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Contribution Details

Type Working Paper
Scope Discipline-based scholarship
Title Optimal estimation of a large-dimensional covariance matrix under Stein’s loss
Organization Unit
Authors
  • Olivier Ledoit
  • Michael Wolf
Language
  • English
Institution University of Zurich
Series Name Working paper series / Department of Economics
Number 122
ISSN 1664-7041
Number of Pages 63
Date 2017
Abstract Text This paper introduces a new method for deriving covariance matrix estimators that are decision-theoretically optimal within a class of nonlinear shrinkage estimators. The key is to employ large-dimensional asymptotics: the matrix dimension and the sample size go to infinity together, with their ratio converging to a finite, nonzero limit. As the main focus, we apply this method to Stein’s loss. Compared to the estimator of Stein (1975, 1986), ours has five theoretical advantages: (1) it asymptotically minimizes the loss itself, instead of an estimator of the expected loss; (2) it does not necessitate post-processing via an ad hoc algorithm (called “isotonization”) to restore the positivity or the ordering of the covariance matrix eigenvalues; (3) it does not ignore any terms in the function to be minimized; (4) it does not require normality; and (5) it is not limited to applications where the sample size exceeds the dimension. In addition to these theoretical advantages, our estimator also improves upon Stein’s estimator in terms of finite-sample performance, as evidenced via extensive Monte Carlo simulations. To further demonstrate the effectiveness of our method, we show that some previously suggested estimators of the covariance matrix and its inverse are decision-theoretically optimal in the large-dimensional asymptotic limit with respect to the Frobenius loss function.
Official URL http://www.econ.uzh.ch/static/wp/econwp122.pdf
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Keywords Large-dimensional asymptotics, nonlinear shrinkage estimation, random matrix theory, rotation equivariance, Stein’s loss, Kovarianzmatrix, Statistik, nichtlineare Analysis
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