Testing linear hypotheses of mean vectors for high-dimension data with unequal covariance matrices

• Published in: Journal of statistical planning and inference Vol. 143; no. 11; pp. 1898 - 1911
• Main Authors: Nishiyama, Takahiro, Hyodo, Masashi, Seo, Takashi, Pavlenko, Tatjana
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.11.2013
• Subjects: (N,p)-asymptotics; Cornish-Fisher transform; Dempster trace criterion; High dimensionality; Multivariate Behrens-Fisher problem; p)-asymptotics; Dempster trace criterion; (N,p)-asymptotics; Multivariate Behrens–Fisher problem; High dimensionality; Cornish–Fisher transform
• ISSN: 0378-3758; 1873-1171; 1873-1171
• DOI: 10.1016/j.jspi.2013.07.008

We propose a new test procedure for testing linear hypothesis on the mean vectors of normal populations with unequal covariance matrices when the dimensionality, p exceeds the sample size N, i.e. p/N→c<∞. Our procedure is based on the Dempster trace criterion and is shown to be consistent in high dimensions. The asymptotic null and non-null distributions of the proposed test statistic are established in the high dimensional setting and improved estimator of the critical point of the test is derived using Cornish–Fisher expansion. As a special case, our testing procedure is applied to multivariate Behrens–Fisher problem. We illustrate the relevance and benefits of the proposed approach via Monte-Carlo simulations which show that our new test is comparable to, and in many cases is more powerful than, the tests for equality of means presented in the recent literature. •We propose a new test procedure for testing linear hypothesis on the mean vectors of normal populations with unequal covariance matrices for high dimension data.•The asymptotic null and non-null distributions of the proposed test statistic are established in the high dimensional setting.•We improved estimator of the critical point of the test that is derived using Cornish–Fisher expansion.•We applied our testing procedure to multivariate Behrens–Fisher problem.•We illustrate the relevance and benefits of the proposed approach via Monte-Carlo simulations.