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

Aoshima, Makoto; Yata, Kazuyoshi

doi:10.1007/s11009-013-9370-7

Cited by 25 publications

(20 citation statements)

References 13 publications

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“…Note that Var(x ij,A ) = Σ i,A for i = 1, 2. Then, from (S5.3), by using Theorem 5 given in Aoshima and Yata (2015), we can obtain the result when (A-iv) is met.…”

Section: S5 Appendix Amentioning

confidence: 96%

“…Note that (2.1) includes the case that Γ i = H i Λ 1/2 i and w ij = z ij . Refer to Bai and Saranadasa (1996), Chen and Qin (2010) and Aoshima and Yata (2015) for the details of the model. As for w ij = (w i1j , ..., w ir i j ) T , we assume the following assumption for π i , i = 1, 2, as necessary:…”

Section: Asymptotic Properties Of T (A)mentioning

confidence: 99%

“…When A = I p , it is not necessary to estimate A and it is quite flexible for high-dimension, non-Gaussian data. See Section 2 of Aoshima and Yata (2015) for the details.…”

Section: Here Let Us Consider (mentioning

confidence: 99%

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Two-sample tests for high-dimension, strongly spiked eigenvalue models

Yin

Cai

2018

STAT SINICA

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Motivated by applications in genomics, we consider in this paper global and multiple testing for the comparisons of two high-dimensional linear regression models. A procedure for testing the equality of the two regression vectors globally is proposed and shown to be particularly powerful against sparse alternatives. We then introduce a multiple testing procedure for identifying unequal coordinates while controlling the false discovery rate and false discovery proportion. Theoretical justifications are provided to guarantee the validity of the proposed tests and optimality results are established under sparsity assumptions on the regression coefficients. The proposed testing procedures are easy to implement. Numerical properties of the procedures are investigated through simulation and data analysis. The results show that the proposed tests maintain the desired error rates under the null and have good power under the alternative at moderate sample sizes. The procedures are applied to the Framingham Offspring study to investigate the interactions between smoking and cardiovascular related genetic mutations important for an inflammation marker.

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“…Note that Var(x ij,A ) = Σ i,A for i = 1, 2. Then, from (S5.3), by using Theorem 5 given in Aoshima and Yata (2015), we can obtain the result when (A-iv) is met.…”

Section: S5 Appendix Amentioning

confidence: 96%

Section: Asymptotic Properties Of T (A)mentioning

confidence: 99%

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Two-sample tests for high-dimension, strongly spiked eigenvalue models

Yin

Cai

2018

STAT SINICA

Self Cite

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“…In unpublished correspondence John Kent pointed out, using a Gaussian scale mixture example, that more than univariate moment conditions are needed. Conditions that are especially appealing because they are based only on the covariance matrix (with no assumption of Gaussianity) have been developed in a series of papers (Yata and Aoshima, 2009, 2010b; Aoshima and Yata, 2011; Yata and Aoshima, 2012a; Aoshima and Yata, 2015). A non-Gaussian condition that makes intuitive sense based on the types of data found in genomics can be found in Jung and Marron (2009).…”

Section: Hdlss Backroundmentioning

confidence: 99%

“…We consider multiple component spike models with distinguishable population eigenvalues in Section 5.1.1 and with indistinguishable eigenvalues in Section 5.1.2. Moreover, we vary d from d ≪ n , through the random matrix version with d ~ n , all the way to the high dimension medium sample size (HDMSS) asymptotics of Cabanski et al (2010); Yata and Aoshima (2012b); Aoshima and Yata (2015) with d ≫ n → ∞. Aoshima and Yata (2015) improves the results of Yata and Aoshima (2012b) under mild conditions.…”

Section: Deeper Conical Behaviormentioning

confidence: 99%

The statistics and mathematics of high dimension low sample size asymptotics

Shen¹,

Shen²,

Zhu³

et al. 2017

STAT SINICA

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The aim of this paper is to establish several deep theoretical properties of principal component analysis for multiple-component spike covariance models. Our new results reveal an asymptotic conical structure in critical sample eigendirections under the spike models with distinguishable (or indistinguishable) eigenvalues, when the sample size and/or the number of variables (or dimension) tend to infinity. The consistency of the sample eigenvectors relative to their population counterparts is determined by the ratio between the dimension and the product of the sample size with the spike size. When this ratio converges to a nonzero constant, the sample eigenvector converges to a cone, with a certain angle to its corresponding population eigenvector. In the High Dimension, Low Sample Size case, the angle between the sample eigenvector and its population counterpart converges to a limiting distribution. Several generalizations of the multi-spike covariance models are also explored, and additional theoretical results are presented.

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A unified approach to testing mean vectors with large dimensions

Ahmad

2018

AStA Adv Stat Anal

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A unified testing framework is presented for large-dimensional mean vectors of one or several populations which may be non-normal with unequal covariance matrices. Beginning with one-sample case, the construction of tests, underlying assumptions and asymptotic theory, is systematically extended to multi-sample case. Tests are defined in terms of U-statistics-based consistent estimators, and their limits are derived under a few mild assumptions. Accuracy of the tests is shown through simulations. Real data applications, including a five-sample unbalanced MANOVA analysis on count data, are also given. Keyword High-dimensional inference • Behrens-Fisher problem • MANOVA • U-statistics 1 Introduction Let X k = (X k1 ,. .. , X kp) ∼ F, k = 1,. .. , n be iid random vectors, where F denotes a p-variate distribution, with E(X k) = μ ∈ R p and Cov(X k) = ∈ R p× p >0. A hypothesis of foremost interest to be tested in this setup is H 0 : μ = 0 against an appropriate alternative, say H 1 : Not H 0. For an extension to g ≥ 2 samples, let X ik = (X ik1 ,. .. , X ikp) ∼ F i be iid random vectors with E(X ik) = μ i ∈ R p , Cov(X ik

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

Cited by 25 publications

References 13 publications

Two-sample tests for high-dimension, strongly spiked eigenvalue models

Two-sample tests for high-dimension, strongly spiked eigenvalue models

The statistics and mathematics of high dimension low sample size asymptotics

A unified approach to testing mean vectors with large dimensions

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