Robust bounded-influence tests in linear models

A robust test that we call an aligned generalized M test for testing subhypotheses in general linear models is developed, and its asymptotic properties are studied. The test is a robustification of the well known F test, and it is an elegant alternative to Ronchetti’s (1982) class of τ tests, p-values associated with it can be approximated readily using existing chi-square tables. The test is based on an appropriately constructed quadratic form and uses the generalized M estimators of the parameters in the reduced model. Under the null hypothesis the asymptotic distribution is a central chi square, and under contiguous alternatives it is a noncentral chi square with the same degrees of freedom. The test can be viewed as a generalization of Sen’s (1982) M test for linear models. The influence function of the test is bounded. The bound not only applies to the influence of residuals but to the influence of position in the factor space as well. On the other hand, Sen’s test has bounded influence only in residuals.