Confidence regions for level sets

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