Statistica Sinica 16(2006), 1335-1365

STRONG LIMIT THEOREMS ON MODEL SELECTION IN

GENERALIZED LINEAR REGRESSION WITH

BINOMIAL RESPONSES

Guoqi Qian and Yuehua Wu

University of Melbourne and York University

Abstract: We prove a law of iterated logarithm for the maximum likelihood estimator of the parameters in a generalized linear regression model with binomial response. This result is then used to derive an asymptotic bound for the difference between the maximum log-likelihood function and the true log-likelihood. It is further used to establish the strong consistency of some penalized likelihood based model selection criteria. We have shown that, under some general conditions, a model selection criterion will select the simplest correct model almost surely if the penalty term is an increasing function of the model dimension and has an order between $O(\log\log n)$ and $O(n)$. Cases involving the commonly used link functions are discussed for illustration of the results.

Key words and phrases: Generalized linear models, law of the iterated logarithm, maximum likelihood estimator, model selection, strong consistency.