Generalized Poisson Research Articles

Generalized Poisson random variable: its distributional properties and actuarial applications

Abstract Generalized Poisson (GP) distribution was introduced in Consul & Jain ((1973). Technometrics, 15(4), 791–799.). Since then it has found various applications in actuarial science and other areas. In this paper, we focus on the distributional properties of GP and its related distributions. In particular, we study the distributional properties of distributions in the $\mathcal{H}$ family, which includes GP and generalized negative binomial distributions as special cases. We demonstrate that the moment and size-biased transformations of distributions within the $\mathcal{H}$ family remain in the same family, which significantly extends the results presented in Ambagaspitiya & Balakrishnan ((1994). ASTINBulletin: the Journal of the IAA, 24(2), 255–263.) and Ambagaspitiya ((1995). Insurance Mathematics and Economics, 2(16), 107–127.). Such findings enable us to provide recursive formulas for evaluating risk measures, such as Value-at-Risk and conditional tail expectation of the compound GP distributions. In addition, we show that the risk measures can be calculated by making use of transform methods, such as fast Fourier transform. In fact, the transformation method showed a remarkable time advantage over the recursive method. We numerically compare the risk measures of the compound sums when the primary distributions are Poisson and GP. The results illustrate the model risk for the loss frequency distribution.

Modeling Zero-inflated Count Data Using Generalized Poisson and Ordinal Logistic Regression Models in Medical Research.

In medical research, the study's design and statistical methods are pivotal, as they guide interpretation and conclusion. Selecting appropriate statistical models hinges on the distribution of the outcome measure. Count data, frequently used in medical research, often exhibit over-dispersion or zero inflation. Occasionally, count data are considered ordinal (with a maximum outcome value of 5), and this calls for the application of ordinal regression models. Various models exist for analyzing over-dispersed data such as negative binomial, generalized Poisson (GP), and ordinal regression model. This study aims to examine whether the GP model is a superior alternative to the ordinal logistic regression (OLR) model, specifically in the context of zero-inflated Poisson models using both simulated and real-time data. Simulated data were generated with varied estimates of regression coefficients, sample sizes, and various proportions of zeros. The GP and OLR models were compared using fit statistics. Additionally, comparisons were made using real-time datasets. The simulated results consistently revealed lower bias and mean squared error values in the GP model compared to the OLR model. The same trend was observed in real-time datasets, with the GP model consistently demonstrating lower standard errors. Except when the sample size was 1000 and the proportions of zeros were 30% and 40%, the Bayesian information criterion consistently favored the GP model over the OLR model. This study establishes that the proposed GP model offers a more advantageous alternative to the OLR model. Moreover, the GP model facilitates easier modeling and interpretation when compared to the OLR model.

A generalized Poisson model to predict host-seeking female Aedes aegypti marked by dusted Metarhizium anisopliae-exposed males.

We developed a biological control method directed toward Aedes aegypti using the release of Metarhizium anisopliae-contaminated males to spread the fungus to wild females. A generalized Poisson model was used to relate Ae. aegypti marked females (MKF) to M. anisopliae-exposed males (FEM). In a mark-recapture parallel arm trial, FEM release was a better predictor than unexposed male (UM) releases to forecast MKF by FEM. Total females (TF), marked males (MKM), and wild males (WM) as predictors were counted in human-landings in 15 households treated with 40 FEM each, vs 40 UM released/household/week in 15 households for eight weeks. Fit of MKF to standard, generalized Poisson (GP), and negative binomial models/arm built by TF, MKM, WM, and interactions as predictors were computed. In both arms, MKF was better modeled by GP, which in treated, all but one of the eight observed data fell within the confidence intervals predicted by the model. However, the control GP had two outliers and MKM as a single predictor. Likewise, the pseudo-R2 measures of 95% and 46% for treated and control groups also showed that the GP with FEM was more suitable to predict MKF. It should thus be possible to use the GP model to indirectly estimate that an increase of one TF or one fungus-exposed male would increase the number of marked-females by 8% or 9%, respectively, while wild males were an irrelevant predictor to the model.

