Qualitative robustness of statistical functionals under strong mixing

TL;DR

This paper introduces a new concept of qualitative robustness for statistical estimators under dependence, extending Hampel's theorem to strongly mixing laws and establishing uniform convergence properties for various metrics.

Contribution

It develops a novel framework for robustness of estimators with dependent data and proves the UGC property for a broad class of strongly mixing processes using a new uniform weak LLN.

Findings

Hampel's theorem extends to dependent observations under strong mixing.

UGC property established for Kolmogorov and Lévy metrics in real-valued cases.

A new uniform weak LLN for strongly mixing variables is proven.

Abstract

A new concept of (asymptotic) qualitative robustness for plug-in estimators based on identically distributed possibly dependent observations is introduced, and it is shown that Hampel's theorem for general metrics $d$ still holds. Since Hampel's theorem assumes the UGC property w.r.t. $d$, that is, convergence in probability of the empirical probability measure to the true marginal distribution w.r.t. $d$ uniformly in the class of all admissible laws on the sample path space, this property is shown for a large class of strongly mixing laws for three different metrics $d$. For real-valued observations, the UGC property is established for both the Kolomogorov $\phi$-metric and the L\'{e}vy $\psi$-metric, and for observations in a general locally compact and second countable Hausdorff space the UGC property is established for a certain metric generating the $\psi$-weak topology. The key is a new uniform weak LLN for strongly mixing random variables. The latter is of independent interest and relies on Rio's maximal inequality.