Markov-Correlated Poisson Processes

In this article, we present a simple generalization of the Bernoulli trials model to a Markov chain with an additional parameter that measures dependence. We then formulate a Markov correlated Poisson process which, due to its flexibility, has great potential for analyzing many practical processes including those for long-term survival analysis.     

What is the effective sample size of a spatial point process?

Point process models are a natural approach for modelling data that arise as point events. In the case of Poisson counts, these may be fitted easily as a weighted Poisson regression. Point processes lack the notion of sample size. This is problematic for model selection, because various classical criteria such as the Bayesian information criterion (BIC) are a function of the sample size, n, and are derived in an asymptotic framework where n tends to infinity. In this paper, we develop an asymptotic result for Poisson point process models in which the observed number of point events, m, plays the role that sample size does in the classical regression context. Following from this result, we derive a version of BIC for point process models, and when fitted via penalised likelihood, conditions for the LASSO penalty that ensure consistency in estimation and the oracle property. We discuss challenges extending these results to the wider class of Gibbs models, of which the Poisson point process model is a special case.     

A backward construction and simulation of correlated Poisson processes

In this paper, we consider a generalisation of the backward simulation method of Duch et al. [New approaches to operational risk modeling. IBM J Res Develop. 2014;58:1–9] to build bivariate Poisson processes with flexible time correlation structures, and to simulate the arrival times of the processes. The proposed backward construction approach uses the Marshall–Olkin bivariate binomial distribution for the conditional law and some well-known families of bivariate copulas for the joint success probability in lieu of the typical conditional independence assumption. The resulting bivariate Poisson process can exhibit various time correlation structures which are commonly observed in real data.     

Exponentially weighted moving average charts for correlated multivariate Poisson processes

In this article, we study exponentially weighted moving average (EWMA) control schemes to monitor the multivariate Poisson distribution with a general covariance structure, so that the practitioner can simultaneously monitor multiple correlated attribute processes more effectively. The statistical performance of the charts is assessed in terms of the run length properties and compared against other mainstream attribute control schemes. The application of the proposed methods to real-life and simulated datasets is demonstrated.     

Bridging the Gap: A Generalized Stochastic Process for Count Data

The Bernoulli and Poisson processes are two popular discrete count processes; however, both rely on strict assumptions. We instead propose a generalized homogenous count process (which we name the Conway–Maxwell–Poisson or COM-Poisson process) that not only includes the Bernoulli and Poisson processes as special cases, but also serves as a flexible mechanism to describe count processes that approximate data with over- or under-dispersion. We introduce the process and an associated generalized waiting time distribution with several real-data applications to illustrate its flexibility for a variety of data structures. We consider model estimation under different scenarios of data availability, and assess performance through simulated and real datasets. This new generalized process will enable analysts to better model count processes where data dispersion exists in a more accommodating and flexible manner.     

Prediction in a mixed Poisson cluster model

Motivated by insurance applications, a mixed Poisson cluster model is considered, where the cluster center process is a mixed Poisson process and descendant processes are additive processes. Each point of the center process represents a claim’s reported time and descendant processes are interpreted as processes of the corresponding payments or number of payments. In this study, we focus on the process aggregating all separate claim’s payment processes. Given the past observations, we study prediction of future increments and their mean-squared errors, also revealing the dependency between future increments from non-reported (IBNR) claims and the past available information. In the existing literature, they are independent since models were considered with a purely Poissonian center process. We derive computationally reasonable expressions for predictors and their variances.     

Modeling spatial overdispersion via the generalized Waring point process

Modeling spatial overdispersion requires point process models with finite‐dimensional distributions that are overdisperse relative to the Poisson distribution. Fitting such models usually heavily relies on the properties of stationarity, ergodicity, and orderliness. In addition, although processes based on negative binomial finite‐dimensional distributions have been widely considered, they typically fail to simultaneously satisfy the three required properties for fitting. Indeed, it has been conjectured by Diggle and Milne that no negative binomial model can satisfy all three properties. In light of this, we change perspective and construct a new process based on a different overdisperse count model, namely, the generalized Waring (GW) distribution. While comparably tractable and flexible to negative binomial processes, the GW process is shown to possess all required properties and additionally span the negative binomial and Poisson processes as limiting cases. In this sense, the GW process provides an approximate resolution to the conundrum highlighted by Diggle and Milne.     

A circular test of association between a point process and a continuous time series

A test of association between a point process and a continuous time series is proposed. The test is exact for a general class of point processes, including Poisson processes. Simulation results for a Poisson point process are reported.     

On a score test for pure birth processes and a poisson dispersion test

We consider a generalization of a standard test for overdispersion (underdispersion) of possibly Poison data. Under the null hypothesis observed counts are increments of Poisson processes. Particular applications are toa random sample of identically distributed processes and a single observed process. The test has intuitive appeal beyond the specific alternatives considered.     

Process capability analysis for serially dependent processes of Poisson counts

Process capability indices evaluate the actual compliance of a process with given external specifications in a single number. For the case of a process of independent and identically distributed Poisson counts, two types of index have been proposed and investigated in the literature. The assumption of serial independence, however, is quite unrealistic for practice. We consider the case of an underlying Poisson INAR(1) process which has an AR(1)-like autocorrelation structure. We show that the performance of the estimated indices is degraded heavily if serial dependence is ignored. Therefore, we develop approaches for estimating the process capability (both for the observation and innovation process), which explicitly consider the observed degree of autocorrelation. For this purpose, we introduce a new unbiased estimator of the innovations’ mean of a Poisson INAR(1) process and derive its exact as well as asymptotic stochastic properties. In this context, we also present new explicit expressions for the third- and fourth-order moments of a Poisson INAR(1) process. Then the capability indices and the performance of their estimators are analysed and recommendations for practice are given.     

