论文标题
Wishart法律和Fisher信息的随机分组
The randomization by Wishart laws and the Fisher information
论文作者
论文摘要
考虑带有协方差矩阵$σ的中心高斯矢量$ x $ in $ \ r^n $。$随机$σ$，以便$σ^{ - 1} $具有带状参数$ p>（n-1）/2 $和平均$pσ的愿望分布。当$σ$是参数时，模型$（f_ {p，σ}）$。为了使用Cramér-Rao不平等，我们还计算了$ i_p（σ）$的倒数。本说明的重要点是，此逆是在对称矩阵空间上的两个简单操作员的线性组合，即$¶（σ）（s）=σsσ$和$（σ\ outimesσ）（s）=σ\，\，\，\，\ mathrm / tracerm c）$。渔民信息本身是一种线性组合$¶（σ^{ - 1}）$和$σ^{ - 1} \ otimesσ^{ - 1}。$最后，通过随机化$σ$本身，我们将$ n $ $ nimimise $ compime $ nime $ compime $ compime $ compime $ compime $ commime $ n linimiest of linforations of linforations of linives of linives of linives of linige of linige of linige compine of linige compime of linecome（U） U $出现在结果中。
Consider the centered Gaussian vector $X$ in $\R^n$ with covariance matrix $ Σ.$ Randomize $Σ$ such that $ Σ^{-1}$ has a Wishart distribution with shape parameter $p>(n-1)/2$ and mean $pσ.$ We compute the density $f_{p,σ}$ of $X$ as well as the Fisher information $I_p(σ)$ of the model $(f_{p,σ} )$ when $σ$ is the parameter. For using the Cramér-Rao inequality, we also compute the inverse of $I_p(σ)$. The important point of this note is the fact that this inverse is a linear combination of two simple operators on the space of symmetric matrices, namely $¶(σ)(s)=σs σ$ and $(σ\otimes σ)(s)=σ\, \mathrm{trace}(σs)$. The Fisher information itself is a linear combination $¶(σ^{-1})$ and $σ^{-1}\otimes σ^{-1}.$ Finally, by randomizing $σ$ itself, we make explicit the minoration of the second moments of an estimator of $σ$ by the Van Trees inequality: here again, linear combinations of $¶(u)$ and $u\otimes u$ appear in the results.
