Please use this identifier to cite or link to this item: https://hdl.handle.net/10419/149789 
Year of Publication: 
2016
Series/Report no.: 
cemmap working paper No. CWP43/16
Publisher: 
Centre for Microdata Methods and Practice (cemmap), London
Abstract: 
Modern construction of uniform confidence bands for nonparametric densities (and other functions) often relies on the classical Smirnov-Bickel-Rosenblatt (SBR) condition; see, for example, Giné 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 sufficient condition is derived from an anti-concentration property of the supremum of the approximating Gaussian process, and we derive an inequality leading 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 boot-strap procedure for constructing honest confidence bands for non-parametric density estimators (this result can be applied in other nonparametric problems as well). An essential advantage of our approach is that it applies generically even in those cases where the limit distribution of the supremum of the studentized empirical process does not exist (or is unknown). This is of particular importance in problems where resolution levels or other tuning parameters have been chosen in a data-driven fashion, which is needed for adaptive constructions of the confidence bands. 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.
Subjects: 
anti-concentration of separable Gaussian processes
honest confidence bands
Lepski's method
multiplier method
non-Donsker empirical processes
Persistent Identifier of the first edition: 
Document Type: 
Working Paper

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