Robustness Property of Robust-BD Wald-Type Test for Varying-Dimensional General Linear Models

An important issue for robust inference is to examine the stability of the asymptotic level and power of the test statistic in the presence of contaminated data. Most existing results are derived in finite-dimensional settings with some particular choices of loss functions. This paper re-examines this issue by allowing for a diverging number of parameters combined with a broader array of robust error measures, called “robust-BD”, for the class of “general linear models”. Under regularity conditions, we derive the influence function of the robust-BD parameter estimator and demonstrate that the robust-BD Wald-type test enjoys the robustness of validity and efficiency asymptotically. Specifically, the asymptotic level of the test is stable under a small amount of contamination of the null hypothesis, whereas the asymptotic power is large enough under a contaminated distribution in a neighborhood of the contiguous alternatives, thus lending supports to the utility of the proposed robust-BD Wald-type test.