On additive and multiplicative damage models and the characterizations of linear and logarithmic exponential families

Rao (1963) has formulated a damage model which we call an additive damage model. A suitable damage model, which we call a multiplicative damage model, has been considered by Krishnaji (1970) for income-related problems. In these models, an original observation is subjected to damage, e.g., death or under-reporting, according to a specified probability law. Within the framework of an additive damage model, with a special form of damage, characterizations of the linear and logarithmic exponential families are formulated using regression properties of the damaged part on the undamaged part. The characterizations of the gamma and Pareto distributions that have been found of some use in the theory of income distributions, are obtained as special cases. Similar results are investigated within the framework of the multiplicative damage model.     

Characterizations for the original observations with linearity of regression in additive damage models

This paper investigates the characterizations of certain discrete distributions within the framework of multivariate additive damage models. The univariate case for such models appoared in an article by N. Krishnaii (1974) and Rao and Rubin (1964). In this paper the survival distriution in specified and it is shown that linearity of the regression of the undamaged part on the damaged part, or the damaged part on the undamaged part leads to the characterizations of independent binomials, independent negative binomials, independent Poissons, multinomial and negative multinomial for the original p-dimensional observation.     

Analysis of hypoglycemic events using negative binomial models

Negative binomial regression is a standard model to analyze hypoglycemic events in diabetes clinical trials. Adjusting for baseline covariates could potentially increase the estimation efficiency of negative binomial regression. However, adjusting for covariates raises concerns about model misspecification, in which the negative binomial regression is not robust because of its requirement for strong model assumptions. In some literature, it was suggested to correct the standard error of the maximum likelihood estimator through introducing overdispersion, which can be estimated by the Deviance or Pearson Chi‐square. We proposed to conduct the negative binomial regression using Sandwich estimation to calculate the covariance matrix of the parameter estimates together with Pearson overdispersion correction (denoted by NBSP). In this research, we compared several commonly used negative binomial model options with our proposed NBSP. Simulations and real data analyses showed that NBSP is the most robust to model misspecification, and the estimation efficiency will be improved by adjusting for baseline hypoglycemia. Copyright © 2013 John Wiley & Sons, Ltd.     

A geometric time-series model with an alternative dependent Bernoulli counting series

In this article, we first introduce an alternative way for construction of the generalized binomial thinning operator with dependent counting series. Some properties of this thinning operator are derived and discussed. Then, by using this thinning operator, we introduce an integer-valued time-series model with geometric marginals. Some conditional and unconditional properties of this model are derived and discussed. Some estimation methods are considered and for some of them, asymptotic properties of the obtained estimates are derived. Performances of the estimates are discussed through some simulations. Finally, a real data example is considered and the goodness-of-fit of this model is compared with the models based on the binomial, negative binomial, and dependent binomial thinning operators.     

A note on the EM algorithm for estimation in the destructive negative binomial cure rate model

In this note we present a modification in the EM algorithm for the destructive negative binomial cure rate model. This alteration enables us to obtain the estimates of the whole parameter vector from the complete log-likelihood function, avoiding the corresponding observed log-likelihood function, which is more involved. To achieve this goal, we resort to the mixture representation of the negative binomial distribution in terms of the Poisson and gamma distributions.     

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.     

On the beta-correlated binomial (bcb) distribution - a three parameter generalization of the binomial distribution

A modification of the beta-correlated binomial (BCB) distribution of Paul (1985) is proposed. This modification over- comes some theoretical difficulties encountered in the original BCB distribution.     

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.     

Inflated modified power series distributions with applications

In certain applications involving discrete data, it is sometimes found that X = 0 is observed with a frequency significantly higher than predicted by the assumed model. Zero inflated Poisson, binomial and negative binomial models have been employed in some clinical trials and in some regression analysis problems.

In this paper, we study the zero inflated modified power series distributions (IMPSD) which include among others the generalized Poisson and the generalized negative binomial distributions and hence the Poisson, binomial and negative binomial distributions. The structural properties along with the distribution of the sum of independent IMPSD variables are studied. The maximum likelihood estimation of the parameters of the model is examined and the variance-covariance matrix of the estimators is obtained. Finally, examples are presented for the generalized Poisson distribution to illustrate the results.     

The two-sided power-binomial distribution for modeling binary count data

The two-sided power (TSP) distribution is a flexible two-parameter distribution having uniform, power function and triangular as sub-distributions, and it is a reasonable alternative to beta distribution in some cases. In this work, we introduce the TSP-binomial model which is defined as a mixture of binomial distributions, with the binomial parameter p having a TSP distribution. We study its distributional properties and demonstrate its use on some data. It is shown that the newly defined model is a useful candidate for overdispersed binomial data.     

The optimal group size using inverse binomial group testing considering misclassification

ABSTRACT

Inverse binomial sampling is preferred for quick report. It is also recommended when the population proportion is really small to ensure a positive sample is contained. Group testing has been discussed extensively under binomial model, but not so much under negative binomial model. In this study, we investigate the problem of how to determine the group size using inverse binomial group testing. We propose to choose the optimal group size by minimizing asymptotic variance of the estimator or the cost relative to Fisher information. We show the good performance of our estimator by applying to the data of Chlamydia.     

