Modified nonparametric kernel estimates of a regression function and their consistencies with rates

Summary Two sets of modified kernel estimates of a regression function are proposed: one when a bound on the regression function is known and the other when nothing of this sort is at hand. Explicit bounds on the mean square errors of the estimators are obtained. Pointwise as well as uniform consistency in mean square and consistency in probability of the estimators are proved. Speed of convergence in each case is investigated. Major work of this research was completed during the first author's two visits (November–December, 1983 and August–September 1984) to the second author at the Universite du Quebec a Montreal. Part of the work of the second author was supported by the Air Force Office of Scientific Research under contract F49620-85-C-0008 while he was at the University of Pittsburgh during Spring in 1985.