COM-Poisson Distribution Research Articles

Unifying mortality forecasting model: an investigation of the COM–Poisson distribution in the GAS model for improved projections

Forecasting mortality rates is crucial for evaluating life insurance company solvency, especially amid disruptions caused by phenomena like COVID-19. The Lee–Carter model is commonly employed in mortality modelling; however, extensions that can encompass count data with diverse distributions, such as the Generalized Autoregressive Score (GAS) model utilizing the COM–Poisson distribution, exhibit potential for enhancing time-to-event forecasting accuracy. Using mortality data from 29 countries, this research evaluates various distributions and determines that the COM–Poisson model surpasses the Poisson, binomial, and negative binomial distributions in forecasting mortality rates. The one-step forecasting capability of the GAS model offers distinct advantages, while the COM–Poisson distribution demonstrates enhanced flexibility and versatility by accommodating various distributions, including Poisson and negative binomial. Ultimately, the study determines that the COM–Poisson GAS model is an effective instrument for examining time series data on mortality rates, particularly when facing time-varying parameters and non-conventional data distributions.

Conway-Maxwell-Poisson profile monitoring with rk-Shewhart control chart: a comparative study

A control chart is an essential tool in Statistical Quality Control for monitoring the production process. It provides a visual means of identifying process irregularities. In this study, we focus on the Shewhart control chart based on the rk-deviance residuals, namely rk-Shewhart control charts to examine the Conway–Maxwell–Poisson (COM-Poisson) profile, which is used to model the count data with varying degrees of dispersion. The primary goal of this study is to identify the biasing parameter that produces the best result among newly presented biasing parameters developed based on existing ones. It provides a short overview of the COM-Poisson distribution, its modeling, and rk parameter estimation in the case of multicollinearity, as well as the construction of the deviance-residual-based Shewhart chart. To evaluate the performance of the rk-Shewhart, we conduct an analysis using a real-life data set, considering various shift sizes. By employing different biasing parameters, we examine the effectiveness of the rk-Shewhart control chart. The performance evaluation outcomes of the rk-Shewhart charts are compared to the ML-deviance-based Shewhart chart and within themselves based on the biasing parameters. The results demonstrate the advantage of the rk-Shewhart charts over the ML-deviance-based control chart in detecting out-of-control signals. Among the considered biasing parameters, the rk-Shewhart chart utilizing the adjusted biasing parameter k_4 shows the best performance based on the ARL metric.

Multivariate Generalized Linear Mixed Models for Count Data

Univariate regression models have rich literature for counting data. However, this is not the case for multivariate count data. Therefore, we present the Multivariate Generalized Linear Mixed Models framework that deals with a multivariate set of responses, measuring the correlation between them through random effects that follows a multivariate normal distribution. This model is based on a GLMM with a random intercept and the estimation process remains the same as a standard GLMM with random effects integrated out via Laplace approximation. We efficiently implemented this model through the TMB package available in R. We used Poisson, negative binomial (NB), and COM-Poisson distributions. To assess the estimator properties, we conducted a simulation study considering four different sample sizes and three different correlation values for each distribution. We achieved unbiased and consistent estimators for Poisson and NB distributions; for COM-Poisson estimators were consistent, but biased, especially for dispersion, variance, and correlation parameter estimators. These models were applied to two datasets. The first concerns a sample from 30 different sites collected in Australia where the number of times each one of the 41 different ant species was registered; which results in an impressive 820 variance-covariance and 41 dispersion parameters are estimated simultaneously, let alone the regression parameters. The second is from the Australia Health Survey with 5 response variables and 5190 respondents. These datasets can be considered overdispersed by the generalized dispersion index. The COM-Poisson model overcame the other two competitors considering three goodness-of-fit indexes, AIC, BIC, and maximized log-likelihood values. As a result, it estimated parameters with smaller standard errors and a greater number of significant correlation coefficients. Therefore, the proposed model is capable of dealing with multivariate count data, either under- equi- or overdispersed responses, and measuring any kind of correlation between them taking into account the effects of the covariates.

