Confidence regions for level sets

Abstract

This paper discusses a universal approach to the construction of confidence regions for level sets {h(x) ≥ 0} ⊂ R d 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 ∆n | h n (x) − h(x)| for appropriate sets ∆ n ⊂ R d . 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.

Confidence Regions for Multivariate Quantiles

“…This method constructs confidence regions which are centered at the distribution of points on a level set. In contrast to that, the approaches by Mammen and Polonik [14] and Chen et al [15] yield confidence regions that bound the multivariate quantile. Thus, the techniques are principally incomparable to one another.…”

“…In this section, we introduce the approaches by Mammen and Polonik [14] and Chen et al [15]. These construct confidence regions for estimated level sets.…”

“…These construct confidence regions for estimated level sets. The Mammen and Polonik [14] method is applicable to level sets of any functions. On the other hand, the method by Chen et al [15] is developed for densities.…”

“…First, we extend two recently developed approaches for construction of level set confidence regions by Mammen and Polonik [14] and Chen et al [15] to the estimation problem at hand. Note that the multivariate quantiles considered here are level sets at specific levels of the copula.…”

“…The confidence region approaches by Mammen and Polonik [14] and Chen et al [15] are discussed in Section 3. Moreover, they are extended to multivariate quantile of X.…”