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.     

Bivariate Negative Binomial Generalized Linear Models for Environmental Count Data

We propose a new bivariate negative binomial model with constant correlation structure, which was derived from a contagious bivariate distribution of two independent Poisson mass functions, by mixing the proposed bivariate gamma type density with constantly correlated covariance structure (Iwasaki & Tsubaki, 2005), which satisfies the integrability condition of McCullagh & Nelder (1989, p. 334). The proposed bivariate gamma type density comes from a natural exponential family. Joe (1997) points out the necessity of a multivariate gamma distribution to derive a multivariate distribution with negative binomial margins, and the luck of a convenient form of multivariate gamma distribution to get a model with greater flexibility in a dependent structure with indices of dispersion. In this paper we first derive a new bivariate negative binomial distribution as well as the first two cumulants, and, secondly, formulate bivariate generalized linear models with a constantly correlated negative binomial covariance structure in addition to the moment estimator of the components of the matrix. We finally fit the bivariate negative binomial models to two correlated environmental data sets.     

Negative binomial time series models based on expectation thinning operators

The study of count data time series has been active in the past decade, mainly in theory and model construction. There are different ways to construct time series models with a geometric autocorrelation function, and a given univariate margin such as negative binomial. In this paper, we investigate negative binomial time series models based on the binomial thinning and two other expectation thinning operators, and show how they differ in conditional variance or heteroscedasticity. Since the model construction is in terms of probability generating functions, typically, the relevant conditional probability mass functions do not have explicit forms. In order to do simulations, likelihood inference, graphical diagnostics and prediction, we use a numerical method for inversion of characteristic functions. We illustrate the numerical methods and compare the various negative binomial time series models for a real data example.     

A Bivariate Generalization of the Noncentral Negative Binomial Distribution

This article proposes a bivariate generalization of the noncentral negative binomial distribution which arises as a model in photon and neural counting. This bivariate generalization is derived as a mixed shifted bivariate negative binomial distribution. Various properties and parameter estimation, especially by a minimum distance method based on the probability generating function, are considered. To show the practical usefulness of the bivariate distribution proposed, an application to model low-flux astronomical images is discussed and a real data set has been analyzed.     

A New Markov Binomial Distribution

In this article, we introduce a two-state homogeneous Markov chain and define a geometric distribution related to this Markov chain. We define also the negative binomial distribution similar to the classical case and call it NB related to interrupted Markov chain. The new binomial distribution is related to the interrupted Markov chain. Some characterization properties of the geometric distributions are given. Recursion formulas and probability mass functions for the NB distribution and the new binomial distribution are derived.     

Confidence regions for two proportions from independent negative binomial distributions

The negative binomial distribution offers an alternative view to the binomial distribution for modeling count data. This alternative view is particularly useful when the probability of success is very small, because, unlike the fixed sampling scheme of the binomial distribution, the inverse sampling approach allows one to collect enough data in order to adequately estimate the proportion of success. However, despite work that has been done on the joint estimation of two binomial proportions from independent samples, there is little, if any, similar work for negative binomial proportions. In this paper, we construct and investigate three confidence regions for two negative binomial proportions based on three statistics: the Wald (W), score (S) and likelihood ratio (LR) statistics. For large-to-moderate sample sizes, this paper finds that all three regions have good coverage properties, with comparable average areas for large sample sizes but with the S method producing the smaller regions for moderate sample sizes. In the small sample case, the LR method has good coverage properties, but often at the expense of comparatively larger areas. Finally, we apply these three regions to some real data for the joint estimation of liver damage rates in patients taking one of two drugs.     

Discrete associated kernels method and extensions

Discrete kernel estimation of a probability mass function (p.m.f.), often mentioned in the literature, has been far less investigated in comparison with continuous kernel estimation of a probability density function (p.d.f.). In this paper, we are concerned with a general methodology of discrete kernels for smoothing a p.m.f. f. We give a basic of mathematical tools for further investigations. First, we point out a generalizable notion of discrete associated kernel which is defined at each point of the support of f and built from any parametric discrete probability distribution. Then, some properties of the corresponding estimators are shown, in particular pointwise and global (asymptotical) properties. Other discrete kernels are constructed from usual discrete probability distributions such as Poisson, binomial and negative binomial. For small samples sizes, underdispersed discrete kernel estimators are more interesting than the empirical estimator; thus, an importance of discrete kernels is illustrated. The choice of smoothing bandwidth is classically investigated according to cross-validation and, novelly, to excess of zeros methods. Finally, a unification way of this method concerning the general probability function is discussed.     