Modeling Under-Dispersed Count Data by the Generalized Poisson Distribution via Two New MM Algorithms

Under-dispersed count data often appear in clinical trials, medical studies, demography, actuarial science, ecology, biology, industry and engineering. Although the generalized Poisson (GP) distribution possesses the twin properties of under- and over-dispersion, in the past 50 years, many authors only treat the GP distribution as an alternative to the negative binomial distribution for modeling over-dispersed count data. To our best knowledge, the issues of calculating maximum likelihood estimates (MLEs) of parameters in GP model without covariates and with covariates for the case of under-dispersion were not solved up to now. In this paper, we first develop a new minimization–maximization (MM) algorithm to calculate the MLEs of parameters in the GP distribution with under-dispersion, and then we develop another new MM algorithm to compute the MLEs of the vector of regression coefficients for the GP mean regression model for the case of under-dispersion. Three hypothesis tests (i.e., the likelihood ratio, Wald and score tests) are provided. Some simulations are conducted. The Bangladesh demographic and health surveys dataset is analyzed to illustrate the proposed methods and comparisons with the existing Conway–Maxwell–Poisson regression model are also presented.

A penalized likelihood approach for dealing with separation in count data regression model

Separation or monotone likelihood can be observed in fitting process of both Poisson or generalized Poisson (GP) regression, particularly in case of small and/or sparse count data, using maximum likelihood estimation (MLE) when one or more regression coefficients diverge to infinity. The study investigates the consequence of separation in the MLE based standard Poisson or GP model and addressed the problems by introducing a penalized likelihood approach. The penalized likelihood function is derived by adding a penalty term to the standard maximum likelihood function, which was originally proposed by Firth (1993) for reducing first order bias in MLE. The corresponding penalized likelihood score equation has shown to achieve convergence and provide finite estimate of the regression coefficient, which was not possible for the maximum likelihood method score equation. The simulation study, with different forms of separation, showed that penalized Poisson model or penalized GP outperform the standard Poisson and even the Zero-inflated Poisson (ZIP) in the presence of complete or quasi-complete separation by achieving convergence and providing finite estimate of the regression coefficients. Even in the presence of near-to-quasi-complete separation, which is very common in practice, the penalized method showed better results than the standard Poisson, GP and ZIP in all simulation scenarios. The method was illustrated using antenatal care data extracted from database of the Bangladesh demographic health survey 2018.

Count Regression Models with an Application to Zoological Data Containing Structural Zeros

Recently, count regression models have been used to model over dispersed and zero-inflated count response variable that is affected by one or more covariates. Generalized Poisson (GP) and negative binomial (NB) regression models have been suggested to deal with over-dispersion. Zero inflated count regression models such as the zero-inflated Poisson (ZIP), zero-inflated negative binomial (ZINB) and zero-inflated generalized Pois son (ZIGP) regression models have been used to handle count data with many zeros. The aim of this study is to model the number of C. caretta hatchlings dying from exposure to the sun. We present an evaluation frame work to the suitability of applying the Poisson, NB, GP, ZIP and ZIGP to zoological data set where the count data may exhibit evidence of many zeros and over-dispersion. Estimation of the model parameters using the method of maximum likelihood (ML) is provided. Based on the score test and the goodness of fit measure for zoological data, the GP regression model performs better than other count regression models.

