Local Convergence Rate of Mean Squared Error in Density Estimation

Abstract

The local convergence rate of a multivariate density estimators based on the certain delta-sequence is studied. In contrast to known results, the conditions on the density are formulated in terms of the modulus of continuity. The main contribution of this study is relaxing the corresponding smoothing conditions in terms of arbitrary modulus of continuity type majorant. In particular, when the density f is an element of L(p)(R(d)) satisfies Lipschitz condition of order gamma = 1 at x, the rate of convergency contains terms with logarithm, which is the best possible convergency rate.