Set estimation and nonparametric detection

This paper considers the estimation of a set S < from a random sample of n points, using the \naive" estimator Ŝn (the simplest one for the non-convex case) de ned as a union of balls, centered at the sample points, with common radius n. In particular, we focus on further exploring a method, rst proposed by Devroye and Wise (1980), for the following problem: given a sample X1; : : : ; Xn from an unknown density f on < , we want to decide whether or not an additional observation Xn+1 has been also drawn from f . According to these authors' proposal, we decide that a change has occurred when Xn+1 = 2 Ŝn, where the set estimator Ŝn is based on X1; : : : ; Xn. If the considered problem is placed in the setup of statistical quality control, the proposed procedure could be seen as a multivariate and completely nonparametric alternative to the classical Shewhart methodology based on tolerance regions. Convergence rates for the probability of false alarm are obtained. We also show that the smoothing parameter n can be used to incorporate information on the shape of S: e.g., if S is assumed to be a connected set, then n can be chosen in such a way that Ŝn is a connected consistent estimator. Two methods, relying on cross validation and smoothed bootstrap ideas, are suggested to select n. A simulation study and a real data example are presented.