Optimal design for curve estimation by local linear smoothing

Abstract

The integral of the mean squared error of an estimator of a regression function is used as a criterion for defining an optimal design measure in the context of local linear regression, when the bandwidth is chosen in a locally optimal manner. An algorithm is proposed that constructs a sequence of piecewise uniform designs with the help of current estimates of the integral of the mean squared error. These estimates do not require direct estimation of the second derivative of the regression function. Asymptotic properties of the algorithm are established and numerical results illustrate the gains that can be made, relative to a uniform design, by using the optimal design or sub-optimal, piecewise uniform designs. The behaviour of the algorithm in practice is also illustrated.