Confidence regions for level sets

This paper discusses a universal approach to the construction of confidence regions for level sets {h(x)≥0}⊂Rq{h(x)≥0}⊂Rq of a function hh of interest. The proposed construction is based on a plug-in estimate of the level sets using an appropriate estimate ĥn of hh. 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 supΔn|ĥn(x)−h(x)| for appropriate sets Δn⊂RqΔn⊂Rq. In contrast to related work from the literature, the existence of a weak limit for an appropriately normalized process {ĥn(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.