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

Let (X, Y), X [set membership, variant] Rp, Y [set membership, variant] R1 have the regression function r(x) = E(Y|X = x). We consider the kernel nonparametric estimate rn(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.