High dimensional correlation matrices: the central limit theorem and its applications
Jiti Gao, Xiao Han, Guangming Pan, Yanrong Yang
Journal of the Royal Statistical Society: Series B (Statistical Methodology) · 2017 · Volume 79 · Issue 3
Abstract
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 dimensionpand the sample sizenare 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 forprandom variables, and then an equivalence test forpfactor loadings andnfactors 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.
Keywords
Central limit theorem, Equivalence test, High dimensional correlation matrix, Independence test, Linear spectral statistics
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