ON THE ALMOST EVERYWHERE PROPERTIES OF THE KERNEL REGRESSION ESTIMATE

MIROSŁAW PAWLAK

Department of Electrical and Computer Engineering, The University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada

(Received August 29, 1988; revised October 20, 1989)

Abstract. The regression m(x) = E{Y | X = x} is estimated by the kernel regression estimate m̂(x) calculated from a sequence (X_{1}, Y_{1}), …, (X_{n}, Y_{n}) of independent identically distributed random vectors from R^{d} × R. The second order asymptotic expansions for Em̂(x) and var m̂(x) are derived. The expansions hold for almost all (μ) x ∈ R^{d}, μ is the probability measure of X. No smoothing conditions on μ and m are imposed. As a result, the asymptotic distribution-free normality for a stochastic component of m̂(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, bandwidthselection, bias adjustment.