Qualitative robustness of statistical functionals under strong mixing. (English) Zbl 1388.62071

Zähle, Henryk

Bernoulli 21, No. 3, 1412-1434 (2015).

Summary: 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 Lévy \(\psi\)-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.

MSC:

62F35	Robustness and adaptive procedures (parametric inference)
62G35	Nonparametric robustness
60F05	Central limit and other weak theorems

Keywords:

\(\psi\)-weak topology; function bracket; Hampel’s theorem; Kolmogorov \(\varphi\)-metric; Lévy \(\psi\)-metric; locally compact and second countable Hausdorff space; plug-in estimator; qualitative robustness; Rio’s maximal inequality; strong mixing; uniform Glivenko-Cantelli theorem; uniform weak law of large numbers

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