Abstract The problem of estimating the scale matrix Σ in a multivariate additive model, with elliptical noise, is considered from a decision-theoretic point of view. As the natural estimators of the form Σ ˆ a = a S (where S is the sample covariance matrix and a is a positive constant) perform poorly, we propose estimators of the general form Σ ˆ a , G = a ( S + S S + G ( Z , S ) ) , where S + is the Moore–Penrose inverse of S and G ( Z , S ) is a correction matrix. We provide conditions on G ( Z , S ) such that Σ ˆ a , G improves over Σ ˆ a under the quadratic loss L ( Σ , Σ ˆ ) = tr ( Σ ˆ Σ − 1 − I p ) 2 . We adopt a unified approach to the two cases where S is invertible and S is singular. To this end, a new Stein–Haff type identity and calculus on eigenstructure for S are developed. Our theory is illustrated with a large class of estimators which are orthogonally invariant.

中文翻译：

---

椭圆对称分布的高低维尺度矩阵估计

摘要 从决策理论的角度考虑在具有椭圆噪声的多元加性模型中估计尺度矩阵 Σ 的问题。由于 Σ ˆ a = a S 形式的自然估计量（其中 S 是样本协方差矩阵，a 是正常数）表现不佳，我们提出了一般形式 Σ ˆ a , G = a ( S + SS + G ( Z , S ) ) ，其中 S + 是 S 的 Moore-Penrose 逆，G ( Z , S ) 是校正矩阵。我们在 G ( Z , S ) 上提供条件使得 Σ ˆ a , G 在二次损失 L ( Σ , Σ ˆ ) = tr ( Σ ˆ Σ − 1 − I p ) 2 下优于 Σ ˆ a 。我们对 S 可逆和 S 奇异的两种情况采用统一的方法。为此，开发了新的 Stein-Haff 型恒等式和 S 的特征结构演算。