AISM 53, 691-707
© 2001 ISM

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.

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