The Annals of Statistics

Tests alternative to higher criticism for high-dimensional means under sparsity and column-wise dependence

Ping-Shou Zhong, Song Xi Chen, and Minya Xu

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Abstract

We consider two alternative tests to the Higher Criticism test of Donoho and Jin [Ann. Statist. 32 (2004) 962–994] for high-dimensional means under the sparsity of the nonzero means for sub-Gaussian distributed data with unknown column-wise dependence. The two alternative test statistics are constructed by first thresholding $L_{1}$ and $L_{2}$ statistics based on the sample means, respectively, followed by maximizing over a range of thresholding levels to make the tests adaptive to the unknown signal strength and sparsity. The two alternative tests can attain the same detection boundary of the Higher Criticism test in [Ann. Statist. 32 (2004) 962–994] which was established for uncorrelated Gaussian data. It is demonstrated that the maximal $L_{2}$-thresholding test is at least as powerful as the maximal $L_{1}$-thresholding test, and both the maximal $L_{2}$ and $L_{1}$-thresholding tests are at least as powerful as the Higher Criticism test.

Article information

Source
Ann. Statist., Volume 41, Number 6 (2013), 2820-2851.

Dates
First available in Project Euclid: 17 December 2013

Permanent link to this document
https://projecteuclid.org/euclid.aos/1387313391

Digital Object Identifier
doi:10.1214/13-AOS1168

Mathematical Reviews number (MathSciNet)
MR3161449

Zentralblatt MATH identifier
1294.62128

Subjects
Primary: 62H15: Hypothesis testing
Secondary: 62G20: Asymptotic properties 62G32: Statistics of extreme values; tail inference

Keywords
Large deviation large $p$ small $n$ optimal detection boundary sparse signal thresholding weak dependence

Citation

Zhong, Ping-Shou; Chen, Song Xi; Xu, Minya. Tests alternative to higher criticism for high-dimensional means under sparsity and column-wise dependence. Ann. Statist. 41 (2013), no. 6, 2820--2851. doi:10.1214/13-AOS1168. https://projecteuclid.org/euclid.aos/1387313391


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References

  • Andrews, D. and Pollard, D. (1994). An introduction to functional central limit theorems for dependent stochastic processes. Int. Statist. Rev. 62 119–132.
  • Arias-Castro, E., Bubeck, S. and Lugosi, G. (2012a). Detection of correlations. Ann. Statist. 40 412–435.
  • Arias-Castro, E., Bubeck, S. and Lugosi, G. (2012b). Detecting positive correlations in a multivariate sample. Available at arXiv:1202.5536v1 [math.ST].
  • Bai, Z. and Saranadasa, H. (1996). Effect of high dimension: By an example of a two sample problem. Statist. Sinica 6 311–329.
  • Bradley, R. C. (2005). Basic properties of strong mixing conditions. A survey and some open questions. Probab. Surv. 2 107–144.
  • Cai, T. T., Jeng, X. J. and Jin, J. (2011). Optimal detection of heterogeneous and heteroscedastic mixtures. J. R. Stat. Soc. Ser. B Stat. Methodol. 73 629–662.
  • Cai, T. and Wu, Y. (2012). Optimal detection for sparse mixtures. Unpublished manuscript.
  • Chen, S. X. and Qin, Y.-L. (2010). A two-sample test for high-dimensional data with applications to gene-set testing. Ann. Statist. 38 808–835.
  • Delaigle, A. and Hall, P. (2009). Higher criticism in the context of unknown distribution, no-nindependence and classification. In Perspectives in Mathematical Sciences. I. Stat. Sci. Interdiscip. Res. 7 109–138. World Sci. Publ., Hackensack, NJ.
  • Delaigle, A., Hall, P. and Jin, J. (2011). Robustness and accuracy of methods for high dimensional data analysis based on Student’s $t$-statistic. J. R. Stat. Soc. Ser. B Stat. Methodol. 73 283–301.
  • Donoho, D. and Jin, J. (2004). Higher criticism for detecting sparse heterogeneous mixtures. Ann. Statist. 32 962–994.
  • Donoho, D. and Jin, J. (2008). Higher criticism thresholding: Optimal feature selection when useful features are rare and weak. Proc. Natl. Acad. Sci. USA 105 14790–14795.
  • Donoho, D. L. and Johnstone, I. M. (1994). Ideal spatial adaptation by wavelet shrinkage. Biometrika 81 425–455.
  • Doukhan, P. (1994). Mixing: Properties and Examples. Lecture Notes in Statistics 85. Springer, New York.
  • Fan, J. (1996). Test of significance based on wavelet thresholding and Neyman’s truncation. J. Amer. Statist. Assoc. 91 674–688.
  • Hall, P. and Jin, J. (2008). Properties of higher criticism under strong dependence. Ann. Statist. 36 381–402.
  • Hall, P. and Jin, J. (2010). Innovated higher criticism for detecting sparse signals in correlated noise. Ann. Statist. 38 1686–1732.
  • Ingster, Y. I. (1997). Some problems of hypothesis testing leading to infinitely divisible distributions. Math. Methods Statist. 6 47–69.
  • Jing, B.-Y., Shao, Q.-M. and Zhou, W. (2008). Towards a universal self-normalized moderate deviation. Trans. Amer. Math. Soc. 360 4263–4285.
  • Joe, H. (1997). Multivariate Models and Dependence Concepts. Monographs on Statistics and Applied Probability 73. Chapman & Hall, London.
  • Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.
  • Petrov, V. V. (1995). Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Oxford Studies in Probability 4. Oxford Univ. Press, New York.
  • Pisier, G. (1983). Some applications of the metric entropy condition to harmonic analysis. In Banach Spaces, Harmonic Analysis, and Probability Theory (Storrs, Conn., 1980/1981). Lecture Notes in Math. 995 123–154. Springer, Berlin.
  • Shao, Q.-M. (1997). Self-normalized large deviations. Ann. Probab. 25 285–328.
  • Sibuya, M. (1960). Bivariate extreme statistics. I. Ann. Inst. Statist. Math. Tokyo 11 195–210.
  • Tukey, J. W. (1976). T13 N: The higher criticism. Course Notes, Statistics 411, Princeton Univ.
  • van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, New York.
  • Wang, Q. and Hall, P. (2009). Relative errors in central limit theorems for Student’s $t$ statistic, with applications. Statist. Sinica 19 343–354.
  • Zhong, P. S., Chen, S. X. and Xu, M. (2013). Supplement to “Tests alternative to higher criticism for high dimensional means under sparsity and column-wise dependence.” DOI:10.1214/13-AOS1168SUPP.

Supplemental materials

  • Supplementary material: A supplement to “Tests alternative to higher criticism for high-dimensional means under sparsity and column-wise dependence”. The supplementary material contains proofs for Proposition 1 and Theorem 1 in Section 2.