论文标题
在高维度和低维度的基于数据的损失下的比例矩阵估计
Scale matrix estimation under data-based loss in high and low dimensions
论文作者
论文摘要
我们考虑在理论决策的观点下，估算Additif模型$ y_ {p \ times n} = m + \ mathcal {e} $的比例矩阵$σ$的问题。在这里，$ p $是变量的数量，$ n $是观察值的数量，$ m $是未知参数的矩阵，级别$ q <p $和$ \ mathcal {e} $是一个随机噪声，其分布与椭圆形的与covariance矩阵相称，与$ i_n \ i_n \ yotimesσ$ \ imimeσ$ \ \ \ \ \ \ \ \ \ \ \\。我们处理该模型的规范形式，其中$ y $在两个矩阵中分解，即$ z_ {q \ times p} $，总结了$ m $中包含的信息，$ m $ $ y_ {m \ times p} $，其中$ m = n-q $，总结了足够的信息以估算$ n-q $ n-q $。作为$ {\ hatσ} _a = a \，s $的自然估计器（其中$ s = u^{t} \，u $，u $和$ a $是一个正常的常数），当$ p> m $（s不可逆转）时，我们提出了形式$ $ $ $ {\hatς}} _ _ _ {a g}的估计器， {s^{+} \，g（z，s）} \ big）$，其中$ {s^{+}} $是$ s $的moore-penrose倒数（与$ s^{ - 1} $相吻合时$ s $时，$ s $是$ s $的）。我们提供校正矩阵$ ss^{+} {g（z，s）} $的条件s^ {+}σ\，（{\hatς} \，σ^ { - 1} - {i} _ {p}）^ {2} \ big）$。我们采用了两种情况下的统一方法，其中$ s $是可逆的（$ p \ leq m $），而$ s $是不可变的（$ p> m $）。
We consider the problem of estimating the scale matrix $Σ$ of the additif model $Y_{p\times n} = M + \mathcal{E}$, under a theoretical decision point of view. Here, $ p $ is the number of variables, $ n$ is the number of observations, $ M $ is a matrix of unknown parameters with rank $q<p$ and $ \mathcal {E}$ is a random noise, whose distribution is elliptically symmetric with covariance matrix proportional to $ I_n \otimes Σ$\,. We deal with a canonical form of this model where $Y$ is decomposed in two matrices, namely, $Z_{q\times p}$ which summarizes the information contained in $ M $, and $ U_{m\times p}$, where $m=n-q$, which summarizes the sufficient information to estimate $ Σ$. As the natural estimators of the form ${\hat Σ}_a=a\, S$ (where $ S=U^{T}\,U$ and $a$ is a positive constant) perform poorly when $p >m$ (S non-invertible), we propose estimators of the form ${\hatΣ}_{a, G} = a\big( S+ S \, {S^{+}\,G(Z,S)}\big)$ where ${S^{+}}$ is the Moore-Penrose inverse of $ S$ (which coincides with $S^{-1}$ when $S$ is invertible). We provide conditions on the correction matrix $SS^{+}{G(Z,S)}$ such that ${\hat Σ}_{a, G}$ improves over ${\hat Σ}_a$ under the data-based loss $L _S( Σ, \hat { Σ}) ={\rm tr} \big ( S^{+}Σ\,({\hatΣ} \, Σ ^ {- 1} - {I}_ {p} )^ {2}\big) $. We adopt a unified approach of the two cases where $ S$ is invertible ($p \leq m$) and $ S$ is non-invertible ($p>m$).
