Boundary estimation based on set-indexed empirical processes

We observe independent random variables taken at the nodes of a grid in the d-dimensional unit cube. It is assumed that there exists a partition of the unit cube into two disjoint regions. Observations with nodes in a particular region stem from a common distribution, whereas observations from different regions differ in distribution. The problem is to estimate the common topological boundary of the two regions. Our estimator is defined as maximizer of a certain set-indexed empirical process. It induces a partition from which we show that the number of misclassified data is stochastically bounded as the sample sizes increase to infinity.