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Abstract:
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When testing the structure of the regression coefficients matrix in multivariate linear regressions, likelihood ratio test (LRT) is one of the most popular approaches in practice. Despite its popularity, it is known that the classical chi-square approximations for LRTs often fail in high-dimensional settings, where the dimensions of responses and predictors (m, p) are allowed to grow with the sample size n. Though various corrected LRTs and other test statistics have been proposed in the literature, the fundamental question of when the classic LRT starts to fail is less studied. We first give the asymptotic boundary where the classic LRT fails and develops the corrected limiting distribution of the LRT for a general asymptotic regime. We then study the test power of the LRT in the high-dimensional setting, and develops a power-enhanced LRT. Lastly, when p>n, where the LRT is not well-defined, we propose a two-step testing procedure by first performing dimension reduction and then applying the proposed LRT. Theoretical properties are developed to ensure the validity of the proposed method. Numerical studies are also presented to demonstrate its good performance.
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