Non–zero inflated modified power series distributions

In this paper, we study the class of inflated modified power series distributions (IMPSD) where inflation occurs at any of the support points. This class include among other the generalized Poisson, the generalized negative binomial, the generalized logarithmic series and the lost games distributions. We give expressions for the moments, factorial moments and central moments of the IMPSD. The maximum likelihood estimation of the parameters of the IMPSD and the variance – covariance matrix of the estimators is obtained. We derive these estimators and their information matrices for mentioned above particular members of IMPSD class. The second part of this paper deals with the distribution of sum of independent and identically distributed random variables taking values s, s+1. s + 2, …, s ≥ 0, with modified power series distributions inflated at the point s.     

On some inflated generalized discrete distributions

Generalized discrete distributions such as the double Poisson and the double binomial family of Lagrange distributions are considered when the probabilities are inflated by a constant λ (0 < λ < 1). In each of the above cases, the effect of inflation on the variance is discussed. Also, the Bayesian estimate of inflation as well as those of the parameters are attempted. A maximum likelihood method is also suggested.     

Bayesian estimation for non zero inflated modified power series distribution under linex and generalized entropy loss functions

Abstract

In this paper, we derive Bayesian estimators of the parameters of modified power series distributions inflated at any of a support point under linex and general entropy loss function. We assume that the prior information can be summarized by a uniform, Beta, two-sided power, Gamma or generalized Pareto distributions. The obtained results are demonstrated on the generalized Poisson and the generalized negative binomial distribution inflated at a given point.     

Incomplete moments of modified power series districutions with applications

The recurrence relations between the incomplete moments and the factorial incomplete moments of the modified power series distributions (MPSD) are derived. These relations are employed to obtain the experessions for the incomplete moments and the incomplete factorial moments of some particular members of the MPSD class such as the generalized negative binomial, the generalized Poisson, the generalized logrithmic series, the lost game distribution and the distribution of the number of customers served in a busy period. An application of the incomplete moments of the generalized Poisson distribution is provided in the economic selection of a manufactured product. A numerical example is provided using the Poisson distribution and the Generalized Poisson distribution. The example illustrates the difference in results using the two models     

Modeling spatial overdispersion via the generalized Waring point process

Modeling spatial overdispersion requires point process models with finite‐dimensional distributions that are overdisperse relative to the Poisson distribution. Fitting such models usually heavily relies on the properties of stationarity, ergodicity, and orderliness. In addition, although processes based on negative binomial finite‐dimensional distributions have been widely considered, they typically fail to simultaneously satisfy the three required properties for fitting. Indeed, it has been conjectured by Diggle and Milne that no negative binomial model can satisfy all three properties. In light of this, we change perspective and construct a new process based on a different overdisperse count model, namely, the generalized Waring (GW) distribution. While comparably tractable and flexible to negative binomial processes, the GW process is shown to possess all required properties and additionally span the negative binomial and Poisson processes as limiting cases. In this sense, the GW process provides an approximate resolution to the conundrum highlighted by Diggle and Milne.     

The Generalized Waring Process and Its Application

The generalized Waring distribution is a discrete distribution with a wide spectrum of applications in areas such as accident statistics, income analysis, environmental statistics, etc. It has been used as a model that better describes such practical situations as opposed to the Poisson distribution or the negative binomial distribution. Associated to both the Poisson and negative binomial distributions are the well-known Poisson and Pólya processes. In this article, the generalized Waring process is defined. Two models have been shown to lead to the generalized Waring process. One is related to a Cox process, while the other is a compound Poisson process. The defined generalized Waring process is shown to be a stationary, but non homogenous Markov process. Several properties are studied and the intensity, individual intensity, and Chapman–Kolmogorov differential equations of it are obtained. Moreover, the Poisson and Pólya processes are shown to arise as special cases of the generalized Waring process. Using this fact, some known results and some properties of them are obtained.     

Minimum variance unbiased estimation in a modified power series distribution and some of its applications

In this paper we study the minimum variance unbiased estimation in the modified power series distribution introduced by the author (1974a). Necessary and sufficient conditions for the existence of minimum variance unbiased estimate (MVUE) of the parameter based on sufficient statistics are obtained. These results are, then, applied to obtain MVUE of θ^r (r ≥ 1) for the generalized negative binomial and the decapitated generalized negative binomial distributions (Jain and Consul, 1971). Similar estimates are obtained for the generalized Poisson (Consul and Jain, 1973a) and the generalized logarithmic series distributions (Jain and Gupta, 1973). Several of the well-known results follow trivially from the results obtained here.     

