ON THE ALMOST EVERYWHERE PROPERTIES OF THE KERNEL
REGRESSION ESTIMATE

MIROS\L AW 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 ^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.

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