Linear pooling of sample covariance matrices

Abstract

We consider covariance matrix estimation in a setting, where there are multiple classes (populations). We propose to estimate each class covariance matrix as a linear combination of all of the 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 similar structure. We develop an effective method for estimating the minimum mean squared error coefficients for the linear combination when the samples are drawn from (unspecified) elliptically symmetric distributions with finite fourth-order moments. To this end, we utilize the spatial sign covariance matrix, which we show (under rather general conditions) to be a consistent estimator of the trace normalized covariance matrix as both the sample size and the dimension grow 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, where it is shown to outperform several state-of-the-art methods.

Regularized tapered sample covariance matrix

“…Next note that Λ sgn = E[ Λ] = Λ + o( Λ F ) when (A) holds by [24,Theorem 2] This fact together with (55) and (56) imply that…”

“…Proof. Follows from Theorem 1 after substituting the values of E [ W • S 2 F given in Lemma 1 and of E tr(S) 2 given in (24) into the denominator of β o in (15) and simplifying the expression.…”

“…The Ell1-estimator is based on the spatial sign covariance matrix (SSCM), which has been popular for constructing robust estimates of the sphericity [22], [23]. This estimator was theoretically studied in [24] and we propose here its adaptation to the sphericity of the tapered covariance matrix W • Σ.…”

“…Recently, it was shown in [24] 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 [24,Prop. 3].…”