A test of sphericity for high-dimensional data and its application for detection of divergently spiked noise

• Published in: Sequential analysis Vol. 37; no. 3; pp. 397 - 411
• Main Authors: Yata, Kazuyoshi, Aoshima, Makoto, Nakayama, Yugo
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 03.07.2018; Taylor & Francis Ltd
• Subjects: Asymptotic methods; Asymptotic properties; Covariance matrix; Cross-data-matrix method; Dimensional analysis; Economic models; Estimating techniques; Gene expression; gene expression data; HDLSS; noise detection; noise reduction method; Normality; sphericity; Statistical analysis
• ISSN: 0747-4946; 1532-4176
• DOI: 10.1080/07474946.2018.1548850

In this article, we consider a test of the sphericity for high-dimensional covariance matrices. We produce a test statistic by using the extended cross-data-matrix (ECDM) methodology. We show that the ECDM test statistic is based on an unbiased estimator of a sphericity measure. In addition, the ECDM test statistic enjoys consistency properties and the asymptotic normality in high-dimensional settings. We propose a new test procedure based on the ECDM test statistic and evaluate its asymptotic size and power theoretically and numerically. We give a two-stage sampling scheme so that the test procedure can ensure a prespecified level both for the size and power. We apply the test procedure to detect divergently spiked noise in high-dimensional statistical analysis. We analyze gene expression data by the proposed test procedure.