Qualitative robustness of statistical functionals under strong mixing

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.Comment: 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

Functional delta-method for the bootstrap of quasi-Hadamard differentiable functionals

The functional delta-method provides a convenient tool for deriving the asymptotic distribution of a plug-in estimator of a statistical functional from the asymptotic distribution of the respective empirical process. Moreover, it provides a tool to derive bootstrap consistency for plug-in estimators from bootstrap consistency of empirical processes. It has recently been shown that the range of applications of the functional delta-method for the asymptotic distribution can be considerably enlarged by employing the notion of quasi-Hadamard differentiability. Here we show in a general setting that this enlargement carries over to the bootstrap. That is, for quasi-Hadamard differentiable functionals bootstrap consistency of the plug-in estimator follows from bootstrap consistency of the respective empirical process. This enlargement often requires convergence in distribution of the bootstrapped empirical process w.r.t.\ a nonuniform sup-norm. The latter is not problematic as will be illustrated by means of examples

Continuous mapping approach to the asymptotics of UU- and VV-statistics

We derive a new representation for $U$- and $V$-statistics. Using this representation, the asymptotic distribution of $U$- and $V$-statistics can be derived by a direct application of the Continuous Mapping theorem. That novel approach not only encompasses most of the results on the asymptotic distribution known in literature, but also allows for the first time a unifying treatment of non-degenerate and degenerate $U$- and $V$-statistics. Moreover, it yields a new and powerful tool to derive the asymptotic distribution of very general $U$- and $V$-statistics based on long-memory sequences. This will be exemplified by several astonishing examples. In particular, we shall present examples where weak convergence of $U$- or $V$-statistics occurs at the rate $a_n^3$ and $a_n^4$, respectively, when $a_n$ is the rate of weak convergence of the empirical process. We also introduce the notion of asymptotic (non-) degeneracy which often appears in the presence of long-memory sequences.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ508 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm