The asymptotic distributions of the largest entries of sample correlation matrices

By Tiefeng Jiang. 2004

160 citations recorded by the source corpus.

Let X_n=(x_{ij}) be an n by p data matrix, where the n rows form a random sample of size n from a certain p-dimensional population distribution. Let R_n=(\rho_{ij}) be the p\times p sample correlation matrix of X_n; that is, the entry \rho_{ij} is the usual Pearson's correlation coefficient between the ith column of X_n and jth column of X_n. For contemporary data both n and p are large. When the population is a multivariate normal we study the test that H_0: the p variates of the population are uncorrelated. A test statistic is chosen as L_n=max_{i\ne j}|\rho_{ij}|. The asymptotic distribution of L_n is derived by using the Chen-Stein Poisson approximation method. Similar results for the non-Gaussian case are also derived.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/0ffe11bf29da765efdc98b17b79719c5b96be216

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In Condensed Matter Physics

1. Wang Zhou
2. Mathias Drton
3. Bala Rajaratnam
4. A. Hero
5. Fan Yang
6. Tommaso Cai
7. Weidong Liu
8. Qi-Man Shao

In Mathematical Physics

1. Mathias Drton
2. Bala Rajaratnam
3. A. Hero
4. Fan Yang
5. Weidong Liu
6. Wang Zhou
7. Fang Han
8. Hongjian Shi

In Statistics and Probability

1. Tommaso Cai
2. Mathias Drton
3. Fang Han
4. Zhao Ren
5. Hongjian Shi
6. Harrison H. Zhou
7. Weidong Liu
8. Wei Biao Wu

Citations in the corpus

• Uniform change point tests in high dimension (2015, background): “Note that high moment assumptions are common in such a context, we refer to [12, 32, 35, 51], where sometimes up to 30 moments and more are required.”
• The limiting distributions of eigenvalues of sample correlation matrices (2004, methodology): “In an earlier work, the test statistic maxi^^xp \pij\ for testing Hq : R = I is studied by Jiang (2004) According to the PCA, the largest eigenvalue is a natural choice for a test statistic.”
• Asymptotic distribution of the largest off-diagonal entry of correlation matrices (2007, methodology): “assumption, Jiang [7] found an asymptotic distribution of Ln.”
• Two-Sample Covariance Matrix Testing and Support Recovery in High-Dimensional and Sparse Settings (2013, unknown): “The limiting behavior of M n is similar to that of the largest off-diagonal entry L n of the sample correlation matrix in Jiang (2004), wherein the article derived the asymptotic distribution of L n under the assumption that the components X 1 , . . . , X p are independent.”
• Testing independence in high dimensions with sums of rank correlations (2015, background): “Jiang (2004) proved that, under suitable centering and scaling, the null distribution of the statistic (1.2) max 1≤p q≤m ( r(pq) )2 converges to an extreme value distribution of type 1 when m/n converges to a constant γ ∈ (0,∞) as m,n −→ ∞.”
• Large-Scale Correlation Screening (2011, methodology): “Tests of diagonal covariance structure are often based on the maximum sample correlation coefficient, and has been mainly studied in the large p regime as n → ∞ (Jiang 2004; Li and Rosalsky 2006; Zhou 2007; Liu, Lin, and Shao 2008; Li, Liu, and Rosalsky 2009; Cai and Jiang 2011).”
• Estimating structured high-dimensional covariance and precision matrices: Optimal rates and adaptive estimation (2016, background): “Jiang (2004) studied the asymptotic distribution of the test statistic Ln = max 1≤i<j≤p |ρ̂i,j | , (44)”
• Testing mutual independence in high dimensions with sums of squares of rank correlations (2015, background): “Jiang (2004) proved that, up to appropriate renormalization, the null distribution of the statistic”
• Global Testing and Large-Scale Multiple Testing for High-Dimensional Covariance Structures (2017, background): “Jiang (2004) showed that nL2n,p has asymptotically the extreme value distribution of type I as in (4), under the conditions E |Xi|30+ε ∞ and p/n → γ ∈ (0,∞).”
• Variance estimation in high-dimensional linear models (2014, background): “Jiang, T. (2004). The asymptotic distributions of the largest entries of sample correlation matrices. Ann. Appl. Probab. 14 865–880. Lafferty, J. and Wasserman, L. (2008). Statistical analysis of semi-supervised regression. Adv. Neur. In. 20 801–808. Letac, G. and Massam, H. (2004). All invariant mo”
• Gaussian Approximation for High Dimensional Time Series (2015, background): “inference on jk;1 j;kp, one needs to derive asymptotic distribution of the maximum deviation max j;kpj^ jk jkjor the normalized version max j;kpj^ jk jkj=˝ jk; cf Equation (2) inXiao and Wu(2013).Jiang (2004) established the Gumbel convergence of the maximum deviation assuming that all 7 entries of ”
• Approximation of Haar distributed matrices and limiting distributions of eigenvalues of Jacobi ensembles (2009, background): “They are proved in Jiang[44].”
• THE ASYMPTOTIC DISTRIBUTION AND BERRY-ESSEEN BOUND OF A NEW TEST FOR INDEPENDENCE IN HIGH DIMENSION WITH AN APPLICATION TO STOCHASTIC OPTIMIZATION (2008, methodology): “Write Xk = (Xk,1,Xk,2, . . . ,Xk,p), 1 ≤ k ≤ n and let L̃n = max 1≤i j≤p |ρ̃(n)i,j |, where ρ̃ (n) i,j = ∑n k=1(Xk,i − X̄ (n) i )(Xk,j − X̄ (n) j ) ( ∑n k=1(Xk,i − X̄ (n) i ) 2)1/2( ∑n k=1(Xk,j − X̄ (n) j ) 2)1/2 and X̄ (n) i = ∑n k=1 Xk,i/n. Jiang [14] proves the following limit theorem con- cernin”
• High-dimensional consistent independence testing with maxima of rank correlations (2018, methodology): “, Jiang, D., Yao, J.-F., and Zheng, S. (2009). Corrections to LRT on large-dimensional covariance matrix by RMT.”
• Theoretical Results on Sparse Representations of Multiple-Measurement Vectors (2006, background): “In another work of Jiang [ 37 ], the limit distribution of the maximal off diagonal entry in a correlation matrix was derived.”
• Maxima of entries of Haar distributed matrices (2005, background): “On the other hand, the maxima of the entries of sample correlation matrices is treated by Jiang [ 23 ] due to a statistical testing problem.”
• Likelihood Ratio Tests for High‐Dimensional Normal Distributions (2013, methodology): “For considering the construction of compressed sensing matrices, the statistical testing problems, the covariance structures of normal distributions, high dimensional regression in statistics and a wide range of applications including signal processing, medical imaging, and seismology, the largest o”