Second Order Asymptotic Risks of Smoothed Linear Estimators in Nonparametric Regression Models

Abstract

We consider experiments with fixed design points and replicated observations at these points, and investigate smoothed linear estimators of the mean vector of these observations. We propose some methods to choose the degree of smoothing in a data-dependent way. To assess the performance of such estimators we approximate the mean square error by second order power series in n ⁻¹ and σ ², respectively, where n is the sample size and σ ² is the common variance of the observations. At least for special cases we see that the adaptive smoothed estimator can improve the least squares estimator asymptotically. Furthermore, we show the second order equivalence of the proposed adaptive estimators.

AMS 1980 subject classifications::

• Primary 62 G 05
• secondary 62 F12

Keywords:

• Smoothed estimators
• kernel estimators
• mean square error
• second order risk comparison