ESTIMATION OF NON-SHARP SUPPORT BOUNDARIES

Let X(1),...,X(n) be independent identically distributed observations from an unknown probability density f(.), such that its support G = su pp f is a subset of the unit square in R(2). We consider the problem o f estimating G from the sample X(1),..., X(n), under the assumption th at the boundary of G is a function of smoothness gamma and that the va lues of density f decrease to 0 as the power alpha of the distance fro m the boundary. We show that a certain piecewise-polynomial estimator of G has optimal rate of convergence (namely, the rate n(-gamma/((alph a+1)gamma+1))) within this class of densities. (C) 1995 Academic Press , Inc.