Abstract:

Testing the equality of two means is a fundamental inference problem. For high-dimensional data, the Hotelling's T-square test either performs poorly or becomes inapplicable. Several modifications have been proposed to address this issue. However, most of them are based on asymptotic normality of the null distributions of their test statistics which inevitably requires strong assumptions on the covariance matrix. We study this problem thoroughly and propose an L2-norm based test that works under mild conditions and even when there are fewer observations than the dimension. Specifically, to cope with  general non-normality of the null distribution we employ the Welch-Satterthwaite chi-square approximation. We derive a sharp upper bound on the approximation error and use it to justify that the chi-square approximation is preferred to normal approximation. Simple ratio-consistent estimators for the parameters in the  chi-square approximation are given. Importantly, our test can cope with singularity or near singularity of the unknown covariance which is commonly seen in high dimensions and is the main cause of non-normality. The power of the proposed test  is also investigated. Extensive simulation studies and an application show that our test outperforms several existing tests in terms of size control, and the powers are comparable when their sizes are comparable.

Biography:

Ming-Yen Cheng is currently a Professor at Department of Mathematics of the Hong Kong Baptist University. Her past work experiences include Chair of Statistics at Department of Statistical Science of the University College London (2008-2010) and Distinguished Professor at Department of Mathematics of the National Taiwan University (2006-2008 and 2010-2017). Her research interests and contributions lies in nonparametric and semiparametric models, high-dimensional data, change-points, classification and clustering, etc. She was elected to Fellow of the Institute of Mathematical Statistics (IMS) in 2007 and Fellow of the American Statistical Association (ASA) in 2009. She is active in editorial service, such as associate editor of Annals of Statistics, Journal of the American Statistical Association and so on, as well as conference organization.