Modelling Time Series of Counts in Epidemiology

We review generalized dynamic models for time series of count data. Usually temporal counts are modelled as following a Poisson distribution, and a transformation of the mean depends on parameters which evolve smoothly with time. We generalize the usual dynamic Poisson model by considering continuous mixtures of the Poisson distribution. We consider Poisson‐gamma and Poisson‐log‐normal mixture models. These models have a parameter for each time t which captures possible extra‐variation present in the data. If the time interval between observations is short, many observed zeros might result. We also propose zero inflated versions of the models mentioned above. In epidemiology, when a count is equal to zero, one does not know if the disease is present or not. Our model has a parameter which provides the probability of presence of the disease given no cases were observed. We rely on the Bayesian paradigm to obtain estimates of the parameters of interest, and discuss numerical methods to obtain samples from the resultant posterior distribution. We fit the proposed models to artificial data sets and also to a weekly time series of registered number of cases of dengue fever in a district of the city of Rio de Janeiro, Brazil, during 2001 and 2002.     

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

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.     

Detecting overdispersion in INARCH(1) processes

A statistical test for the degree of overdispersion of count data time series based on the empirical version of the (Poisson) index of dispersion is considered. The test design relies on asymptotic properties of this index of dispersion, which in turn have been analyzed for time series stemming from a compound Poisson (Poisson‐stopped sum) INAR(1) model. This approach is extended to the popular Poisson INARCH(1) model, which exhibits unconditional overdispersion but has an (equidispersed) conditional Poisson distribution. The asymptotic distribution of the index of dispersion if applied to time series stemming from such a model is derived. These results allow us to investigate the ability of the dispersion test to discriminate between Poisson INAR(1) and INARCH(1) models. Furthermore, the question is considered if the index of dispersion could be used to test the null of a Poisson INARCH(1) model against the alternative of an INARCH(1) model with additional conditional overdispersion.     

Description of purchase incidence by multivariate heterogeneous Poisson processes

In this paper the application of bivariate Poisson heterogeneous models to budget data is studied. This study was motivated from inconsistencies that we encountered when univariate Poisson based models were applied to cumulative data sets. Application of a multivariate Poisson based model is a possible solution to this problem. In this paper we will study the feasibility of estimators based on these models.     

The median of the poisson distribution

The purpose of this paper is twofold: first, to provide a closed form expression for the median of the Poisson distribution and, second, to improve the known estimates of the difference between the median and the mean of the Poisson distribution. We use elementary techniques based on the monotonicity of certain sequences involving tail probabilities of the Poisson distribution and the Central Limit TheoremReceived November 2002     

Opmerkingen en berekeningen betreffende "Poisson-proces met interrupties" van J. C. A. Zaat

Summary
In an earlier paper by J.C. A. ZAAT an interrupted Poisson process was studied. Here we clarify some of the results there obtained. The process is obtained from a stationary Poisson process on the real axis by covering the axis with a sequence of adjoining intervals which have alternatively length a and b, the first left-hand endpoint of an a-interval to the right of 0 being chosen in a stochastic point with If has a rectangular distribution on [0, a + b] the interrupted Poisson process is stationary, but the distribution of the length of an interval between two successive points in the interrupted process is now different for different intervals. For each distribution ofy over [0, a + b] the distribution of the length of the interval between the nth and the (n + I)st point to the right ofO in the interrupted Poisson process tends to the distribution function G(y) as n tends to infinity. Somehow ZAAT based his calculations exclusively on this stationary distribution.     

Investigating the impact of excess zeros on hurdle‐generalized Poisson regression model with right censored count data

Typically, a Poisson model is assumed for count data. In many cases, there are many zeros in the dependent variable, thus the mean is not equal to the variance value of the dependent variable. Therefore, Poisson model is not suitable anymore for this kind of data because of too many zeros. Thus, we suggest using a hurdle‐generalized Poisson regression model. Furthermore, the response variable in such cases is censored for some values because of some big values. A censored hurdle‐generalized Poisson regression model is introduced on count data with many zeros in this paper. The estimation of regression parameters using the maximum likelihood method is discussed and the goodness‐of‐fit for the regression model is examined. An example and a simulation will be used to illustrate the effects of right censoring on the parameter estimation and their standard errors.     

On a study of the exponential and Poisson characteristics of the Poisson process

Based on the exponential and Poisson characteristics of the Poisson process, in this work we present some characterizations of the Poisson process as a renewal process. More precisely, let γ_t be the residual life at time t of the renewal process A={A(t),t≥0 }, under suitable condition, we prove that if Var(γ_t)=E ² (γ_t),∀t≥0, then A is a Poisson process. Secondly, we show that if Var (A(t)) is proportional to E (A(t)), then A is a Poisson process also, and Var (A(t))=E (A(t)). Received: August 1999     

Quick consistency of quasi-maximum likelihood estimators in a multivariate poisson process

Recent results by the present author on quick consistency of quasi-maximum likelihood estimators of parameters in a multivariate Poisson process are strengthened at the cost of replacing the inverse continuity assumption with the strong uniform identifiability condition.     

Testing for the bivariate Poisson distribution

This paper studies goodness-of-fit tests for the bivariate Poisson distribution. Specifically, we propose and study several Cramér–von Mises type tests based on the empirical probability generating function. They are consistent against fixed alternatives for adequate choices of the weight function involved in their definition. They are also able to detect local alternatives converging to the null at a certain rate. The bootstrap can be used to consistently estimate the null distribution of the test statistics. A simulation study investigates the goodness of the bootstrap approximation and compares their powers for finite sample sizes. Extensions for testing goodness-of-fit for the multivariate Poisson distribution are also discussed.     

