Qualitative robustness of statistical functionals under strong mixing
Abstract
A new concept of (asymptotic) qualitative robustness for plugin 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 realvalued 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:

arXiv eprints
 Pub Date:
 March 2012
 arXiv:
 arXiv:1203.5245
 Bibcode:
 2012arXiv1203.5245Z
 Keywords:

 Mathematics  Statistics Theory
 EPrint:
 Published at http://dx.doi.org/10.3150/14BEJ608 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)