On boundary detection

Abstract

Given a sample of a random variable supported by a smooth compact manifold M⊂Rd, we propose a test to decide whether the boundary of M is empty or not with no preliminary support estimation. The test statistic is based on the maximal distance between a sample point and the average of its kn-nearest neighbors. We prove that the level of the test can be estimated, that, with probability one, its power is one for n large enough, and that there exists a consistent decision rule. Heuristics for choosing a convenient value for the kn parameter and identifying observations close to the boundary are also given. We provide a simulation study of the test.