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High-dimensional tests for spherical location and spiked covariance

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  • Ley, Christophe
  • Paindaveine, Davy
  • Verdebout, Thomas

Abstract

This paper mainly focuses on one of the most classical testing problems in directional statistics, namely the spherical location problem that consists in testing the null hypothesis H0:θ=θ0 under which the (rotational) symmetry center θ is equal to a given value θ0. The most classical procedure for this problem is the so-called Watson test, which is based on the sample mean of the observations. This test enjoys many desirable properties, but its asymptotic theory requires the sample size n to be large compared to the dimension p. This is a severe limitation, since more and more problems nowadays involve high-dimensional directional data (e.g., in genetics or text mining). In the present work, we derive the asymptotic null distribution of the Watson statistic as both n and p go to infinity. This reveals that (i) the Watson test is robust against high dimensionality, and that (ii) it allows for (n,p)-asymptotic results that are universal, in the sense that p may go to infinity arbitrarily fast (or slowly) as a function of n. Turning to Euclidean data, we show that our results also lead to a test for the null that the covariance matrix of a high-dimensional multinormal distribution has a “θ0-spiked” structure. Finally, Monte Carlo studies corroborate our asymptotic results and briefly explore non-null rejection frequencies.

Suggested Citation

  • Ley, Christophe & Paindaveine, Davy & Verdebout, Thomas, 2015. "High-dimensional tests for spherical location and spiked covariance," Journal of Multivariate Analysis, Elsevier, vol. 139(C), pages 79-91.
    • Handle: RePEc:eee:jmvana:v:139:y:2015:i:c:p:79-91
    DOI: 10.1016/j.jmva.2015.02.019
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    References listed on IDEAS

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    1. Tony Cai, T. & Jiang, Tiefeng, 2012. "Phase transition in limiting distributions of coherence of high-dimensional random matrices," Journal of Multivariate Analysis, Elsevier, vol. 107(C), pages 24-39.
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    3. Davy Paindaveine & Thomas Verdebout, 2013. "Optimal Rank-Based Tests for the Location Parameter of a Rotationally Symmetric Distribution on the Hypersphere," Working Papers ECARES ECARES 2013-36, ULB -- Universite Libre de Bruxelles.
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    Cited by:

    1. Davy Paindaveine & Thomas Verdebout, 2017. "Detecting the Direction of a Signal on High-dimensional Spheres: Non-null and Le Cam Optimality Results," Working Papers ECARES ECARES 2017-40, ULB -- Universite Libre de Bruxelles.
    2. Anders Bredahl Kock & David Preinerstorfer, 2019. "Power in High‐Dimensional Testing Problems," Econometrica, Econometric Society, vol. 87(3), pages 1055-1069, May.
    3. Feng, Long & Sun, Fasheng, 2015. "A note on high-dimensional two-sample test," Statistics & Probability Letters, Elsevier, vol. 105(C), pages 29-36.
    4. Christine Cutting & Davy Paindaveine & Thomas Verdebout, 2015. "Tests of Concentration for Low-Dimensional and High-Dimensional Directional Data," Working Papers ECARES ECARES 2015-05, ULB -- Universite Libre de Bruxelles.
    5. Christine Cutting & Davy Paindaveine & Thomas Verdebout, 2015. "Testing Uniformity on High-Dimensional Spheres against Contiguous Rotationally Symmetric Alternatives," Working Papers ECARES ECARES 2015-04, ULB -- Universite Libre de Bruxelles.
    6. Arthur Pewsey & Eduardo García-Portugués, 2021. "Recent advances in directional statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 1-58, March.

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