Generalized Poisson Distribution Research Articles

Generalised Poisson-Modified Mishra distribution

This distribution is a compound probability distribution constructed by compounding generalised Poisson distribution of Consul and Jain (1973) with Modified Mishra distribution of Sah (2022). All the important characteristics required for this distribution have been obtained and explained. In order to check the validity of the theoretical work, goodness of fit has been applied to some secondary data which have been used by others. It has been observed that this distribution gives a better fit than Poisson-Lindley distribution of Sankaran (1970) as well as Poisson-Modified Mishra distribution of Sah and Sahani (2023).

Parameter estimation for strict arcsine distribution

The two-parameter strict arcsine distribution as a member of the natural exponential family with cubic variance function has been shown to be a viable candidate for statistical analysis of count data. Efficient methods of parameter estimation will be essential in practical applications of the distribution. In this paper we examine some methods of parameter estimation. Due to the simple expression of the probability generating function, a probability generating function-based estimation procedure is considered and compared with other estimation procedures. Since the accuracy of the parameter estimation procedure affects the probability of correct selection in choosing the correct probability distribution, we extend the investigation by examining the discrimination between strict arcsine and generalized Poisson distributions in which both have cubic variance functions.

Circadian characteristics of term and preterm labors

The labor is a physiological event considered to have its own circadian (diurnal) rhythm, but some of the data remain conflicting, especially for preterm births. In this retrospective study, we analyzed the circadian trends of labor onset times in the Slovenian birth cohort from 1990 to 2018 with over 550,000 cases of singleton births. The number of term and preterm labor onsets was calculated for each hour in a day and circadian trends were evaluated for each of the study groups by modeling with a generalized Poisson distribution linked with the cosinor regression model using logarithmic link function. The induced labors were taken as the control group since the timing of labor depends mostly on the working schedule of personnel and not on the intrinsic rhythmic characteristics. For induced labors, the main peak in the number of labor cases was observed in the late morning hours (around 10 AM) for all gestational ages. The prominence of this peak becomes smaller in spontaneous premature labors with gradually disrupting rhythmicity in very preterm and extremely preterm cases. Labors starting with spontaneous contractions peak between 6 and 7 AM and lose the rhythmicity at 35 weeks of gestation while labors starting with a spontaneous rupture of membranes peak at 1 AM and lose the rhythmicity at 31 weeks of gestation, suggesting differences in underlying mechanisms. According to our knowledge, this is the first study that shows differences of circadian trends between different types of spontaneous labors, i.e., labors initiated with contraction and labors initiated with a spontaneous rupture of membranes. Moreover, the obtained results represent evidence of gradual disruption of rhythmicity from mild to extreme prematurity.

Bayesian modeling of spatial integer-valued time series

Many infectious diseases spread through person-to-person contact, either directly or indirectly. The proposal incorporates spatial-temporal patterns into multivariate integer-valued GARCH models with either a generalized Poisson distribution or zero-inflated generalized Poisson distribution in order to describe these features of the data. By considering the neighboring locations of the target series, the set-up incorporates a flexible and continuous conceptualization of distance to present the spatial components, thereby highlighting for the non-separability of space and time. Such an approach eliminates the need to pre-assign a spatial weight matrix. Newly designed models are utilized to investigate time-series counts of infectious diseases, enabling inference, prediction, and model selection within a Bayesian framework through Markov chain Monte Carlo (MCMC) algorithms. As an illustration, design simulation studies and multivariate weekly dengue cases are scrutinized for the performance of the Bayesian methods. The proposed models successfully capture the characteristics of spatial dependency, over-dispersion, and a large portion of zeros, providing a comprehensive model for the observed phenomena in the data.

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.

Feasibility of skewness-based characterization of SiPMs with unresolved spectra

Conventional methods of SiPM characterization are mostly based on a resolved spectrum of photoelectron or dark electron detection events which represents a random number of fired SiPM cells. However, such methods are not applicable for unresolved spectra caused, for example, by a low single electron response to noise ratio or a high dark count rate, for example of irradiated SiPMs. This challenge is anticipated in modern developments of radiation-intense facilities like CERN CMS and applications of large-area SiPMs and SiPM arrays with noisy readout like Cherenkov Telescope Array.The method of statistical moments is a well-known technique for the estimation of parameters of a random variable distribution using sample moments of raw data. The first two moments have already been used for estimation of the gain and the number of photoelectrons by the mean and the variance of detected charges relying on a priori knowledge of the Excess Noise Factor of the SiPM or PMT. The third moment or skewness of the measured charge can also be involved to estimate three main SiPM parameters including crosstalk.The skewness-based characterization presumes Generalized Poisson distribution of fired cells which was found to be in good agreement with many experiments. Initial theoretical and experimental evaluations of the skewness method have shown promising results. This study focuses on the feasibility of using such method to characterize the parameters of SiPM with unresolved charge-amplitude spectrum.

