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Consistency for the negative binomial regression with fixed covariate

Author

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  • Rafael Weißbach

    (University of Rostock)

  • Lucas Radloff

    (University of Rostock)

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.

Suggested Citation

  • Rafael Weißbach & Lucas Radloff, 2020. "Consistency for the negative binomial regression with fixed covariate," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 83(5), pages 627-641, July.
    • Handle: RePEc:spr:metrik:v:83:y:2020:i:5:d:10.1007_s00184-019-00750-5
    DOI: 10.1007/s00184-019-00750-5
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    References listed on IDEAS

    as
    1. Voß, Sebastian & Weißbach, Rafael, 2014. "A score-test on measurement errors in rating transition times," Journal of Econometrics, Elsevier, vol. 180(1), pages 16-29.
    2. Gourieroux,Christian & Monfort,Alain, 1995. "Statistics and Econometric Models," Cambridge Books, Cambridge University Press, number 9780521471626.
    3. Weißbach, Rafael & Walter, Ronja, 2010. "A likelihood ratio test for stationarity of rating transitions," Journal of Econometrics, Elsevier, vol. 155(2), pages 188-194, April.
    4. Gourieroux,Christian & Monfort,Alain, 1995. "Statistics and Econometric Models," Cambridge Books, Cambridge University Press, number 9780521477451.
    5. Rafael Weißbach, Rafael & Voß, Sebastian, 2014. "A score-test on measurement errors in rating transition times," VfS Annual Conference 2014 (Hamburg): Evidence-based Economic Policy 100532, Verein für Socialpolitik / German Economic Association.
    6. Gourieroux,Christian & Monfort,Alain, 1995. "Statistics and Econometric Models," Cambridge Books, Cambridge University Press, number 9780521405515.
    7. Gourieroux,Christian & Monfort,Alain, 1995. "Statistics and Econometric Models," Cambridge Books, Cambridge University Press, number 9780521477444.
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    Cited by:

    1. Shaomin Li & Haoyu Wei & Xiaoyu Lei, 2021. "Heterogeneous Overdispersed Count Data Regressions via Double Penalized Estimations," Papers 2110.03552, arXiv.org, revised Feb 2022.
    2. Rafael Weißbach & Dominik Wied, 2022. "Truncating the exponential with a uniform distribution," Statistical Papers, Springer, vol. 63(4), pages 1247-1270, August.
    3. Rafael Weißbach & Yongdai Kim & Achim Dörre & Anne Fink & Gabriele Doblhammer, 2021. "Left-censored dementia incidences in estimating cohort effects," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 27(1), pages 38-63, January.
    4. Shaomin Li & Haoyu Wei & Xiaoyu Lei, 2022. "Heterogeneous Overdispersed Count Data Regressions via Double-Penalized Estimations," Mathematics, MDPI, vol. 10(10), pages 1-25, May.

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