The Annals of Statistics

Limiting behavior of eigenvalues in high-dimensional MANOVA via RMT

Zhidong Bai, Kwok Pui Choi, and Yasunori Fujikoshi

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Abstract

In this paper, we derive the asymptotic joint distributions of the eigenvalues under the null case and the local alternative cases in the MANOVA model and multiple discriminant analysis when both the dimension and the sample size are large. Our results are obtained by random matrix theory (RMT) without assuming normality in the populations. It is worth pointing out that the null and nonnull distributions of the eigenvalues and invariant test statistics are asymptotically robust against departure from normality in high-dimensional situations. Similar properties are pointed out for the null distributions of the invariant tests in multivariate regression model. Some new formulas in RMT are also presented.

Article information

Source
Ann. Statist., Volume 46, Number 6A (2018), 2985-3013.

Dates
Received: May 2017
Revised: October 2017
First available in Project Euclid: 7 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.aos/1536307240

Digital Object Identifier
doi:10.1214/17-AOS1646

Mathematical Reviews number (MathSciNet)
MR3851762

Zentralblatt MATH identifier
06968606

Subjects
Primary: 62H10: Distribution of statistics
Secondary: 62E20: Asymptotic distribution theory

Keywords
Asymptotic distribution eigenvalues discriminant analysis high-dimensional case MANOVA nonnormality RMT test statistics

Citation

Bai, Zhidong; Choi, Kwok Pui; Fujikoshi, Yasunori. Limiting behavior of eigenvalues in high-dimensional MANOVA via RMT. Ann. Statist. 46 (2018), no. 6A, 2985--3013. doi:10.1214/17-AOS1646. https://projecteuclid.org/euclid.aos/1536307240


