Minimax Optimal Estimation in Partially Linear Additive Models under High Dimension

TL;DR

This paper establishes minimax estimation rates for high-dimensional partially linear additive models, revealing how sparsity, dimensionality, and smoothness influence the estimation difficulty and demonstrating near-optimal estimators.

Contribution

It derives minimax rates for both parametric and nonparametric parts in high-dimensional models and shows that penalized least squares can nearly attain these bounds.

Findings

Minimax lower bounds depend on sparsity, dimensionality, and smoothness.

Estimation rates for nonparametric components can be slowed to sparse rates.

Penalized least squares estimators nearly achieve the minimax bounds.

Abstract

In this paper, we derive minimax rates for estimating both parametric and nonparametric components in partially linear additive models with high dimensional sparse vectors and smooth functional components. The minimax lower bound for Euclidean components is the typical sparse estimation rate that is independent of nonparametric smoothness indices. However, the minimax lower bound for each component function exhibits an interplay between the dimensionality and sparsity of the parametric component and the smoothness of the relevant nonparametric component. Indeed, the minimax risk for smooth nonparametric estimation can be slowed down to the sparse estimation rate whenever the smoothness of the nonparametric component or dimensionality of the parametric component is suffciently large. In the above setting, we demonstrate that penalized least square estimators can nearly achieve minimax lower bounds.