Scalar-Invariant Test for High-Dimensional Regression Coefficients

TL;DR

This paper introduces a new scalar-invariant test for high-dimensional linear regression coefficients that remains valid when the number of variables exceeds the sample size, addressing limitations of existing tests.

Contribution

A novel scalar-invariant test statistic for high-dimensional regression coefficients, with proven asymptotic normality under mild conditions.

Findings

Test performs well in various simulation scenarios.

Addresses invariance issues of previous tests.

Applicable when p > n in high-dimensional settings.

Abstract

This article is concerned with simultaneous tests on linear regression coefficients in high-dimensional settings. When the dimensionality is larger than the sample size, the classic $F$-test is not applicable since the sample covariance matrix is not invertible. In order to overcome this issue, both Goeman, Finos and van Houwelingen (2011) and Zhong and Chen (2011) proposed their test procedures after excluding the $(\X^{'}\X)^{-1}$ term in $F$-statistics. However, both these two test are not invariant under the group of scalar transformations. In order to treat those variables in a `fair' way, we proposed a new test statistic and establish its asymptotically normal under certain mild conditions. Simulation studies showed that our test procedure performs very well in many cases.