Nonparametric estimation of a regression function

SummaryConsider the regression model Yi*=g(xi*)+ei*, i=1,2,...,n, where xi*'s denote unordered design variables, and g is an unknown function defined on the interval [0,1]. Assume {ei*} are iid random variables with zero mean and finite variance. Priestley and Chao (1972) and Clark (1977) proposed estimators g2nand g3n, respectively for g. In this paper, the asymptotic behavior of g2nand g3nis studied utilizing the properties of the new estimator g1n. It is shown that g1n, g2n, g3nare 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 {Βn} is any sequence of constants tending to ∞ as n→∞, then for all $$0 < a \leqq b < 1,({{n^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} } \mathord{\left/ {\vphantom {{n^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} } {\beta _n \log n)\mathop {\sup }\limits_{a \leqq x \leqq b} |g_{1n} (x) - g(x)|\xrightarrow{{w.p.1}}0,}}} \right. \kern-\nulldelimiterspace} {\beta _n \log n)\mathop {\sup }\limits_{a \leqq x \leqq b} |g_{1n} (x) - g(x)|\xrightarrow{{w.p.1}}0,}}$$ , as n→∞. Finally, when g is monotone, a strong consistent isotonic estimator gn*is considered.