On bivariate generalized binomial and negative binomial distributions

The present paper deals with two types of generalized general binomial (binomial or negative binomial) distributions: (i) a univariate general binomial generalized by a bivariate distribution and (ii) a bivariate general binomial generalized by two independent univariate distributions. The probabilities, moments, conditional distributions and regression functions for these distributions are obtained in terms of bipartitional polynomials. Moreover recurrence relations for the probabilities and moments, independent of the bipartitional polynomials, are given. Finally these general results are applied to the (i) Binomial-Bivariate Poisson and (ii) Bivariate Binomial-Poissons distributions.     

A new bivariate generalized Poisson distribution

In this paper, a new bivariate generalized Poisson distribution (GPD) that allows any type of correlation is defined and studied. The marginal distributions of the bivariate model are the univariate GPDs. The parameters of the bivariate distribution are estimated by using the moment and maximum likelihood methods. Some test statistics are discussed and one numerical data set is used to illustrate the applications of the bivariate model.     

Bivariate exponentiated‐exponential geometric regression model

A bivariate exponentiated‐exponential geometric regression model that allows negative, zero, or positive correlation is defined and studied. The model can accommodate under‐ or over‐dispersed count data. The regression model is based on the univariate exponentiated‐exponential geometric distribution, and the marginal means of the bivariate model are functions of the explanatory variables. The parameters of the bivariate regression model are estimated by using the maximum likelihood method. Some test statistics including goodness of fit are discussed. A simulation study is conducted to compare the model with the bivariate generalized Poisson regression model. One numerical data set is used to illustrate the application of the regression model.     

Description of purchase incidence by multivariate heterogeneous Poisson processes

In this paper the application of bivariate Poisson heterogeneous models to budget data is studied. This study was motivated from inconsistencies that we encountered when univariate Poisson based models were applied to cumulative data sets. Application of a multivariate Poisson based model is a possible solution to this problem. In this paper we will study the feasibility of estimators based on these models.     

Investigating the impact of excess zeros on hurdle‐generalized Poisson regression model with right censored count data

Typically, a Poisson model is assumed for count data. In many cases, there are many zeros in the dependent variable, thus the mean is not equal to the variance value of the dependent variable. Therefore, Poisson model is not suitable anymore for this kind of data because of too many zeros. Thus, we suggest using a hurdle‐generalized Poisson regression model. Furthermore, the response variable in such cases is censored for some values because of some big values. A censored hurdle‐generalized Poisson regression model is introduced on count data with many zeros in this paper. The estimation of regression parameters using the maximum likelihood method is discussed and the goodness‐of‐fit for the regression model is examined. An example and a simulation will be used to illustrate the effects of right censoring on the parameter estimation and their standard errors.     

A unified approach to bivariate discrete distributions

Here we introduce a bivariate generalized hypergeometric factorial moment distribution (BGHFMD) through its probability generating function (p.g.f.) whose marginal distributions are the generalized hypergeometric factorial moment distributions introduced by Kemp and Kemp (Bull Int Stat Inst 43:336–338,1969). Well-known bivariate versions of distributions such as binomial, negative binomial and Poisson are special cases of this distribution. A genesis of the distribution and explicit closed form expressions for the probability mass function of the BGHFMD, its factorial moments and the p.g.f.’s of its conditional distributions are derived here. Certain recurrence relations for probabilities, moments and factorial moments of the bivariate distribution are also established.     

On some properties of the linear function Poisson binomial and negative binomial distributions

Summary A general model in fluctuations of sums of random variables leading, under certain assumptions, to each of the generalized and linear function Poisson, binomial and negative binomial distributions is presented. Moreover the generating functions and the factorial moments of the linear function Poisson, binomial and negative binomial distributions are obtained in close forms and certain distributional properties are discussed.     

ON THE DISTRIBUTION OF ORDER STATISTICS FOR A RANDOM SAMPLE SIZE

The distributions of the life lengths of a parallel and of a series system with a random number of components have been studied in reliability theory. In this paper we obtain the distributions of the i'th order statistics and the range, assuming the sample size to be random, with a generalized negative binomial, a generalized Poisson and a generalized logarithmic series distribution. The results of Raghunandanan and Patil (1972) follow immediately from our results.     