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References

  • Amemiya, Y. (1990). A note on the limiting distribution of certain characteristic roots. Statist. Probab. Lett. 9 465–470.
  • Anderson, T. W. (1951). The asymptotic distribution of certain characteristic roots and vectors. In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950 103–130. Univ. California Press, Berkeley, CA.
  • Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis, 3rd ed. Wiley, Hoboken, NJ.
  • Bai, Z. D. (1985). A note on the limiting distribution of the eigenvalues of a class of random matrices. J. Math. Res. Exposition 5 113–118.
  • Bai, Z., Choi, K. P. and Fujikoshi, Y. (2018). Supplement to “Limiting behavior of eigenvalues in high-dimensional MANOVA via RMT.” DOI:10.1214/17-AOS1646SUPP.
  • Bai, Z. D., Liu, H. X. and Wong, W. K. (2011). Asymptotic properties of eigenmatrices of a large sample covariance matrix. Ann. Appl. Probab. 21 1994–2015.
  • Bai, Z. D. and Saranadasa, H. (1996). Effect of high dimension: By an example of a two sample problem. Statist. Sinica 6 311–329.
  • Bai, Z. D. and Silverstein, J. W. (2004). CLT for linear spectral statistics of large-dimensional sample covariance matrices. Ann. Probab. 32 535–605.
  • Bai, Z. D. and Zhou, W. (2008). Large sample covariance matrices without independence structures in columns. Statist. Sinica 5 425–442.
  • Bai, Z., Jiang, D., Yao, J.-F. and Zheng, S. (2009). Corrections to LRT on large-dimensional covariance matrix by RMT. Ann. Statist. 37 3822–3840.
  • Cai, T. T. and Xia, Y. (2014). High-dimensional sparse MANOVA. J. Multivariate Anal. 131 174–196.
  • 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.
  • Chiani, M. (2016). Distribution of the largest root of a matrix for Roy’s test in multivariate analysis of variance. J. Multivariate Anal. 143 467–471.
  • Dempster, A. P. (1958). A high dimensional two sample significance test. Ann. Math. Statist. 29 995–1010.
  • Dempster, A. P. (1960). A significance test for the separation of two highly multivariate small samples. Biometrics 16 41–50.
  • Fujikoshi, Y. (1977). Asymptotic expansions for the distributions of the latent roots in MANOVA and the canonical correlations. J. Multivariate Anal. 7 386–396.
  • Fujikoshi, Y., Himeno, T. and Wakaki, H. (2004). Asymptotic results of a high-dimensional MANOVA test and power comparison when the dimension is large compared to the sample size. J. Japan Statist. Soc. 34 19–26.
  • Fujikoshi, Y., Himeno, T. and Wakaki, H. (2008). Asymptotic results in canonoical discriminat analysis when the dimension is large compared to the sample. J. Statist. Plann. Inference 138 3457–3466.
  • Glynn, W. J. and Muirhead, R. J. (1978). Inference in canonical correlation analysis. J. Multivariate Anal. 8 468–478.
  • Hsu, P. L. (1941). On the limiting distribution of roots of a determinantal equation. J. Lond. Math. Soc. 16 183–194.
  • Hu, J. and Bai, Z. D. (2016). A review of 20 years of naive tests of significance for high-dimensional mean vectors and covariance matrices. Sci. China Math. 59 2281–2300.
  • Johnstone, I. M. (2008). Multivariate analysis and Jacobi ensembles: Largest eigenvalue, Tracy–Widom limits and rate of convergence. Ann. Statist. 36 2638–2716.
  • Krishnaiah, P. R. and Chang, T. C. (1971). On the exact distributions of the extreme roots of the Wishart and $\mathrm{MANOVA}$ matrices. J. Multivariate Anal. 1 108–117.
  • Li, J. and Chen, S. X. (2012). Two sample tests for high-dimensional covariance matrices. Ann. Statist. 40 908–940.
  • Muirhead, R. J. (1978). Latent roots and matrix variates. Ann. Statist. 6 5–33.
  • Muirhead, R. J. (1982). Aspect of Multivariate Statistical Theory. Wiley, New York.
  • Pan, G. M. and Zhou, W. (2011). Central limit theorem for Hotelling’s $T^{2}$ statistic under large dimension. Ann. Appl. Probab. 21 1860–1910.
  • Schott, J. R. (2007). Some high-dimensional tests for a one-way MANOVA. J. Multivariate Anal. 98 1825–1839.
  • Seo, T., Kanda, T. and Fujikoshi, Y. (1994). Asymptotic distributions of the sample roots in MANOVA models under nonnormality. J. Japan Statist. Soc. 24 133–140.
  • Silverstein, J. (1995). Strong convergence of empirical distribution of eigenvlaues of large dimensional random matrices. J. Multivariate Anal. 55 331–339.
  • Srivastava, M. S. and Du, M. (2008). A test for the mean vector with fewer observations than the dimension under non-normality. J. Multivariate Anal. 99 386–402.
  • Srivastava, M. S. and Fujikoshi, Y. (2006). Multivariate analysis of variance with fewer observations than the dimension. J. Multivariate Anal. 97 1927–1940.
  • Srivastava, M. S. and Kubokawa, T. (2013). Tests for multivariate analysis of variance in high dimension under non-normality. J. Multivariate Anal. 115 204–216.
  • Sugiura, N. (1976). Asymptotic expansions of the distributions of the latent roots and latent vectors of the Wishart and multivariate $F$ matrices. J. Multivariate Anal. 6 500–525.
  • Ullah, I. and Jones, B. (2015). Regularised MANOVA for high-dimensional data. Aust. N. Z. J. Stat. 57 377–389.
  • Wakaki, H., Fujikoshi, Y. and Ulyanov, V. (2014). Asymptotic expansions of the distributions of MANOVA test statistics when the dimension is large. Hiroshima Math. J. 44 247–259.
  • Wang, L., Peng, B. and Li, R. (2015). A high-dimensional nonparamteric multivariate test for mean vector. J. Amer. Statist. Assoc. 110 1658–1669.
  • Zheng, S. (2012). Central limit theorems for linear spectral statistics of large-dimensional $F$-matrices. Ann. Inst. Henri Poincaré Probab. Stat. 47 444–476.

Supplemental materials

  • Supplement to “Limiting behavior of eigenvalues in high-dimensional MANOVA via RMT”. In the supplementary material, we presented (i) the truncation and normalization techniques as mentioned at the beginning of Section 7 of this paper; (ii) details of reparametrization for assumptions in (4.1) hold; (iii) proofs of (5.9) and (5.10); (iv) more plots of the empirical sizes of the three invariant tests considered in this paper; (v) the proofs of Lemmas 7.1 to 7.8; and (vi) the proofs of Lemmas 8.1 and 8.2.