On Com-Negative Binomial Distributions

Abstract

In this paper we contrast the two forms of the Com-negative binomial distributions presented in Chakraborty and Ong (2016) and Zhang et al. (2018) as applied to over-dispersed count data. These two forms are designated COMNB and CMNB respectively in this paper. We also contrast the performances of the Negative binomial (NB) and the Com-Poisson (CMP) distributions to the former two distributions. These distributions are applied to example data sets that exhibit overdispersed and utrahigh zero-inflated count data. Because there are no closed form
expressions for the means and variances of COMNB, CMNB and the CMP models, the method of moments is proposed to generate these moments and consequently, we are able to generate Wald’s test statistics for these distributions rather than the AIC or -2LL as a measure of goodness-of fit. The zero-inflated forms of these distributions are further extended to fit zero-inflated models to some of these data sets. All the models are implemented in SAS PROC NLMIXED. For each distribution considered, MLE estimation based on the log-likelihood functions are obtained using the Adaptive Gaussian Quadrature (usually with 32 q-points) and then optimized by using the Newton-Raphson algorithm. Starting values are obtained from those obtained from employing the Poisson or Negative binomial models. Our results indicate that the COMNB performs much better than the newly proposed CMNB.

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