SIMPLE TRANSFORMATION TECHNIQUES FOR IMPROVED NONPARAMETRIC REGRESSION

We propose and investigate two new methods for achieving less bias in nonparametric regression. We show that the new methods have bias of or der h(4), where h is a smoothing parameter, in contrast to the basic k ernel estimator's order h(2). The methods are conceptually very simple . At the first stage, perform an ordinary non-parametric regression on {x(i), Y-i} to obtain (m) over cap(x) (we use local linear fitting). In the first method, at the second stage, repeat the non-parametric re gression but on the transformed dataset {(m) over cap(x(i)), Y-i}, tak ing the estimator at x to be this second stage estimator at (m) over c ap(x). In the second, and more appealing, method, again perform non-pa rametric regression on {(m) over cap(x(i)), Y-i}, but this time make t he kernel weights depend on the original x scale rather than using the (m) over cap(x) scale. We concentrate more of our effort in this pape r on the latter because of its advantages over the former. Our emphasi s is largely theoretical, but we also show that the latter method has practical potential through some simulated examples.