Qualitative Robustness for General Stochastic Processes.

Abstract : This paper generalizes Hampel's definition of robustness and pi robustness of a sequence of estimators to the case of non i.i.d. stochastic processes, using appropriate metrics on the space of finite and infinite dimensional samples. Also presented is a different approach to qualitative robustness based on uniform insensitivity of the sequence of estimators when the sample is affected by round-off errors or by a small fraction of outliers. Given are two definitions based on this approach: strong and weak pointwise robustness. The authors show that for estimating a finite dimensional real parameter, robustness is equivalent to weak pointwise robustness and at least in the i.i.d. case is also equivalent to strong pointwise robustness. Finally it is shown that the continuity condition given by Papantoni-Kazakos and Gray is sufficient for strong pointwise robustness. This implies the strong pointwise robustness of GM-estimates for autoregressive models.