A hybrid model of spatial autoregressive-multivariate adaptive generalized Poisson regression spline

Several Multivariate Adaptive Regression Spline (MARS) approaches are available to model categorical and numerical (especially continuous) data. Currently, there are other numerical data types—discrete or count data—that call for specific consideration in modeling. Additionally, spatially correlated count data is frequently observed. This has been seen in the case of health data, for example, the number of newborn fatalities, tuberculosis patients, hospital visitors, etc. However, currently no structurally consistent nonparametric regression and MARS model for count data incorporating spatial lag autocorrelation. The SAR-MAGPRS estimator (Spatial Autoregressive - Multivariate Adaptive Generalized Poisson Regression Spline) is developed to fill this gap. Although it can be applied to different count distributions, the estimator was developed in this study under the assumption of a Generalized Poisson distribution. This paper provides an information-theoretic framework for incorporating knowledge of the spatial structure and non-parametric regression models, especially MARS for the count data types. Moreover, the proposed method can assist in modeling the number of diseases while health policies are being developed. The framework presents an application of the Penalized Least Square (PLS) method to estimate the SAR – MAGPRS model.

Application of a count data model to evaluate the anti-metastatic efficacy of QAP14 in 4T1 breast cancer allografts

Dopamine D1 receptor (D1DR) is proved to be a promising target to prevent tumor metastasis, and our previous studies showed that QAP14, a potent anti-cancer agent, exerted inhibitory effect on lung metastasis via D1DR activation. Therefore, the purpose of the study was to establish count data models to quantitatively characterize the disease progression of lung metastasis and assess the anti-metastatic effect of QAP14. Data of metastatic progression were collected in 4T1 tumor-bearing mice. Generalized Poisson distribution best described the variability of metastasis counts among the individuals. An empirical PK/PD model was developed to establish mathematical relationships between steady plasma concentrations of QAP14 and metastasis growth dynamics. The latency period of metastasis was estimated to be 12 days after tumor implantation. Our model structure also fitted well to other D1DR agonists (fenoldopam and l-stepholidine) which had inhibitory impact on breast cancer lung metastasis likewise. QAP14 40 mg/kg showed the best inhibitory efficacy, for it provided the longest prolongation of metastasis-free periods compared with fenoldopam or l-stepholidine. This study provides a quantitative method to describe the lung metastasis progression of 4T1 allografts, as well as an alternative PD model structure to evaluate anti-metastatic efficacy.

Transactional Interpretation and the Generalized Poisson Distribution

The aim of this paper is to study the quantum-like approach to the description of the market in the context of the principle of minimum Fisher information. We wish to investigate the validity of using squeezed coherent states as market strategies. For this purpose, we focus on the representation of any squeezed coherent state with respect to the basis of the eigenvectors of the observable of market risk. We derive a formula for the probability of being the squeezed coherent state in one of these states. The distribution that we call generalized Poisson establishes the relation between the squeezed coherent states and their description in the language of risk in quantum terms. We provide a formula specifying the total risk of squeezed coherent strategy. Then, we propose a risk of risk concept that is in fact the second central moment of the generalized Poisson distribution. This is an important numerical characterization of squeezed coherent strategies. We provide its interpretations on the basis of the uncertainty relation for time and energy.

Moving Average Control Chart for Generalized Poisson Distribution

This study aim to propose a moving average control chart explicit formula based on the Generalized Poisson distribution (MAGP ) and to compare the performance of the MAGP control chart with that of the Shewhart control chart (CGP ). The average run length (ARL) is used to consider the performance. The ARL results indicated that the MAGP control chart is superior to the CGP control chart based on the Generalized Poisson distribution for detecting small shifts. Practitioners will find this explicit formula to be accurate as well as simple to apply and use.

