Robustness of Estimators on Stationary Observations

Outlier Resistant Time Series Operations via Qualitative Robustness and Saddle-Point Game Formalizations- A Review: Filtering and Smoothing

“…The definition of qualitative robustness, in conjunction with the Prohorov and Rho-Bar or Vasershtein distances lead to constructive sufficient conditions that data operations should satisfy [2], [6], [7] and [10]. These conditions are included in Theorem 1 below, whose proof can be found in [2].…”

“…The definition was extended to include processes with memory, first by Papantoni- Kazakos and Gray (1979) and then by Cox (1978), Bustos et al (1984) and Papantoni- Kazakos (1984aKazakos ( , 1984bKazakos ( , 1987. Solutions for outlier resistant prediction, filtering and smoothing were first developed by Tsaknakis et al (1988Tsaknakis et al ( , 1986, while an overview of the theory can be found in Kazakos et al (1990).…”

Performance Bounds in Robust Filtering and Smoothing.

“…For the class of memoryless processes, these sufficient conditions were provided by Hampel and they can be found in [1]. For the class of stationary processes with memory, the formalization of qualitative robustness and the subsequent sufficient conditions can be found in [10]. A first qualitative formalization of the filtering problem is presented in [12].…”

“…Furthermore, this solution corresponds to mean-square error minimization in the worst case. It does not necessarily guarantee performance continuity within the class M. For such continuity the sufficient conditions of the qualitative robustness [10,11,12] should be satisfied. That can be accomplished through the appropriate selection of the operation gZ, and it may also require that only a subset of the set F(k,gt) be considered in the filter or smoother selection.…”

“…Furthermore, the first bound is exactly equal to the mean square error induced by the Gaussian measure on F. Then, the properties of qualitative robustness [10], [111, [12] are in general violated if the filters or smoothers that correspond to the above lower bounds are selected. This is so, because boundness and asymptotic continuity (as in 110]) are not in general satisfied then.…”