AISM 54, 879-899

© 2002 ISM

## 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.

**Source**
(TeX , DVI )