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ON THE ALMOST EVERYWHERE PROPERTIES OF THE KERNEL

REGRESSION ESTIMATE

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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
(*X*_{1}, *Y*_{1}),...., (*X*_{n}, *Y*_{n}) of independent identically distributed
random vectors from *R*^{d} × *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 *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*
(*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.

**Source**
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