Confidence Regions for Level Sets

Mammen, Enno ; Polonik, Wolfgang

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Document Type: Article
Year of publication: 2013
The title of a journal, publication series: Journal of Multivariate Analysis : JMVA
Volume: 122
Page range: 202-214
Place of publication: Amsterdam [u.a.]
Publishing house: Elsevier
ISSN: 0047-259X , 1095-7243
Publication language: English
Institution: Außerfakultäre Einrichtungen > SFB 884
Subject: 310 Statistics
Keywords (English): Level sets ; Nonparametric curve estimation ; Kernel density estimation ; Smooth bootstrap
Abstract: 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 View the MathML sourceĥn 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 View the MathML sourcesupΔn|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 View the MathML source{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.

Dieser Eintrag ist Teil der Universitätsbibliographie.

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