ON THE UNIMODALITY OF GENERALIZED POISSON DISTRIBUTION

Abstract. The unimodality property is very important in many statistical problems. In this paper, it is shown that the generalized Poisson distribution is unimodal. Upper and lower bounds to the mode are given.     

Estimating a generalized correlation coefficient for a generalized bivariate probit model

In this paper we consider semiparametric estimation of a generalized correlation coefficient in a generalized bivariate probit model. The generalized correlation coefficient provides a simple summary statistic measuring the relationship between the two binary decision processes in a general framework. Our semiparametric estimation procedure consists of two steps, combining semiparametric estimators for univariate binary choice models with the method of maximum likelihood for the bivariate probit model with nonparametrically generated regressors. The estimator is shown to be consistent and asymptotically normal. The estimator performs well in our simulation study.     

Poisson Regression for Clustered Data

We compare five methods for parameter estimation of a Poisson regression model for clustered data: (1) ordinary (naive) Poisson regression (OP), which ignores intracluster correlation, (2) Poisson regression with fixed cluster‐specific intercepts (FI), (3) a generalized estimating equations (GEE) approach with an equi‐correlation matrix, (4) an exact generalized estimating equations (EGEE) approach with an exact covariance matrix, and (5) maximum likelihood (ML). Special attention is given to the simplest case of the Poisson regression with a cluster‐specific intercept random when the asymptotic covariance matrix is obtained in closed form. We prove that methods 1–5, except GEE, produce the same estimates of slope coefficients for balanced data (an equal number of observations in each cluster and the same vectors of covariates). All five methods lead to consistent estimates of slopes but have different efficiency for unbalanced data design. It is shown that the FI approach can be derived as a limiting case of maximum likelihood when the cluster variance increases to infinity. Exact asymptotic covariance matrices are derived for each method. In terms of asymptotic efficiency, the methods split into two groups: OP & GEE and EGEE & FI & ML. Thus, contrary to the existing practice, there is no advantage in using GEE because it is substantially outperformed by EGEE and FI. In particular, EGEE does not require integration and is easy to compute with the asymptotic variances of the slope estimates close to those of the ML.     

ON THE DISTRIBUTIONS OF ORDER STATISTICS FOR A RANDOM SAMPLE SIZE*

The probability distribution of the i –th and j–th order statistics and of the range R of a sample of size n, taken from a population with probability density function f (x) have been obtained when the sample size n is a random variable N and has: (i) a generalized Poisson distribution; and (ii) a generalized negative bonimial distribution. Specific results are then obtained; (a) when f (x) is uniform over (0,1); and (b) when f(x) is exponential. All the results for N, being a Poisson, binomial and negative binomial rv follow as special cases.     

On the Generalized Bootstrap for Sample Surveys with Special Attention to Poisson Sampling

We study the generalized bootstrap technique under general sampling designs. We focus mainly on bootstrap variance estimation but we also investigate the empirical properties of bootstrap confidence intervals obtained using the percentile method. Generalized bootstrap consists of randomly generating bootstrap weights so that the first two (or more) design moments of the sampling error are tracked by the corresponding bootstrap moments. Most bootstrap methods in the literature can be viewed as special cases. We discuss issues such as the choice of the distribution used to generate bootstrap weights, the choice of the number of bootstrap replicates, and the potential occurrence of negative bootstrap weights. We first describe the generalized bootstrap for the linear Horvitz‐Thompson estimator and then consider non‐linear estimators such as those defined through estimating equations. We also develop two ways of bootstrapping the generalized regression estimator of a population total. We study in greater depth the case of Poisson sampling, which is often used to select samples in Price Index surveys conducted by national statistical agencies around the world. For Poisson sampling, we consider a pseudo‐population approach and show that the resulting bootstrap weights capture the first three design moments of the sampling error. A simulation study and an example with real survey data are used to illustrate the theory.     

Modelling Time Series of Counts in Epidemiology

We review generalized dynamic models for time series of count data. Usually temporal counts are modelled as following a Poisson distribution, and a transformation of the mean depends on parameters which evolve smoothly with time. We generalize the usual dynamic Poisson model by considering continuous mixtures of the Poisson distribution. We consider Poisson‐gamma and Poisson‐log‐normal mixture models. These models have a parameter for each time t which captures possible extra‐variation present in the data. If the time interval between observations is short, many observed zeros might result. We also propose zero inflated versions of the models mentioned above. In epidemiology, when a count is equal to zero, one does not know if the disease is present or not. Our model has a parameter which provides the probability of presence of the disease given no cases were observed. We rely on the Bayesian paradigm to obtain estimates of the parameters of interest, and discuss numerical methods to obtain samples from the resultant posterior distribution. We fit the proposed models to artificial data sets and also to a weekly time series of registered number of cases of dengue fever in a district of the city of Rio de Janeiro, Brazil, during 2001 and 2002.     

