Maximin Estimation of Multidimensional Boundaries

We consider the problem of estimating the location and size of a discontinuity in an otherwise smooth multidimensional regression function. The boundary or location of the discontinuity is assumed to be a closed curve respective surface, and we aim to estimate this closed set. Our approach utilizes the uniform convergence of multivariate kernel estimators for directional limits. Differences of such limits converge to zero under smoothness assumptions, and to the jump size along the discontinuity. This leads to the proposal of a maximin estimator, which selects the boundary for which the minimal estimated directional difference among all points belonging to this boundary is maximized. It is shown that this estimated boundary is almost surely enclosed in a sequence of shrinking neighborhoods around the true boundary, and corresponding rates of convergence are obtained.