Marked and Weighted Empirical Processes of Residuals with Applications to Robust Regressions

A new class of marked and weighted empirical processes of residuals is introduced. The framework is general enough to accommodate both stationary and non-stationary regressions as well as a wide class of estimation procedures with applications in misspecification testing and robust statistics. Two applications are presented. First, we analyze the relationship between truncated moments and linear statistical functionals of residuals. In particular, we show that the asymptotic behaviour of these functionals, expressed as integrals with respect to their empirical distribution functions, can be easily analyzed given the main theorems of the paper. In our context the integrands can be unbounded provided that the underlying distribution meets certain moment conditions. A general first order asymptotic approximation of the statistical functionals is derived and then applied to some cases of interest. Second, the consequences of using the standard cumulant based normality test for robust regressions are analyzed. We show that the rescaling of the moment based statistic is case dependent, i.e., it depends on the truncation and the estimation method being used. Hence, using the standard least squares normalizing constants in robust regressions will lead to incorrect inferences. However, if appropriate normalizations, which we derive, are used then the test statistic is asymptotically chi-square.