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Abstract:
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We propose a new regression interpretation of the Cholesky factor of the covariance matrix, as opposed to the well known regression interpretation of the Cholesky factor of the inverse covariance, which leads to a new class of regularized covariance estimators suitable for high-dimensional problems. Regularizing the Cholesky factor of the covariance via this regression interpretation always results in a positive definite estimator. In particular, one can obtain a positive definite banded estimator of the covariance matrix at the same computational cost as the popular banded estimator proposed by Bickel and Levina (2008), which is not guaranteed to be positive definite. We also establish theoretical connections between banding Cholesky factors of the covariance matrix and its inverse and constrained maximum likelihood estimation under the banding constraint.
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