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Testing in High-Dimensional Spiked Models

Author

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  • Johnstone, I. M
  • Onatski, A.

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 {code} 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.

Suggested Citation

  • Johnstone, I. M & Onatski, A., 2018. "Testing in High-Dimensional Spiked Models," Cambridge Working Papers in Economics 1806, Faculty of Economics, University of Cambridge.
  • Handle: RePEc:cam:camdae:1806
    Note: ao319
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    File URL: http://www.econ.cam.ac.uk/research-files/repec/cam/pdf/cwpe1806.pdf
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    Cited by:

    1. Hafner, Christian M. & Linton, Oliver B. & Tang, Haihan, 2020. "Estimation of a multiplicative correlation structure in the large dimensional case," Journal of Econometrics, Elsevier, vol. 217(2), pages 431-470.

    More about this item

    Keywords

    Likelihood ratio test; hypergeometric function; principal components analysis; canonical correlations; matrix denoising; multiple response regression;
    All these keywords.

    JEL classification:

    • E20 - Macroeconomics and Monetary Economics - - Consumption, Saving, Production, Employment, and Investment - - - General (includes Measurement and Data)

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