The generalized Poisson-binomial distribution and the computation of its distribution function

The Poisson-binomial distribution is useful in many applied problems in engineering, actuarial science and data mining. The Poisson-binomial distribution models the distribution of the sum of independent but non-identically distributed random indicators whose success probabilities vary. In this paper, we extend the Poisson-binomial distribution to a generalized Poisson-binomial (GPB) distribution. The GPB distribution corresponds to the case where the random indicators are replaced by two-point random variables, which can take two arbitrary values instead of 0 and 1 as in the case of random indicators. The GPB distribution has found applications in many areas such as voting theory, actuarial science, warranty prediction and probability theory. As the GPB distribution has not been studied in detail so far, we introduce this distribution first and then derive its theoretical properties. We develop an efficient algorithm for the computation of its distribution function, using the fast Fourier transform. We test the accuracy of the developed algorithm by comparing it with enumeration-based exact method and the results from the binomial distribution. We also study the computational time of the algorithm under various parameter settings. Finally, we discuss the factors affecting the computational efficiency of the proposed algorithm and illustrate the use of the software package.     

Some bivariate families of lagrangian probability distributions

The bivariate Lagrange expansion, given by Poincare (1986), has been explained and slightly modified which gives bivariate Lagrangian probability models. A generalized bivariate Lagrangian Poisson distribution with six parameters has been obtained and studied. Also, the bivariate Lagrangian binomial, bivariate Lagrangian negative binomial and bivariate Lagrangian logarithmic series distribution have been obtained.     

Using the predictive distribution to determine control limits for the Bayesian MEWMA chart

Bayesian control charts have been proposed for monitoring multivariate processes with the multivariate exponentially weighted moving average (MEWMA) statistic. It has been suggested that we use limits based on the predictive distribution of the MEWMA statistic. This analysis, however is based on the erroneous result that the average run length (ARL) is a function of the exceedance probability, that is, the probability that the first point exceeds the control limit. We show how this result can be corrected and we discuss how the Bayesian MEWMA chart with limits based on the predictive distribution compares with other multivariate control chart procedures.     

ESTIMATING THE NUMBER OF VISITS TO THE DOCTOR

The frequency of doctor consultations has direct consequences for health care budgets, yet little statistical analysis of the determinants of doctor visits has been reported. We consider the distribution of the number of visits to the doctor and, in particular, we model its dependence on a number of demographic factors. Examination of the Australian 1995 National Health Survey data reveals that generalized linear Poisson or negative binomial models are inadequate for modelling the mean as a function of covariates, because of excessive zero counts, and a mean‐variance relationship that varies enormously over covariate values. A negative binomial model is used, with parameter values estimated in subgroups according to the discrete combinations of the covariate values. Smoothing splines are then used to smooth and interpolate the parameter values. In effect the mean and the shape parameters are each modelled as (different) functions of gender, age and geographical factors. The estimated regressions for the mean have simple and intuitive interpretations. However, the dependence of the (negative binomial) shape parameter on the covariates is more difficult to interpret and is subject to influence by extreme observations. We illustrate the use of the model by estimating the distribution of the number of doctor consultations in the Statistical Local Area of Ryde, based on population numbers from the 1996 census.     

Estimation of probabilities in the class of modified power series distributions

In this paper we consider the class of modified power series distribution introduced by GUPTA (1974) and derive a minimum variance unbiased estimator (MVUE) of the probability function for this class. these results are then applied to obtain MVUE of the probability function for the generalized negative binomial distributions, the generalized poisson distribution, the generalized logarithmic series distribution and the lost game distribution. A large number of results in the literature follow trivially from out results as special cases.     

Statistical Analysis for the Inverse Trinomial Distribution

This article considers parameter estimation, goodness of fit, likelihood ratio and score tests, and model selection by Akaike information criterion for the inverse trinomial (IT) distribution, a classical one-dimensional random walk distribution. The IT distribution has a cubic variance function of the mean and is a generalization of the negative binomial distribution. Basic distributional properties and expressions for the probability mass function, recurrence formula, moments, and score functions are also presented.     

