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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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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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.