A Characterization For Gpsd Signals In Additive Noise

In this paper we consider a convoluted generalized power series distibution and characterize the distributions by soiutions to system of differential equations. Characterization results are derived for Poisson, binomial, geometric and Pascal (negative binomial) as special cases and later unified with Samaniego [1976, 1980] and Samaniego and Gong [1979]

receiveddate="Oct1985" reviseddate="Jun1986"     

Monitoring correlated processes with binomial marginals

Few approaches for monitoring autocorrelated attribute data have been proposed in the literature. If the marginal process distribution is binomial, then the binomial AR(1) model as a realistic and well-interpretable process model may be adequate. Based on known and newly derived statistical properties of this model, we shall develop approaches to monitor a binomial AR(1) process, and investigate their performance in a simulation study. A case study demonstrates the applicability of the binomial AR(1) model and of the proposed control charts to problems from statistical process control.     

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 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.     

Extended binomial AR(1) processes with generalized binomial thinning operator

Abstract

In this article, we introduce an extended binomial AR(1) model based on the generalized binomial thinning operator. This operator relaxes the independence assumption of the binomial thinning operator and contains dependent Bernoulli counting series. The new model contains the binomial AR(1) model as a particular case. Some probabilistic and statistical properties are explored. Estimators of the model parameters are derived by conditional maximum likelihood (CML), conditional least squares (CLS) and weighted conditional least squares (WCLS) methods. Some asymptotic properties and numerical results of the estimators are studied. The good performance of the new model is illustrated, among other competitive models in the literature, by an application to the monthly drunken driving counts.     

Characterization of generalized markov-polya and generalized polya-eggenberger distributions

A discrete model is considered where the original observation is subjected to partial destruction according to the Generalized Markov-Polya (GMP) damage model. A characterization of the Generalized Polya-Eggenberger distribution (GPED) is given in the context of the Rao-Rubin condition. More specifically, if the probability that an observation n of a non-negative integer valued r.v.X is reduced to an integer k during a damage, process is given by the GMPD, and if the resulting r.v.Y is such thatrit satisfies the RR-conditlon, then X has a GPED. Secondly, if N = A + B, where B is the missing part and A is the recorded part such that the conditional distribution P(A= x|N=n) is the GMPD, then the r.v.'s A and B are independent if, and only if, N has a GPED. Several other characterizations are also given for these two distributions. The results of Rao-Rubin ‘1964’, Patil-Ratnaparkhi (1977) and Consul (1975) follow as special cases.     

Estimation in a bivariate integer-valued autoregressive process

ABSTRACT

A bivariate integer-valued autoregressive time series model is presented. The model structure is based on binomial thinning. The unconditional and conditional first and second moments are considered. Correlation structure of marginal processes is shown to be analogous to the ARMA(2, 1) model. Some estimation methods such as the Yule–Walker and conditional least squares are considered and the asymptotic distributions of the obtained estimators are derived. Comparison between bivariate model with binomial thinning and bivariate model with negative binomial thinning is given.     

Quasi-binomial zero-inflated regression model suitable for variables with bounded support

In recent years, a variety of regression models, including zero-inflated and hurdle versions, have been proposed to explain the case of a dependent variable with respect to exogenous covariates. Apart from the classical Poisson, negative binomial and generalised Poisson distributions, many proposals have appeared in the statistical literature, perhaps in response to the new possibilities offered by advanced software that now enables researchers to implement numerous special functions in a relatively simple way. However, we believe that a significant research gap remains, since very little attention has been paid to the quasi-binomial distribution, which was first proposed over fifty years ago. We believe this distribution might constitute a valid alternative to existing regression models, in situations in which the variable has bounded support. Therefore, in this paper we present a zero-inflated regression model based on the quasi-binomial distribution, taking into account the moments and maximum likelihood estimators, and perform a score test to compare the zero-inflated quasi-binomial distribution with the zero-inflated binomial distribution, and the zero-inflated model with the homogeneous model (the model in which covariates are not considered). This analysis is illustrated with two data sets that are well known in the statistical literature and which contain a large number of zeros.     

A family of distributions to model load sharing systems

The main characteristic of a load sharing system is that after the failure of one component the surviving components have to shoulder extra load and hence are prone to failure at an earlier time than what is expected under the original model. In others, the failure of one component may release extra resources to the survivors, thus delaying the system failure. In this paper we consider such m component systems and some observation schemes and identifiability issues under them. Then we construct a general semiparametric multivariate family of distributions which explicitly models this phenomenon through proportional conditional hazards. We suggest estimates for the constant of proportionality. We propose a nonparametric test for the hypothesis that the failures take place independently according to the common distribution against the alternative hypothesis that the second failure takes place earlier than warranted, study its properties and illustrate its use.