Abstract: This paper investigates new aspects of robust inference for general linear models, calling for a broader array of error measures, beyond the conventional notion of quasi-likelihood, and allowing for a diverging number of parameters. We propose a class of robust error measures, called robust-BD, based on the notion of Bregman divergence (BD). That includes the (negative) quasi-likelihood and many other commonly used error measures as special cases, and we introduce the robust-BD estimators of parameters. We re-examine the classical likelihood ratio-type test statistic, constructed by replacing the negative log-likelihood with the robust-BD, and find that its asymptotic null distribution is a sum of weighted χ² with weights relying on unknown quantities, thus is not asymptotically distribution free. We propose a robust version of the Wald-type test statistic, based on the robust-BD estimator, and show that it is asymptotically χ² (central) under the null, thus distribution free, and χ² (noncentral) under the contiguous alternatives. Numerical examples are presented to illustrate the computational simplicity and effectiveness of the proposed estimator and test in the presence of outliers.

Key words and phrases: Generalized linear model, hypothesis test, influence function, quasi-likelihood, robustness.