A Lagrangian Non Central Negative Binomial Distribution of the First Kind

A Lagrangian probability distribution of the first kind is proposed. Its probability mass function is expressed in terms of generalized Laguerre polynomials or, equivalently, a generalized hypergeometric function. The distribution may also be formulated as a Charlier series distribution generalized by the generalizing Consul distribution and a non central negative binomial distribution generalized by the generalizing Geeta distribution. This article studies formulation and properties of the distribution such as mixture, dispersion, recursive formulas, conditional distribution and the relationship with queuing theory. Two illustrative examples of application to fitting are given.     

The non-central negative binomial distribution: Further properties and applications

Abstract

The non-central negative binomial distribution is both a mixed and compound Poisson distribution with applications in photon and neural counting, statistical optics, astronomy and a stochastic reversible counter system. In this paper various important probabilistic properties of the non-central negative binomial distribution in practical applications like log-concavity, discrete self-decomposability, unimodality, asymptotic behavior and tail length of the probability distribution have been derived. The construction as a mixed Poisson process by specifying a joint distribution for the inter-arrival times and its application is illustrated by a fit to real life data set.     

Multivariate discrete exponential family of distributions and their properties

The paper generalizes the univariate discrete exponential family of distributions to the multivariate situation, and this generalization includes the multivariate power series distributions, the multivariate Lagrangian distributions, and the modified multivariate power-series distributions. This provides a unified approach for the study of these three classes of distributions. We obtain recurrence relations for moments and cumulants, and the maximum likelihood estimation for the discrete exponential family. These results are applied to some multivariate discrete distributions like the Lagrangian Poisson, Lagrangian (negative) multinomial, logarithmic series distributions and multivariate Lagrangian negative binomial distribution.     

On binomial and circular binomial distributions of orderk forl-overlapping success runs of lengthk

The number ofl-overlapping success runs of lengthk inn trials, which was introduced and studied recently, is presently reconsidered in the Bernoulli case and two exact formulas are derived for its probability distribution function in terms of multinomial and binomial coefficients respectively. A recurrence relation concerning this distribution, as well as its mean, is also obtained. Furthermore, the number ofl-overlapping success runs of lengthk inn Bernoulli trials arranged on a circle is presently considered for the first time and its probability distribution function and mean are derived. Finally, the latter distribution is related to the first, two open problems regarding limiting distributions are stated, and numerical illustrations are given in two tables. All results are new and they unify and extend several results of various authors on binomial and circular binomial distributions of orderk.     

Multivariate models for correlated count data

In this study, we deal with the problem of overdispersion beyond extra zeros for a collection of counts that can be correlated. Poisson, negative binomial, zero-inflated Poisson and zero-inflated negative binomial distributions have been considered. First, we propose a multivariate count model in which all counts follow the same distribution and are correlated. Then we extend this model in a sense that correlated counts may follow different distributions. To accommodate correlation among counts, we have considered correlated random effects for each individual in the mean structure, thus inducing dependency among common observations to an individual. The method is applied to real data to investigate variation in food resources use in a species of marsupial in a locality of the Brazilian Cerrado biome.     

On a characterization of the dispersion matrix based on the properties of regression

This paper provides a partial solution to a problem posed by J. Neyman (1965) regarding the characterization of multivariate negative binomial distribution based on the properties of regression. It is shown that some of the properties of regression characterize the form of the nonsingular dispersion matrix of the parent distribution, which, interestingly enough, corresponds to only two types viz. those of positive and negative multivariate binomial distributions.     

Some Theoretical Comparisons of Negative Binomial and Zero-Inflated Poisson Distributions

In this article, we compare the zero-inflated Poisson (ZIP) and negative binomial (NB) distributions based on three most important criteria: the probability of zero, the mean value, and the variance. Our results show that with same mean value and variance, the ZIP distribution always has a larger probability of zeros; with same mean value and probability of zeros, the NB distribution always has a larger variance; and with same variance and probability of zeros, the ZIP distribution always has a larger mean value. We also study the properties of Vuong test in model selection in three cases by simulations.