(Received August 29, 1988; revised October 20, 1989)
Abstract. The regression m(x) = E{Y | X = x} is estimated by the kernel regression estimate ^m(x) calculated from a sequence (X1, Y1),...., (Xn, Yn) of independent identically distributed random vectors from Rd × R. The second order asymptotic expansions for E^m(x) and var ^m(x) are derived. The expansions hold for almost all (mu) x \in Rd, 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 (x) is established. Also some bandwidth-selection rule is suggested and bias adjustment is proposed.
Key words and phrases: Regression function, kernel estimate, asymptotic expansions, distribution-free properties, asymptotic normality, bandwidth-selection, bias adjustment.