Nonparametric estimation of a regression function

Summary Consider the regression model Y_i^*=g(x_i^*)+e_i^*, i=1,2,...,n, where x_i^*'s denote unordered design variables, and g is an unknown function defined on the interval [0,1]. Assume {e_i^*} are iid random variables with zero mean and finite variance. Priestley and Chao (1972) and Clark (1977) proposed estimators g_2nand g_3n, respectively for g. In this paper, the asymptotic behavior of g_2nand g_3nis studied utilizing the properties of the new estimator g_1n. It is shown that g_1n, g_2n, g_3nare asymptotically equivalent in various senses. Moreover, consistency results are established and rates of uniform convergence obtained. For example, if E¦e^*¦³<, if g is Lipschitz of order 1, and if {_n} is any sequence of constants tending to as n, then for all , as n. Finally, when g is monotone, a strong consistent isotonic estimator g_n^*is considered.