ON THE ALMOST EVERYWHERE PROPERTIES OF THE KERNEL REGRESSION ESTIMATE

The regression m(x) = E{Y \ X = x} is estimated by the kernel regression estimate m triple-overdot (x) calculated from a sequence (X1, Y1),..., (X(n), Y(n)) of independent identically distributed random vectors from R(d) x R. The second order asymptotic expansions for Em triple-overdot (x) and var m triple-overdot (x) are derived. The expansions hold for almost all (mu) x is-an-element-of R(d), mu is the probability measure of X. No smoothing conditions on mu and m are imposed. As a result, the asymptotic distribution-free normality for a stochastic component of m triple-overdot (x) is established. Also some bandwidth-selection rule is suggested and bias adjustment is proposed.