A F-type test on linear hypothesis testing of high-dimensional means

Abstract

Abstract This paper proposes a new F-type test for linear hypotheses about mean vectors in high-dimensional settings with heteroscedasticity. While numerous tests exist for high-dimensional mean problems, many rely on restrictive covariance conditions to ensure asymptotic normality under the null. To relax these constraints, we develop a test whose statistic, under normality and the null, follows an F-type mixture distribution, defined as the ratio of two independent chi-square-type mixtures. We establish that, under mild regularity conditions, the test statistic has the same normal and non-normal limiting distributions as this F-type mixture. The null distribution is approximated via the Welch–Satterthwaite chi-square method applied to the mixture components. Simulations confirm that the proposed test maintains accurate size control. Finally, we demonstrate its practical utility through a real-data application.