Consistency for the negative binomial regression with fixed covariate

We model an overdispersed count as a dependent measurement, by means of the Negative Binomial distribution. We consider quantitative regressors that are fixed by design. The expectation of the dependent variable is assumed to be a known function of a linear combination involving regressors and their coefficients. In the NB1-parametrization of the negative binomial distribution, the variance is a linear function of the expectation, inflated by the dispersion parameter, and not a generalized linear model. We apply a general result of Bradley and Gart (1962) to derive weak consistency and asymptotic normality of the maximum likelihood estimator for all parameters. To this end, we show (i) how to bound the logarithmic density by a function that is linear in the outcome of the dependent variable, independently of the parameter. Furthermore (ii) the positive definiteness of the matrix related to the Fisher information is shown with the Cauchy-Schwarz inequality