Integrating Regularized Covariance Matrix Estimators Seminar

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Abstract: When the dimension p of a covariance matrix is close to or even larger than the sample size n, regularizing the sample covariance matrix is the key to obtaining a satisfactory covariance matrix estimator. One branch of regularization assumes specific structures, like sparse, banded, or a having a factor structure for the true covariance matrix. Another branch regularize on the eigenvalues directly without assuming these structures. The former makes sense when one is confident in a specific structure for S0, while the latter is sound when no specific structures are known. Under the more practical scenario where one is not 100\% certain of which regularization method to use or what specific structure to assume for S0, we introduce an integration covariance matrix estimator which is a linear combination of a rotation-equivariant estimator and a regularized covariance matrix estimator assuming a specific structure for S0. We estimate the weights in the linear combination and show that they asymptotically almost surely go to the true underlying weights. In particular, if S0 does have a specific structure so that the corresponding regularized estimator is converging to S0 in the spectral norm, we show that the weight for the rotation-equivariant estimator tends to 0, while that for the regularized estimator tends to 1 asymptotically. To generalize, we can put two regularized estimators into the linear combination, each assumes a specific structure for S0. Our estimated weights can then be shown to go to the true weights too, and if one regularized estimator is converging to S0 in the spectral norm, the corresponding weight then tends to 1 and others tend to 0 asymptotically. We demonstrate the superior performance of our estimator when compared to other state-of-the-art estimators through extensive simulation studies and a real data analysis.