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Correlation tests for high-dimensional data using extended cross-data-matrix methodology

JOURNAL OF MULTIVARIATE ANALYSIS 2013年 117卷 313-331页

作者： Yata, Kazuyoshi Aoshima, Makoto Univ Tsukuba Inst Math Ibaraki 3058571 Japan

In this paper, we consider tests of correlation when the sample size is much lower than the dimension. We propose a new estimation methodology called the extended cross-data-matrix methodology. By applying the method, we give a new test statistic for high-dimensional correlations. We show that the test statistic is asymptotically normal when p -> infinity and n -> infinity. We propose a test procedure along with sample size determination to ensure both prespecified size and power for testing high-dimensional correlations. We further develop a multiple testing procedure to control both family wise error rate and power. Finally, we demonstrate how the test procedures perform in actual data analyses by using two microarray data sets. (C) 2013 Elsevier Inc. All rights reserved.

关键词： cross-data-matrix methodology Graphical modeling HDLSS High-dimensional regression Pathway analysis Two-stage procedure

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Hypothesis tests for high-dimensional covariance structures

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ANNALS OF THE INSTITUTE OF STATISTICAL MATHEMATICS 2021年第3期73卷 599-622页

作者： Ishii, Aki Yata, Kazuyoshi Aoshima, Makoto Tokyo Univ Sci Dept Informat Sci 2641 Yamazaki Noda Chiba 2788510 Japan Univ Tsukuba Inst Math 1-1-1 Tennodai Tsukuba Ibaraki 3058571 Japan

We consider hypothesis testing for high-dimensional covariance structures in which the covariance matrix is a (i) scaled identity matrix, (ii) diagonal matrix, or (iii) intraclass covariance matrix. Our purpose is to systematically establish a nonparametric approach for testing the high-dimensional covariance structures (i)-(iii). We produce a new common test statistic for each covariance structure and show that the test statistic is an unbiased estimator of its corresponding test parameter. We prove that the test statistic establishes the asymptotic normality. We propose a new test procedure for (i)-(iii) and evaluate its asymptotic size and power theoretically when both the dimension and sample size increase. We investigate the performance of the proposed test procedure in simulations. As an application of testing the covariance structures, we give a test procedure to identify an eigenvector. Finally, we demonstrate the proposed test procedure by using a microarray data set.

关键词： cross-data-matrix methodology Diagonal structure HDLSS Intraclass correlation model Test of eigenvector Unbiased estimate

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

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methodology AND COMPUTING IN APPLIED PROBABILITY 2015年第2期17卷 419-439页

作者： Aoshima, Makoto Yata, Kazuyoshi Univ Tsukuba Inst Math Ibaraki 3058571 Japan

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.

关键词： Asymptotic normality Confidence region cross-data-matrix methodology Large p small n Microarray Two-stage procedure

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PCA consistency for the power spiked model in high-dimensional settings

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JOURNAL OF MULTIVARIATE ANALYSIS 2013年 122卷 334-354页

作者： Yata, Kazuyoshi Aoshima, Makoto Univ Tsukuba Inst Math Ibaraki 3058571 Japan

In this paper, we propose a general spiked model called the power spiked model in high-dimensional settings. We derive relations among the data dimension, the sample size and the high-dimensional noise structure. We first consider asymptotic properties of the conventional estimator of eigenvalues. We show that the estimator is affected by the high-dimensional noise structure directly, so that it becomes inconsistent. In order to overcome such difficulties in a high-dimensional situation, we develop new principal component analysis (PCA) methods called the noise-reduction methodology and the cross-data-matrix methodology under the power spiked model. We show that the new PCA methods can enjoy consistency properties not only for eigenvalues but also for PC directions and PC scores in high-dimensional settings. (C) 2013 Elsevier Inc. All rights reserved.

关键词： cross-data-matrix methodology HDLSS Large p small n Microarray data Noise-reduction methodology

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