Improved Estimators in Nonparametric Regression Problems

Abstract

Linear estimators of multivariate means are considered. Generalizations of some well-known theorems about admissibility of linear estimators are given. The results then are applied to show that commonly used kernel-type estimators in nonparametric regression problems can be constructively improved in a simple way. An asymptotic result is described that gives a quantitative measure of the maximum improvement to be gained in certain situations. A theoretical bound shows that gains are achievable in the relative risk of up to 58.6% (rectangular kernel) or 29.2% (Epanechnikov kernel). Some examples of smaller sample size are also investigated, and these show relative risk gains ranging up to 18% in realistic settings.