Skewness-based characterization of silicon photomultipliers

Characterization of SiPMs is an objective of high importance in almost any research, development, and application of these unique photon detectors. Two decades of characterization method developments resulted in a comprehensive and elegant methodology based on a precisely resolved number of fired SiPM cells or the SiPM spectrum. Spectrum fitting procedures are very sensitive to a proper selection of many initial guess values including parameters of minor importance and happen to be unreliable. Moreover, conventional methods are useless for degraded or unresolved SiPM spectra. This challenge was anticipated for SiPM-based detectors of high-luminosity particle calorimeters where a severe radiation degradation of the SiPMs would co-exist with their acceptable performance. To address it, one can measure a mean and variance of SiPM charge responses relying on a priori known Excess Noise Factor of the SiPM but its stability during SiPM degradation is uncertain. This study proposes and evaluates a new approach for the SiPM characterization based on the first three statistical moments (mean, variance, and skewness) of its charge response including skewness. It assumes the Generalized Poisson distribution of the number of fired cells and provides simple closed-form analytical expressions for the parameter estimation. It allows determining a gain, a number of detected photons, and a probability of correlated events directly from raw data. The skewness-based characterization is anticipated to be especially useful for mass testing and continuous monitoring of SiPMs in large-scale experiments due to its simplicity and robustness. Initial evaluations of the method show promising results.

Multivariate zero-inflated Bell distribution and its inference and applications

In this paper we introduce a multivariate family of distributions for multivariate count data with excess zeros, which is a multivariate extension of the univariate zero-inflated Bell distribution. We derive various general properties of this multivariate distribution. In particular, the marginal distributions are univariate zero-inflated Bell distributions. The model parameters are estimated using the traditional maximum likelihood estimation method. In addition, we develop a simple EM algorithm to compute the maximum likelihood estimates of the parameters of the new multivariate distribution with closed-form expressions for the maximum likelihood estimators. Empirical applications that employ real multivariate count data are considered to illustrate the usefulness of the new class of multivariate distributions, and comparisons with the multivariate zero-inflated Poisson distribution, multivariate zero-adjusted Poisson distributions, and multivariate zero-inflated generalized Poisson distribution are made.

Discrimination between Some Over Dispersed Count Distributions

The Poisson inverse Gaussian and generalized Poisson distributions are widely used in modelling overdispersed count data which are commonly found in healthcare, insurance, engineering, econometric and ecology. The inverse trinomial distribution is a relatively new count distribution arising from a one-dimensional random walk model (Shimizu & Yanagimoto, 1991). The Poisson inverse Gaussian distribution is a popular count model that has been proposed as an alternative to the negative binomial distribution. The inverse trinomial and generalized Poisson models possess a common characteristic of having a cubic variance function, while the Poisson inverse Gaussian has a quadratic variance function. The nature of the variance function seems to be an important property in modelling overdispersed count data. Hence it is of interest to be able to select among the three models in practical applications. This paper considers discrimination of three models based on the likelihood ratio statistic and computes via Monte Carlo simulation the probability of correct selection.

Exposure-Bleeding Count Modeling of Emicizumab for the Prophylaxis of Bleeding in Persons with Hemophilia A with/Without Inhibitors Against Factor VIII.

Background and ObjectiveEmicizumab is a monoclonal antibody that bridges activated coagulation factor IX and factor X to restore effective hemostasis in persons with hemophilia A. It is indicated for routine prophylaxis of bleeding episodes in persons with hemophilia A. The aim of the present study is to describe the exposure–response relationship between emicizumab concentrations and bleeding frequency, and to confirm adequate bleeding control of the investigated dosing regimens 1.5 mg/kg once weekly, 3 mg/kg every 2 weeks, and 6 mg/kg every 4 weeks.MethodsTreated bleeding events were pooled from 445 persons with hemophilia A with and without inhibitors against factor VIII, participating in six clinical studies. Emicizumab concentrations were predicted using a previously developed population pharmacokinetic model. A count model was used to quantify the exposure–response relationship. These models were used to illustrate the relationship between emicizumab concentrations and cumulative count of bleeding over 1 year (annualized bleeding rate).ResultsThe final exposure–response model, based on a generalized Poisson distribution and an inhibitory Emax relationship, adequately describes the relationship between daily emicizumab concentrations and daily bleed frequency. A significant effect of factor VIII prophylaxis among persons with hemophilia A without inhibitors was found. Annualized bleeding rate simulations show that the three emicizumab dosing regimens maintain the concentrations close to the plateau of the effect. At the average steady-state concentration across all regimens (53.5 µg/mL), the predicted mean annualized bleeding rate is 1.28, corresponding to a 94.0% reduction from baseline.ConclusionsThese results confirm that average emicizumab concentrations achieved with all three emicizumab dosing regimens provide adequate bleeding control.Supplementary InformationThe online version contains supplementary material available at 10.1007/s40262-021-01006-0.

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.