Mixed Poisson Distribution Research Articles

Bootstrap Confidence Intervals for the Poisson-Pranav Distribution Parameter with an Application to COVID-19 Data

The Poisson distribution is commonly applied in statistical modeling to represent the count of events, assuming that the events are independent and occur at a constant rate. This assumption, however, may not always hold true in real-life situations. The Poisson distribution may be inadequate if the underlying rate of occurrence is not constant. The mixed Poisson distribution is proposed to solve this limitation because it permits the rate parameter of the Poisson distribution to be random rather than fixed. The Poisson-Pranav distribution, which is classified as a type of mixed Poisson distribution, has been commonly utilized to analyze count data that exhibits over-dispersion over many domains. However, no research has been done into building confidence intervals for the parameter of the Poisson-Pranav distribution using the bootstrap method. Using Monte Carlo simulation, the coverage probabilities and average lengths of the percentile bootstrap (PB), basic bootstrap (BB), and biased-corrected and accelerated (BCa) bootstrap’s interval-estimation performances were compared. The bootstrap method was not appropriate for achieving the desired nominal confidence level with small sample sizes. In addition, the performance of the bootstrap confidence intervals did not differ significantly when the sample size was increased considerably. In each case studied, the BCa bootstrap confidence intervals performed better than the others. The effectiveness of bootstrap confidence intervals was demonstrated by applying them to the number of COVID-19 deaths in Belgium. The computations substantially supported the proposed bootstrap confidence intervals.

A Statistical Model for Count Data Analysis and Population Size Estimation: Introducing a Mixed Poisson–Lindley Distribution and Its Zero Truncation

Count data consists of both observed and unobserved events. The analysis of count data often encounters overdispersion, where traditional Poisson models may not be adequate. In this paper, we introduce a tractable one-parameter mixed Poisson distribution, which combines the Poisson distribution with the improved second-degree Lindley distribution. This distribution, called the Poisson-improved second-degree Lindley distribution, is capable of effectively modeling standard count data with overdispersion. However, if the frequency of the unobserved events is unknown, the proposed distribution cannot be directly used to describe the events. To address this limitation, we propose a modification by truncating the distribution to zero. This results in a tractable zero-truncated distribution that encompasses all types of dispersions. Due to the unknown frequency of unobserved events, the population size as a whole becomes unknown and requires estimation. To estimate the population size, we develop a Horvitz–Thompson-like estimator utilizing truncated distribution. Both the untruncated and truncated distributions exhibit desirable statistical properties. The estimators for both distributions, as well as the population size, are asymptotically unbiased and consistent. The current study demonstrates that both the truncated and untruncated distributions adequately explain the considered medical datasets, which are the number of dicentric chromosomes after being exposed to different doses of radiation and the number of positive Salmonella. Moreover, the proposed population size estimator yields reliable estimates.

Bayesian estimation to deconvolute single-particle ICP-MS data with a mixed Poisson distribution

Model parameters (mean particle-event intensity and frequency, etc.) were directly estimated using the Bayesian method without summing particle-event intensities. Model parameters were transformed to NP information.

Estimates of the Convergence Rate in the Generalized Rényi Theorem with a Structural Digamma Distribution Using Zeta Metrics

This paper considers a generalization of the Rényi theorem to the case of a structural distribution with a scale parameter. In terms of the zeta metric, some estimates of the convergence rate in the generalized Rényi theorem are obtained when the structural mixed Poisson distribution of the summation index is a scale mixture of the generalized gamma distribution. Estimates of the convergence rate for the structural digamma distribution are given as a special case. The paper extends the results previously obtained for the generalized gamma distribution.

A New Three-Parameter Mixed Poisson Transmuted Weighted Exponential Distribution with Applications to Insurance Data

The classical Poisson distribution assumes equidispersion and this is hardly the case when given count observations. To improve efficiency and general applicability for dispersed data, this paper obtains a new mixing probability distribution using quadratic transmutation map on the weighed exponential distribution. The mixing distribution is used to obtain a new three parameter mixed Poisson distribution. The new distribution has unimodal shape with positive skewness and tends to zero speedily. Basic properties of the new distribution like the distribution function, moments, and dispersion index are obtained. Maximum likelihood estimates of the parameters of the distribution are obtained. Applications are made using four claims frequencies from different insurance companies. Results show that the new distribution gives a good fit in comparisons with some referred discrete distributions.

