Asymptotic normality in linear regression with approximately sparse structure

DOI10.3390/MATH10101657arXiv2203.04154MaRDI QIDQ6393142

Author name not available (Why is that?)

Publication date: 8 March 2022

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,

m a t h b b X

, and outcome variable,

Y

, i.e. the statistic

| m a t h b b X^{'} Y |_{2}^{2}

. Additionally, we consider a specific case of approximate sparsity of the model parameter vector

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.

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