Testing high-dimensional linear hypotheses through spectral shrinkage

Debashis Paul, University of California, Davis

Friday, January 31, 2020 - 3:30pm

We consider the problem of testing linear hypotheses associated with a high-dimensional linear model under the setting where the dimensionality of the response is comparable to the sample size. Classical likelihood ratio tests for such problems suffer from significant loss of power within this asymptotic framework. We propose regularization schemes that modify the likelihood ratio statistic by applying nonlinear shrinkage to the eigenvalues of the empirical covariance matrix of the residuals. We make the structural assumption that the spectral measure of the noise covariance converges to a nontrivial limit. We show that the proposed tests are able to significantly improve on the performance of the classical likelihood ratio test. We address the problem of finding the optimal regularization parameter within a decision-theoretic framework by adopting a probabilistic formulation of the alternatives. 

(This is a joint work with Haoran Li, Alexander Aue, Jie Peng and Pei Wang).

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