Trigonometric series regression estimators with an application to partially linear models

Abstract

Let μ be a function defined on an interval [ a, b] of finite length. Suppose that y 1, …, y n are uncorrelated observations satisfying E( y j ) = μ( t j ) and var( y j ) = σ 2, j = 1, …, n, where the t j 's are fixed design points. Asymptotic (as n → ∞) approximations of the integrated mean squared error and the partial integrated mean squared error of trigonometric series type estimators of μ are obtained. Our integrated squared bias approximations closely parallel those of Hall in the setting of density estimation. Estimators that utilize only cosines are shown to be competitive with the so-called cut-and-normalized kernel estimators. Our results for the cosine series estimator are applied to the problem of estimating the linear part of a partially linear model. An efficient estimator of the regression coefficient in this model is derived without undersmoothing the estimate of the nonparametric component. This differs from the result of Rice whose nonparametric estimator was a partial spline.