Statistical Inference Of Negative Binomial Regression Model

The counting random data are often used in practical problems. The count data regression models should be studied, in order to reveal the relation between the counting random data and the response variables. The basic commonly-used counting data regression models are the Poisson regression. Due to the complexity of counting data, the real data are not always fit for the basic Poisson. The deviation often appears greater (over-dispersion) or smaller (under-dispersion) than the expected situation. If the basic regression models were used for the data analysis, the results are often very poor. The assumption of the Poisson models is that the mean and variance are equal, while the variance is greater than the mean in practical problems. The negative binomial regression models can be a very good solution to this problem, which are successfully applied in several fields, such as biological statistics, econometrics, and veterinary epidemiology. The goodness-of-fit for the negative binomial regression models should be studied. Statistical diagnosis is an important research for data analysis, which is an important method of identifying outliers and influential points. Since the statistical analysis method of diagnosis of regression models is proposed by Cook, this method attracts many interests of domestic and foreign scholars. The method is simple, effective, and it has been successfully applied to some regression models, such as Poisson regression, negative binomial regression models, and the Poisson-inverse Gauss mixture model.This paper will systematically study the negative binomial regression models, including NB-1model, NB-2model, NB-P model, NB-H model, NB-C model. Firstly, we derived from the Poisson-Gamma mixed models with two and three parameters, i.e. NB-1, NB-2, NB-P models, based on Gamma distribution as the weight. The data of German economic society as an example is researched. Secondly, the NB-H and NB-C models as the extensions of the negative binomial regression models are discussed, which have rarely been studied in domestic and foreign literatures. The simulations of NB-H and NB-C models are made, and the data of German economic society is analyzed by the above models. After analysis and research on the above several models, the goodness of fit of the models is discussed by the AIC statistics and BIC statistics. The results illustrate that NB-P models are most effective. At last, the diagnostics on the NB-2and the NB-P regression models based on the case-deletion models are proposed. The plots of Cook statistics on NB-2and NB-P models are made.