An Alternative Count Distribution for Modeling Dispersed Observations

In most cases, count data have higher variances than means; hence using the Poisson distribution to model such observations is misleading because of the equality of the Poisson mean and variance. This study proposes a new two-parameter mixed Poisson distribution for modeling dispersed count observations. The exponential distribution is transmuted to obtain a new mixing distribution for the new proposition. Different moment-based mathematical properties of the new proposition are obtained. Applications using dispersed count observations with excess zero are made. Comparisons with related distributions for modeling dispersed observation reveal that the new distribution performs creditably well.

On the Poisson transmuted Ailamujia distribution with applications to dispersed and skewed count data

Assuming classical Poisson distribution for skewed and dispersed count observations may be misleading since it assumes equality for mean and variance. A new count distribution suitable for modelling dispersed and skewed count observations is proposed in this paper. This is achieved by assuming transmuted Ailamujia distribution for the parameter of the Poisson distribution and used mixed Poisson distribution process to obtain a new distribution. Mathematical properties of the new distribution are obtained and some measures of dispersion are assessed by assuming different parameter combinations for the distribution. Using the MLE for parameter estimation, the performance of the new proposition is assessed on four referred dispersed count data. Results obtained show that the new proposition provide better fit for all the datasets considered.

Zero-modified Poisson-Modification of Quasi Lindley distribution and its application

Abstract The Poisson-Modification of Quasi Lindley (PMQL) distribution is a newly introduced mixed Poisson distribution for over-dispersed count data. The aim of this article is to introduce the Zero-modified PMQL (ZMPMQL) distribution as an alternative to the PMQL distribution in order to accommodate zero inflation/deflation. The method of obtaining the ZMPMQL distribution jointly with some of its important properties, namely the probability mass and distribution functions, mean, variance, index of dispersion, and quantile function are presented. Furthermore, some of its special cases are discussed. The maximum likelihood (ML) estimation method is used for the unknown parameter estimation. A simulation study is conducted in order to evaluate the asymptotic theory of the ML estimation method and to show the superiority of the ML method over the method of moments estimation. The applicability of the introduced distribution is illustrated by using a real-world data set.

Max-Compound Cox Processes. III

Extreme values are considered in samples with random size that have a mixed Poisson distribution being generated by a doubly stochastic Poisson process. We prove some inequalities providing bounds on the rate of convergence in limit theorems for the distributions of max-compound Cox processes.

Bayesian estimation of the effect of health inequality in disease detection

BackgroundMeasuring health inequality is essential to ensure that everyone has equal accessibility to health care. Studies in the past have continuously presented and showed areas or groups of people affected by various inequality in accessing the health resources and services to help improve this matter. Alongside, disease prevention is as important to minimise the disease burden and improve health and quality of life. These aspects are interlinked and greatly contributes to one’s health.MethodIn this study, the Gini coefficient and Lorenz curve are used to give an indication of the overall health inequality. The impact of this inequality in granular level is demonstrated using Bayesian estimation for disease detection. The Bayesian estimation used a two-component modelling approach that separates the case detection process and incidence rate using a mixed Poisson distribution while capturing underlying spatio-temporal characteristics. Bayesian model averaging is used in conjunction with the two-component modelling approach to improve the accuracy of estimates by incorporating many candidate models into the analysis instead of using fixed component models. This method is applied to an infectious disease, influenza, in Victoria, Australia between 2013 and 2016 and the corresponding primary health care of the state.ResultThere is a relatively equal distribution of health resources and services pertaining to general practitioners (GP) and GP clinics in Victoria, Australia. Roughly 80 percent of the population shares 70 percent of the number of GPs and GP clinics. The Bayesian estimation with model averaging revealed that access difficulty to health services impacts both case detection probability and incidence rate. Minimal differences are recorded in the observed and estimated incidence of influenza cases considering social deprivation factors. In most years, areas in Victoria’s southwest and eastern parts have potential under-reported cases consistent with their relatively lower number of GP or GP clinics.ConclusionThe Bayesian model estimated a slight discrepancy between the estimated incidence and the observed cases of influenza in Victoria, Australia in 2013-2016 period. This is consistent with the relatively equal health resources and services in the state. This finding is beneficial in determining areas with potential under-reported cases and under-served health care. The proposed approach in this study provides insight into the impact of health inequality in disease detection without requiring costly and time-extensive surveys and relying mainly on the data at hand. Furthermore, the application of Bayesian model averaging provided a flexible modelling framework that allows covariates to move between case detection and incidence models.

