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Abstract |
This talk is concerned with test of significance on high dimensional covariancestructures, and aims to develop a unified framework for testing commonly-used linearcovariance structures including independence structure, compound symmetric structure, banded structure and factor model structure. We first propose estimating theparameters involved in the linear covariance structure by the squared loss. This estimation procedure yields a consistent estimate of the covariance parameters. We developtwo tests for these covariance structures based on the entropy loss and quadratic lossused for covariance matrix estimation in the classical multivariate analysis. Some existing tests for a specific covariance structure are special cases of these two tests. Tostudy the asymptotic properties of the proposed tests, we study related high dimensional random matrix theory, and establish several highly useful asymptotic resultsfor high-dimensional random matrix. Using these asymptotic results, we derive thelimiting null distributions of these two tests, and their asymptotic distributions underthe alternative hypothesis. The asymptotic distribution enables us to derive the powerfunction of the proposed tests. We further show that the quadratic loss based test isasymptotically unbiased. We conduct Monte Carlo simulation study to examine the finite sample performance of the two tests. Our simulation results show that the limitingnull distributions approximate their null distributions quite well, and the correspondingasymptotic critical values keep Type I error rate very well. Our numerical comparisonimplies that the proposed tests outperform existing ones in terms of controlling TypeI error rate and power. Our simulation indicates that the test based on quadratic lossseems to have better power than the test based on entropy loss. We illustrate theproposed testing procedure by an empirical analysis of Chinese stock market data. |
