Strong uniform convergence of the recursive regression estimator under φ-mixing conditions

Suppose the observations (X <Subscript> i </Subscript>, Y <Subscript> i </Subscript>) taking values in R <Superscript> d </Superscript>×R, [InlineMediaObject not available: see fulltext.] are φ-mixing. Compared with the i.i.d. case, some known strong uniform convergence results for the estimators of the regression function r(x)=E(Y <Subscript> i </Subscript>|X <Subscript> i </Subscript>=x) need strong moment conditions under φ-mixing setting. We consider the following improved kernel estimators of r(x) suggested by Cheng (1983): [InlineMediaObject not available: see fulltext.] Qian and Mammitzsch (2000) investigated the strong uniform convergence and convergence rate for [InlineMediaObject not available: see fulltext.] to r(x) under weaker moment conditions than those of the others in the literature, and the optimal convergence rate can be attained under almost the same conditions as stated in Theorem 3.3.2 of Györfi et al. (1989). In this paper, under the similar conditions of Qian and Mammitzsch (2000), we study the strong uniform convergence and convergence rates for [InlineMediaObject not available: see fulltext.] (j=2,3) to r(x), which have not been discussed by Qian and Mammitzsch (2000). In contrast to [InlineMediaObject not available: see fulltext.], our estimators [InlineMediaObject not available: see fulltext.] and [InlineMediaObject not available: see fulltext.] are recursive, which is highly desirable for practical computation. Copyright Springer-Verlag 2004