On boundary detection

Given a sample of a random variable supported by a smooth compact manifold $M\subset \mathbb{R}^d$, 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 $k_n$-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 $k_n$ parameter and identifying observations close to the boundary are also given. We provide a simulation study of the test.