Optimal Shrinkage Covariance Matrix Estimation Under Random Sampling From Elliptical Distributions

Abstract

This paper considers the problem of estimating a high-dimensional (HD) covariance matrix when the sample size is smaller, or not much larger, than the dimensionality of the data, which could potentially be very large. We develop a regularized sample covariance matrix (RSCM) estimator which can be applied in commonly occurring sparse data problems. The proposed RSCM estimator is based on estimators of the unknown optimal (oracle) shrinkage parameters that yield the minimum mean squared error (MMSE) between the RSCM and the true covariance matrix when the data is sampled from an unspecified elliptically symmetric distribution. We propose two variants of the RSCM estimator which differ in the approach in which they estimate the underlying sphericity parameter involved in the theoretical optimal shrinkage parameter. The performance of the proposed RSCM estimators are evaluated with numerical simulation studies. In particular when the sample sizes are low, the proposed RSCM estimators often show a significant improvement over the conventional RSCM estimator by Ledoit and Wolf (2004). We further evaluate the performance of the proposed estimators in classification and portfolio optimization problems with real data wherein the proposed methods are able to outperform the benchmark methods.

A Robust LCMP Beamformer with Limited Snapshots

“…However, GLC's perfromramce degrades when the number of sensors is relatively large [11]. A recent covariance matrix estimation technique for data sampled from elliptical symmetric distribution is proposed in [12]. It is also based on the estimation of the optimal shrinkage parameters that minimizes the mean squared error.…”

Computation‐constrained spectrum sensing in IoT‐based scenarios

“…Usually, SS works in a local or cooperative fashion. Classical local sensing schemes for SS include energy detector (ED) [6], cyclostationary feature detector [7], covariance matrix detector [8] etc. Cooperative SS (CSS) [9] is designed to alleviate the hidden terminal issue with multiple CR equipment located in different locations.…”

Fusing Eigenvalues

“…When the sample size n is not orders of magnitude larger than the dimensionality, p, it has long been recognized that larger eigenvalues of the SCM tend to overestimate, whereas the smaller eigenvalues tend to underestimate the true eigenvalues. Consequently, regularized or penalized estimators of CM have been introduced in a series of papers [1][2][3][4][5][6][7][8][9][10].…”

“…A regularized estimator of CM may be an optimally weighted average of the SCM and a well-structured target estimator, which determines what type of structure is imposed on the estimator. The weight parameter controls how much structure is required [2,[9][10][11]. Another approach in regularizing the SCM is to shrink the eigenvalues towards each other, and not towards a predefined target value.…”

“…Despite numerical stability, most RSCM estimators in the literature are severely biased, i.e., the penalized eigenvalues significantly deviate from the true values [12,16]. Except in specific scenarios [10,17,18], optimum tuning parameters η may not be analytically derived in general, without making prior assumptions on the distribution of the data or model parameters. Our numerical example illustrates that the proposed eFusion estimator, has significantly smaller bias than the benchmark method, the elasso estimator [12].…”