Constructing Generalized FGM Copulas by Means of Certain Univariate Distributions

In this paper we focus on specific generalized Fairlie- Gumbel-Morgenstern (or Sarmanov) copulas which are generated by a single function (so-called generator or generator function) defined on the unit interval. In particular, we introduce a class of generators based on density-quantile functions of certain univariate distributions. Many of the generator functions from the literature are recovered as special cases. Moreover, two new generators are suggested, implying to new copulas. Finally, the opposite way around, it is shown how to calculate the univariate distribution which belongs to a given copula generator function.     

An application of the generalized Poisson difference distribution to the Bayesian modelling of football scores

The analysis of sports data, in particular football match outcomes, has always produced an immense interest among the statisticians. In this paper, we adopt the generalized Poisson difference distribution (GPDD) to model the goal difference of football matches. We discuss the advantages of the proposed model over the Poisson difference (PD) model, which was also used for the same purpose. The GPDD model, like the PD model, is based on the goal difference in each game that allows us to account for the correlation without explicitly modelling it. The main advantage of the GPDD model is its flexibility in the tails by considering shorter as well as longer tails than the PD distribution. We carry out the analysis in a Bayesian framework in order to incorporate external information, such as historical knowledge or data, through the prior distributions. We model both the mean and the variance of the goal difference and show that such a model performs considerably better than a model with a fixed variance. Finally, the proposed model is fitted to the 2012–2013 Italian Serie A football data, and various model diagnostics are carried out to evaluate the performance of the model.     

Assessing the robustness of estimators when fitting Poisson inverse Gaussian models

The generalized linear mixed model (GLMM) extends classical regression analysis to non-normal, correlated response data. Because inference for GLMMs can be computationally difficult, simplifying distributional assumptions are often made. We focus on the robustness of estimators when a main component of the model, the random effects distribution, is misspecified. Results for the maximum likelihood estimators of the Poisson inverse Gaussian model are presented.     

GENERALIZED AUTOREGRESSIVE SCORE MODELS WITH APPLICATIONS

We propose a class of observation‐driven time series models referred to as generalized autoregressive score (GAS) models. The mechanism to update the parameters over time is the scaled score of the likelihood function. This new approach provides a unified and consistent framework for introducing time‐varying parameters in a wide class of nonlinear models. The GAS model encompasses other well‐known models such as the generalized autoregressive conditional heteroskedasticity, autoregressive conditional duration, autoregressive conditional intensity, and Poisson count models with time‐varying mean. In addition, our approach can lead to new formulations of observation‐driven models. We illustrate our framework by introducing new model specifications for time‐varying copula functions and for multivariate point processes with time‐varying parameters. We study the models in detail and provide simulation and empirical evidence. Copyright © 2012 John Wiley & Sons, Ltd.     

Estimation in the generalized Poisson model via robust testing

An estimation method is presented which compromises robust efficiency with computational feasibility in the case of the generalized Poisson model. The formal setup is built on flexible nonparametric extensions of the underlying model. The estimation efficiency is expressed via minimax properties of tests resulting from expansions of estimators. The nonparametric neighborhoods related to the proposed score function are exemplified and a real data case is analysed. The resulting method balances several qualitative features of statistical inference: strong differentiability (asymptotic derivations are more accurate), efficiency and natural model extension (quality of formal basic assumptions).     

A local maximum likelihood estimator for Poisson regression

A local maximum likelihood estimator based on Poisson regression is presented as well as its bias, variance and asymptotic distribution. This semiparametric estimator is intended to be an alternative to the Poisson, negative binomial and zero-inflated Poisson regression models that does not depend on regularity conditions and model specification accuracy. Some simulation results are presented. The use of the local maximum likelihood procedure is illustrated on one example from the literature. This procedure is found to perform well. This research was partially supported by Calouste Gulbenkian Foundation and PRODEP III.