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

The local polynomial estimator is particularly affected by the curse of dimensionality. So, the potentialities of such a tool become ineffective for large dimensional applications. Motivated by this, we propose a new estimation procedure based on the local linear estimator and a nonlinearity sparseness condition, which focuses on the number of covariates for which the gradient is not constant. Our procedure, called BID for Bias-Inflation-Deflation, is automatic and easily applicable to models with many covariates without any additive assumption to the model. It simultaneously gives a consistent estimation of a) the optimal bandwidth matrix, b) the multivariate regression function and c) the multivariate, bias-corrected, confidence bands. Moreover, it automatically identify the relevant covariates and it separates the nonlinear from the linear effects. We do not need pilot bandwidths. Some theoretical properties of the method are discussed in the paper. In particular, we show the nonparametric oracle property. For linear models, the BID automatically reaches the optimal rate $Op(n^{-1/2})$, equivalent to the parametric case. A simulation study shows a good performance of the BID procedure, compared with its direct competitor.