A $$U$$ -statistic approach for a high-dimensional two-sample mean testing problem under non-normality and Behrens–Fisher setting

Ahmad, M. Rauf

doi:10.1007/s10463-013-0404-2

Cited by 25 publications

(12 citation statements)

References 24 publications

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“…Assumption deviates from its usual form in the literature where such trace ratios are assumed to vanish. It can be shown that assumption holds for commonly used covariance structures as

p \to \infty

(Ahmad, ). All assumptions are only required for the distributions under the alternative.…”

Section: Test Statistics For Two Populationsmentioning

confidence: 99%

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Location‐invariant Multi‐sample U‐tests for Covariance Matrices with Large Dimension

Ahmad

2017

Scandinavian J Statistics

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For two or more multivariate distributions with common covariance matrix, test statistics for certain special structures of the common covariance matrix are presented when the dimension of the multivariate vectors may exceed the number of such vectors. The test statistics are constructed as functions of location‐invariant estimators defined as U‐statistics, and the corresponding asymptotic theory is used to derive the limiting distributions of the proposed tests. The properties of the test statistics are established under mild and practical assumptions, and the same are numerically demonstrated using simulation results with small or moderate sample sizes and large dimensions.

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“…Assumption deviates from its usual form in the literature where such trace ratios are assumed to vanish. It can be shown that assumption holds for commonly used covariance structures as

p \to \infty

(Ahmad, ). All assumptions are only required for the distributions under the alternative.…”

Section: Test Statistics For Two Populationsmentioning

confidence: 99%

“…It can be shown that assumption 2 holds for commonly used covariance structures as p ! 1 (Ahmad, 2014a). All assumptions are only required for the distributions under the alternative.…”

Section: Test Statistics For Two Populationsmentioning

confidence: 99%

Location‐invariant Multi‐sample U‐tests for Covariance Matrices with Large Dimension

Ahmad

2017

Scandinavian J Statistics

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“…For the treatment of degenerate kernels in similar contexts under different conditions of degeneracy, see Ahmad et al, [18] and Ahmad. [19] Now, for valid application of Lehmann's theorem, we need to set the following assumptions.…”

Section: Test Statistic For H 01mentioning

confidence: 99%

Tests for high-dimensional covariance matrices using the theory ofU-statistics

Ahmad

Rosen

2014

Journal of Statistical Computation and Simulation

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Test statistics for sphericity and identity of the covariance matrix are presented, when the data are multivariate normal and the dimension, p, can exceed the sample size, n. Under certain mild conditions mainly on the traces of the unknown covariance matrix, and using the asymptotic theory of U-statistics, the test statistics are shown to follow an approximate normal distribution for large p, also when p n. The accuracy of the statistics is shown through simulation results, particularly emphasizing the case when p can be much larger than n. A real data set is used to illustrate the application of the proposed test statistics.

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“…To overcome the singularity problem in Hotelling's

T^{2}

test, Bai and Saranadasa () replaced the sample covariance matrix in with the identity matrix, so that their test statistic is essentially the same as

{false(\overline{X} - \overline{Y} false)}^{T} false(\overline{X} - \overline{Y} false)

. Following their method, Chen and Qin () and Ahmad ) proposed some U ‐statistics for testing whether two mean vectors are equal. These test methods were referred to as the unscaled Hotelling's tests in Dong et al () As an alternative, Chen et al () and Li et al () proposed replacing the inverse sample covariance matrix

S^{- 1}

with the regularized estimator

{false(S + λ I_{p} false)}^{- 1}

in Hotelling's test statistic, where

I_{p}

is the identity matrix and

λ > 0

is a regularization parameter.…”

Section: Introductionmentioning

confidence: 99%

Diagonal Likelihood Ratio Test for Equality of Mean Vectors in High-Dimensional Data

Tong

Genton

2018

Biometrics

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We propose a likelihood ratio test framework for testing normal mean vectors in high-dimensional data under two common scenarios: the one-sample test and the two-sample test with equal covariance matrices. We derive the test statistics under the assumption that the covariance matrices follow a diagonal matrix structure. In comparison with the diagonal Hotelling's tests, our proposed test statistics display some interesting characteristics. In particular, they are a summation of the log-transformed squared -statistics rather than a direct summation of those components. More importantly, to derive the asymptotic normality of our test statistics under the null and local alternative hypotheses, we do not need the requirement that the covariance matrices follow a diagonal matrix structure. As a consequence, our proposed test methods are very flexible and readily applicable in practice. Simulation studies and a real data analysis are also carried out to demonstrate the advantages of our likelihood ratio test methods.

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A $$U$$ -statistic approach for a high-dimensional two-sample mean testing problem under non-normality and Behrens–Fisher setting

Cited by 25 publications

References 24 publications

Location‐invariant Multi‐sample U‐tests for Covariance Matrices with Large Dimension

Location‐invariant Multi‐sample U‐tests for Covariance Matrices with Large Dimension

Tests for high-dimensional covariance matrices using the theory ofU-statistics

Diagonal Likelihood Ratio Test for Equality of Mean Vectors in High-Dimensional Data

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