Robust Second Order Accurate Inference for Generalized Linear Models

In the framework of generalized linear models, the nonrobustness of classical estimators and tests for the parameters is a well known problem and alternative methods have been proposed in the literature. These methods are robust and can cope with deviations from the assumed distribution. However, they are based on first order asymptotic theory and their accuracy in moderate to small samples is still an open question. In this paper we propose a test statistic which combines robustness and good accuracy for moderate to small sample sizes. We combine results from Cantoni and Ronchetti (2001) and Robinson, Ronchetti and Young (2003) to obtain a robust test statistic for hypothesis testing and variable selection which is asymptotically chi-square distributed as the three classical tests but with a relative error of order O(1/n). This leads to reliable inference in the presence of small deviations from the assumed model distribution and to accurate testing and variable selection even in moderate to small samples.