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Asymptotic Normality for Inference on Multisample, High-Dimensional Mean Vectors Under Mild Conditions

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  • Makoto Aoshima

    (University of Tsukuba)

  • Kazuyoshi Yata

    (University of Tsukuba)

Abstract

In this paper, we consider the asymptotic normality for various inference problems on multisample and high-dimensional mean vectors. We verify that the asymptotic normality of concerned statistics is proved under mild conditions for high-dimensional data. We show that the asymptotic normality can be justified theoretically and numerically even for non-Gaussian data. We introduce the extended cross-data-matrix (ECDM) methodology to construct an unbiased estimator at a reasonable computational cost. With the help of the asymptotic normality, we show that the concerned statistics given by ECDM can ensure consistency properties for inference on multisample and high-dimensional mean vectors. We give several applications such as confidence regions for high-dimensional mean vectors, confidence intervals for the squared norm and the test of multisample mean vectors. We also provide sample size determination so as to satisfy prespecified accuracy on inference. Finally, we give several examples by using a microarray data set.

Suggested Citation

  • Makoto Aoshima & Kazuyoshi Yata, 2015. "Asymptotic Normality for Inference on Multisample, High-Dimensional Mean Vectors Under Mild Conditions," Methodology and Computing in Applied Probability, Springer, vol. 17(2), pages 419-439, June.
    • Handle: RePEc:spr:metcap:v:17:y:2015:i:2:d:10.1007_s11009-013-9370-7
    DOI: 10.1007/s11009-013-9370-7
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    References listed on IDEAS

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    1. Chen, Song Xi & Qin, Yingli, 2010. "A Two Sample Test for High Dimensional Data with Applications to Gene-set Testing," MPRA Paper 59642, University Library of Munich, Germany.
    2. Yata, Kazuyoshi & Aoshima, Makoto, 2013. "Correlation tests for high-dimensional data using extended cross-data-matrix methodology," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 313-331.
    3. Kazuyoshi Yata & Makoto Aoshima, 2012. "Inference on High-Dimensional Mean Vectors with Fewer Observations Than the Dimension," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 459-476, September.
    4. N. Mukhopadhyay & T. Solanky, 1997. "Estimation after sequential selection and ranking," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 45(1), pages 95-106, January.
    5. Yata, Kazuyoshi & Aoshima, Makoto, 2010. "Effective PCA for high-dimension, low-sample-size data with singular value decomposition of cross data matrix," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 2060-2077, October.
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    Citations

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    Cited by:

    1. Yata, Kazuyoshi & Aoshima, Makoto, 2016. "High-dimensional inference on covariance structures via the extended cross-data-matrix methodology," Journal of Multivariate Analysis, Elsevier, vol. 151(C), pages 151-166.
    2. Aki Ishii & Kazuyoshi Yata & Makoto Aoshima, 2021. "Hypothesis tests for high-dimensional covariance structures," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 73(3), pages 599-622, June.
    3. Ishii, Aki & Yata, Kazuyoshi & Aoshima, Makoto, 2022. "Geometric classifiers for high-dimensional noisy data," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    4. Jun Li, 2018. "Asymptotic normality of interpoint distances for high-dimensional data with applications to the two-sample problem," Biometrika, Biometrika Trust, vol. 105(3), pages 529-546.
    5. M. Rauf Ahmad, 2019. "A unified approach to testing mean vectors with large dimensions," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 103(4), pages 593-618, December.
    6. Zhang, Jin-Ting & Zhou, Bu & Guo, Jia, 2022. "Linear hypothesis testing in high-dimensional heteroscedastic one-way MANOVA: A normal reference L2-norm based test," Journal of Multivariate Analysis, Elsevier, vol. 187(C).

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