TESTING IN HIGH-DIMENSIONAL SPIKED MODELS

• 成果类型: Article
• 署名作者: Johnstone, Iain M.; Onatski, Alexei
• 署名单位: Stanford University; University of Cambridge
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/18-AOS1697
• 发表日期: 2020
• 页码: 1231-1254
• 关键词: linear spectral statistics; Low-rank Matrix; LARGEST EIGENVALUE; latent roots; CONVERGENCE; algorithm; limit

摘要

We consider the five classes of multivariate statistical problems identified by James (Ann. Math. Stat. 35 (1964) 475-501), which together cover much of classical multivariate analysis, plus a simpler limiting case, symmetric matrix denoising. Each of James' problems involves the eigenvalues of E-1 H where H and E are proportional to high-dimensional Wishart matrices. Under the null hypothesis, both Wisharts are central with identity covariance. Under the alternative, the noncentrality or the covariance parameter of H has a single eigenvalue, a spike, that stands alone. When the spike is smaller than a case-specific phase transition threshold, none of the sample eigenvalues separate from the bulk, making the testing problem challenging. Using a unified strategy for the six cases, we show that the log likelihood ratio processes parameterized by the value of the subcritical spike converge to Gaussian processes with logarithmic correlation. We then derive asymptotic power envelopes for tests for the presence of a spike.