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

摘要

Suppose the observations (X_i, Y_i) taking values in R~d×R, i=1, …, n 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_i|X_i=x) need strong moment conditions under φ-mixing setting. We consider the following improved kernel estimators of r(x) suggested by Cheng (1983): r-circumflex_n~(1) (x) = Σ from i=1 to n of Y_iI(|Y_i| < b_n) K((x-X_i)/h_n)/Σ from j=1 to n of K((x-X_j)/h_n), r-circumflex_n~(2)(x) = Σ from i=1 to n of Y_iI(|Y_i| < B_i)h_i~(-d)K((x-X_i)/h_i)/Σ from j=1 to n of h_j~(-d)K((x-X_j)/h_j), r-circumflex_n~(3)(x) = Σ from i=1 to n of Y_iI(|Y_i| < b_i) K ((x-X_i)/h_i)/Σ from j=1 to n of K((x-X_j)/h_j). Qian and Mammitzsch (2000) investigated the strong uniform convergence and convergence rate for r-circumflex_n~(1) 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 Gyorfi 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 r-circumflex_n~(j) (j=2,3) to r(x), which have not been discussed by Qian and Mammitzsch (2000). In contrast to r-circumflex_n~(1), our estimators r-circumflex_n~(2) and r-circumflex_n~(3) are recursive, which is highly desirable for practical computation.