Estimation of non-sharp support boundaries

Let X₁, …, X_n 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, …, X_n, 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.