Sparse covariance estimation using Stein's unbiased risk estimator (SURE)
- The problem of estimating an unknown covariance matrix is among the most widely-studied problems in multivariate statistics. A popular estimator was proposed by Stein which regularizes the sample covariance matrix by shrinking its eigenvalues together. Although numerical studies have found that Stein's estimator performs well relative to other estimators, its somewhat complicated functional form makes it difficult to analyze theoretically, and therefore few theoretical properties of Stein's estimator have been established. We undertake a detailed investigation of Stein's covariance estimator in both the finite-sample and asymptotic regimes. We also suggest alternative estimators based on the same approach which can sometimes lead to significant risk reductions over Stein's proposed estimator, and which can also generalize Stein's methodology for other loss functions, including those for the inverse covariance (precision) matrix. Finally, we propose a novel framework for constructing sparse estimates of the covariance matrix by augmenting the objective function minimized by Stein's estimator with a sparsity-inducing penalty function; the resulting optimization problem is convex and can be solved efficiently via a simple block coordinate descent approach. We demonstrate that the resulting estimator performs well via numerical simulations and establish conditions under which it yields an asymptotically consistent estimate of the population covariance parameter.
|Type of resource
|electronic; electronic resource; remote
|1 online resource.
|Stanford University, Institute for Computational and Mathematical Engineering.
|Statement of responsibility
|Submitted to the Institute for Computational and Mathematical Engineering.
|Thesis (Ph.D.)--Stanford University, 2015.
- © 2015 by Brett A Naul
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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