Two Sample Tests for High Dimensional Covariance Matrices

By Jun Yu Li, Songxi Chen. 2012

249 citations recorded by the source corpus.

We propose two tests for the equality of covariance matrices between two high-dimensional populations. One test is on the whole variance-covariance matrices, and the other is on offdiagonal sub-matrices which define the covariance between two non-overlapping segments of the high-dimensional random vectors. The tests are applicable (i) when the data dimension is much larger than the sample sizes, namely the “large p, small n” situations and (ii) without assuming parametric distributions for the two populations. These two aspects surpass the capability of the conventional likelihood ratio test. The proposed tests can be used to test on covariances associated with gene ontology terms.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/88c16d834b68de0fef473b9481c76dbb8850ed5e

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. Shu-rong Zheng
2. Jianfeng Yao
3. Jiang Hu
4. David Ruppert
5. Tommaso Cai
6. Zongming Ma
7. Fan Yang
8. Chen Wang

In Statistics and Probability

1. Tommaso Cai
2. Jianfeng Yao
3. Shu-rong Zheng
4. Debashis Paul
5. Z. Bai
6. Tiefeng Jiang
7. Alexander Aue
8. Fan Yang

Citations in the corpus

• Likelihood Ratio Tests for High‐Dimensional Normal Distributions (2013, background): “(iii) Recently, some authors study similar problems under the nonparametric setting, for example, the works of Cai et al. (2013), Cai & Ma (2013), Chen et al. (2010), Li & Chen (2012), Qiu & Chen (2012) and Xiao & Wu (2013).”
• Optimal hypothesis testing for high dimensional covariance matrices (2012, methodology): “Motivated by an unbiased estimator of the Frobenius norm of Σ 1 − Σ 2 , Chen and Li [6] proposed a test using a linear combination of U -statistics and studied its power.”
• Change-point detection in high-dimensional covariance structure (2016, methodology): “We also thank Jun Li and Haeran Cho for providing the implementation of the approach proposed in [30] and in [12] respectively.”
• Global Testing and Large-Scale Multiple Testing for High-Dimensional Covariance Structures (2017, methodology): “Qiu and Chen (2012) proposed a test based on a U -statistic which is an unbiased estimator of ∑ |i−j|≥k σ 2 i,j for testing bandedness of Σ. Compared with the test proposed in Cai and Jiang (2011), which is powerful for sparse alternatives, the test introduced in Qiu and Chen (2012) is powerful for…”
• On testing the equality of high dimensional mean vectors with unequal covariance matrices (2014, background): “Remark 5 By assumption (a), the unbiasedness of estimators ̂tr 2i and ̂tr i j can be easily proved and their ratio-consistency can be found in Li and Chen (2012).”
• Two-Sample Covariance Matrix Testing and Support Recovery in High-Dimensional and Sparse Settings (2013, unknown): “For reasons of space, these simulation results are given in the supplementary material (Cai, Liu, and Xia 2012).”
• RAPTT: An Exact Two-Sample Test in High Dimensions Using Random Projections (2014, background): “…on simultaneous testing of means based on marginal tests in high dimensions; Benjamini and Hochberg (1995) worked on controlling false discovery rates in multiple testing; Ledoit and Wolf (2002); Chen et al. (2010); Li and Chen (2012) considered testing of covariance matrices in high dimensions.”
• Substitution principle for CLT of linear spectral statistics of high-dimensional sample covariance matrices with applications to hypothesis testing (2014, methodology): “To our best knowledge, few procedures exist for such testing problems on large sample covariance matrices, two exceptions being [12, 15]; see also [11] on a minimax study for the identity test.”