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Likelihood Ratio Tests for High-Dimensional Normal Distributions

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  • Tiefeng Jiang
  • Yongcheng Qi

Abstract

type="main" xml:id="sjos12147-abs-0001"> In their recent work, Jiang and Yang studied six classical Likelihood Ratio Test statistics under high-dimensional setting. Assuming that a random sample of size n is observed from a p-dimensional normal population, they derive the central limit theorems (CLTs) when p and n are proportional to each other, which are different from the classical chi-square limits as n goes to infinity, while p remains fixed. In this paper, by developing a new tool, we prove that the mentioned six CLTs hold in a more applicable setting: p goes to infinity, and p can be very close to n. This is an almost sufficient and necessary condition for the CLTs. Simulations of histograms, comparisons on sizes and powers with those in the classical chi-square approximations and discussions are presented afterwards.

Suggested Citation

  • Tiefeng Jiang & Yongcheng Qi, 2015. "Likelihood Ratio Tests for High-Dimensional Normal Distributions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(4), pages 988-1009, December.
    • Handle: RePEc:bla:scjsta:v:42:y:2015:i:4:p:988-1009
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    File URL: http://hdl.handle.net/10.1111/sjos.12147
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    References listed on IDEAS

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    1. Banzato, Erika & Chiogna, Monica & Djordjilović, Vera & Risso, Davide, 2023. "A Bartlett-type correction for likelihood ratio tests with application to testing equality of Gaussian graphical models," Statistics & Probability Letters, Elsevier, vol. 193(C).
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    3. Kathryn Stewart, 2020. "Total Variation Approximation of Random Orthogonal Matrices by Gaussian Matrices," Journal of Theoretical Probability, Springer, vol. 33(2), pages 1111-1143, June.
    4. Mingyue Hu & Yongcheng Qi, 2023. "Limiting distributions of the likelihood ratio test statistics for independence of normal random vectors," Statistical Papers, Springer, vol. 64(3), pages 923-954, June.
    5. Dörnemann, Nina, 2023. "Likelihood ratio tests under model misspecification in high dimensions," Journal of Multivariate Analysis, Elsevier, vol. 193(C).
    6. Tang, Ping & Lu, Rongrong & Xie, Junshan, 2022. "Asymptotic distribution of the maximum interpoint distance for high-dimensional data," Statistics & Probability Letters, Elsevier, vol. 190(C).
    7. Niu, Zhenzhen & Hu, Jiang & Bai, Zhidong & Gao, Wei, 2019. "On LR simultaneous test of high-dimensional mean vector and covariance matrix under non-normality," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 338-344.
    8. Yongcheng Qi & Fang Wang & Lin Zhang, 2019. "Limiting distributions of likelihood ratio test for independence of components for high-dimensional normal vectors," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(4), pages 911-946, August.
    9. Zhendong Wang & Xingzhong Xu, 2021. "High-dimensional sphericity test by extended likelihood ratio," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(8), pages 1169-1212, November.
    10. Yinqiu He & Zi Wang & Gongjun Xu, 2021. "A Note on the Likelihood Ratio Test in High-Dimensional Exploratory Factor Analysis," Psychometrika, Springer;The Psychometric Society, vol. 86(2), pages 442-463, June.
    11. Jiayu Lai & Xiaoyi Wang & Kaige Zhao & Shurong Zheng, 2023. "Block-diagonal test for high-dimensional covariance matrices," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 32(1), pages 447-466, March.
    12. Dette, Holger & Dörnemann, Nina, 2020. "Likelihood ratio tests for many groups in high dimensions," Journal of Multivariate Analysis, Elsevier, vol. 178(C).

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