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Random Matrix Theory Location: 1.012 Session Chair: Nestor Parolya | |
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Nonlinear higher-order shrinkage estimation of the large dimensional covariance and precision matrices 1: Delft University of Technology, Netherlands, The; 2: Linköping University, Sweden In this paper, we develop nonlinear higher-order shrinkage estimators for both covariance and precision matrices. Our framework applies to settings in which the sample size n is either larger or smaller than p, the dimensionality of the data-generating process. The proposed estimators incorporate higher-order moments up to an arbitrary order and therefore encompass linear shrinkage estimators as special cases. We derive recursive representations of these higher-order nonlinear shrinkage estimators using partial exponential Bell polynomials. Through simulation studies, the proposed methods are compared with the oracle nonlinear shrinkage estimator and are shown to be particularly effective in settings where no closed-form expressions for nonlinear shrinkage estimators are available. The theoretical derivations rely on mild assumptions on the underlying model, including the existence of fourth moments and a bounded spectrum of the true population covariance matrix. The finite-sample performance of the proposed estimators is evaluated in an extensive simulation study and benchmarked against existing approaches. Our main finding is that the higher-order shrinkage estimators can outperform well-established nonlinear shrinkage methods, particularly when the concentration ratio p/n is large. | |
