UNIVERSAL CONSISTENCY OF LOCAL POLYNOMIAL KERNEL REGRESSION ESTIMATES

MICHAEL KOHLER

Mathematisches Institut A, Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany, e-mail: kohler@mathematik.uni-stuttgart.de

(Received December 18, 2000; revised May 25, 2001)

Abstract. Regression function estimation from independent and identically distributed data is considered. The L_{2} error with integration with respect to the design measure is used as an error criterion. It is shown that suitably defined local polynomial kernel estimates are weakly and strongly universally consistent, i.e., it is shown that the L_{2} errors of these estimates converge to zero almost surely and in L_{1} for all distributions.

Key words and phrases: Local polynomial kernel estimates, regression estimates, weak and strong universal consistency.