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first time to use this website, hope I made it correctly.

Suppose $\underline{X}\sim n(\underline{\mu},\Sigma)$ where $\Sigma= (\begin{matrix}&\Sigma_{11} &\Sigma_{12}\\&\Sigma_{21} & \Sigma_{22}\end{matrix})$where $\Sigma_{11}$ is (n-1)*(n-1) dimensional, $\Sigma_{12}$ is (n-1)1 dimensional, $\Sigma_{21}$ is 1(n-1) dimensional, $\Sigma_{22}$ is 1*1 dimensional. Now we already know that $X_1 = x_1, X_2 = x_2, \cdots, X_{n-1} = x_{n-1}$, thus the minimum MSE of $X_n$ is $\hat{X_n} = E[X_n|X_1 = x_1, X_2 = x_2, \cdots, X_{n-1} = x_{n-1}] = \mu_n + \Sigma_{21}\Sigma_{11}^{-1}(\begin{matrix}&x_1-\mu_1\\ &x_2-\mu_2\\ &\cdots\\ &x_{n-1}-\mu_{n-1})\end{matrix}$.

My question is: how to calculate the corresponding MSE? I'm not very familiar with the matrix calculation. enter image description here

West
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  • what does $\Sigma_{11}^{-1}(x_1-\mu_1, x_2-\mu_2, \cdots, x_{n-1}-\mu_{n-1})$ mean ? Also what is $\Sigma_{21}$ ? – Ahmad Bazzi Oct 07 '18 at 17:57
  • Sorry about the confusion, the covariance matrix: $\Sigma=(\Sigma_{11} \Sigma_{12} \Sigma_{21} \Sigma_{22})$(I don't know how to start a new line for $\Sigma_{21} \Sigma_{22})$ ) where$\Sigma_{11}$ is (n-1)(n-1) dimensional, $\Sigma_{12}$ is (n-1)1 dimensional, $\Sigma_{21}$ is 1(n-1) dimensional, $\Sigma_{22}$ is11 dimensional. Hope this help explain it well. – West Oct 07 '18 at 18:00
  • Please include all details (i.e. define your expressions) in your post itself by clicking on 'edit'. – StubbornAtom Oct 07 '18 at 18:08
  • I've added a picture to the original question, is that helpful? – West Oct 07 '18 at 18:17
  • @West Since you already used MathJax, why not write the entire post that way? It is always better than pictures anyway. – StubbornAtom Oct 07 '18 at 18:18
  • Sorry about that, I have made some changes to the original question, is that clear now? – West Oct 07 '18 at 18:29

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