Generalized Poisson Distribution Research Articles

Generalized Poisson distributions for systems with two-particle interactions

In a cosmological context, observational best fits for galaxies’ distributions in the Universe have been tackled by recourse to different distribution functions. We provide here arguments favoring the formulation of a rather general distribution function (DF), of Poisson origin, describing galaxy clustering. The DF should be useful irrespective of distances or temperatures. We will be discussing distribution function for gravitational interactions.

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.

Monitoring count data with Shewhart control charts based on the Touchard model

AbstractThe most common control chart used to monitor count data is based on Poisson distribution, which presents a strong restriction: The mean is equal to the variance. To deal with under‐ or overdispersion, control charts considering other count distributions as Negative Binomial (NB) distribution, hyper‐Poisson, generalized Poisson distribution (GPD), Conway–Maxwell–Poisson (COM‐Poisson), Poisson–Lindley, new generalized Poisson–Lindley (NGPL) have been developed and can be found in the literature. In this paper we also present a Shewhart control chart to monitor count data developed on Touchard distribution, which is a three‐parameter extension of the Poisson distribution (Poisson distribution is a particular case) and in the family of weighted Poisson models. Its normalizing constant is related to the Touchard polynomials, hence the name of this model. It is a flexible distribution that can account for both under‐ or overdispersion and concentration of zeros that are frequently found in non‐Poisson count data. Consequences in terms of speed to signal departures of stability of the parameters are obtained when incorrect control limits based on non‐Touchard distribution (like Poisson, NB or COM‐Poisson) are used to monitor count data generated by a Touchard distribution. Numerical examples illustrate the current proposal.

Inference Procedures on the Generalized Poisson Distribution from Multiple Samples: Comparisons with Nonparametric Models for Analysis of Covariance (ANCOVA) of Count Data

Count data that exhibit over dispersion (variance of counts is larger than its mean) are commonly analyzed using discrete distributions such as negative binomial, Poisson inverse Gaussian and other models. The Poisson is characterized by the equality of mean and variance whereas the Negative Binomial and the Poisson inverse Gaussian have variance larger than the mean and therefore are more appropriate to model over-dispersed count data. As an alternative to these two models, we shall use the generalized Poisson distribution for group comparisons in the presence of multiple covariates. This problem is known as the ANCOVA and is solved for continuous data. Our objectives were to develop ANCOVA using the generalized Poisson distribution, and compare its goodness of fit to that of the nonparametric Generalized Additive Models. We used real life data to show that the model performs quite satisfactorily when compared to the nonparametric Generalized Additive Models.

On a Generalised Exponential-Lindley Mixture of Generalised Poisson Distribution

Background: A mixture distribution arises when some or all parameters in a mixing distribution vary according to the nature of original distribution. A generalised exponential-Lindley distribution (GELD) was obtained by Mishra and Sah (2015). In this paper, generalized exponential- Lindley mixture of generalised Poisson distribution (GELMGPD) has been obtained by mixing generalised Poisson distribution (GPD) of Consul and Jain’s (1973) with GELD. In the proposed distribution, GELD is the original distribution and GPD is a mixing distribution. Generalised exponential- Lindley mixture of Poisson distribution (GELMPD) was obtained by Sah and Mishra (2019). It is a particular case of GELMGPD. 
 Materials and Methods: GELMGPD is a compound distribution obtained by using the theoretical concept of some continuous mixtures of generalised Poisson distribution of Consul and Jain (1973). In this mixing process, GELD plays a role of original distribution and GPD is considered as mixing distribution. 
 Results: Probability mass of function (pmf) and the first four moments about origin of the generalised exponential-Lindley mixture of generalised Poisson distribution have been obtained. The method of moments has been discussed to estimate parameters of the GELMGPD. This distribution has been fitted to a number of discrete data-sets which are negative binomial in nature. P-value of this distribution has been compared to the PLD of Sankaran (1970) and GELMPD of Sah and Mishra (2019) for similar type of data-sets. 
 Conclusion: It is found that P-value of GELMGPD is greater than that in each case of PLD and GELMPD. Hence, it is expected to be a better alternative to the PLD of Sankaran and GELMPD of Sah and Mishra for similar types of discrete data-sets which are negative binomial in nature. It is also observed that GELMGPD gives much more significant result when the value of is negative.

Parameter Estimation and Hypothesis Testing of Geographically Weighted Multivariate Generalized Poisson Regression

We introduce a new multivariate regression model based on the generalized Poisson distribution, which we called geographically-weighted multivariate generalized Poisson regression (GWMGPR) model, and we present a maximum likelihood step-by-step procedure to obtain parameters for it. We use the maximum likelihood ratio test to examine the significance of the regression parameters and to define their critical region.