A Family Which Integrates the Generalized Charlier Series and Extended Noncentral Negative Binomial Distributions

The generalized Charlier series distribution includes the binomial distribution, and the noncentral negative binomial distribution extends the negative binomial distribution. The present article proposes a family of counting distributions, which contains both the generalized Charlier series and extended noncentral negative binomial distributions. Compound and mixture formulations of the proposed distribution are given. The probability mass function is expressible in terms of the confluent hypergeometric function as well as the Gauss hypergeometric function. Recursive formulae for probability mass function have been studied by Panjer, Sundt and Jewell, Schröter, Sundt, and Kitano et al. in the context of insurance risk. This article explores horizontal, vertical, triangular, and diagonal recursions. Recursive formulae as well as exact expressions for descending factorial moments are studied. The proposed distribution allows overdispersion or underdispersion relative to a Poisson distribution. An illustrative example of data fitting is given.     

On some distributions arising in inverse cluster sampling

In this paper, distributions of items sampled inversely in clusters are derived. In particular, negative binomial type of distributions are obtained and their properties are studied. A logarithmic series type of distribution is also defined as a limiting form of the obtained generalized negative binomial distribution.     

Another generalization of the logarithmic series and the geometric distributions

A new generalization of the logarithmic series distribution is presented based on a generalized negative binomial distribution obtained from a generalized Poisson distribution compounded with the truncated gamma distribution. By length biasing this generalized log-series distribution, another generalized geometric distribution is uresented. For the generalized log-series distribution, maximum likelihood estimators are developed and an example is presented for illustration.     

Doubly inflated Poisson model using Gaussian copula

Multivariate data are present in many research areas. Its analysis is challenging when assumptions of normality are violated and the data are discrete. The Poisson discrete data can be thought of as very common discrete type, but the inflated and the doubly inflated correspondence are gaining popularity (Sengupta, Chaganty, and Sabo 2015; Lee, Jung, and Jin 2009; Agarwal, Gelfand, and Citron-Pousty 2002).

Our aim is to build a statistical model that can be tractable and used to estimate the model parameters for the multivariate doubly inflated Poisson. To keep the correlation structure, we incorporate ideas from the copula distributions. A multivariate doubly inflated Poisson distribution using Gaussian copula is introduced. Data simulation and parameter estimation algorithms are also provided. Residual checks are carried out to assess any substantial biases. The model dimensionality has been increased to test the performance of the provided estimation method. All results show high-efficiency and promising outcomes in the modeling of discrete data and particularly the doubly inflated Poisson count type data, under a novel modified algorithm.     

On the Correlation Structure of Gaussian Copula Models for Geostatistical Count Data

We describe a class of random field models for geostatistical count data based on Gaussian copulas. Unlike hierarchical Poisson models often used to describe this type of data, Gaussian copula models allow a more direct modelling of the marginal distributions and association structure of the count data. We study in detail the correlation structure of these random fields when the family of marginal distributions is either negative binomial or zero‐inflated Poisson; these represent two types of overdispersion often encountered in geostatistical count data. We also contrast the correlation structure of one of these Gaussian copula models with that of a hierarchical Poisson model having the same family of marginal distributions, and show that the former is more flexible than the latter in terms of range of feasible correlation, sensitivity to the mean function and modelling of isotropy. An exploratory analysis of a dataset of Japanese beetle larvae counts illustrate some of the findings. All of these investigations show that Gaussian copula models are useful alternatives to hierarchical Poisson models, specially for geostatistical count data that display substantial correlation and small overdispersion.     

Regression for doubly inflated multivariate Poisson distributions

Dependent multivariate count data occur in several research studies. These data can be modelled by a multivariate Poisson or Negative binomial distribution constructed using copulas. However, when some of the counts are inflated, that is, the number of observations in some cells are much larger than other cells, then the copula-based multivariate Poisson (or Negative binomial) distribution may not fit well and it is not an appropriate statistical model for the data. There is a need to modify or adjust the multivariate distribution to account for the inflated frequencies. In this article, we consider the situation where the frequencies of two cells are higher compared to the other cells and develop a doubly inflated multivariate Poisson distribution function using multivariate Gaussian copula. We also discuss procedures for regression on covariates for the doubly inflated multivariate count data. For illustrating the proposed methodologies, we present real data containing bivariate count observations with inflations in two cells. Several models and linear predictors with log link functions are considered, and we discuss maximum likelihood estimation to estimate unknown parameters of the models.     

Bayesian Estimation for Deformed Modified Power Series Distribution

In this article, posterior distribution, posterior moments, and predictive distribution for the modified power series distributions deformed at any of a support point under linex and generalized entropy loss function are derived. It is assumed that the prior information can be summarized by a uniform, Beta, two-sided power, Gamma, or generalized Pareto distributions. The obtained results are demonstrated on the generalized Poisson and the generalized negative binomial distribution deformed at a given point.     