Counting models for overdispersed data: A review with application to tuberculosis data

The present work reviews distributions for counting data: Poisson; Negative Binomial; COM-Poisson and Generalized Poisson, and their regression models. Aspects such as parameter estimation and model choice criteria are presented. And as an application example, we use the regression models of these distributions to explain the relationship between tuberculosis notifications with the HDI Human Development Index of the 102 cities in the state of Alagoas. The existing relationship between notifications of tuberculosis with HDI is significant and overdispersion at the level α = 5% of probability, and the COM-Poisson distribution regression model was the best fit data, according to the Akaike AIC and Bayesian BIC information criteria.

A Family of Finite Mixture Distributions for Modelling Dispersion in Count Data

This paper considers the construction of a family of discrete distributions with the flexibility to cater for under-, equi- and over-dispersion in count data using a finite mixture model based on standard distributions. We are motivated to introduce this family because its simple finite mixture structure adds flexibility and facilitates application and use in analysis. The family of distributions is exemplified using a mixture of negative binomial and shifted negative binomial distributions. Some basic and probabilistic properties are derived. We perform hypothesis testing for equi-dispersion and simulation studies of their power and consider parameter estimation via maximum likelihood and probability-generating-function-based methods. The utility of the distributions is illustrated via their application to real biological data sets exhibiting under-, equi- and over-dispersion. It is shown that the distribution fits better than the well-known generalized Poisson and COM–Poisson distributions for handling under-, equi- and over-dispersion in count data.

An alternative hyper-Poisson integer-valued GARCH model with application to polio, internet protocol and COVID-19 data

<abstract> <p>Time series of counts are observed widely in actuarial science, finance, epidemiology and biology. These time series may exhibit over-, equi- and under-dispersion. The Poisson distribution is commonly used in count time series models, but it is restricted by the equality of mean and variance. Other distributions such as the generalized Poisson, double Poisson, hyper-Poisson, and COM-Poisson distributions have been proposed to replace the Poisson distribution to model the different levels of dispersion in time series of counts. These models have certain limitations such as complex expressions for the mean and variance which complicate the formulation as GARCH models. In this study, we propose an alternative hyper-Poisson (AHP) distribution, with simple forms of conditional mean and variance, for an integer-valued GARCH (INGARCH) model for time series of counts that also exhibit the different levels of dispersion. We demonstrate that the AHP-INGARCH model is comparable to some existing INGARCH models. Additionally, the model can cover a wider range of dispersion. The maximum likelihood estimation can be used to estimate the parameters of the proposed model. Applications to three real-life data sets related to polio, internet protocol and daily COVID-19 new deaths underscore the usefulness of the proposed model in studying both over-dispersed and under-dispersed time series of counts.</p> </abstract>

New statistical process control charts for overdispersed count data based on the Bell distribution.

Poisson distribution is a popular discrete model used to describe counting information, from which traditional control charts involving count data, such as the c and u charts, have been established in the literature. However, several studies recognize the need for alternative control charts that allow for data overdispersion, which can be encountered in many fields, including ecology, healthcare, industry, and others. The Bell distribution, recently proposed by Castellares et al. (2018), is a particular solution of a multiple Poisson process able to accommodate overdispersed data. It can be used as an alternative to the usual Poisson (which, although not nested in the Bell family, is approached for small values of the Bell distribution) Poisson, negative binomial, and COM-Poisson distributions for modeling count data in several areas. In this paper, we consider the Bell distribution to introduce two new exciting, and useful statistical control charts for counting processes, which are capable of monitoring count data with overdispersion. The performance of the so-called Bell charts, namely Bell-c and Bell-u charts, is evaluated by the average run length in numerical simulation. Some artificial and real data sets are used to illustrate the applicability of the proposed control charts.

Regression-based imputation of explanatory discrete missing data

Imputation of missing values is a strategy for handling non-responses in surveys or data loss in measurement processes, which may be more effective than ignoring the losses and omitting them. The characteristics of variables presenting missing values must be considered when choosing the imputation method to be used; in particular when the variable is a count the literature dealing with this issue is scarce. If the variable has an excess of zeros it is necessary to consider models including parameters for handling zero-inflation. Likewise, if problems of over- or under-dispersion are observed, generalizations of the Poisson, such as the Hermite or Conway–Maxwell Poisson distributions are recommended for carrying out imputation. The aim of this study was to assess the performance of various regression models in the imputation of a discrete variable based on Poisson generalizations, in comparison with classical counting models, through a comprehensive simulation study considering a variety of scenarios and a real data example. To do so we compared the results of estimations using only complete data, and using imputations based on the most common count models. The COMPoisson distribution provides in general better results in any dispersion scenario, especially when the amount of missing information is large.

