Optimal estimation of a large-dimensional covariance matrix under Stein’s loss

By Olivier Ledoit, Michael Wolf. Bernoulli, 2018

85 citations recorded by the source corpus.

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.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/ba8af1c17f2f4df9a258a192042f19c9fe71629f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Statistics and Probability

Citations in the corpus

Quadratic Shrinkage for Large Covariance Matrices (2020, methodology): “Following Ledoit and Wolf (2018), we extend the class of rotation-equivariant covariance matrix estimators from Section 2 into the realm of large-dimensional asymptotics.”
Nonlinear Shrinkage of the Covariance Matrix for Portfolio Selection: Markowitz Meets Goldilocks (2017, unknown): “This result is established formally by Ledoit and Wolf (2012, 2013b).”
Nonparametric eigenvalue-regularized precision or covariance matrix estimator (2016, methodology): “With respect to the Stein’s loss function, the inverse of our estimator is asymptotically the optimal precision matrix estimator, with optimality proved through what [1] has found.”
The Power of (Non-)Linear Shrinking: A Review and Guide to Covariance Matrix Estimation (2020, unknown): “…loss, defined as LS St R̂T ;RT :¼ 1 2N Tr R 1T R̂T þ RT R̂ 1 T 1: (3.16) These two loss functions yield oracle shrinkage formulas different from Equation (4.8), and also different from each other; for the sake of space, the interested reader is referred to Ledoit and Wolf (2018) for the details.”