Statistica Sinica 27 (2017), 1419-1436
Abstract: We consider high-dimensional location test problems in which the number of variables 𝒫 may exceed the sample size 𝓃. The classical 𝑇2 test does not work well because the contamination bias in estimating the covariance matrix grows rapidly with 𝒫. Unlike most existing remedies abandoning all the correlation information, the composite 𝑇2 test developed here makes use of them in a practical and effcient way. Under mild conditions, the proposed test statistic is asymptotically normal, and allows the dimensionality to almost exponentially increase in 𝓃. The test inherits certain appealing features of the classical 𝑇2 test and does not suffer from large bias contamination. Due to incorporating much correlation information, the proposed test can deliver more robust performance than existing methods in many cases. Simulation studies demonstrate the validity of asymptotic analysis.
Key words and phrases: Asymptotic normality, composite 𝑇2 test, high-dimensional data, large-𝒫-small-𝓃.