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High dimensional correlation matrices: the central limit theorem and its applications
Author(s) -
Gao Jiti,
Han Xiao,
Pan Guangming,
Yang Yanrong
Publication year - 2017
Publication title -
journal of the royal statistical society: series b (statistical methodology)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 6.523
H-Index - 137
eISSN - 1467-9868
pISSN - 1369-7412
DOI - 10.1111/rssb.12189
Subject(s) - central limit theorem , mathematics , independence (probability theory) , statistic , random matrix , test statistic , equivalence (formal languages) , dimension (graph theory) , statistics , limit (mathematics) , matrix (chemical analysis) , sample size determination , sample (material) , multivariate random variable , statistical hypothesis testing , distance correlation , random variable , mathematical analysis , pure mathematics , eigenvalues and eigenvectors , physics , materials science , quantum mechanics , composite material , thermodynamics
Summary Statistical inferences for sample correlation matrices are important in high dimensional data analysis. Motivated by this, the paper establishes a new central limit theorem for a linear spectral statistic of high dimensional sample correlation matrices for the case where the dimension p and the sample size n are comparable. This result is of independent interest in large dimensional random‐matrix theory. We also further investigate the sample correlation matrices of a high dimensional vector whose elements have a special correlated structure and the corresponding central limit theorem is developed. Meanwhile, we apply the linear spectral statistic to an independence test for p random variables, and then an equivalence test for p factor loadings and n factors in a factor model. The finite sample performance of the test proposed shows its applicability and effectiveness in practice. An empirical application to test the independence of household incomes from various cities in China is also conducted.