The Annals of Statistics

A Cramér moderate deviation theorem for Hotelling’s $T^{2}$-statistic with applications to global tests

Weidong Liu and Qi-Man Shao

Full-text: Open access

Abstract

A Cramér moderate deviation theorem for Hotelling’s $T^{2}$-statistic is proved under a finite $(3+\delta)$th moment. The result is applied to large scale tests on the equality of mean vectors and is shown that the number of tests can be as large as $e^{o(n^{1/3})}$ before the chi-squared distribution calibration becomes inaccurate. As an application of the moderate deviation results, a global test on the equality of $m$ mean vectors based on the maximum of Hotelling’s $T^{2}$-statistics is developed and its asymptotic null distribution is shown to be an extreme value type I distribution. A novel intermediate approximation to the null distribution is proposed to improve the slow convergence rate of the extreme distribution approximation. Numerical studies show that the new test procedure works well even for a small sample size and performs favorably in analyzing a breast cancer dataset.

Article information

Source
Ann. Statist., Volume 41, Number 1 (2013), 296-322.

Dates
First available in Project Euclid: 26 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aos/1364302744

Digital Object Identifier
doi:10.1214/12-AOS1082

Mathematical Reviews number (MathSciNet)
MR3059419

Zentralblatt MATH identifier
1347.62032

Subjects
Primary: 62E20: Asymptotic distribution theory
Secondary: 62H15: Hypothesis testing 60F10: Large deviations

Keywords
Cramér moderate deviation Hotelling’s $T^{2}$-statistic global tests simultaneous hypothesis tests FDR brain structure gene selection

Citation

Liu, Weidong; Shao, Qi-Man. A Cramér moderate deviation theorem for Hotelling’s $T^{2}$-statistic with applications to global tests. Ann. Statist. 41 (2013), no. 1, 296--322. doi:10.1214/12-AOS1082. https://projecteuclid.org/euclid.aos/1364302744


Export citation

References

  • Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis, 3rd ed. Wiley, Hoboken, NJ.
  • Arratia, R., Goldstein, L. and Gordon, L. (1989). Two moments suffice for Poisson approximations: The Chen–Stein method. Ann. Probab. 17 9–25.
  • Cai, T., Liu, W. and Xia, Y. (2012). Two-sample test of high dimensional means under dependency. Technical report.
  • Cao, J. and Worsley, K. J. (1999). The detection of local shape changes via the geometry of Hotelling’s $T^2$ fields. Ann. Statist. 27 925–942.
  • 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.
  • de la Peña, V. H., Lai, T. L. and Shao, Q.-M. (2009). Self-Normalized Processes: Limit Theory and Statistical Applications. Springer, Berlin.
  • Dembo, A. and Shao, Q.-M. (2006). Large and moderate deviations for Hotelling’s $T^2$-statistic. Electron. Commun. Probab. 11 149–159 (electronic).
  • Donoho, D. and Jin, J. (2004). Higher criticism for detecting sparse heterogeneous mixtures. Ann. Statist. 32 962–994.
  • Fan, J., Hall, P. and Yao, Q. (2007). To how many simultaneous hypothesis tests can normal, Student’s $t$ or bootstrap calibration be applied? J. Amer. Statist. Assoc. 102 1282–1288.
  • Gerardina, E., Chételatd, G., Chupin, M., Cuingnet, R., Desgranges, B., Kime, H. S., Niethammer, M., Dubois, B., Stéphane Lehéricy, S., Line Garnero, L., Eustache, F. and Colliot, O. (2009). Multidimensional classification of hippocampal shape features discriminates Alzheimer’s disease and mild cognitive impairment from normal aging. NeuroImage 47 1476–1486.
  • Hall, P. and Jin, J. (2010). Innovated higher criticism for detecting sparse signals in correlated noise. Ann. Statist. 38 1686–1732.
  • Hall, P. and Wang, Q. (2010). Strong approximations of level exceedences related to multiple hypothesis testing. Bernoulli 16 418–434.
  • Jing, B.-Y., Shao, Q.-M. and Wang, Q. (2003). Self-normalized Cramér-type large deviations for independent random variables. Ann. Probab. 31 2167–2215.
  • Lin, Z. and Liu, W. (2009). On maxima of periodograms of stationary processes. Ann. Statist. 37 2676–2695.
  • Liu, W.-D., Lin, Z. and Shao, Q.-M. (2008). The asymptotic distribution and Berry–Esseen bound of a new test for independence in high dimension with an application to stochastic optimization. Ann. Appl. Probab. 18 2337–2366.
  • Liu, W. and Shao, Q.-M. (2010). Cramér-type moderate deviation for the maximum of the periodogram with application to simultaneous tests in gene expression time series. Ann. Statist. 38 1913–1935.
  • Liu, W. and Shao, Q. M. (2013). Supplement to “A Cramér moderate deviation theorem for Hotelling’s $T^2$-statistic with applications to global tests.” DOI:10.1214/12-AOS1082SUPP.
  • Martens, J. W. M., Nimmrich, I., Koenig, T., Look, M. P., Harbeck, N., Model, F., Kluth, A., de Vries, J. B., Sieuwerts, A. M., Portengen, H., Meijer-Van Gelder, M. E., Piepenbrock, C., Olek, A., Höfler, H., Kiechle, M., Klijn, J. G. K., Schmitt, M., Maier, S. and Foekens, J. A. (2005). Association of DNA methylation of Phosphoserine Aminotransferase with response to endocrine therapy in patients with recurrent breast cancer. Cancer Research 65 4101–4117.
  • Sakhanenko, A. I. (1991). Berry–Esseen type estimates for large deviation probabilities. Sib. Math. J. 32 647–656.
  • Shao, Q.-M. (1999). A Cramér type large deviation result for Student’s $t$-statistic. J. Theoret. Probab. 12 385–398.
  • Storey, J. D. and Tibshirani, R. (2001). Estimating false discovery rates under dependence, with applications to DNA microarrays. Technical report.
  • Styner, M., Oguz, I., Xu, S., Brechbühler, C., Pantazis, D., Levitt, J. J., Shenton, M. E. and Gerig, G. (2006). Framework for the statistical shape analysis of brain structures using SPHARM-PDM. Insight Journal 1071 242–250.
  • Taylor, J. E. and Worsley, K. J. (2008). Random fields of multivariate test statistics, with applications to shape analysis. Ann. Statist. 36 1–27.
  • Zaĭtsev, A. Y. (1987). On the Gaussian approximation of convolutions under multidimensional analogues of S. N. Bernstein’s inequality conditions. Probab. Theory Related Fields 74 535–566.
  • Zhao, Z., Taylor, W. D., Styner, M., Steffens, D. C., Krishnan, K. R. R. and MacFall, J. R. (2008). Hippocampus shape analysis and late-life depression. PLoS One 3 e1837.

Supplemental materials

  • Supplementary material: Supplement to “A Cramér moderate deviation theorem for Hotelling’s $T^{2}$-statistic with applications to global tests”. The supplement material includes the moderate deviation result by Sakhanenko (1991), the proof of Theorem 3.3 and the simulation results in Section 4.