Convergence rates in nonparametric estimation of level sets

✍ Scribed by Amparo Baı́llo; Juan A. Cuesta-Albertos; Antonio Cuevas

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 125 KB
• Volume: 53
• Category: Article
• ISSN: 0167-7152
• DOI: 10.1016/s0167-7152(01)00006-2

✦ Synopsis

A level set of type {f6c} (where f is a density on R d and c is a positive value) can be estimated by its empirical version { f n 6c}, where f n denotes a nonparametric (kernel) density estimator. We analyze, from two di erent points of view, the asymptotic behavior of the probability content of { f n 6c}.

Our results are motivated by applications in cluster analysis and outlier detection. Although the mathematical treatment is quite di erent in both cases, the conclusions are basically coincident. Roughly speaking, we show that the convergence rates are at most of type n - 1=(d+2) . For the univariate case d = 1 this would be in the same spirit of the classical cube-root results found in some nonparametric setups.