Consistency for the negative binomial regression with fixed covariate
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Date
2018
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Abstract
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.
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Keywords
negative binomial regression, asymptotic normality, weak consistency, fixed covariates, NB1-parametrization