Bayesian regression models for ecological count data in PyMC3

Ecological inferences need structurally flexible statistical models to accommodate complex ecological phenomena. PyMC3 is a Probabilistic Programming Language (PPL) and allows for custom statistical distributions to build complex statistical models. This study used PyMC3 to implement Bayesian generalized Poisson (GP), zero-inflated GP, and hurdle GP regression models for over- and under-dispersed counts. The Bayesian GP regression models were fitted to simulated counts and real-world counts of over- and under-dispersion, respectively. Coefficient estimates of the Bayesian regression models were consistent with the known values used in the simulations and those of published work or models. Simulations demonstrated that Bayesian GP regression models with the NUTS sampler worked correctly for under-dispersed counts if the number of non-zero frequency classes was five or more. PyMC3 is not only flexible for building complex statistical models using custom likelihood functions, but also syntactically concise. The programming flexibility of PyMC3 can provide ecologists and environmental scientists with flexible, robust Bayesian computational platforms.

Comparing of Car-Bym, Generalized Poisson, and Negative Binomial Models on Tuberculosis Data in Banyumas Districs

In 2019 the number of people with TB (Tuberculosis) in Banyumas, Central Java, is high (1,910 people have been detected with TB). The number of people infected Tuberculosis (TB) in Banyumas is the count data and it is also the area data. In modeling, the parameter estimation and characteristic of the data need to be considered. Here, we studied comparing Generalized Poisson (GP), negative binomial (NB), and Poisson and CAR.BYM model for TB cases in Banyumas. Here, we use two methods for parameter estimation, maximum likelihood estimation (MLE) and Bayes. The MLE is used for GP and NB models, whereas Bayes is used for Poisson and CAR-BYM. The results showed that Poisson model detected overdispersion where deviance value is 67.38 for 22 degrees of freedom. Therefore, ratio of deviance to degrees of freedom is 3.06 (>1). This indicates that there was overdispersion. The folowing GP, NB, Poisson-Bayes and CAR-BYM are used to modeling TB data in Banyumas and we compare their RMSE. With refer to RMES criteria, we found that CAR-BYM is the best model for modeling TB in Banyumas because its RMSE is smallest.

Estimation in zero-inflated Generalized Poisson distribution

Overdispersion is a common phenomenon in Poisson modelling. The generalized Poisson (GP) distribution accommodates both overdispersion and under dispersion in count data. In this paper, we briefly overview different overdispersed and zero-inflated regression models. To study the impact of fitting inaccurate model to data simulated from some other model, we simulate data from ZIGP distribution and fit Poisson, Generalized Poisson (GP), Zero-inflated Poisson (ZIP), Zero-inflated Generalized Poisson (ZIGP) and Zero-inflated Negative Binomial (ZINB) model. We compare the performance of the estimates of Poisson, GP, ZIP, ZIGP and ZINB through mean square error, bias and standard error when the samples are generated from ZIGP distribution. We propose estimators of parameters of ZIGP distribution based on the first two sample moments and proportion of zeros referred to as MOZE estimator and compare its performance with maximum likelihood estimate (MLE) through a simulation study. It is observed that MOZE are almost equal or even more efficient than that of MLE of the parameters of ZIGP distribution.

Estimation of safety performance functions for urban intersections using various functional forms of the negative binomial regression model and a generalized Poisson regression model

Intersections are established dangerous entities of a highway system due to the challenging and unsafe roadway environment they are characterized for drivers and other road users. In efforts to improve safety, an enormous interest has been shown in developing statistical models for intersection crash prediction and explanation. The selection of an adequate form of the statistical model is of great importance for the accurate estimation of crash frequency and the correct identification of crash contributing factors. Using a six-year crash data, road infrastructure and geometric design data, and traffic flow data of urban intersections, we applied three different functional forms of negative binomial models (i.e., NB-1, NB-2, NB-P) and a generalized Poisson (GP) model to develop safety performance functions (SPF) by crash severity for signalized and unsignalized intersections. This paper presents the relationships found between the explanatory variables and the expected crash frequency. It reports the comparison of different models for total, injury & fatal, and property damage only crashes in order to obtain ones with the maximum estimation accuracy. The comparison of models was based on the goodness of fit and the prediction performance measures.The fitted models showed that the traffic flow and several variables related to road infrastructure and geometric design significantly influence the intersection crash frequency. Further, the goodness of fit and the prediction performance measures revealed that the NB-P model outperformed other models in most crash severity levels for signalized intersections. For the unsignalized intersections, the GP model was the best performing model. When only the NB models were compared, the functional form NB-P performed better than the traditional NB-1 and, more specifically, the NB-2 models. In conclusion, our findings suggest a potential improvement in the estimation accuracy of the SPFs for urban intersections by applying the NB-P and GP models.

