Bias-corrected inference for multivariate nonparametric regression: Model selection and oracle property

Abstract

The local polynomial estimator is particularly affected by the curse of dimensionality, which reduces the potential of this tool for large-dimensional applications. We propose an estimation procedure based on the local linear estimator and a sparseness condition that focuses on nonlinearities in the model. Our procedure, called BID (bias inflation–deflation), is automatic and easily applicable to models with many covariates without requiring any additivity assumption. It is an extension of the RODEO method, and introduces important new contributions: consistent estimation of the multivariate optimal bandwidth (the tuning parameter of the estimator); consistent estimation of the multivariate bias-corrected regression function and confidence bands; and automatic identification and separation of nonlinear and linear effects. Some theoretical properties of the method are discussed. In particular, we show the nonparametric oracle property. For linear models, BID automatically reaches the optimal rate O p ( n − 1 / 2 ) , equivalent to the parametric case. A simulation study shows the performance of the procedure for finite samples.