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, the power is one for $n$ large enough and that there exists consistent decision rule. A heuristic for choosing a convenient value for the $k_n$ parameter is also given. Finally we provide a simulation study of the test.