Linear and nonlinear signal detection and estimation in high-dimensional nonparametric regression under weak sparsity

Abstract

The partially linear model provides an effective tool to combat the curse of dimensionality in nonparametric regression. Its applicability is, however, compromised by the need for correct distinction between linear and nonlinear components. Existing solutions are restricted to low dimensions or regression functions endowed with special structures. This paper considers a general nonparametric regression framework under which signal strength is embedded in a continuous spectrum scaled by asymptotic orders. Under a weak sparsity condition which allows for the presence of many weak, non-detectable, signals, a novel penalised regression procedure is proposed for detection and estimation of strong linear and nonlinear signals under high dimensions. The procedure applies bandwidth regularisation and SCAD penalisation to select nonlinear and linear signals, respectively, under a partially linear model setting. Theoretical results are established for its consistency in detecting strong signals and its error rate in estimating the regression function. Numerical examples are presented to illustrate its performance.