Negative binomial control limits for count data with extra‐Poisson variation

Traditional techniques for calculating control limits for processes with discrete responses are based on the Poisson distribution. However, for many processes, the assumption of a Poisson distribution is violated. In such cases, use of traditional Poisson control limits may result in an inflated risk of Type I error. The negative binomial distribution is a natural extension of the Poisson distribution and allows for over‐dispersion relative to the Poisson distribution. A simple approach to calculating exact and approximate control limits for count data based on the negative binomial distribution is described. The approach is illustrated by application to water bacteria count data taken from a water purification system. Copyright © 2003 John Wiley & Sons, Ltd.     

A generalized Bayesian nonlinear mixed‐effects regression model for zero‐inflated longitudinal count data in tuberculosis trials

In this paper, we investigate Bayesian generalized nonlinear mixed‐effects (NLME) regression models for zero‐inflated longitudinal count data. The methodology is motivated by and applied to colony forming unit (CFU) counts in extended bactericidal activity tuberculosis (TB) trials. Furthermore, for model comparisons, we present a generalized method for calculating the marginal likelihoods required to determine Bayes factors. A simulation study shows that the proposed zero‐inflated negative binomial regression model has good accuracy, precision, and credibility interval coverage. In contrast, conventional normal NLME regression models applied to log‐transformed count data, which handle zero counts as left censored values, may yield credibility intervals that undercover the true bactericidal activity of anti‐TB drugs. We therefore recommend that zero‐inflated NLME regression models should be fitted to CFU count on the original scale, as an alternative to conventional normal NLME regression models on the logarithmic scale.     

Ordering Comparison of Logarithmic Series Random Variables with their Mixtures

The problem of comparing some known distributions in various types of stochastic orderings has been of interest to many authors. In particular, several authors have been recently concerned with the comparison of Poisson, binomial, and negative binomial distributions with their respective mixtures. Incidentally, these distributions are among the four well-known distributions of the family of generalized power series distributions (GPSD's). The remaining distribution is the logarithmic series distribution. In this paper, we shall be concerned with comparing this remaining distribution of the class GPSD with its mixture in terms of various types of stochastic orderings such as the simple stochastic, likelihood ratio, uniformly more variable, convex, hazard rate and expectation orderings. Derivation of the results in this case prove to be computationally trickier than the other three. The special case when the means of the two distributions are the same is also discussed. Finally, an illustrative explicit example is provided.     

New polya and inverse polya distributions of order k

New Polya and inverse Polya distributions of order k are derived by means of generalized urn models and by compounding the binomial and negative binomial distributions of order k of Philippou (1986, 1983) with the beta distribution. It i s noted that the present Polpa distribution of order k includes as special cases a new hypergeometric distribution of order k, a negative one,an inverse one, and a discrete uniform of the same order. The probability generating functions, means and variances of the new distributions are obtained, and five asymptotic results are established relating them to the abovedmentioned binomial and negative binomial distributions of order k, and to the Poisson distribution of the same order of Philippou (1983).Moment estimates are also given and applications are indicated.     

Generalized Poisson–Lindley linear model for count data

The purpose of this paper is to develop a new linear regression model for count data, namely generalized-Poisson Lindley (GPL) linear model. The GPL linear model is performed by applying generalized linear model to GPL distribution. The model parameters are estimated by the maximum likelihood estimation. We utilize the GPL linear model to fit two real data sets and compare it with the Poisson, negative binomial (NB) and Poisson-weighted exponential (P-WE) models for count data. It is found that the GPL linear model can fit over-dispersed count data, and it shows the highest log-likelihood, the smallest AIC and BIC values. As a consequence, the linear regression model from the GPL distribution is a valuable alternative model to the Poisson, NB, and P-WE models.     

Joint (k,l)-hurdle random effects models for mixed longitudinal power series and normal outcomes

A random effects model for analyzing mixed longitudinal normal and count outcomes with and without the possibility of non ignorable missing outcomes is presented. The count response is inflated in two points (k and l) and the (k, l)-Hurdle power series is used as its distribution. The new distribution contains, as special submodels, several important distributions which are discussed, such as (k, l)-Hurdle Poisson and (k, l)-Hurdle negative binomial and (k, l)-Hurdle binomial distributions among others. Random effects are used to take into account the correlation between longitudinal outcomes and inflation parameters. A full likelihood-based approach is used to yield maximum likelihood estimates of the model parameters. A simulation study is performed in which for count outcome (k, l)-Hurdle Poisson, (k, l)-Hurdle negative binomial and (k, l)-Hurdle binomial distributions are considered. To illustrate the application of such modelling the longitudinal data of body mass index and the number of joint damage are analyzed.