Linear Pooling of Sample Covariance Matrices

Abstract

We consider the problem of estimating highdimensional covariance matrices of K-populations or classes in the setting where the samples sizes are comparable to the data dimension. We propose estimating each class covariance matrix as a distinct linear combination of all class sample covariance matrices. This approach is shown to reduce the estimation error when the sample sizes are limited, and the true class covariance matrices share a somewhat similar structure. We develop an effective method for estimating the coefficients in the linear combination that minimize the mean squared error under the general assumption that the samples are drawn from (unspecified) elliptically symmetric distributions possessing finite fourth-order moments. To this end, we utilize the spatial sign covariance matrix, which we show (under rather general conditions) to be an unbiased estimator of the normalized covariance matrix as the dimension grows to infinity. We also show how the proposed method can be used in choosing the regularization parameters for multiple target matrices in a single class covariance matrix estimation problem. We assess the proposed method via numerical simulation studies including an application in global minimum variance portfolio optimization using real stock data.

Regularized Tapered Sample Covariance Matrix

“…Next note that Λ sgn = E[ Λ] = Λ + o(∥Λ∥ F ) when (A) holds by [28,Theorem 2]. This fact together with (61) and (62) imply that…”

“…The robust properties of SSCM comes from the fact that it is distributionfree under elliptical models and has the highest possible breakdown point [26], [27]. The Ell1-estimator was theoretically studied in [28] and we propose here its generalization to the sphericity of the tapered covariance matrix W • Σ.…”

“…Recently, it was shown in [28] that the following estimate of sphericity based on the SSCM (when µ is known),…”

“…In other words, E[γ] → γ as p → ∞ when (A) holds. We note that Assumption (A) is sufficiently general and holds for many covariance matrix models as shown in [28,Prop. 3].…”

“…Implicitly, this often means that the eigenvalues of Σ tend to be similar as p grows. In [28,Theorem 2] it was shown that the bias of the SSCM Λ is of the order of the sphericity, γ = tr(Λ 2 )/p, and becomes negligible when γ = o(p).…”