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Consistency for the negative binomial regression with fixed covariate

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Abstract

We model an overdispersed count as a dependent measurement, by means of the Negative Binomial distribution. We consider a quantitative covariate that is fixed by design. The expectation of the dependent variable is assumed to be a known function of a linear combination involving the possibly multidimensional covariate and its 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 the distribution not a generalized linear model. For the maximum likelihood estimator for all parameters we apply a general result of Bradley and Gart (Biometrika 49:205–214, 1962) to derive weak consistency and asymptotic normality and a technique in Fahrmeir and Kaufmann (Ann Stat 13:342–368, 1985) for strong consistency. To this end, we show (1) how to bound the logarithmic density by a function that is linear in the outcome of the dependent variable, independently of the parameter. Furthermore (2) the positive definiteness of the matrix related to the Fisher information is shown with the Cauchy–Schwarz inequality.

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Authors and Affiliations

  1. Faculty for Economic and Social Sciences, University of Rostock, 18051, Rostock, Germany

    Rafael Weißbach & Lucas Radloff

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  1. Rafael Weißbach
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  2. Lucas Radloff
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Correspondence to Rafael Weißbach.

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The financial support from the Deutsche Forschungsgemeinschaft (DFG) is gratefully acknowledged (Grant WE3573/3-1 “Multi-state, multi-time, multi-level analysis of health-related demographic events: Statistical aspects and applications” and CRC 823 “Statistical modelling of nonlinear dynamic processes”, Project A1: Dynamic Dependence Structures in Risky Asset Returns).

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Weißbach, R., Radloff, L. Consistency for the negative binomial regression with fixed covariate. Metrika 83, 627–641 (2020). https://doi.org/10.1007/s00184-019-00750-5

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  • DOI: https://doi.org/10.1007/s00184-019-00750-5

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