On a generalization of Poisson sampling

In real-time sampling, the units of a population pass a sampler one by one. Alternatively the sampler may successively visit the units of the population. Each unit passes only once and at that time it is decided whether or not it should be included in the sample. The goal is to take a sample and efficiently estimate a population parameter. The list sequential sampling method presented here is called correlated Poisson sampling. The method is an alternative to Poisson sampling, where the units are sampled independently with given inclusion probabilities. Correlated Poisson sampling uses weights to create correlations between the inclusion indicators. In that way it is possible to reduce the variation of the sample size and to make the samples more evenly spread over the population. Simulation shows that correlated Poisson sampling improves the efficiency in many cases.     

Estimation of the intensity function of an inhomogeneous Poisson process with a change‐point

Recent work on point processes includes studying posterior convergence rates of estimating a continuous intensity function. In this article, convergence rates for estimating the intensity function and change‐point are derived for the more general case of a piecewise continuous intensity function. We study the problem of estimating the intensity function of an inhomogeneous Poisson process with a change‐point using non‐parametric Bayesian methods. An Markov Chain Monte Carlo (MCMC) algorithm is proposed to obtain estimates of the intensity function and the change‐point which is illustrated using simulation studies and applications. The Canadian Journal of Statistics 47: 604–618; 2019 © 2019 Statistical Society of Canada     

On multiple change-point estimation for Poisson process

This work is devoted to the problem of change-point parameter estimation in the case of the presence of multiple changes in the intensity function of the Poisson process. It is supposed that the observations are independent inhomogeneous Poisson processes with the same intensity function and this intensity function has two jumps separated by a known quantity. The asymptotic behavior of the maximum-likelihood and Bayesian estimators are described. It is shown that these estimators are consistent, have different limit distributions, the moments converge and that the Bayesian estimators are asymptotically efficient. The numerical simulations illustrate the obtained results.     

Prediction Performance of Correlated Error Procedures for Spatial Counts

Abstract

Few guidelines exist for the application of geostatistical methods to spatial counts and the prediction to unsampled areas is an important aspect of experimental field research. The prediction performances of kriging and a correlated errors Poisson model are compared through simulation. Counts with a known spatial covariance structure are generated in an investigation involving several factors: area size, overall mean, range of correlation, spatial covariance function, and the presence of trend. The correlated errors Poisson model generally gives superior prediction performance when an exponential covariance structure is used.     

A weighted Poisson distribution and its application to cure rate models

The family of weighted Poisson distributions offers great flexibility in modeling discrete data due to its potential to capture over/under-dispersion by an appropriate selection of the weight function. In this paper, we introduce a flexible weighted Poisson distribution and further study its properties by using it in the context of cure rate modeling under a competing cause scenario. A special case of the new distribution is the COM-Poisson distribution which in turn encompasses the Bernoulli, Poisson, and geometric distributions; hence, many of the well-studied cure rate models may be seen as special cases of the proposed model. We focus on the estimation, through the maximum likelihood method, of the cured proportion and the properties of the failure time of the susceptibles/non cured individuals; a profile likelihood approach is also adopted for estimating the parameters of the weighted Poisson distribution. A Monte Carlo simulation study demonstrates the accuracy of the proposed inferential method. Finally, as an illustration, we fit the proposed model to a cutaneous melanoma data set.     

Serial dependence and regression of Poisson INARMA models

Time series of counts occur in many fields of practice, with the Poisson distribution as a popular choice for the marginal process distribution. A great variety of serial dependence structures of stationary count processes can be modelled by the INARMA family. In this article, we propose a new approach to the INMA(q) family in general, including previously known results as special cases. In the particular case of Poisson marginals, we will derive new results concerning regression properties and the serial dependence structure of INAR(1) and INMA(q) models. Finally, we present explicit expressions for the distribution of jumps in such processes.     

Estimation of poisson means under weighted squared error loss

In estimating the means of several independent Poisson distributions, we show that the maximum likelihood estimator is inadmissible when general weighted squared error loss is the criterion. Using this result, we extend the known results on estimation of several Poisson means (Peng 1975, Hudson 1978) to the case where possibly more than one observation is taken from each Poisson distribution and the samples are not necessarily of the same size.     

An Immigration-Death Process with a Time-Dependent Delay

In this article, a non homogeneous immigration-death process with delay in the death rate parameter is considered. A process with a time-dependent delay is developed and shown to result in a non homogeneous Poisson process.     

Isotonic estimation of the intensity of a nonhomogeneous Poisson process: The multiple realization setup

In the 1950s Brunk and Van Eeden each obtained maximum-likelihood estimators of a finite product of probability density functions under partial or complete ordering of their parameters. Their results play an important role in the general theory of inference under order restrictions and lead to an isotonic estimator of the intensity of a nonhomogeneous Poisson process. Here an elementary derivation of the maximum likelihood estimator (m.l.e.) for the intensity of a nonhomogeneous Poisson process is given when several (possibly censored) realizations are available. Boswell obtained the m.l.e. based on a single realization as well as a conditional m.l.e. under the same conditions. An example is given to show that in the multirealization setup a conditional m.l.e. may not exist; the proofs are, we believe, new and elementary. An illustrative application is given.