EFFECT OF HIGH DIMENSION: BY AN EXAMPLE OF A TWO SAMPLE PROBLEM

By Z. Bai. 1999

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Adaptivity and Computation-Statistics Tradeoffs for Kernel and Distance based High Dimensional Two Sample Testing (2015, methodology): “Assumption [A1] implies that X, Y have means µ 1 , µ 2 and covariances Σ , like in Bai and Saranadasa [1996].”
Revisiting Classifier Two-Sample Tests (2016, background): “The power bound in Theorem ?? has an optimal order of magnitude for multi-dimensional problems ( ???”
A two sample test in high dimensional data (2013, methodology): “A simple generalization of the test statistic proposed by Bai and Saranadasa [1] is given by Qn = [(x̄1 − x̄2)′(x̄1 − x̄2)− N−11 tr S1 − N −1 2 tr S2]/p 1/2.”
Testing Many Moment Inequalities (2013, result): “Moreover, the results of Bai and Saranadasa (1996) suggest that the statistic T ′ or its variants may lead to higher power than T ′′ even when p is smaller than but close to n.”
Independence test for high dimensional data based on regularized canonical correlation coefficients (2015, background): “There have been a substantial body of research work dealing with high dimensional data, for example, [2, 4, 7, 9, 10, 12], etc.”
Multivariate Theory for Analyzing High Dimensional Data (2007, methodology): “The asymptotic distribution of the FD test under the alternative that µ = 0, is given by lim n,p→∞ [ P {( r 2 )1/2 (FD − 1) > z1−α | µ = (nN)−1/2δ } − Φ ( −z1−α + nµ′µ√ 2pa2 )] = 0, see Bai and Saranadasa (1996).”
On Sliced Inverse Regression With High-Dimensional Covariates (2006, background): “It is impossible to uniformly formulate the asymptotic results for different ratios λ = p/n when p diverges to infinity as n tends to infinity (see Bai and Saranadasa 1996; Bai 1999 for details).”
Identifying Cointegration by Eigenanalysis (2015, methodology): “The form zt = Bet in Condition 2 (i), has been used by Bai and Saranadasa (1996) and Chen and Qin (2010).”
Test for Bandedness of High Dimensional Covariance Matrices with Bandwidth Estimation (2012, methodology): “To facilitate our analysis, as Bai and Saranadasa (1996) and Chen, Zhang and Zhong (2010), we assume a multivariate model for the high-dimensional data.”
A More Powerful Two-Sample Test in High Dimensions using Random Projection (2011, background): “In Section 3.3, we demonstrate an optimality property of the choice k = ⌊ n/ 2 ⌋ , which is valid in moderate or high-dimensions, i.e., p ≥ ⌊ n/ 2 ⌋ , and we restrict our attention to this case in Theorems 2 and 3.”
Statistical Query Lower Bounds for Robust Estimation of High-Dimensional Gaussians and Gaussian Mixtures (2016, methodology): “) The classical test for this task is Hotelling’s T-squared statistic [Hot31], which is unfortunately not defined when the sample size is smaller than the dimension [ZB96].”
Two-Stage Procedures for High-Dimensional Data (2011, background): “Bai and Saranadasa (1996), Srivastava (2007), and Chen and Qin (2010) considered testing hypothesis (3.”
Likelihood Ratio Tests for High‐Dimensional Normal Distributions (2013, methodology): “Bai and Saranadasa [4] further showed that even when the Hotelling’s T 2 statistic is well defined, Dempster’s F test is more powerful than the Hotelling’s T 2 test if the dimension is proportionally close to the within sample degree of freedom.”
Inference on Causal and Structural Parameters using Many Moment Inequalities (2013, result): “Moreover, the results of Bai and Saranadasa (1996) suggest that the statistic T ′ or its variants may lead to higher power than T ′′ even when p is smaller than but close to n.”
Selective review of offline change point detection methods (2020, background): “Gaussian signals with arbitrary variance/covariance 947 matrix Σ (cost function cM withM = Σ −1), more robust replacements to similar 948 conventional statistics were introduced [147, 148].”
Tests alternative to higher criticism for high-dimensional means under sparsity and column-wise dependence (2013, methodology): “In addition to the connection to the HC test, the maximal Lγ-thresholding test, by its nature of formulation, is related to the high-dimensional multivariate testing procedures, for instance, the tests proposed by Bai and Saranadasa (1996) and Chen and Qin (2010). While these tests can maintain accu”
High Dimensional Classification Using Features Annealed Independence Rules. (2007, background): “(2004), and Bair et al.(2004); partial least squares in Nguyen and Rocke (2002 ), Huang and Pan (2003), and Boulesteix(2004); and sliced inverse regression in Chiaromonte and Martinelli (2002), Antoniadis et al.”