Why European biodiversity reporting is not reliable.

The Convention on Biological Diversity (CBD) aims to end the loss of biodiversity, which is one of the greatest ecological challenges of our time. The lack of success in biodiversity policy implementation is partly related to gaps in biodiversity monitoring. Our overall objective is to contribute to the preparation of the upcoming post 2020 period by a review of biodiversity indicator choices in European CBD reports and hence in national monitoring systems. Negative binary generalized models and poisson generalized linear models prove that through free indicator choice in CBD reporting, countries do not choose biodiversity indicators according to their national geographic and socioeconomic characteristics. Moreover, species and ecosystem diversity indicators were chosen with a disproportionate frequency compared to that of genetic diversity indicators. Consequently, trends derived from national CBD reports and monitoring systems in Europe are not reliable, which should be an alarming signal concerning biodiversity policy implementation. Finally, a flow chart to revise national biodiversity monitoring systems is proposed.

REGRESI GENERALIZED POISSON UNTUK MEMODELKAN JUMLAH PENDERITA GIZI BURUK PADA BALITA DI SURABAYA

The expansion of Poisson regression model which is used to solve the underdispersion data or overdispersion data known as Generalized Poisson (GP) regression model. The purpose of this final project is getting the parameter estimator of generalized linear model with response for GP distribution using maximum likelihood. This GP regression model can be applied on the data of number of Marasmus Kwashiorkorpatients in 25 subdistrict in Surabaya city in 2010. The variable response is the number of Marasmus Kwashiorkor patients, where as the predictor responses are the number of people who married at early age , the number of family heads who not graduated elementary school, the number of children who participated posyandu, the number of medical , the number of visits BKIA, and the number of poor population . The result of the GP regression model with statistic test can be concluded that the number of Marasmus Kwashiorkor patientsaffected by the number of visits BKIA and education levels of parents.The expansion of Poisson regression model which is used to solve the underdispersion data or overdispersion data known as Generalized Poisson (GP) regression model. The purpose of this final project is getting the parameter estimator of generalized linear model with response for GP distribution using maximum likelihood. This GP regression model can be applied on the data of number of Marasmus Kwashiorkorpatients in 25 subdistrict in Surabaya city in 2010. The variable response is the number of Marasmus Kwashiorkor patients, where as the predictor responses are the number of people who married at early age , the number of family heads who not graduated elementary school, the number of children who participated posyandu, the number of medical , the number of visits BKIA, and the number of poor population . The result of the GP regression model with statistic test can be concluded that the number of Marasmus Kwashiorkor patientsaffected by the number of visits BKIA and education levels of parents.

Statistical Models of Overdispersed Spatial Defects for Predicting the Yield of Integrated Circuits

Defects generated in semiconductor manufacturing processes have serious effects on the yield of integrated circuits (ICs). Accurate modeling of the defect counts on IC chips is crucial for predicting the yield. The conventional Poisson yield model tends to underestimate the true yield by ignoring overdispersed patterns of defects on the wafer. This article uses various models based on the generalized Poisson (GP) distribution and/or HZ distributions to explore the overdispersed defect counts on semiconductor wafers. Real wafer map data are used to compare the performance of both nonregression and regression modeling approaches in terms of the log-likelihood, AIC, and relative bias for yield estimation. Analytical results indicate that the GP distribution is a competitive alternative to the negative binomial (NB) distribution for modeling defect counts on IC chips because the GP distribution can model overdispersion, underdispersion, or no dispersion. In particular, HZ models based on the NB and GP distributions show good potential for predicting the yield of IC chips on wafers.

