On the convergence rate of maximal deviation distribution for kernel regression estimates

Let (X, Y), X R^p, Y R¹ have the regression function r(x) = E(Y¦X = x). We consider the kernel nonparametric estimate r_n(x) of r(x) and obtain a sequence of distribution functions approximating the distribution of the maximal deviation with power rate. It is shown that the distribution of the maximal deviation tends to double exponent (which is a conventional form of such theorems) with logarithmic rate and this rate cannot be improved.