On Generalized Binomial and Negative Binomial Distributions for Dependent Bernoulli Variables

We study the distributions of the random variables S_n and V_r related to a sequence of dependent Bernoulli variables, where S_n denotes the number of successes in n trials and V_r the number of trials necessary to obtain r successes. The purpose of this article is twofold: (1) Generalizing some results on the “nature” of the binomial and negative binomial distributions we show that S_n and V_r can follow any prescribed discrete distribution. The corresponding joint distributions of the Bernoulli variables are characterized as the solutions of systems of linear equations. (2) We consider a specific type of dependence of the Bernoulli variables, where the probability of a success depends only on the number of previous successes. We develop some theory based on new closed-form representations for the probability mass functions of S_n and V_r which enable direct computations of the probabilities.     

On Probabilistic Proofs of Certain Binomial Identities

This short note gives a simple statistical proof of a binomial identity, by evaluating the Laplace transform of the maximum of n independent exponential random variables in two different ways. As a by-product, we obtain a rigorous proof of an interesting result concerning the exponential distribution. The connections between a probabilistic approach and our approach are discussed. In the process, several new binomial identities are also obtained.     

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.     

An Extended Binomial Distribution with Applications

Based on Skellam (Poisson difference) distribution, an extended binomial distribution is introduced as a byproduct of extending Moran's characterization of Poisson distribution to the Skellam distribution. Basic properties of the distribution are investigated. Also, estimation of the distribution parameters is obtained. Applications with real data are also described.     

Goodness-of-fit tests for binomial AR(1) processes

The binomial AR(1) model describes a nonlinear process with a first-order autoregressive (AR(1)) structure and a binomial marginal distribution. To develop goodness-of-fit tests for the binomial AR(1) model, we investigate the observed marginal distribution of the binomial AR(1) process, and we tackle its autocorrelation structure. Motivated by the family of power-divergence statistics for handling discrete multivariate data, we derive the asymptotic distribution of certain categorized power-divergence statistics for the case of a binomial AR(1) process. Then we consider Bartlett's formula, which is widely used in time series analysis to provide estimates of the asymptotic covariance between sample autocorrelations, but which is not applicable when the underlying process is nonlinear. Hence, we derive a novel Bartlett-type formula for the asymptotic distribution of the sample autocorrelations of a binomial AR(1) process, which is then applied to develop tests concerning the autocorrelation structure. Simulation studies are carried out to evaluate the size and power of the proposed tests under diverse alternative process models. Several real examples are used to illustrate our methods and findings.     

Binomial AR(1) processes with innovational outliers

Abstract

Binomial integer-valued AR processes have been well studied in the literature, but there is little progress in modeling bounded integer-valued time series with outliers. In this paper, we first review some basic properties of the binomial integer-valued AR(1) process and then we introduce binomial integer-valued AR(1) processes with two classes of innovational outliers. We focus on the joint conditional least squares (CLS) and the joint conditional maximum likelihood (CML) estimates of models’ parameters and the probability of occurrence of the outlier. Their large-sample properties are illustrated by simulation studies. Artificial and real data examples are used to demonstrate good performances of the proposed models.     

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.     

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.     

Two Conditionally Specified Mixed Binomial Distributions

Two generalized hypergeometric distributions are identified as mixed binomial distributions by conditional specification. Both distributions show profiles that are not possible in other mixed binomial distributions such as the beta-binomial distribution. A simulation study illustrates that beta-binomial distribution is more precise to fit data with usual profiles but the two distributions presented can improve the capability of fitting data in other less common scenes.     

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.     

Property of bivariate poisson distribution and its application to stochastic processes

Assuming that the random vectors X ₁ and X ₂ have independent bivariate Poisson distributions, the conditional distribution of X ₁ given X ₁?+?X ₂?=?n is obtained. The conditional distribution turns out to be a finite mixture of distributions involving univariate binomial distributions and the mixing proportions are based on a bivariate Poisson (BVP) distribution. The result is used to establish two properties of a bivariate Poisson stochastic process which are the bivariate extensions of the properties for a Poisson process given by Karlin, S. and Taylor, H. M. (1975). A First Course in Stochastic Processes, Academic Press, New York.     