Multivariate Conway-Maxwell-Poisson Distribution: Sarmanov Method and Doubly Intractable Bayesian Inference

ABSTRACT In this article, a multivariate count distribution with Conway-Maxwell (COM)-Poisson marginals is proposed. To do this, we develop a modification of the Sarmanov method for constructing multivariate distributions. Our multivariate COM-Poisson (MultCOMP) model has desirable features such as (i) it admits a flexible covariance matrix allowing for both negative and positive nondiagonal entries; (ii) it overcomes the limitation of the existing bivariate COM-Poisson distributions in the literature that do not have COM-Poisson marginals; (iii) it allows for the analysis of multivariate counts and is not just limited to bivariate counts. Inferential challenges are presented by the likelihood specification as it depends on a number of intractable normalizing constants involving the model parameters. These obstacles motivate us to propose Bayesian inferential approaches where the resulting doubly intractable posterior is handled with via the noisy exchange algorithm or the Grouped Independence Metropolis–Hastings algorithm. Numerical experiments based on simulations are presented to illustrate the proposed Bayesian approach. We demonstrate the potential of the MultCOMP model through a real data application on the numbers of goals scored by the home and away teams in the English Premier League from 2018 to 2021. Here, our interest is to assess the effect of a lack of crowds during the COVID-19 pandemic on the well-known home team advantage. A MultCOMP model fit shows that there is evidence of a decreased number of goals scored by the home team, not accompanied by a reduced score from the opponent. Hence, our analysis suggests a smaller home team advantage in the absence of crowds, which agrees with the opinion of several football experts. Supplementary materials for this article are available online.

A Conditional Maximum Likelihood Estimation of the COM-Poisson Distribution and its Uniqueness and Existence

A Conditional Maximum Likelihood Estimation of the COM-Poisson Distribution and its Uniqueness and Existence

Development of New Performance Measures Based on Data Mining Weights for Hotspot Identification

In this study, new performance measures are proposed for hotspot identification in urban intersections. These measures reflect severity factor weights, which are determined based on data mining. To estimate the severity factor weights of crashes at urban intersections, the study utilizes tree-based random forest (RF) and extreme gradient boosting (XGB) methods. The importance of variables in the severity classification model is standardized and utilized for calculating the score of each crash, which is aggregated into intersections. The aggregated score is used as a dependent variable for the safety performance functions (SPFs) in the network screening process. To illustrate the under-dispersed severity score aggregation data, SPFs that follow the COM-Poisson distribution as well as the negative binomial (NB) are developed. Independent variables in SPFs set up intersection geometry elements that can be collected from online GIS services. Four additional performance measures are proposed, each reflecting a severity weight. Data about a total of 42,513 intersection crashes from 2017 to 2018 in South Korea were collected for crash injury severity analysis. Hotspot identification was performed on 81 intersections, and three consistency tests were conducted to validate the four measures. Tests show that the RF-based weighted [Formula: see text] and [Formula: see text] have the best consistency. Since the severity factor weights of each crash are reflected, intersections vulnerable to dangerous crashes can be analyzed in more detail. Using this method, it is expected that effective safety improvement project plans can be established with the input of safety managers in the future.

Piecewise linear approximations of baseline under proportional hazards based COM-Poisson cure models

Cure models are widely popular in modeling time-to-event data that are characterized by cure fraction owing to long-term survivors. Mixture cure models are perhaps the most studied cure models in the literature. In this article, however, we consider a competing risks scenario where the number of competing causes is modeled by flexible Conway-Maxwell (COM) Poisson distribution, and the lifetimes corresponding to the competing causes are assumed to be independently distributed following proportional hazards model. The baseline hazard function is modeled by a piecewise linear function, and hence, estimated non parametrically. Probability of obtaining zero competing causes is used to estimate the cure rate. Collectively, the resultant cure model is exceedingly general and flexible. The estimation of the parameters is carried out using maximum likelihood (ML) method by implementing the expectation-maximization (EM) algorithm, except for the dispersion parameter of the COM-Poisson distribution, which is estimated by the profile likelihood method. The performance of the model is tested under various settings of censoring rate, sample size and mean lifetime. Discrimination of models is performed and carried out with likelihood-based and information-based criteria. Performance of the proposed model is further illustrated using a real-world data on cutaneous melanoma.

