Asymptotic Normality in Linear Regression with Approximately Sparse Structure

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, X, and outcome variable, Upsilon, i.e., the statistic parallel to X'Y parallel to(2)(2), under rather unrestrictive assumptions for the model parameters beta(j). We employ variance-gamma distribution in order to derive the results, which, along with the asymptotic results, allows us to easily define the exact distribution of the statistic. 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.

Automated summary

This paper studies the asymptotic normality in high-dimensional linear regression, focusing on the case where the covariance matrix of the regression variables has a KMS structure. The main result is the derivation of the exact asymptotic distribution for the squared norm of the product between predictor matrix X and outcome variable Y, under rather unrestrictive assumptions for the model parameters. A Monte Carlo simulation study is conducted for a specific case of approximate sparsity of the model parameter vector beta.