Optimal partial ridge estimation in restricted semiparametric regression models

This paper is concerned with the ridge estimation of the parameter vector β in partial linear regression model yi=xiβ+f(ti)+ϵi,1≤i≤n, with correlated errors, that is, when Cov(ϵ)=σ2V, with a positive definite matrix V and ϵ=(ϵ1,…,ϵn), under the linear constraint Rβ=r, for a given matrix R and a given vector r. The partial residual estimation method is used to estimate β and the function f(⋅)f(⋅). Under appropriate assumptions, the asymptotic bias and variance of the proposed estimators are obtained. A generalized cross validation (GCV) criterion is proposed for selecting the optimal ridge parameter and the bandwidth of the kernel smoother. An extension of the GCV theorem is established to prove the convergence of the GCV mean. The theoretical results are illustrated by a real data example and a simulation study.