Estimation of Count Data using Bivariate Negative Binomial Regression Models

Negative binomial regression model (NBR) is a popular approach for modeling overdispersed count data with covariates. Several parameterizations have been performed for NBR, and the two well-known models, negative binomial-1 regression model (NBR-1) and negative binomial-2 regression model (NBR-2), have been applied. Another parameterization of NBR is negative binomial-P regression model (NBR-P), which has an additional parameter and the ability to nest both NBR-1 and NBR-2. This paper introduces several forms of bivariate negative binomial regression model (BNBR) which can be fitted to bivariate count data with covariates. The main advantages of having several forms of BNBR are that they are nested and allow likelihood ratio test to be performed for choosing the best model, they have flexible forms of mean-variance relationship, they can be fitted to bivariate count data with positive, zero or negative correlations, and they allow overdispersion of the two dependent variables. Applications of several forms of BNBR are illustrated on two sets of count data; Australian health care and Malaysian motor insurance.