High-dimensional linear regression inference via $\ell^2$ weak convergence

TL;DR

This paper establishes weak convergence for high-dimensional linear regression estimators in Hilbert spaces, enabling asymptotic normality and advanced inference methods for complex hypotheses.

Contribution

It introduces a novel weak convergence framework for high-dimensional regression estimators, allowing for diverging sparsity and facilitating new hypothesis testing and confidence band procedures.

Findings

Asymptotic normality of estimators under diverging sparsity.

Development of tests for finitely many linear hypotheses.

Construction of simultaneous confidence bands for the regression function.

Abstract

We prove weak convergence in a separable Hilbert space for estimators of high-dimensional regression coefficients, which yields asymptotic normality and enables direct use of standard asymptotic tools such as the continuous mapping theorem. The approach permits diverging sparsity with many small nonzero coefficients, while requiring that only finitely many have moderate magnitude. As applications, we develop a test for finitely many linear hypotheses and, via a Scheff\'{e}-type approach, simultaneous inference for infinitely many linear hypotheses, yielding both a global test and simultaneous confidence bands for the regression function. The limiting distributions are given by weighted sums of independent chi-squared variables, and plug-in critical values achieve asymptotically correct size.