RECENT DEVELOPMENTS IN COUNT DATA MODELLING: THEORY AND APPLICATION

Abstract. This paper deals with statistical methods for modelling individual behavior when the endogenous variable is a nonnegative integer. Examples are the number of children, the number of job changes or the number of shopping trips in a given period. Several approaches—Poisson, robust Poisson, negative binomial (NEGBIN), NEGBIN _k , hurdle Poisson, truncated-at-zero Poisson—are discussed with a focus on specification, estimation, and testing. An application to labor mobility data illustrates the gain obtained by carefully taking into account the specific structure of the data.     

A local maximum likelihood estimator for Poisson regression

A local maximum likelihood estimator based on Poisson regression is presented as well as its bias, variance and asymptotic distribution. This semiparametric estimator is intended to be an alternative to the Poisson, negative binomial and zero-inflated Poisson regression models that does not depend on regularity conditions and model specification accuracy. Some simulation results are presented. The use of the local maximum likelihood procedure is illustrated on one example from the literature. This procedure is found to perform well. This research was partially supported by Calouste Gulbenkian Foundation and PRODEP III.     

高速公路收费站车辆到达分布的拟合检验

以1分钟为单位时间间隔,以1小时为1个样本观测时段,通过实地调查获取了柳忠高速公路天水北路收费站连续13个小时的车辆到达数据。在此基础上,对高速公路收费站的车辆到达过程服从泊松分布这一假设进行了χ2拟合检验,检验结果表明:高速公路收费站的车辆到达过程服从泊松分布。     

浅析二项分布、泊松分布和正态分布之间的关系

二项分布、泊松分布和正态分布一直是学习和研究概率统计的基础。在一定条件下,这三个分布之间存在着密切关系。文章通过求极限分布,研究了二项分布与泊松分布、二项分布与正态分布之间的关系,并利用特征函数和分布函数相互唯一确定这一性质,分析了泊松分布和正态分布之间的关系。     

An axiomatic foundation for poisson frequency analyses of weakly interacting populations

An axiomatic foundation is developed for Poisson frequency analyses of population processes. In particular it is shown that for a wide range of population processes, Poisson frequencies can be characterized in terms of two simple independence axioms: (i) independence of the individual states of population members for any given population size, and (ii) independence of the population frequencies within any given partition of states. Hence, for all independent population processes satisfying these two conditions, one must necessarily employ the Poisson distribution for all frequency analyses. More generally, it is shown that for a variety of weakly interacting populations in which these conditions are approximately satisfied, the Poisson distribution continues to provide a natural framework for analysis. In addition, it is shown that this notion of independent population processes yields a new characterization of classical spatial Poisson processes. Finally, these results are applied to a specific class of population processes involving spatial flow phenomena.     

A multivariate Poisson mixture model for marketing applications

This paper describes a multivariate Poisson mixture model for clustering supermarket shoppers based on their purchase frequency in a set of product categories. The multivariate nature of the model accounts for cross-selling effects between the purchases made in different product categories. However, for computational reasons, most multivariate approaches limit the covariance structure by including just one common interaction term, or by not including any covariance at all. Although this reduces the number of parameters significantly, it is often too simplistic as typically multiple interactions exist on different levels. This paper proposes a theoretically more complete variance/covariance structure of the multivariate Poisson model, based on domain knowledge or preliminary statistical analysis of significant purchase interaction effects in the data. Consequently, the model does not contain more parameters than necessary, whilst still accounting for the existing covariance in the data. Practically, retail category managers can use the model to devise customized merchandising strategies.     

Estimating Nested Count Data Models

Count data models have found a wide variety of applications not only in applied economics and finance but also in diverse fields ranging from biometrics to political science. Poisson and negative binomial (NB) models have been extensively used in count data analysis. Two particular NB model specifications, NBI and NBII, have been especially popular. However, these models impose arbitrary restrictions on the relation between the conditional mean and variance of the dependent variable, limiting their generality. This study proposes tests for selection among the Poisson and NB models by formally demonstrating that the log likelihood function (LLF) of a general NB model parametrically nests the LLF of the Poisson, NBI and NBII as testable special cases. It also proposes estimation of the general NB model since it allows greater flexibility in the relationship between the mean and variance of the dependent variable than NBI and NBII. The empirical application, which uses micro-level data on recreational boating, provides support for the paper's main theme. Tests clearly reject not only the Poisson, but also NBI and NBII, in favour of a different NB model, underscoring the importance of the general model specification.     

Testing the compounding structure of the CP-INARCH model

A statistical test to distinguish between a Poisson INARCH model and a Compound Poisson INARCH model is proposed, based on the form of the probability generating function of the compounding distribution of the conditional law of the model. For first-order autoregression, the normality of the test statistics’ asymptotic distribution is established, either in the case where the model parameters are specified, or when such parameters are consistently estimated. As the test statistics’ law involves the moments of inverse conditional means of the Compound Poisson INARCH process, the analysis of their existence and calculation is performed by two approaches. For higher-order autoregressions, we use a bootstrap implementation of the test. A simulation study illustrating the finite-sample performance of this test methodology in what concerns its size and power concludes the paper.     

On the equivalence of location choice models: Conditional logit, nested logit and Poisson

It is well understood that the two most popular empirical models of location choice - conditional logit and Poisson - return identical coefficient estimates when the regressors are not individual specific. We show that these two models differ starkly in terms of their implied predictions. The conditional logit model represents a zero-sum world, in which one region’s gain is the other regions’ loss. In contrast, the Poisson model implies a positive-sum economy, in which one region’s gain is no other region’s loss. We also show that all intermediate cases can be represented as a nested logit model with a single outside option. The nested logit turns out to be a linear combination of the conditional logit and Poisson models. Conditional logit and Poisson elasticities mark the polar cases and can therefore serve as boundary values in applied research.