论文标题
基于数据损失的协方差矩阵估计
Covariance matrix estimation under data-based loss
论文作者
论文摘要
在本文中，我们考虑了估计多元线性回归模型的$ p \ times p $ scale矩阵$σ$ $ y = x \，β+ \ mathcal {e} \，当观察到的矩阵$ y $ y属于大型椭圆形的对称分布时，$。在此模型的规范表格$（z^t u^t）^t $之后，通过基于数据的损失tr tr $（s^{+}σ\ \，（σ^{ - σ^{ - 1}}} \hatς-i_p） - $ s^{+} $是其摩尔 - 芬罗倒数。我们为通常的估计器$ a \，s $提供替代性估计器，其中$ a $是一个正常数，它具有较小的相关风险。与通常的二次损失tr $（σ^{ - 1} \hatς -i_p）^2 $相比，我们获得了更大的估计器和更广泛的椭圆形分布类别。一项数值研究说明了理论。
In this paper, we consider the problem of estimating the $p\times p$ scale matrix $Σ$ of a multivariate linear regression model $Y=X\,β+ \mathcal{E}\,$ when the distribution of the observed matrix $Y$ belongs to a large class of elliptically symmetric distributions. After deriving the canonical form $(Z^T U^T)^T$ of this model, any estimator $\hat{ Σ}$ of $Σ$ is assessed through the data-based loss tr$(S^{+}Σ\, (Σ^{-1}\hatΣ - I_p)^2 )\,$ where $S=U^T U$ is the sample covariance matrix and $S^{+}$ is its Moore-Penrose inverse. We provide alternative estimators to the usual estimators $a\,S$, where $a$ is a positive constant, which present smaller associated risk. Compared to the usual quadratic loss tr$(Σ^{-1}\hatΣ - I_p)^2$, we obtain a larger class of estimators and a wider class of elliptical distributions for which such an improvement occurs. A numerical study illustrates the theory.