The Geographically Weighted Multivariate Poisson Inverse Gaussian Regression Model and Its Applications

This study aims to develop a method for multivariate spatial overdispersion count data with mixed Poisson distribution, namely the Geographically Weighted Multivariate Poisson Inverse Gaussian Regression (GWMPIGR) model. The parameters of the GWMPIGR model are estimated locally using the maximum likelihood estimation (MLE) method by considering spatial effects. Therefore, the significance of the regression parameter differs for each location. In this study, four GWMPIGR models are evaluated based on the exposure variable and the spatial weighting function. We compare the performance of those four models in real-world application using data on the number of infant, under-5 and maternal deaths in East Java in 2019 using five predictor variables. In this study, the GWMPIGR model uses one exposure variable and three exposure variables. Compared to the fixed kernel Gaussian weighting function, the GWMPIGR model with the fixed kernel bisquare weighting function and one exposure variable has a better fit based on the AICc value. Furthermore, according to the best GWMPIGR model, there are several regional groups formed based on predictors that significantly affected each event in East Java in 2019.

The Poisson-Transmuted Janardan Distribution for Modelling Count Data

In this paper, we introduce a new mixed Poisson distribution, called the Poisson-transmuted Janardan distribution. The Poisson-Janardan and Poisson-Lindley distributions are sub-model of the proposed distribution. Some mathematical properties of the proposed distribution, including the moments, moment generating function, probability generating function and generation of a Poisson-transmuted Janardan random variable, are presented. The parameter estimation is discussed based on the method of moments and the maximum likelihood estimation. In addition, we illustrated the application of the proposed distribution by fitting with 4 real data sets and comparing it with some other distributions based on the Kolmogorov-Smirnov test for criteria.

The Distribution Patterns of Valency-changing Verbs: An Approach of Quantitative Linguistics

The present study attempts to explore the distribution patterns of the valency-changing verbs from the perspective of quantitative linguistics. We took authentic spoken language data as the research materials. The corpus used in this paper is a self-built spoken English corpus containing about 21,000 words. We half-manually annotated the corpus with the help of SpaCy, a natural language processing tool. According to the annotation results and statistical data, we obtained a total of 217 valency-changing English verbs and 248 sentence components governed by them. After analysis, the current study came to the following conclusions: First, bivalent verbs are most frequent among the three types of valency-changing verbs; second, after fitting all the language data to different probability distributions, we found that the rank-frequency distributions of all the valency-changing English verbs with different numbers of obligatory arguments obey the power law, and the frequencies of bivalent valency-changing verbs obey other kinds of distributions such as the mixed Poisson distribution.

A new two-parameter discrete poisson-generalized Lindley distribution with properties and applications to healthcare data sets

Mixed-Poisson distributions have been used in many fields for modeling the over-dispersed count data sets. To open a new opportunity in modeling the over-dispersed count data sets, we introduce a new mixed-Poisson distribution using the generalized Lindley distribution as a mixing distribution. The moment and probability generating functions, factorial moments as well as skewness, and kurtosis measures are derived. Using the mean-parametrized version of the proposed distribution, we introduce a new count regression model which is an appropriate model for over-dispersed counts. The healthcare data sets are analyzed employing a new count regression model. We conclude that the new regression model works well in the case of over-dispersion.

Altitude Factors Affect Dengue Fever Cases in South Sulawesi: A Study Using Poisson Inverse Gaussian Regression Model

Poisson regression is used to model enumeration data such as data on the number of DHF cases. This model has the assumption that is fulfilled is the average and the variance must have the same value or it is called the equidispersion. But this assumption is not fulfilled because the data on the number of dengue cases experienced violations of this assumption. The violation is that the average value is smaller than the variance value or it is called overdispersion. This results in incorrect conclusions because the prediction standard error is underestimated. The way to prevent this is by combining the Poisson distribution and discrete or continuous distribution, this combination is called Mixed Poisson Distribution. Researchers use one of the Mixed Poisson methods, namely Inverse Gaussian Poisson Regression (PIG) because the method is used when the data is overdispersed and the parameters are known or close form on the likelihood function. Based on the results of the study, it is known that the height of the area is a factor that significantly influences DHF cases in South Sulawesi.