Inference Procedures on the Ratio of Modified Generalized Poisson Distribution Means: Applications to RNA_SEQ Data

The Poisson and the Negative Binomial distributions are commonly used as analytic tools to model count data. The Poisson is characterized by the equality of mean and variance whereas the Negative Binomial has a variance larger than the mean and therefore is appropriate to model over-dispersed count data. The Generalized Poisson Distribution is becoming a popular alternative to the Negative Binomial. We have considered inference procedures on a modified form of this distribution when two samples are available from two independent populations and the target effect size of interest is the ratio of the two population means. The statistical objective is to construct confidence limits on the ratio. We first test the presence of over dispersion and derive several estimators in the single sample situation. When two samples are available, our interest is focused on the estimation of an effect size measured by the ratio of the respective population means. We have compared two methods; namely the Fieller’s and the delta methods in terms of coverage probabilities. We have illustrated the methodologies on published genomic datasets.

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.

Efficient monitoring of overdispersed counts with time-varying population sizes

The monitoring of count data is a major issue in many industrial applications and research activities. Count data often exhibit overdispersion with variance greater than mean, which makes the commonly used Poisson distribution problematic because of its underlying assumption of equal mean and variance. In this article, we consider the surveillance strategy for overdispersed count data, and we also take into account the effect of time-varying population sizes. In particular, to model the data we propose to adapt the generalized Poisson distribution to incorporate the incidence rate, the overdispersion factor, and the non-constant population sizes. The weighted likelihood ratio test is employed for online monitoring. Simulations show that the proposed method is efficient at detecting changes simultaneously in both the incidence rate and the overdispersion factor, and it is robust under different time-varying patterns of the population sizes.

Modelling adoption intensity of improved soybean production technologies in Ghana - a Generalized Poisson approach.

Soybean is an important cash crop especially for farmers in the north of Ghana. However, cultivation of the commodity is dominated by smallholders equipped with traditional tools, coupled with low or no adoption of improved soybean production technologies. Using primary data collected from 300 soybean farmers across northern Ghana, the study employed count data modelling to estimate the determinants of adoption intensity of sustainable soybean production technologies. The study accounted for potential estimation errors due to under-dispersion and over-dispersion, by using a model based on the generalized Poisson distribution. On the average, a farmer adopted 50% of the identified sustainable soybean production technologies. Age, education, extension visits, mass media through radio, and the perception of adoption of soybean production technologies being risky are significant with positive influence on the adoption intensity of sustainable soybean production technologies. The study therefore recommends among others, that various extension programmes should intensify education on the benefits of adopting sustainable soybean production practices. There is the need to set up many technology demonstration farms to give farmers hands-on training during field days.

Generalized Poisson Hidden Markov Model for Overdispersed or Underdispersed Count Data

The most suitable statistical method for explaining serial dependency in time series count data is that based on Hidden Markov Models (HMMs). These models assume that the observations are generated from a finite mixture of distributions governed by the principle of Markov chain (MC). Poisson-Hidden Markov Model (P-HMM) may be the most widely used method for modelling the above said situations. However, in real life scenario, this model cannot be considered as the best choice. Taking this fact into account, we, in this paper, go for Generalised Poisson Distribution (GPD) for modelling count data. This method can rectify the overdispersion and underdispersion in the Poisson model. Here, we develop Generalised Poisson Hidden Markov model (GP-HMM) by combining GPD with HMM for modelling such data. The results of the study on simulated data and an application of real data, monthly cases of Leptospirosis in the state of Kerala in South India, show good convergence properties, proving that the GP-HMM is a better method compared to P-HMM.

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.

On the use of the modified power series family of distributions in a cure rate model context.

In this paper, we propose a generalization of the power series cure rate model for the number of competing causes related to the occurrence of the event of interest. The model includes distributions not yet used in the cure rate models context, such as the Borel, Haight and Restricted Generalized Poisson distributions. The model is conveniently parameterized in terms of the cure rate. Maximum likelihood estimation based on the Expectation Maximization algorithm is discussed. A simulation study designed to assess some properties of the estimators is carried out, showing the good performance of the proposed estimation procedure in finite samples. Finally, an application to a bone marrow transplant data set is presented.

On zero-inflated hyper-Poisson distribution and its applications

Here we consider a zero-inflated version of the hyper-Poisson distribution and study some of its important properties through deriving its probability generating function and expressions for factorial moments, mean, variance, recursion formulae for probabilities, raw moments, and factorial moments. The estimation of the parameters of the zero-inflated hyper-Poisson distribution is attempted. The distribution has been fitted to certain real life data sets and thereby shown that the proposed model gives better fit to the data sets compared to existing models such as zero-inflated Poisson distibution (ZIPD), zero-inflated negative binomial distribution (ZINBD), zero-inflated Conway-Maxwell Poisson distribution (ZICMPD), and zero-inflated generalized Poisson distribution (ZIGPD). Further, Rao’s efficient score test procedure is applied for examining the significance of the parameters and a simulation study is carried out for assessing the performance of the maximum likelihood estimators of the parameters of the distribution.