Statistical modeling of patterns in annual reproductive rates.

Reproduction by individuals is typically recorded as count data (e.g., number of fledglings from a nest or inflorescences on a plant) and commonly modeled using Poisson or negative binomial distributions, which assume that variance is greater than or equal to the mean. However, distributions of reproductive effort are often underdispersed (i.e., variance < mean). When used in hypothesis tests, models that ignore underdispersion will be overly conservative and may fail to detect significant patterns. Here we show that generalized Poisson (GP) and Conway-Maxwell-Poisson (CMP) distributions are better choices for modeling reproductive effort because they can handle both overdispersion and underdispersion; we provide examples of how ecologists can use GP and CMP distributions in generalized linear models (GLMs) and generalized linear mixed models (GLMMs) to quantify patterns in reproduction. Using a new R package, glmmTMB, we construct GLMMs to investigate how rainfall and population density influence the number of fledglings in the warbler Oreothlypis celata and how flowering rate of Heliconia acuminata differs between fragmented and continuous forest. We also demonstrate how to deal with zero-inflation, which occurs when there are more zeros than expected in the distribution, e.g., due to complete reproductive failure by some individuals.

GPLDA: A Generalized Poisson Latent Dirichlet Topic Model

The earliest modification of Latent Dirichlet Allocation (LDA) in terms of words or document attributes is by relaxing its exchangeability assumption via the Bag-of-word (BoW) matrix. Several authors have proposed many modifications of the original LDA by focusing on model that assumes the current topic depends on the words from previous topic. Most of the earlier work ignored the document length distribution since it is assumed that it will fizzle out at the modelling stage. Thus, in this paper, the Poisson document length distribution of LDA model is replaced with Generalized Poisson (GP) distribution which has the strength of capturing complex structures. The main strengths of GP are in capturing overdispersed (variance larger than mean) and under dispersed (variance smaller than mean) count data. The Poisson distribution used by LDA strongly relies on the assumption that the mean and variance of document lengths are equal. This assumption is often unrealistic with most real-life text data where the variance of document length may be greater than or less than their mean. Approximate estimate of the GPLDA model parameters was achieved using Newton-Raphson approximation technique of log-likelihood. Performance and comparative analysis of GPLDA with LDA using accuracy and F1 showed improved results.

A domain decomposition method for the polarizable continuum model based on the solvent excluded surface

In this paper, an efficient solver for the polarizable continuum model in quantum chemistry is developed which takes the solvent excluded surface (the smooth molecular surface) as the solute–solvent boundary. This model requires to solve a generalized Poisson (GP) equation defined in [Formula: see text] with a space-dependent dielectric permittivity function. First, the original GP-equation is transformed into a system of two coupled equations defined in a bounded domain. Then, this domain is decomposed into overlapping balls and the Schwarz domain decomposition method is used. This method involves a direct Laplace solver and an efficient GP-solver to solve the local sub-equations in balls. For each solver, the spherical harmonics are used as basis functions in the angular direction of the spherical coordinate system. A series of numerical experiments are presented to test the performance of this method.

Marginalized zero-inflated generalized Poisson regression

ABSTRACTThe generalized Poisson (GP) regression model has been used to model count data that exhibit over-dispersion or under-dispersion. The zero-inflated GP (ZIGP) regression model can additionally handle count data characterized by many zeros. However, the parameters of ZIGP model cannot easily be used for inference on overall exposure effects. In order to address this problem, a marginalized ZIGP is proposed to directly model the population marginal mean count. The parameters of the marginalized zero-inflated GP model are estimated by the method of maximum likelihood. The regression model is illustrated by three real-life data sets.

