OPTIMAL SHRINKAGE OF EIGENVALUES IN THE SPIKED COVARIANCE MODEL

• 成果类型: Article
• 署名作者: Donoho, David; Gavish, Matan; Johnstone, Iain
• 署名单位: Stanford University; Hebrew University of Jerusalem
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/17-AOS1601
• 发表日期: 2018
• 页码: 1742-1778
• 关键词: Empirical Bayes; random-matrix; normal precision; Minimax Risk; estimators; distance; distributions; statistics; number; roots

摘要

We show that in a common high-dimensional covariance model, the choice of loss function has a profound effect on optimal estimation. In an asymptotic framework based on the spiked covariance model and use of orthogonally invariant estimators, we show that optimal estimation of the population covariance matrix boils down to design of an optimal shrinker eta that acts elementwise on the sample eigenvalues. Indeed, to each loss function there corresponds a unique admissible eigenvalue shrinker eta* dominating all other shrinkers. The shape of the optimal shrinker is determined by the choice of loss function and, crucially, by inconsistency of both eigenvalues and eigenvectors of the sample covariance matrix. Details of these phenomena and closed form formulas for the optimal eigenvalue shrinkers are worked out for a menagerie of 26 loss functions for covariance estimation found in the literature, including the Stein, Entropy, Divergence, Frechet, Bhattacharya/Matusita, Frobenius Norm, Operator Norm, Nuclear Norm and Condition Number losses.