The Annals of Statistics

Consistency and Asymptotic Normality of the Maximum Likelihood Estimator in Generalized Linear Models

Ludwig Fahrmeir and Heinz Kaufmann

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Abstract

Generalized linear models are used for regression analysis in a number of cases, including categorical responses, where the classical assumptions are violated. The statistical analysis of such models is based on the asymptotic properties of the maximum likelihood estimator. We present mild general conditions which, respectively, assure weak or strong consistency or asymptotic normality. Most of the previous work has been concerned with natural link functions. In this case our normality condition, though obtained by a different approach, is closely related to a condition of Haberman (1977a). Examples show how the general conditions reduce to weak requirements for special exponential families. Further, for regressors with a compact range, sufficient conditions are given which do not involve the unknown parameter, and are therefore easy to check in practice. Responses with a bounded range, e.g. categorical responses, and stochastic regressors also are treated.

Article information

Source
Ann. Statist., Volume 13, Number 1 (1985), 342-368.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176346597

Digital Object Identifier
doi:10.1214/aos/1176346597

Mathematical Reviews number (MathSciNet)
MR773172

Zentralblatt MATH identifier
0594.62058

JSTOR
links.jstor.org

Subjects
Primary: 62F12: Asymptotic properties of estimators
Secondary: 62H12: Estimation

Keywords
Generalized linear models categorical response models maximum likelihood estimator consistency asymptotic normality

Citation

Fahrmeir, Ludwig; Kaufmann, Heinz. Consistency and Asymptotic Normality of the Maximum Likelihood Estimator in Generalized Linear Models. Ann. Statist. 13 (1985), no. 1, 342--368. doi:10.1214/aos/1176346597. https://projecteuclid.org/euclid.aos/1176346597


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Corrections

  • See Correction: Ludwig Fahrmeir, Heinz Kaufmann. Correction: Consistency and Asymptotic Normality of the Maximum Likelihood Estimator in Generalized Linear Models. Ann. Statist., Volume 14, Number 4 (1986), 1643--1643.