Kernel Meets Sieve: Post-Regularization Confidence Bands for Sparse Additive Model

TL;DR

This paper introduces a new kernel-sieve hybrid estimator for constructing confidence bands in high-dimensional sparse additive models, combining kernel regression and spline methods to improve inference accuracy.

Contribution

The paper proposes a novel kernel-sieve hybrid estimator that enables simple, honest confidence band construction for sparse additive models, bridging the gap between existing methods.

Findings

Confidence bands are asymptotically honest based on Gaussian process approximation.

Numerical experiments demonstrate the method's effectiveness on synthetic and neuroscience data.

The hybrid estimator outperforms traditional sieve or kernel-based methods in high-dimensional settings.

Abstract

We develop a novel procedure for constructing confidence bands for components of a sparse additive model. Our procedure is based on a new kernel-sieve hybrid estimator that combines two most popular nonparametric estimation methods in the literature, the kernel regression and the spline method, and is of interest in its own right. Existing methods for fitting sparse additive model are primarily based on sieve estimators, while the literature on confidence bands for nonparametric models are primarily based upon kernel or local polynomial estimators. Our kernel-sieve hybrid estimator combines the best of both worlds and allows us to provide a simple procedure for constructing confidence bands in high-dimensional sparse additive models. We prove that the confidence bands are asymptotically honest by studying approximation with a Gaussian process. Thorough numerical results on both synthetic data and real-world neuroscience data are provided to demonstrate the efficacy of the theory.