STRONG UNIVERSAL POINTWISE CONSISTENCY OF RECURSIVE REGRESSION ESTIMATES

HARRO WALK

Mathematisches Institut A, Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany

(Received January 7, 1999; revised February 28, 2000)

Abstract. For semi-recursive and recursive kernel estimates of a regression of Y on X (d-dimensional random vector X, integrable real random variable Y), introduced by Devroye and Wagner and by Révész, respectively, strong universal pointwise consistency is shown, i.e. strong consistency P_{X}-almost everywhere for general distribution of (X, Y). Similar results are shown for the corresponding partitioning estimates.

Key words and phrases: Nonparametric regression estimation, semi-recursive estimation, recursive estimation, kernel estimates, partitioning estimates, strong universal pointwise consistency, strong laws of large numbers, conditional expectations, truncation, covering.