Estimation of Non-sharp Support Boundaries

Let X1, ..., Xn be independent identically distributed observations from an unknown probability density f(·), such that its support G = supp f is a subset of the unit square in R2. We consider the problem of estimating G from the sample X1, ..., Xn, under the assumption that the boundary of G is a function of smoothness γ and that the values of density f decrease to 0 as the power α of the distance from the boundary. We show that a certain piecewise-polynomial estimator of G has optimal rate of convergence (namely, the rate n−γ/((α + 1)γ + 1)) within this class of densities.