NONPARAMETRIC-ESTIMATION OF REGRESSION LEVEL SETS

Let Y-i=f(X-i) + xi(i), i = 1,...,n where f is an unknown regression f unction, (xi(1),...,xi(n)) lid centered gaussian variables independent of the design (X-1,...,X-n), X-i is an element of K = [-1/2, 1/2](2). Consider the problem of estimating the level set G = G(f)(lambda)= {x is an element of K:f(x) greater than or equal to lambda} from (X-1, Y -1),...,(X-n, Y-n) under certain assumptions on the boundary smoothnes s of G. We propose piecewise-polynomial estimators based on the maximi zation of local empirical excess masses. With assumptions on the desig n we show that these estimators have optimal rates of convergence in a n asymptotically minimax meaning, within studied classes of regression s. For ''bad'' design we obtain other, non-optimal, rates. We generali ze these results to the N-dimensional case, N not equal 2.