No Access

A study of two high-dimensional likelihood ratio tests under alternative hypotheses

Huijun Chen

School of Mathematics, Jilin University, Changchun 130021, P. R. China

E-mail Address: hchen@jlu.edu.cn

Search for more papers by this author

and

Tiefeng Jiang

http://orcid.org/0000-0002-1099-8850

School of Statistics, University of Minnesota, 224 Church Street, S.E., MN 55455, USA

E-mail Address: jiang040@umn.edu

Search for more papers by this author

https://doi.org/10.1142/S2010326317500162Cited by:13 (Source: Crossref)

Abstract

Let $N_{p} (μ, Σ)$ be a $p$ -dimensional normal distribution. Testing $Σ$ equal to a given matrix or $(μ, Σ)$ equal to a given pair through the likelihood ratio test (LRT) is a classical problem in the multivariate analysis. When the population dimension $p$ is fixed, it is known that the LRT statistics go to $χ^{2}$ -distributions. When $p$ is large, simulation shows that the approximations are far from accurate. For the two LRT statistics, in the high-dimensional cases, we obtain their central limit theorems under a big class of alternative hypotheses. In particular, the alternative hypotheses are not local ones. We do not need the assumption that $p$ and $n$ are proportional to each other. The condition $n - 1 > p \to \infty$ suffices in our results.

Keywords:

AMSC: 62H15, 62H10

References

1. T. Anderson, An Introduction to Multivariate Statistical Analysis, 3rd edn. (Wiley-Interscience, 2003). Google Scholar
2. Z. Bai, D. Jiang, J. Yao and S. Zheng, Corrections to LRT on large-dimensional covariance matrix by RMT, Ann. Stat. 37 (2009) 3822–3840. Crossref, Google Scholar
3. P. Billingsley, Probability and Measure, 2nd edn., Wiley Series in Probability and Mathematical Statistics (John Wiley & Sons, 1986). Google Scholar
4. T. Cai and Z. Ma, Optimal hypothesis testing for high dimensional covariance matrices, Bernoulli 19(5B) (2013) 2359–2388. Crossref, Google Scholar
5. S. Chen, L. Zhang and P. Zhong, Tests for high dimensional covariance matrices, J. Amer. Stat. Assoc. 105 (2010) 810–819. Crossref, Google Scholar
6. M. Eaton, Multivariate Statistics: A Vector Space Approach, Wiley Series in Probability and Statistics (John Wiley & Sons, 1983). Google Scholar
7. D. Jiang, T. Jiang and F. Yang, Likelihood ratio tests for covariance matrices of high-dimensional normal distributions, J. Stat. Plann. Inference 142 (2012) 2241–2256. Crossref, Google Scholar
8. T. Jiang and Y. Qi, Likelihood ratio tests for high-dimensional normal distributions, Scand. J. Stat. 42(4) (2015) 988–1009. Crossref, Google Scholar
9. T. Jiang and F. Yang, Central limit theorems for classical likelihood ratio tests for high-dimensional normal distributions, Ann. Stat. 41(4) (2013) 2029–2074. Crossref, Google Scholar
10. O. Ledoit and M. Wolf, Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size, Ann. Stat. 30 (2002) 1081–1102. Crossref, Google Scholar
11. R. J. Muirhead, Aspects of Multivariate Statistical Theory (Wiley, New York, 1982). Crossref, Google Scholar
12. A. Onatski, M. Moreira and M. Hallin, Signal detection in high dimension: The multispiked case, Ann. Statist. 42(1) (2014) 225–254. Crossref, Google Scholar
13. J. R. Schott, Some tests for the equality of covariance matrices, J. Stat. Plann. Inference 94 (2001) 25–36. Crossref, Google Scholar
14. J. R. Schott, Testing for complete independence in high dimensions, Biometrika 92 (2005) 951–956. Crossref, Google Scholar
15. J. R. Schott, A test for the equality of covariance matrices when the dimension is large relative to the sample sizes, Comput. Statist. Data Anal. 51 (2007) 6535–6542. Crossref, Google Scholar
16. M. S. Srivastava, Some tests concerning the covariance matrix in high-dimensional data, J. Japan Statist. Soc. 35 (2005) 251–272. Crossref, Google Scholar
17. S. Zheng, Z. Bai and J. Yao, Substitution principle for CLT of linear spectral statistics of high-dimensional sample covariance matrices with applications to hypothesis testing, Ann. Stat. 43 (2015) 546–591. Crossref, Google Scholar