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

Suppose the observations ðXi; YiÞ taking values in Rd R; i 1⁄4 1; . . . ; n are u-mixing. Compared with the i.i.d. case, some known strong uniform convergence results for the estimators of the regression function rðxÞ 1⁄4 EðYijXi 1⁄4 xÞ need strong moment conditions under u-mixing setting. We consider the following improved kernel estimators of rðxÞ suggested by Cheng (1983): brð1Þ n ðxÞ 1⁄4 Xn i1⁄41 YiIðjYij < bnÞK x Xi hn Xn j1⁄41 K x Xj hn , ; brð2Þ n ðxÞ 1⁄4 Xn i1⁄41 YiIðjYij < biÞh d i K x Xi hi Xn j1⁄41 h d j K x Xj hj , ; brð3Þ n ðxÞ 1⁄4 Xn i1⁄41 YiIðjYij < biÞK x Xi hi Xn j1⁄41 K x Xj hj , : Qian and Mammitzsch (2000) investigated the strong uniform convergence and convergence rate for br n to rðxÞ 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 br n ðj 1⁄4 2; 3Þ to rðxÞ, which have not been discussed by Qian and Mammitzsch (2000). In contrast to br n , our estimators br ð2Þ n and br ð3Þ n are recursive, which is highly desirable for practical computation.