Convergence rates in nonparametric estimation of level sets

A level set of type {f less than or equal to c} (where f is a density on R-d and c is a positive value) can be estimated by its empirical version {(f) over cap (n) less than or equal to c}, where (f) over cap (n) denotes a nonparametric (kernel) density estimator. We analyze, from two different points of view, the asymptotic behavior of the probability content of {(f) over cap (n) less than or equal to c}. Our results are motivated by applications in cluster analysis and outlier detection. Although the mathematical treatment is quite different 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. (C) 2001 Elsevier Science B.V. All rights reserved.