Kernel estimation of density level sets

Abstract

Let f be a multivariate density and f n be a kernel estimate of f drawn from the n-sample X 1 , . . . , X n of i.i.d. random variables with density f. We compute the asymptotic rate of convergence towards 0 of the volume of the symmetric difference between the t-level set {f t} and its plug-in estimator {f n t}. As a corollary, we obtain the exact rate of convergence of a plug-in-type estimate of the density level set corresponding to a fixed probability for the law induced by f.

Nonparametric estimation of regression level sets using kernel plug-in estimator

“…Besides, we introduce H the (d − 1)-dimensional Hausdor measure (Evans and Gariepy [23]). Recall that H agrees with ordinary (k -1)-dimensional surface area on nice sets (Proposition A.1 in [1]). Finally, we set K = K 2 dλ.…”

“…Using a kernel estimator for r, we get a rate of convergence equivalent to the one obtained by Cadre [1] for the density function.…”

“…One can cite the works of Cadre [1], Cuevas and Fraiman [2], Hartigan [3], Polonik [4], Tsybakov [5], Walther [6]. This large number of works on this subject is motivated by the high number of possible applications.…”

Highest Density Difference Region Estimation with Application to Flow Cytometric Data

“…Contributions include Hartigan (1987), Müller & Sawitzki (1991), Polonik (1995), Tsybakov (1997), Baíllo, Cuesta-Albertos & Cuevas (2001), Cadre (2006) and Jang (2006). Alternative terminology includes estimation of the density contours, density level sets and excess mass regions.…”

A graph-based estimator of the number of clusters

“…, X n drawn from f . A rough analysis suggests first to estimate the level sets of the probability density f (Polonik [16], Tsybakov [18], Cadre [3]), and then to evaluate the number of connected components of the resulting set estimate. However, this does not seem to be a promising strategy, especially because it requires assessing the level sets, which is, in the present context, a superfluous operation.…”