Asymptotic normality in linear regression with approximately sparse structure

TL;DR

This paper derives the exact asymptotic distribution of a key statistic in high-dimensional linear regression with KMS-structured covariance matrices, especially under approximate sparsity, supported by simulations.

Contribution

It provides the first derivation of the asymptotic distribution of the squared norm of the predictor-outcome product in high-dimensional settings with KMS covariance structure.

Findings

The statistic converges quickly to its limiting distribution even with high correlation.

Simulation results support the theoretical findings under various conditions.

Potential for developing statistical tests based on the asymptotic distribution.

Abstract

In this paper we study the asymptotic normality in high-dimensional linear regression. We focus on the case where the covariance matrix of the regression variables has a KMS structure, in asymptotic settings where the number of predictors, $p$, is proportional to the number of observations, $n$. The main result of the paper is the derivation of the exact asymptotic distribution for the suitably centered and normalized squared norm of the product between predictor matrix, $\mathbb{X}$, and outcome variable, $Y$, i.e. the statistic $\|\mathbb{X}'Y\|_{2}^{2}$. Additionally, we consider a specific case of approximate sparsity of the model parameter vector $\beta$ and perform a Monte-Carlo simulation study. The simulation results suggest that the statistic approaches the limiting distribution fairly quickly even under high variable multi-correlation and relatively small number of observations, suggesting possible applications to the construction of statistical testing procedures for the real-world data and related problems.