skip to content

Faculty of Economics

Journal Cover

Johnstone, I. M. and Onatski, A.

Testing in High-dimensional Spiked Models

Annals of Statistics

Vol. 48 no. 3 pp. 1231-1254 (2018)

Abstract: We consider the five classes of multivariate statistical problems identified by James (1964), which together cover much of classical multivariate analysis, plus a simpler limiting case, symmetric matrix denoising. Each of James’ problems involves the eigenvalues of H -1 E 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 non-centrality 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 sub-critical spike converge to Gaussian processes with logarithmic correlation. We then derive asymptotic power envelopes for tests for the presence of a spike.

Keywords: likelihood ratio test, hypergeometric function, principal components analysis, canonical correlations, matrix denoising, multiple response regression

Author links: Alexey Onatskiy  

Publisher's Link: https://doi.org/10.1214/18-AOS1697



Cambridge Working Paper in Economics Version of Paper: Testing in High-Dimensional Spiked Models, Johnstone, I. M. and Onatski, A., (2018)

Papers and Publications



Recent Publications


Onatski, A. and Wang, C. Spurious Factor Analysis Econometrica [2021]

Jochmans, K. and Weidner, M. Fixed-Effect Regressions on Network Data Econometrica [2019]

Biroli, P., Boneva, T., Raja, A. and Rauh, C. Parental Beliefs About Returns to Child Health Investments Journal of Econometrics [2020]

Bergin, P. R. and Corsetti, G. Beyond Competitive Devaluations: The Monetary Dimensions of Comparative Advantage American Economic Journal: Macroeconomics [2020]