds, but the PM and PM-L seemed clearly poorer than TM and LM. An approximate test rejected the hypothesis that the conditional distribution of NLB was Poisson.

Some Applications of Radial Plots

Abstract: A radial plot is a graphical display for comparing estimates that have differing precisions. It is a scatter plot of standardized estimates against reciprocals of standard errors, possibly with respect to a transformed scale, designed so that the original estimates can be compared and interpreted. The estimates may be means, regression coefficients, proportions, rates, odds ratios, random effects, or indeed any parameter estimates that merit comparison between individuals or groups. This article illustrates some uses of radial plots by discussing a variety of data examples taken from the literature. The statistical application areas include interlaboratory trials, point process event rates, empirical Bayes estimation, modeling of counting data, analysis of overdispersed and underdispersed binomial and Poisson data, mixture modeling and meta-analysis.

Preservation of Certain Classes of Life Distributions under Poisson Shock Models

Abstract: Recently defined classes of life distributions are considered, and some relationships among them are proposed. The life distribution H of a device subject to shocks occurring randomly according to a Poisson process is also considered, and sufficient conditions for H to belong to these classes are discussed.

A proof that uniform dose gives the greatest TCP for fixed integral dose in the planning target volume.

Abstract: In this note it is shown that for a fixed integral dose to the planning target volume, the highest tumour control probability (TCP) arises when the dose is spatially uniform. This 'uniform dose theorem' is proved both for (i) a specific TCP model based on Poisson/independent voxel statistics, and (ii) any model for voxel control probability having a specific shape with respect to increasing dose.

A note on a test for Poisson overdispersion

Abstract: SUMMARY This note discusses an error occurring in a test for Poisson overdispersion suggested by Tiago de Oliveira (1965). The limiting null distribution of the suggested statistic is neither pivotal nor is it standard normal. The error lies in the computation of the asymptotic standard error of the overdispersion estimate, for which a corrected version is given. The corrected version of the test statistic becomes equivalent to the normalized version of Fisher's index of dispersion.

Estimating statistical significance of sequence alignments

Abstract: Algorithms that compare two proteins or DNA sequences and produce an alignment of the best matching segments are widely used in molecular biology. These algorithms produce scores that when comparing random sequences of length n grow proportional to n or to log(n) depending on the algorithm parameters. The Azuma-Hoeffding inequality gives an upper bound on the probability of large deviations of the score from its mean in the linear case. Poisson approximation can be applied in the logarithmic case.

Stein's method for compound Poisson approximation: The local approach

Abstract: In the present paper, compound Poisson approximation by Stein's method is considered. A general theorem analogous to the local approach for Poisson approximation is proved. It is then applied to a reliability problem involving the number of isolated vertices in the rectangular lattice on the torus.

Estimating confidence limits on a standardised mortality ratio when the expected number is not error free.

Abstract: OBJECTIVE--The aim was to demonstrate how the beta distribution may be used to find confidence limits on a standardised mortality ratio (SMR) when the expected number of events is subject to random variation and to compare these limits with those obtained with the standard exact approach used for SMRs and with a Fieller-based confidence interval. DESIGN--The relationship of the binomial and the beta distributions is explained. For cohort studies in which deaths are counted in exposed and unexposed groups exact confidence limits on the relative risk are found conditional on the total number of observed deaths. A similar method for the SMR is justified by analogy between the SMR and the relative risk found from such cohort studies, and the fact that the relevant (beta) distribution does not require integer parameters. SOURCE OF DATA--Illustrative examples of hypothetical data were used, together with a MINITAB macro (see appendix) to perform the calculations. MAIN RESULTS--Exact confidence intervals that include error in the expected number are much wider than those found with the standard exact method. Fieller intervals are comparable with the new exact method provided the observed and expected numbers (taken to be means of Poisson variates) are large enough to approximate normality. As the expected number is increased, the standard method gives results closer to the new method, but may still lead to different conclusions even with as many as 100 expected. CONCLUSIONS--If there is reason to suppose the expected number of deaths in an SMR is subject to sampling error (because of imprecisely estimated rates in the standard population) then exact confidence limits should be found by the methods described here, or approximate Fieller-based limits provided enough events are observed and expected to approximate normality.

Bayesian inference in Markovian queues

Abstract: This paper is concerned with the Bayesian analysis of general queues with Poisson input and exponential service times. Joint posterior distribution of the arrival rate and the individual service rate is obtained from a sample consisting inn observations of the interarrival process andm complete service times. Posterior distribution of traffic intensity inM/M/c is also obtained and the statistical analysis of the ergodic condition from a decision point of view is discussed.

Random Vibration of Beam under Moving Loads

Abstract: Structural response to moving loads is an important area of study for highway bridge designers. In this paper, a new approach for the evaluation of nonstationary stochastic response of a bridge under moving loads of vehicular traffic is presented. The random nature of the force arrivals at bridge beams constitute a Poisson process of events. The proposed approach considers the input process as a filtered Poisson process--the response of a linear undamped oscillator excited by a superposition of two Poisson white noise processes. Through an extension of Ito's differential rule, the cumulant differential equations of every order of the response process are obtained. These equations are first-order linear differential equations with constant forcing function and a closed-form solution of the random stochastic response is provided. The results of a numerical application of the approach is offered in terms of cumulants up to the fourth order and a comparison with those obtained by other means is presented.

On the Compound Generalized Poisson Distributions

Abstract: GOOVAERTS and KAAS (1991) present a recursive scheme, involving Panjer's recursion, to compute the compound generalized Poisson distribution (CGPD). In the present paper, we study the CGPD in detail. First, we express the generating functions in terms of Lambert's W function. An integral equation is derived for the pdf of CGPD, when the claim severities are absolutely continuous, from the basic principles. Also we derive the asymptotic formula for CGPD when.the distribution of claim severity satisfies certain conditions. Then we present a recursive formula somewhat different and easier to implement than the recursive scheme of GOOVAERTS and KAAS (1991), when the distribution of claim severity follows an arithmetic distribution, which can be used to evaluate the CGPD. We illustrate the usage of this formula with a numerical example.