Some permutation tests for high dimensional mean vectors

Abstract For high dimensional data where the dimension p of the observation vector is larger than the group sample sizes $$n_i, i=1,2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , classical approaches based on asymptotic theory are no longer valid. A recent parametric projection test shows a comparable type I error but with a substantial power improvement. However, the asymptotic theory of all such statistics may not hold with small $$n_i$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> , especially, in the presence of strong correlation structures. We here propose the use of a permutation test based on the same parametric projection test statistic. Extensive simulation results show that the permutation versions are more accurate under the null hypothesis than the parametric projection test when sample sizes are small and, especially when there is a strong correlation in the data. A significant test of “Sell in May and Go Away” for the manufacturing sector in China’s A-share market from May 2001 to October 2017 illustrates its application.