Bayesian Inference for Spatial Count Data that May be Over-Dispersed or Under-Dispersed with Application to the 2016 US Presidential Election

We propose a method of spatial prediction using count data that can be reasonably modeled assuming the Conway-Maxwell Poisson distribution (COM-Poisson). The COM-Poisson model is a two parameter generalization of the Poisson distribution that allows for the flexibility needed to model count data that are either over or under-dispersed. The computationally limiting factor of the COM-Poisson distribution is that the likelihood function contains multiple intractable normalizing constants and is not always feasible when using Markov Chain Monte Carlo (MCMC) techniques. Thus, we develop a prior distribution of the parameters associated with the COM-Poisson that avoids the intractable normalizing constant. Also, allowing for spatial random effects induces additional variability that makes it unclear if a spatially correlated Conway-Maxwell Poisson random variable is over or under-dispersed. We propose a computationally efficient hierarchical Bayesian model that addresses these issues. In particular, in our model, the parameters associated with the COM-Poisson do not include spatial random effects (leading to additional variability that changes the dispersion properties of the data), and are then spatially smoothed in subsequent levels of the Bayesian hierarchical model. Furthermore, the spatially smoothed parameters have a simple regression interpretation that facilitates computation. We demonstrate the applicability of our approach using simulated examples, and a motivating application using 2016 US presidential election voting data in the state of Florida obtained from the Florida Division of Elections.

Monitoring circuit boards products in the presence of indeterminacy

In this article a repetitive group sampling control has been introduced for the neutrosophic statistics under the Conway-Maxwell-Poisson (COM-Poisson) distribution. The suggested chart has been compared with the existing plan using simulated data generated from neutrosophic COM-Poisson distribution. The practical implementation of the suggested chart has also been expounded using the data from the manufacturing of the electric circuit boards. Overall, the results demonstrate that the suggested chart will be a proficient addition in the control chart literature. It is also observed that the suggested chart is an ideal chart when applied under appropriate conditions.

Bayesian Inference, Model Selection and Likelihood Estimation using Fast Rejection Sampling: The Conway-Maxwell-Poisson Distribution

Bayesian inference for models with intractable likelihood functions represents a challenging suite of problems in modern statistics. In this work we analyse the Conway-Maxwell-Poisson (COM-Poisson) distribution, a two parameter generalisation of the Poisson distribution. COM-Poisson regression modelling allows the flexibility to model dispersed count data as part of a generalised linear model (GLM) with a COM-Poisson response, where exogenous covariates control the mean and dispersion level of the response. The major difficulty with COM-Poisson regression is that the likelihood function contains multiple intractable normalising constants and is not amenable to standard inference and Markov Chain Monte Carlo (MCMC) techniques. Recent work by Chanialidis et al. (2018) has seen the development of a sampler to draw random variates from the COM-Poisson likelihood using a rejection sampling algorithm. We provide a new rejection sampler for the COM-Poisson distribution which significantly reduces the central processing unit (CPU) time required to perform inference for COM-Poisson regression models. An extension of this work shows that for any intractable likelihood function with an associated rejection sampler it is possible to construct unbiased estimators of the intractable likelihood which proves useful for model selection or for use within pseudo-marginal MCMC algorithms (Andrieu and Roberts, 2009). We demonstrate all of these methods on a real-world dataset of takeover bids.

Mixed EWMA–CUSUM chart for COM-Poisson distribution

In this article, we develop a mixed EWMA-CUSUM control chart for monitoring moderate and/or small shift in the process following the Conway Maxwell Poisson (COM-Poisson) distribution. To determine the ability and performance of the proposed mixed EWMA-CUSUM control chart, we evaluate the average run lengths and compare with different control charts. We will compare the efficiency of the proposed control chart over the existing control charts in terms of average run length. The results show an improvement in ARL when compared with the existing charts. A real example and a simulation study is also added to use the chart in the industry.

