Shrinking in COMFORT

R-NL: Covariance Matrix Estimation for Elliptical Distributions Based on Nonlinear Shrinkage

“…Closest in spirit to our proposal the parametric approach of Hediger and Näf (2022), and the present paper can be seen as a generalization of this approach to the nonparametric class of elliptical distributions, in line with the previous literature cited above. Whereas many of the aforementioned robust linear shrinkage papers have important theoretical results, the empirical examination of their estimators in simulations and real data applications is often limited.…”

“…However, we note that iteration (6) can be seen as a simultaneous iteration over the eigenvalues and eigenvectors, whereby only the former is changed by nonlinear shrinkage. Following the ideas in Hediger and Näf (2022), we instead aim to iterate over the eigenvectors for fixed (shrunken) eigenvalues. That is, after the first iteration, we fix the eigenvalues obtained by nonlinear shrinkage, denoted Λ 0 .…”

“…In the context of covariance matrix estimation, an alternative approach is to use shrinkage estimation for the correlation matrix and to estimate the vector of variances separately, after which one combines the two estimators to obtain a 'final' estimator of the covariance matrix itself. Such an approach is used by Hediger and Näf (2022) in a static setting (that is, for i.i.d. data) and by Engle et al (2019);De Nard et al (2022) in a dynamic setting (that is, for time series data).…”

“…In this paper, we address these problems by developing a simple algorithm based on nonlinear shrinkage (Ledoit and Wolf, 2012, 2020, 2022b, inspired by the above robust approaches and the work of Hediger and Näf (2022). In essence, the algorithm applies the quadratic inverse shrinkage (QIS) method of Ledoit and Wolf (2022b) to appropriately standardized data, thereby greatly increasing its finite-sample performance in heavy-tailed models.…”

“…Thus, we refer to the new method as "Robust Nonlinear Shrinkage" (R-NL). In particular, we extend the methodology of Hediger and Näf (2022) from a parametric model to general elliptical mixture models. This approach includes an iteration over the space of orthogonal matrices, which we prove converges to a stationary point.…”