Convolution type estimators for nonparametric regression

Convolution type kernel estimators such as the Priestley-Chao estimator have been discussed by several authors in the fixed design regression model Yi = g(ti)+ [var epsilon]i, where [var epsilon]i are uncorrelated random errors, ti are fixed design points where measurements are made, and g is the function to be estimated from the noisy measurements Yi. Using properties of order statistics and concomitants, we derive the asymptotic mean squared error of these estimators in the random design case where given i.i.d. bivariate observations (Xi, Yi), i = 1,..., n, the aim is to estimate the regression function m(x) = E(Y/X =x). The comparison with the well-known quotient type Nadaraya-Watson kernel estimators shows that convolution type estimators have a bias behavior which corresponds to that in the fixed design case. This makes possible the straightforward extension to the estimation of derivatives.