Parameterization of the COM-Poisson Distribution through the Mean Using Spectral Algorithms for Solving Nonlinear Equations

The Poisson regression is popularly used to model count data. However, real data often do not satisfy the assumption of equality of the mean and variance which is an important property of the Poisson distribution. The Poisson – Gamma (Negative binomial) distribution and the recent Conway-Maxwell-Poisson (COM-Poisson) distributions are some of the proposed models for over- and under-dispersion respectively. Nevertheless, the parameterization of the COM-Poisson distribution still remains a major challenge in practice as the location parameter of the original COM-Poisson distribution rarely represents the mean of the distribution. As a result, this paper proposes a new parameterization of the COM-Poisson distribution via the central location (mean) so that more easily-interpretable models and results can be obtained. The parameterization involves solving nonlinear equations which do not have analytical solutions. The nonlinear equations are solved using the efficient and fast derivative free spectral algorithm. Implementation of the parameterization in R (R Core Team, 2018) is used to present useful numerical results concerning the relationship between the mean of the COM-Poisson distribution and the location parameter in the original COM-Poisson parameterization. The proposed technique is further used to fit COM-Poisson probability models to real life datasets. It was found that obtaining estimates via this parameterization makes the estimation easier and faster compared to directly maximizing the likelihood function of the standard COM-Poisson distribution.

On some inferential issues for the destructive COM-Poisson-generalized gamma regression cure rate model

In this paper, we assume the initial number of malignant cells to undergo a destructive process and hence what we record is only from the undamaged portion of the original number of malignant cells. This gives a realistic and practical interpretation of the biological mechanism of the occurrence of a cancerous tumor. We propose to model the initial number of malignant cells by the flexible Conway-Maxwell (COM) Poisson distribution, which includes some of the commonly used discrete distributions as its particular cases. Furthermore, we propose to model the time taken by each active malignant cell after an initial treatment by the wider class of generalized gamma distribution, which includes some of the commonly used lifetime distributions as its particular cases. The main contribution is in developing the likelihood inference based on the expectation maximization algorithm for such a flexible and general destructive cure rate model. An extensive simulation study is carried out to demonstrate the performance of the proposed estimation method. The flexibilities of the COM-Poisson distribution and the generalized gamma distribution are also utilized to carry out a two-way model discrimination using some likelihood-based methods. Finally, a melanoma dataset is analyzed for illustrative purposes.

Reparametrization of COM–Poisson regression models with applications in the analysis of experimental data

The COM–Poisson distribution is a two-parameter generalization of the Poisson distribution that can deal with under-, equi- and overdispersed count data. Unfortunately, its location parameter does not correspond to the expectation, which complicates the parameter interpretation. In this article, we propose a straightforward reparametrization of the COM–Poisson distribution based on an approximation to the expectation. Estimation and inference are done using the likelihood paradigm. Simulation studies show that the maximum likelihood estimators are unbiased and consistent for both regression and dispersion parameters. In addition, the nature of the deviance surfaces suggests that these parameters are also orthogonal for most of the parameter space, which is advantageous for interpretation, inference and computational efficiency. Study of the distribution’s properties, through a consideration of dispersion, zero-inflation and heavy tail indexes, together with the results of data analyses show the flexibility over standard approaches. The computational routines and datasets are available in the supplementary material.

A Conway Maxwell Poisson type generalization of the negative hypergeometric distribution

Negative hypergeometric distribution arises as a waiting time distribution when we sample without replacement from a finite population. It has applications in many areas such as inspection sampling and estimation of wildlife populations. However, as is well known, the negative hypergeometric distribution is over-dispersed in the sense that its variance is greater than the mean. To make it more flexible and versatile, we propose a modified version of negative hypergeometric distribution called COM-Negative Hypergeometric distribution (COM-NH) by introducing a shape parameter as in the COM-Poisson and COMP-Binomial distributions. It is shown that under some limiting conditions, COM-NH approaches to a distribution that we call the COM-Negative binomial (COMP-NB), which in turn, approaches to the COM Poisson distribution. For the proposed model, we investigate the dispersion characteristics and shape of the probability mass function for different combinations of parameters. We also develop statistical inference for this model including parameter estimation and hypothesis tests. In particular, we investigate some properties such as bias, MSE, and coverage probabilities of the maximum likelihood estimators for its parameters by Monte Carlo simulation and likelihood ratio test to assess shape parameter of the underlying model. We present illustrative data to provide discussion.