Statistica Sinica 23 (2013), 667-690

JACKKNIFE EMPIRICAL LIKELIHOOD TEST

FOR EQUALITY OF TWO HIGH DIMENSIONAL MEANS

Ruodu Wang, Liang Peng and Yongcheng Qi

University of Waterloo, Georgia Institute of Technology and

University of Minnesota-Duluth

Abstract: There is a long history of testing the equality of two multivariate means. A popular test is the Hotelling $T^2$, but in large dimensions it performs poorly due to the possible inconsistency of sample covariance estimation. Bai and Saranadasa (1996) and Chen and Qin (2010) proposed tests not involving the sample covariance, and derived asymptotic limits, which depend on whether the dimension is fixed or diverges, under a specific multivariate model. In this paper, we propose a jackknife empirical likelihood test that has a chi-square limit independent of the dimension. The conditions are much weaker than those needed in existing methods. A simulation study shows that the proposed new test has a very robust size across dimensions and has good power.

Key words and phrases: High dimensional mean, hypothesis test, Jackknife empirical likelihood.