Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size

• 成果类型: Article
• 署名作者: Ledoit, O; Wolf, M
• 署名单位: University of California System; University of California Los Angeles; Pompeu Fabra University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• 发表日期: 2002
• 页码: 1081-1102
• 关键词: white noise; criteria; CONVERGENCE; signals; number

摘要

This paper analyzes whether standard covariance matrix tests work when dimensionality is large, and in particular larger than sample size, In the latter case, the singularity of the sample covariance matrix makes likelihood ratio tests degenerate, but other tests based on quadratic forms of sample covariance matrix eigenvalues remain well-defined. We study the consistency property and limiting distribution of these tests as dimensionality and sample size go to infinity together, with their ratio converging to a finite nonzero limit. We find that the existing test for sphericity is robust against high dimensionality, but not the test for equality of the covariance matrix to a given matrix. For the latter test, we develop a new correction to the existing test statistic that makes it robust against high dimensionality.