Using Auxiliary Data for Binomial Parameter Estimation with Nonignorable Nonresponse

Nonignorable nonresponse is a nonresponse mechanism that depends on the values of the variable having nonresponse. When an observed data of a binomial distribution suffer missing values from a nonignorable nonresponse mechanism, the binomial distribution parameters become unidentifiable without any other auxiliary information or assumption. To address the problems of non identifiability, existing methods mostly based on the log-linear regression model. In this article, we focus on the model when the nonresponse is nonignorable and we consider to use the auxiliary data to improve identifiability; furthermore, we derive the maximum likelihood estimator (MLE) for the binomial proportion and its associated variance. We present results for an analysis of real-life data from the SARS study in China. Finally, the simulation study shows that the proposed method gives promising results.     

Binomial AR(1) processes: moments,cumulants, and estimation

The modelling and analysis of count-data time series are areas of emerging interest with various applications in practice. We consider the particular case of the binomial AR(1) model, which is well suited for describing binomial counts with a first-order autoregressive serial dependence structure. We derive explicit expressions for the joint (central) moments and cumulants up to order 4. Then, we apply these results for expressing moments and asymptotic distribution of the squared difference estimator as an alternative to the sample autocovariance. We also analyse the asymptotic distribution of the conditional least-squares estimators of the parameters of the binomial AR(1) model. The finite-sample performance of these estimators is investigated in a simulation study, and we apply them to real data about computerized workstations.     

The Least Upper Bound on the Poisson-Negative Binomial Relative Error

In this article, basic mathematical computations are used to determine the least upper bound on the relative error between the negative binomial cumulative distribution function with parameters n and p and the Poisson cumulative distribution function with mean λ =nq = n(1 ? p). Following this bound, it is indicated that the negative binomial cumulative distribution function can be properly approximated by the Poisson cumulative distribution function whenever q is sufficiently small. Five numerical examples are presented to illustrate the obtained result.     

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.     

Analyzing Hybrid Randomized Response Data with a Binomial Selection Procedure

The operating characteristics (OCs) of an indifference-zone ranking and selection procedure are derived for randomized response binomial data. The OCs include tables and figures to facilitate tradeoffs between sample size and a stated probability of a correct selection, i.e., correctly identifying the binomial population (out of k ≥ 2) characterized by the largest probability of success. Measures of efficiency are provided to assist the analyst in selection of an appropriate randomized response design for the collection of the data. A hybrid randomized response model, which includes the Warner model and the Greenberg et al. model, is introduced to facilitate comparisons among a wider range of statistical designs than previously available. An example comparing failure rates of contraceptive methods is used to illustrate the use of these new results.     

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.     

ROBUST ESTIMATION FOR EPIDEMIC MODELS

A robust approach to the analysis of epidemic data is suggested. This method is based on a natural extension of M-estimation for i.i.d. observations where the distribution may be asymmetric. It is discussed initially in the context of a general discrete time stochastic process before being applied to previously studied epidemic models. In particular we consider a class of chain binomial models and models based on time dependent branching processes. Robustness and efficiency properties are studied through simulation and some previously analysed data sets are considered.     

Discrete Linnik Weibull distribution

Abstract

We introduce a new family of distributions using truncated discrete Linnik distribution. This family is a rich family of distributions which includes many important families of distributions such as Marshall–Olkin family of distributions, family of distributions generated through truncated negative binomial distribution, family of distributions generated through truncated discrete Mittag–Leffler distribution etc. Some properties of the new family of distributions are derived. A particular case of the family, a five parameter generalization of Weibull distribution, namely discrete Linnik Weibull distribution is given special attention. This distribution is a generalization of many distributions, such as extended exponentiated Weibull, exponentiated Weibull, Weibull truncated negative binomial, generalized exponential truncated negative binomial, Marshall-Olkin extended Weibull, Marshall–Olkin generalized exponential, exponential truncated negative binomial, Marshall–Olkin exponential and generalized exponential. The shape properties, moments, median, distribution of order statistics, stochastic ordering and stress–strength properties of the new generalized Weibull distribution are derived. The unknown parameters of the distribution are estimated using maximum likelihood method. The discrete Linnik Weibull distribution is fitted to a survival time data set and it is shown that the distribution is more appropriate than other competitive models.     

Modeling Lifetimes by a Stochastic Process Hitting a Critical Point

First hitting times arise naturally in survival analysis where the underlying stochastic counting process represents the strength of the health of an individual. The patient experiences a clinical endpoint when this process reaches a critical point for the first time. We propose a very flexible and unified first hitting time density function in a stochastic carcinogenesis counting process, and its mathematical properties are investigated. The Poisson and negative binomial first hitting time models are addressed and two examples with real data are presented.