Bayesian survival model induced by frailty for lifetime with long‐term survivors

It is introduced the proportional hazards frailty model to allow a discrete distribution for the frailty variable. Frailty zero can be interpreted as being immune or cured. It is defined a class of survival models induced by a discrete frailty having a mixed Poisson distribution, which can account for unobserved dispersion. Further, a new regression to evaluate the effects of covariates in the cure fraction is constructed. Several former cure survival models are special cases of the proposed modeling framework. The inferential approach is based on Bayesian methods. Some simulation results are provided to assess the performance of the new regression. Its importance is illustrated by means of an application to colorectal cancer data.

Correlation between economic and industrial demand and scientific innovation: a case study of thermodynamics discipline statistics of National Natural Science Foundation of China

The thermodynamics discipline has developed rapidly in China in the past 30 years, especially with the integration of nanoparticles and other cutting-edge disciplines. However, it is still needed to discuss how to promote the interdisciplinary scientific innovation in thermal science. This paper makes an empirical analysis of counting data through Poisson regression, and constructs mixed Poisson distribution and panel Poisson distribution through project approval data of Thermodynamics of National Natural Science Foundation of China (NSFC) and China Education Statistics Yearbook from 2015 to 2017. The empirical results show that the four cities of Beijing, Tianjin, Chongqing and Shanghai account for 23% of youth projects and 42% of key projects. The human resources invested by universities and the use of teaching and research resources have a significant positive impact on the project approval of thermodynamics. In addition, empirical evidence shows that general projects and key projects have a higher chance of being approved than youth projects. It is worth noting that the empirical results reveal that the projects are more likely to be approved for applicants with expected deliverables of monographs or handbooks. In our conclusion, this result is related to the importance that NSFC attached to the practicability of technology and the urgency of industry needs. Monographs or handbook can provide more systematic guidance and more detailed illustration of a certain thermal scientific outcome. Therefore, applicants whose expected results are mainly published academic papers have a smaller chance of project approval in China.

The Po-F (Poisson-Frechet) distribution: an overview with respect to Rainfall data

A new mix distribution has been proposed in this paper. We have mixed Poisson distribution with Frechet distribution. Various properties like Reliability analysis, moments, quantile function and order statistics of the new distribution have been discussed. Maximum –likelihood estimation method has been used to estimate the parameters. The confidence interval along with fishar information matrix have been proposed. Finally 57 years rainfall data have been applied in the data analysis part and compared with other distributions. As a result, our new distribution have been fit better than the other distributions.

Bivariate Mixed Poisson and Normal Generalised Linear Models with Sarmanov Dependence—An Application to Model Claim Frequency and Optimal Transformed Average Severity

The aim of this paper is to introduce dependence between the claim frequency and the average severity of a policyholder or of an insurance portfolio using a bivariate Sarmanov distribution, that allows to join variables of different types and with different distributions, thus being a good candidate for modeling the dependence between the two previously mentioned random variables. To model the claim frequency, a generalized linear model based on a mixed Poisson distribution -like for example, the Negative Binomial (NB), usually works. However, finding a distribution for the claim severity is not that easy. In practice, the Lognormal distribution fits well in many cases. Since the natural logarithm of a Lognormal variable is Normal distributed, this relation is generalised using the Box-Cox transformation to model the average claim severity. Therefore, we propose a bivariate Sarmanov model having as marginals a Negative Binomial and a Normal Generalized Linear Models (GLMs), also depending on the parameters of the Box-Cox transformation. We apply this model to the analysis of the frequency-severity bivariate distribution associated to a pay-as-you-drive motor insurance portfolio with explanatory telematic variables.

Moderate parts in regenerative compositions: The case of regular variation

A regenerative random composition of integer n is constructed by allocating n standard exponential points over a countable number of intervals, comprising the complement of the closed range of a subordinator S. Assuming that the Lévy measure of S is infinite and regularly varying at zero of index −α, α∈(0,1), we find an explicit threshold r=r(n), such that the number Kn,r(n) of blocks of size r(n) converges in distribution without any normalization to a mixed Poisson distribution. The sequence (r(n)) turns out to be regularly varying with index α/(α+1) and the mixing distribution is that of the exponential functional of S. The result is derived as a consequence of a general Poisson limit theorem for an infinite occupancy scheme with power-like decay of the frequencies. We also discuss asymptotic behavior of Kn,w(n) in cases when w(n) diverges but grows slower than r(n). Our findings complement previously known strong laws of large numbers for Kn,r in case of a fixed r∈N. As a key tool we employ new Abelian theorems for Laplace–Stiletjes transforms of regularly varying functions with the indexes of regular variation diverging to infinity.