George Tzougas, London School of Economics
An EM type algorithm for ML estimation of the negative binomial-gamma regression model
Mixed Poisson regression models have been massively overused for modeling heterogeneous count data in a wide range of areas, such as sociology, biology, biometrics, genetics, medicine, marketing, applied econometrics and insurance. The negative binomial - gamma regression model can be considered as an alternative to mixed Poisson models since it can adequately capture the stylized characteristics of highly dispersed count data. However, due to the complexity of its likelihood, direct maximization is difficult and has not been addressed in the literature so far. The main achievement is that we propose a simple Expectation-Maximization (EM) type algorithm for maximum likelihood estimation of the model which can overcome the numerical difficulties occurring when standard numerical techniques are used. Moreover, the algorithm we present has the considerable mathematical flexibility for fitting other mixed Negative Binomial regression models stemming from several other mixing distributions. Additionally, the by-products of the algorithm can be useful for further inference. For example, the posterior expectations, which are readily available after the convergence of the EM algorithm, can be employed for Empirical Bayes estimation and can be used to predict future outcomes. Finally, a real data application using motor insurance data is examined and some operating characteristics of the algorithm are discussed.