Suppose the observations (Xi, Yi), i = 1,..., n, are ϕ-mixing. The strong uniform convergence and convergence rate for the estimator of the regression function was studied by several authors, e.g. Collomb and Györfi et al. But the optimal convergence rates are not reached unless the Yi are bounded or the E exp (a|Yi|) are bounded for some a > 0. Compared with the independent, identically distributed case the convergence of the Nadaraya–Watson estimator under ϕ-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. Compared with Theorem A of Mack and Silverman or Theorem 3.3.1 of Györfi et al., 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 as for Theorem 3.3.2 of Györfi et al.