MODELLING ZERO-INFLATED COUNT DATA WITH A SPECIAL CASE OF THE GENERALISED POISSON DISTRIBUTION

AbstractA one-parameter version of the generalised Poisson distribution provided by Consul and Jain (1973) is considered in this paper. The distribution is unimodal with a zero vertex and over-dispersed. A generalised linear model related to this distribution is also presented. Its parameters can be estimated by using a Fisher-Scoring algorithm which is equivalent to iteratively reweighted least squares. Due to its flexibility and capacity to describe highly skewed data with an excessive number of zeros, the model is suitable to be applied in insurance settings as an alternative to the negative binomial and zero-inflated model.

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.

The CPCAT as a novel tool to overcome the shortcomings of NOEC/LOEC statistics in ecotoxicology: a simulation study to evaluate the statistical power

Species reproduction is an important determinant of population dynamics. As such, this is an important parameter in environmental risk assessment. The closure principle computational approach test (CPCAT) was recently proposed as a method to derive a NOEC/LOEC for reproduction count data such as the number of juvenile Daphnia. The Poisson distribution used by CPCAT can be too restrictive as a model of the data-generating process. In practice, the generalized Poisson distribution could be more appropriate, as it allows for inequality of the population mean mu and the population variance sigma ^2. It is of fundamental interest to explore the statistical power of CPCAT and the probability of determining a regulatory relevant effect correctly. Using a simulation, we varied between Poisson distribution (mu =sigma ^2) and generalized Poisson distribution allowing for over-dispersion (mu <sigma ^2) and under-dispersion (mu >sigma ^2). The results indicated that the probability of detecting the LOEC/NOEC correctly was ge 0.8 provided the effect was at least 20% above or below the mean level of the control group and mean reproduction of the control was at least 50 individuals while over-dispersion was missing. Specifically, under-dispersion increased, whereas over-dispersion reduced the statistical power of the CPCAT. Using the well-known Hampel identifier, we propose a simple and straight forward method to assess whether the data-generating process of real data could be over- or under-dispersed.

A Generalised Poisson Mishra Distribution

Background: “Mishra distribution" of B. K. Sah (2015) has been obtained in honor of Professor A. Mishra, Department of Statistics, Patna University, Patna (Sah, 2015). A one parameter Poisson-Mishra distribution (PMD) of B. K. Sah (2017) has been obtained by compounding Poisson distribution with Mishra distribution. It has been found that this distribution gives better fit to all the discrete data sets which are negative binomial in nature used by Sankarn (1970) and others. A generalisation of PMD has been obtained by mixing the generalised Poisson distribution of Consul and Jain (1973) with the Mishra distribution.Materials and Methods: It is based on the concept of the generalisations of some continuous mixtures of Poisson distribution.Results: Probability density function and the first four moments about origin of the proposed distribution have been obtained. The estimation of parameters of this distribution has been discussed by using the first moment about origin and the probability mass function at x = 0 . This distribution has been fitted to a number of discrete data-sets to which earlier Poisson-Lindley distribution (PLD) and PMD have been fitted.Conclusion: P-value of generalised Poisson-Mishra distribution is greater than PLD and PMD. Hence, it provides a better alternative to the PLD of Sankarn and PMD of B. K. Sah.Nepalese Journal of Statistics, Vol. 2, 27-36

A weighted negative binomial Lindley distribution with applications to dispersed data.

A new discrete distribution is introduced. The distribution involves the negative binomial and size biased negative binomial distributions as sub-models among others and it is a weighted version of the two parameter discrete Lindley distribution. The distribution has various interesting properties, such as bathtub shape hazard function along with increasing/decreasing hazard rate, positive skewness, symmetric behavior, and over- and under-dispersion. Moreover, it is self decomposable and infinitely divisible, which makes the proposed distribution well suited for count data modeling. Other properties are investigated, including probability generating function, ordinary moments, factorial moments, negative moments and characterization. Estimation of the model parameters is investigated by the methods of moments and maximum likelihood, and a performance of the estimators is assessed by a simulation study. The credibility of the proposed distribution over the negative binomial, Poisson and generalized Poisson distributions is discussed based on some test statistics and four real data sets.

A comparison of spiking experiments to estimate the detection proportion of qualitative microbiological methods

ABSTRACTThe detection proportion of a qualitative microbiological test method is the probability to detect a single micro-organism. A general expression for the moment estimator of the detection proportion is provided. It depends on the distribution of the spikes used in a validation study through its moment-generating function. Several forms of spiking experiments are compared on their estimation performance using simulations and assuming a generalized Poisson distribution (GPD) for the spikes. The optimal design, which minimizes the mean squared error of our proposed moment estimator, depends on the dispersion parameter of the GPD. The design that uses just one spiked solution instead of multiple solutions is optimal for Poisson and overdispersed Poisson and it is robust against distributions for the spikes.