Qualitative robustness for stochastic processes
Defining appropriate metrics between samples, the authors generalize Hampel's concept of qualitative robustness of a sequence of estimators (\(\pi\)-robustness) to the case of stochastic processes. Also a new approach to qualitative robustness is presented, based on the intuitive concept of resistance, where instead of considering the insensitivity of estimates with respect to small changes in distribution of a process, the insensitivity corresponds to a given sample point \(x\in X^{\infty}\). Resulting concepts of (asymptotic) strong and weak resistance are then related to \(\pi\)-robustness. It is shown, in particular, that when a finite-dimensional parameter is estimated, then \(\pi\)-robustness is equivalent to the weak resistance. The authors also prove that a class of estimates which includes generalized M-estimates (GM-estimates) for linear models and autoregressive processes is strongly resistant.
scientific article; zbMATH DE number 4050761