The Annals of Statistics

Qualitative Robustness for Stochastic Processes

Graciela Boente, Ricardo Fraiman, and Victor J. Yohai

Full-text: Open access


In this paper we generalize Hampel's concept of qualitative robustness of a sequence of estimators to the case of stochastic processes with non-i.i.d. observations, defining appropriate metrics between samples. We also present a different approach to qualitative robustness which formalizes the notion of resistance. We give two definitions based on this approach: strong and weak resistance. We show that for estimating a finite dimensional real parameter, $\pi$-robustness is equivalent to weak resistance and, in the i.i.d. case, is also equivalent to strong resistance. Finally, we prove the strong resistance of a class of estimators which includes common GM-estimates for linear models and autoregressive processes.

Article information

Ann. Statist., Volume 15, Number 3 (1987), 1293-1312.

First available in Project Euclid: 12 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62F35: Robustness and adaptive procedures
Secondary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Qualitative robustness robust estimation GM-estimators stochastic processes autoregressive models


Boente, Graciela; Fraiman, Ricardo; Yohai, Victor J. Qualitative Robustness for Stochastic Processes. Ann. Statist. 15 (1987), no. 3, 1293--1312. doi:10.1214/aos/1176350506.

Export citation