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September 28th, 2017, 04:34 AM   #1
Joined: Sep 2017
From: australia

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variance of ith deleted residual

From linear model, $y = X\beta + \epsilon$, if $\hat{\sigma}^2 = \frac{|y - Xb|^2}{n-p}$ is the variance of error and $\hat{\sigma}_{-i}^2 = \frac{|y_{-i} - X_{-i}b_{-i}|^2}{n-p-1}$ is the estimate of the error variance σ obtained by fitting all the
observations except the i-th. Use the fact that $b - b_{-i} = \frac{(X^TX)^{-1}x_ie_i}{1 - H_{i,i}}$ (can be proved) to show that $\hat{\sigma}_{-i}^2 = \frac{(n - p)\hat{\sigma}^2 - e_i^2/(1-H_{i,i})}{n-p-1}$ where $e_i = y_i - \hat{y_i}$ and $H_{i,i} = x_i(X^TX)^{-1}x_i^T$ is the hat matrix.

So i used the fact $b - b_{-i} = \frac{(X^TX)^{-1}x_ie_i}{1 - H_{i,i}}$, made $b_{-i}$ the subject and sub it in $\hat{\sigma^2_{-i}}$ but no luck simplifying the to desired answer. ANy hints will be helpful.
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