Anti-concentration and honest, adaptive confidence bands

Abstract

Modern construction of uniform condence e and Nickl (2010). This condition requires the existence of a limit distribution of an extreme value type for the supremum of a studentized empirical process (equivalently, for the supremum of a Gaussian process with the same covariance function as that of the studentized empirical process). The principal contribution of this paper is to remove the need for this classical condition. We show that a considerably weaker sucient condi- tion is derived from an anti-concentration property of the supremum of the approximating Gaussian process, and we derive an inequality lead- ing to such a property for separable Gaussian processes. We refer to the new condition as a generalized SBR condition. Our new result shows that the supremum does not concentrate too fast around any value. We then apply this result to derive a Gaussian multiplier bootstrap procedure for constructing honest condence bands for nonparametric density estimators (this result can be applied in other nonparametric problems as well). An essential advantage of our approach is that it ap- plies generically even in those cases where the limit distribution of the supremum of the studentized empirical process does not exist (or is un- known). This is of particular importance in problems where resolution levels or other tuning parameters have been chosen in a data-driven fash- ion, which is needed for adaptive constructions of the condence bands. Furthermore, our approach is asymptotically honest at a polynomial rate { namely, the error in coverage level converges to zero at a fast, polynomial speed (with respect to the sample size). In sharp contrast, the approach based on extreme value theory is asymptotically honest only at a logarithmic rate { the error converges to zero at a slow, loga- rithmic speed. Finally, of independent interest is our introduction of a new, practical version of Lepski's method, which computes the optimal, non-conservative resolution levels via a Gaussian multiplier bootstrap method.