Qualitative robustness for stochastic processes. (English) Zbl 0644.62037

Boente, Graciela; Fraiman, Ricardo; Yohai, Victor J.

Ann. Stat. 15, 1293-1312 (1987).

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.

Reviewer: T.Bednarski

MSC:

62F35	Robustness and adaptive procedures (parametric inference)
62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)

Keywords:

pi-robustness; strong resistance; qualitative robustness; weak resistance; generalized M-estimates; linear models; autoregressive processes