Long Memory Models for Financial Time Series of Counts and Evidence of Systematic Market Participant Trading Behaviour Patterns in Futures on US Treasuries

This paper propose to incorporate the family of Gegenbauer Autoregressive Moving Average (GARMA) models and a special sub family called the Autoregressive Fractionally Integrated Moving Average (ARFIMA) models into the mean functions of count distributions, including Poisson, Negative Binomial (NB), Generalized Poisson (GP) and Double Poisson (DP). These distributions display equi-, over- and under-dispersion features and the additional long memory structure enhances modelling flexibility. Furthermore, we define the error terms of the GARMA family under two different approaches, the parameter-driven and observation-driven approaches. To estimate these models, we adopt a Bayesian approach implemented using R package Rstan. Then Deviance Information Criterion is evaluated to select some best-fitting models to undertake forecasting of these different classes of long memory count time series. The models are applied to analyse 136 individual U.S. Commodity Futures Trading Commission (CFTC) time series of counts which are each related to trade volume for futures-only and futures-and-options crossed classified with four fixed income securities provided by the U.S. federal government: Treasury notes (T-notes) at 2, 5 and 10 years and Treasury bonds (T-bonds) at marturities of 20 or 30 year terms. These represent some of the most active futures contracts traded in financial markets. The analysis developed provides the first detailed study of this kind undertaken by market participant type according to the categories of reportables and non-reportables and sub-categories of positions of their trading behaviours categorised by the buy and sell sides of long-call, short-put and spread being the difference between long-call and short-put. Furthermore, we study the disaggregated reportables data according to the CFTC four categories of traders, namely, dealers, asset managers, leveraged funds and other reportables. The studies performed demonstrate particular long-memory structures are consistently observed in these highly liquid and important financial instruments and in particular demonstrate systematic forecastable patterns in trading behaviour and liquidity by market participant type, which is as yet not reported or identified in financial literature. Such findings are then directly used as forecasting tools to study further relationships between price discovery and price volatility dynamics.

The Applications of Generalized Poisson Distribution in Accident Data Analysis

Accident counts at a site are usually assumed to be Poisson distributed. However, past studies have shown that many variations exist in the accident data, which often results in the data to be either overdispersed or underdispersed. Thus, a Generalized Poisson (GP) distribution is introduced as an alternative distribution for accident counts data. The GP distribution can cater the data that are more or less dispersed than required by Poisson distribution. This paper examines the appropriateness of using GP distribution in accident data analysis and Empirical Bayes (EB) safety estimation method. The results show that the GP distribution is a better fitting model than Poisson distribution such that, generally, the goodness-of-fit statistics studied (SSE and χ²) are significantly improved. When the GP distribution is used to modify the EB method, it is able to improve the prediction of safety as compared to the original EB method.

Functional Form for the Zero-Inflated Generalized Poisson Regression Model

The generalized Poisson (GP) regression is an increasingly popular approach for modeling overdispersed as well as underdispersed count data. Several parameterizations have been performed for the GP regression, and the two well known models, the GP-1 and the GP-2, have been applied. The GP-P regression, which has been recently proposed, has the advantage of nesting the GP-1 and the GP-2 parametrically, besides allowing the statistical tests of the GP-1 and the GP-2 against a more general alternative. In several cases, count data often have excessive number of zero outcomes than are expected in the Poisson. This zero-inflation phenomenon is a specific cause of overdispersion, and the zero-inflated Poisson (ZIP) regression model has been proposed. However, if the data continue to suggest additional overdispersion, the zero-inflated negative binomial (ZINB-1 and ZINB-2) and the zero-inflated generalized Poisson (ZIGP-1 and ZIGP-2) regression models have been considered as alternatives. This article proposes a functional form of the ZIGP which mixes a distribution degenerate at zero with a GP-P distribution. The suggested model has the advantage of nesting the ZIP and the two well known ZIGP (ZIGP-1 and ZIGP-2) regression models, besides allowing the statistical tests of the ZIGP-1 and the ZIGP-2 against a more general alternative. The ZIP and the functional form of the ZIGP regression models are fitted, compared and tested on two sets of count data; the Malaysian insurance claim data and the German healthcare data.