Penalized spline estimation in the partially linear model

Abstract

Penalized spline estimators have received considerable attention in recent years because of their good finite-sample performance, especially when many regressors are employed. In this paper, we propose a penalized B-spline estimator in the context of the partially linear model and study its asymptotic properties under a two-sequence asymptotics: both the number of knots and the penalty factor vary with the sample size. We establish asymptotic distributions of the estimators of both the parametric and nonparametric components in the model. In addition, as a previous step, we obtain the rate of convergence of the estimator of the regression function in a nonparametric model. The results in this paper contribute to the recent theoretical literature on penalized B-spline estimators by allowing for (i) multivariate covariates, (ii) heteroskedasticity of unknown form, (iii) derivative estimation, and (iv) statistical inference in the semi-linear model, under the two-sequence asymptotics. Our main findings rely on some apparently new technical results for splines that may be of independent interest. We also report results from a small-scale simulation study.