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Abstract:
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We study the positive definiteness (PDness) problem in covariance matrix estimation. For high dimensional data, many regularized estimators are proposed under structural assumptions on the true covariance matrix including sparsity. They are shown to be asymptotically consistent and rate-optimal in estimating the true covariance matrix and its structure. However, many of them do not take into account the PDness of the estimator and produce a non-PD estimate. To achieve the PDness, researchers consider additional regularizations on eigenvalues, which make both the asymptotic analysis and computation much harder. In this work, we propose a simple modification of the regularized covariance matrix estimator to make it PD while preserving the support. The proposed modification, denoted by FSPD, is shown to preserve the asymptotic properties of the first-stage estimator. It has a closed form expression and its computation is optimization-free, unlike existing estimators. In addition, the FSPD can be applied to any non-PD matrix including the precision matrix. We discuss its application to multivariate procedures including the choice of working correlation matrix in a longitudinal study.
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