In regression analysis of counts, a lack of simple and efficient algorithms for posterior computation has made Bayesian approaches appear unattractive and thus underdeveloped. We propose a lognormal and gamma mixed negative binomial (NB) regression model for counts, and present efficient closed-form Bayesian inference; unlike conventional Poisson models, the proposed approach has two free param...

We introduce negative binomial matrix factorization (NBMF), a matrix factorization technique specially designed for analyzing over-dispersed count data. It can be viewed as an extension of Poisson matrix factorization (PF) perturbed by a multiplicative term which models exposure. This term brings a degree of freedom for controlling the dispersion, making NBMF more robust to outliers. We show th...

By developing data augmentation methods unique to the negative binomial (NB) distribution, we unite seemingly disjoint count and mixture models under the NB process framework. We develop fundamental properties of the models and derive efficient Gibbs sampling inference. We show that the gamma-NB process can be reduced to the hierarchical Dirichlet process with normalization, highlighting its un...

The Poisson-Gamma model has properties that are very similar to the Poisson model discussed in Appendix C, in which the dependent variable i y is modeled as a Poisson variable with a mean i  where the model error is assumed to follow a Gamma distribution. As it names implies, the Poisson-Gamma is a mixture of two distributions and was first derived by Greenwood and Yule (1920). This mixture di...

This paper demonstrates that the conditional negative binomial model for panel data, proposed by Hausman, Hall and Griliches (1984), is not a true fixed-effects method. This method—which has been implemented in both Stata and LIMDEP—does not, in fact, control for all stable covariates. Three alternative methods are explored. A negative multinomial model yields the same estimator as the conditio...

In this paper, a quasi-negative binomial distribution (QNBD) derived from the class of generalized Lagrangian probability distributions is studied. The negative binomial distribution is a special case of QNBD. Some properties of QNBD, including the upper tail behavior and limiting distributions, are investigated. It is shown that the moments do not exist in some situations and the limiting dist...

This study attempts to explore the viability of dual-state models (i.e., zero-inflated and hurdle models) for traffic analysis zones (TAZs) based pedestrian and bicycle crash frequency analysis. Additionally, spatial spillover effects are explored in the models by employing exogenous variables from neighboring zones. The dual-state models such as zero-inflated negative binomial and hurdle negat...

Ecological river restoration projects aim to revitalize healthy and self-sustaining river systems that can provide irreplaceable benefits to human society. Cheonggyecheon and Anyangcheon are two sites of recent river restoration projects in Korea. To assess the economic value of two rivers, count data was collected to conduct the individual travel cost method (ITCM) in this study. Five statisti...

Abstract: This paper introduces three generalized geometric distributions: the binomial, negative binomial and Poisson distribution of the same order k. The generating functions and probability distributions of them are investigated, and then the corresponding modes of the distributions are discussed. By the Fibonacci sequence, the modes of the negative binomial distribution of order k are deri...