Consistency of modified versions of Bayesian Information Criterion in sparse linear regression with subgaussian errors

TL;DR

This paper proves the consistency of modified Bayesian Information Criteria, like mBIC and mBIC2, for sparse linear regression models with subgaussian errors, extending previous results beyond normally distributed errors.

Contribution

It establishes the theoretical consistency of mBIC and mBIC2 in high-dimensional sparse regression with subgaussian errors, broadening their applicability.

Findings

Proves consistency of mBIC and mBIC2 under subgaussian errors

Extends previous normal-error assumptions to subgaussian case

Supports use of these criteria in high-dimensional sparse models

Abstract

We consider a sparse linear regression model, when the number of available predictors, $p$, is much larger than the sample size, $n$, and the number of non-zero coefficients, $p_0$, is small. To choose the regression model in this situation, we cannot use classical model selection criteria. In recent years, special methods have been proposed to deal with this type of problem, for example modified versions of Bayesian Information Criterion, like mBIC or mBIC2. It was shown that these criteria are consistent under the assumption that both $n$ and $p$ as well as $p_0$ tend to infinity and the error term is normally distributed. In this article we prove the consistency of mBIC and mBIC2 under the assumption that the error term is a subgaussian random variable.