Statistical inference in sparse high-dimensional additive models

Abstract

In this paper, we discuss the estimation of a nonparametric component f1 of a nonparametric additive model Y=f1(X1)+⋯+fq(Xq)+ϵ. We allow the number q of additive components to grow to infinity and we make sparsity assumptions about the number of nonzero additive components. We compare this estimation problem with that of estimating f1 in the oracle model Z=f1(X1)+ϵ, for which the additive components f2,…,fq are known. We construct a two-step presmoothing-and-resmoothing estimator of f1 and state finite-sample bounds for the difference between our estimator and a corresponding smoothing estimator fˆ1(oracle) in the oracle model. In an asymptotic setting, these bounds can be used to show asymptotic equivalence of our estimator and the oracle estimator; the paper thus shows that, asymptotically, under strong enough sparsity conditions, knowledge of f2,…,fq has no effect on estimation accuracy. Our first step is to estimate f1 with an undersmoothed estimator based on near-orthogonal projections with a group Lasso bias correction. In the second step, we construct pseudo responses Yˆ by evaluating this undersmoothed estimator of f1 at the design points and then apply the smoothing method of the oracle estimator fˆ1(oracle) to the nonparametric regression problem with “responses” Yˆ and covariates X1. Our mathematical exposition centers primarily on establishing properties of the presmoothing estimator. We present simulation results demonstrating close-to-oracle performance of our estimator in practical applications.