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

Suppose the observations (X-i, Y-i), i = 1,..., n, are phi -mixing. The strong uniform convergence and convergence rate for the estimator of the regression function was studied by serveral authors, e.g. G. Collomb (1984), L. Gyorfi et al. (1989). But the optimal convergence rates are not reached unless the Y-i are bounded or the Eexp(a\Y-i\) are bounded for some a > 0. Compared with the i.i.d. case the convergence of the Nadaraya-Watson estimator under phi -mixing variables needs strong moment conditions. In this paper we study the strong uniform convergence and convergence rate for the improved kernel estimator of the regression function which has been suggested by Cheng P. (1983). Compared with Theorem A in Y. P. Mack and B. Silverman (1982) or Theorem 3.3.1 in L. Gyorfi et al. (1989), we prove the convergence for this kind of estimators under weaker moment conditions. The optimal convergence rate for the improved kernel estimator is attained under almost the same conditions of Theorem 3.3.2 in L. Gyorfi et al. (1989).