Confidence regions for level sets

This paper discusses a universal approach to the construction of confidence regions for level sets {h(x) ≥ 0} ⊂ Rd of a function h of interest. The proposed construction is based on a plug-in estimate of the level sets using an appropriate estimate �hn of h. The approach provides finite sample upper and lower confidence limits. This leads to generic conditions under which the constructed confidence regions achieve a prescribed coverage level asymptotically. The construction requires an estimate of quantiles of the distribution of sup∆n |�hn(x) − h(x)| for appropriate sets ∆n ⊂ Rd. In contrast to related work from the literature, the existence of a weak limit for an appropriately normalized process {�hn(x), x ∈ D} is not required. This adds significantly to the challenge of deriving asymptotic results for the corresponding coverage level. Our approach is exemplified in the case of a density level set utilizing a kernel density estimator and a bootstrap procedure. This research was support by a grant of the DFG and the NSF AMS 2000 subject classifications. Primary 62G07, Secondary 62G08, 62G09.