Nonlinear Shrinkage Estimation of Large-Dimensional Covariance Matrices

By Olivier Ledoit, Michael Wolf. 2012

244 citations recorded by the source corpus.

Many statistical applications require an estimate of a covariance matrix and/or its inverse. When the matrix dimension is large compared to the sample size, which happens frequently, the sample covariance matrix is known to perform poorly and may suffer from ill-conditioning. There already exists an extensive literature concerning improved estimators in such situations. In the absence of further knowledge about the structure of the true covariance matrix, the most successful approach so far, arguably, has been shrinkage estimation. Shrinking the sample covariance matrix to a multiple of the identity, by taking a weighted average of the two, turns out to be equivalent to linearly shrinking the sample eigenvalues to their grand mean, while retaining the sample eigenvectors. Our paper extends this approach by considering nonlinear transformations of the sample eigenvalues. We show how to construct an estimator that is asymptotically equivalent to an oracle estimator suggested in previous work. As demonstrated in extensive Monte Carlo simulations, the resulting bona fide estimator can result in sizeable improvements over the sample covariance matrix and also over linear shrinkage.

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Citations in the corpus

Geometric median and robust estimation in Banach spaces (2013, methodology): “However, it is possible to use various alternatives in place of the sample covariance, such as the shrinkage estimator [25], banding/tapering estimator [3], or hard thresholding estimator [4] of the covariance matrix, to name a few.”
Using Principal Component Analysis to Estimate a High Dimensional Factor Model with High-Frequency Data (2017, unknown): “A factor model can serve as the reference point for shrinkage estimation (see Ledoit and Wolf (2012) and Ledoit and Wolf (2004)).”
Efficient Estimation of Approximate Factor Models (2012, unknown): “ng problem. Other works on large covariance estimation include Cai and Zhou (2012), Fan et al. (2008), Jung and Marron (2009), Witten, Tibshirani and Hastie (2009), Deng and Tsui (2010), Yuan (2010), Ledoit and Wolf (2012), El Karoui (2008), Pati et al. (2012), Rohde and Tsybakov (2011), Zhou et al.”
Nonlinear Shrinkage of the Covariance Matrix for Portfolio Selection: Markowitz Meets Goldilocks (2017, unknown): “Theorem 2.2 of Ledoit and Wolf (2013b) and Proposition 4.3 of Ledoit and Wolf (2012) imply that ∀x ∈ Supp(F ) d̂T (x) a.s−→ d∗(x), and that this convergence is uniform over x ∈ Supp(F ), apart from arbitrarily small boundary regions of the support.”
Econometric Analysis of Vast Covariance Matrices Using Composite Realized Kernels and Their Application to Portfolio Choice (2016, unknown): “Important alternatives include shrinkage highlighted by, for example, Ledoit and Wolf (2012).”
Random matrix theory in statistics: A review (2014, unknown): “These results have been used by Ledoit and Wolf (2012) to develop shrinkage estimators of the population covariance matrix that are rotationally invariant and whose construction only requires nonlinear shrinkage of the sample eigenvalues.”
Optimal estimation of a large-dimensional covariance matrix under Stein’s loss (2018, unknown): “Although the present paper focuses mainly on Stein’s loss and the nonlinear shrinkage function φ∗, a key innovation relative to Ledoit and Wolf [27,28] is the method of Section 3.”