Nonparametric Estimation of Regression Level Sets

Let where f is an unknown regression function, (ξ1,...,ξn) are iid centered gaussian variables independent of the design (X 1,……X n), Consider the problem of estimating the level set from (X 1, Y 1),….. (X n, Y n)Consider under certain assumptions on the boundary smoothness of G. We propose piecewise-polynomial estimators based on the maximization of local empirical excess masses. With assumptions on the design we show that these estimators have optimal rates of convergence in an asymptotically minimax meaning, within studied classes of regressions. For “bad” design we obtain other, non-optimal, rates. We generalize these results to the N-dimensional case, N ≠ 2.