Qualitative robustness of statistical functionals under strong mixing
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 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.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2012
- DOI:
- 10.48550/arXiv.1203.5245
- arXiv:
- arXiv:1203.5245
- Bibcode:
- 2012arXiv1203.5245Z
- Keywords:
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- Mathematics - Statistics Theory
- E-Print:
- Published at http://dx.doi.org/10.3150/14-BEJ608 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)