Estimation and Uniform Inference in Sparse High-Dimensional Additive Models

TL;DR

This paper introduces a new method for constructing valid confidence bands for a specific component in high-dimensional sparse additive models, using sieve estimation and bootstrap techniques, with proven theoretical guarantees and demonstrated effectiveness.

Contribution

It develops a novel approach combining sieve estimation and Z-estimation for uniform inference in high-dimensional additive models, including bootstrap-based confidence bands.

Findings

Method provides reliable coverage in simulations

Achieves optimal rates for uniform lasso estimation

Applicable in small sample scenarios

Abstract

We develop a novel method to construct uniformly valid confidence bands for a nonparametric component $f_1$ in the sparse additive model $Y=f_1(X_1)+\ldots + f_p(X_p) + \varepsilon$ in a high-dimensional setting. Our method integrates sieve estimation into a high-dimensional Z-estimation framework, facilitating the construction of uniformly valid confidence bands for the target component $f_1$. To form these confidence bands, we employ a multiplier bootstrap procedure. Additionally, we provide rates for the uniform lasso estimation in high dimensions, which may be of independent interest. Through simulation studies, we demonstrate that our proposed method delivers reliable results in terms of estimation and coverage, even in small samples.