Free University of Brussels (VUB), Belgium - Department of Mathematics
Nonparametric density and regression estimation has been the subject of intense investigation for many years and this has led to a large number of methods. One very well-known and commonly used class of estimators consists of the so-called kernel-type estimators.
It is well-known that - in contrast to the choice of the kernel - the choice of the bandwidth is rather problematic, as it is responsible for an important bias/variance trade-off of the resulting kernel-type estimator. Typically, the bandwidth that is most appropriate will vary according to the situation and will depend on the available data. This means that one can no longer investigate the behavior of such â€œoptimalâ€ estimators (namely, estimators based upon data-dependent bandwidth sequences) via the classical results, as they are only applicable to estimators based upon deterministic bandwidth sequences.
In this talk, we will show how the theory of empirical processes can be used to prove (optimal) â€œuniform in bandwidthâ€ consistency results for a wide variety of kernel-type estimators. The term â€œuniform in bandwidthâ€ means that a supremum over suitable ranges of bandwidths is added to the original asymptotic result, permitting for instance to handle kernel-type estimators based upon bandwidths that are functions of the data and/or the location. Additionally, such uniform in bandwidth results can also be obtained for the more general class of conditional U-statistics. The basic tools of this empirical-process-based approach are appropriate exponential deviation inequalities and moment inequalities for empirical processes.