Law of Iterated Logarithm and Strong Consistency in Poisson Regression Model Selection

Abstract

In this paper we first derive a law of iterated logarithm for the maximum likelihood estimator of the parameters in a Poisson regression model. We then use this result to establish the strong consistency of a class of model selection criteria in Poisson regression model selection. We show that under some general conditions, a model selection criterion, which consists of a minus maximum log-likelihood and a penalty term, will select the simplest correct model almost surely if the penalty term increases with model dimension and has an order in between $O(\log\log n)$ and $O(n)$.