variance of ith deleted residual
From linear model, $y = X\beta + \epsilon$, if $\hat{\sigma}^2 = \frac{y  Xb^2}{np}$ is the variance of error and $\hat{\sigma}_{i}^2 = \frac{y_{i}  X_{i}b_{i}^2}{np1}$ is the estimate of the error variance σ obtained by fitting all the
observations except the ith. Use the fact that $b  b_{i} = \frac{(X^TX)^{1}x_ie_i}{1  H_{i,i}}$ (can be proved) to show that $\hat{\sigma}_{i}^2 = \frac{(n  p)\hat{\sigma}^2  e_i^2/(1H_{i,i})}{np1}$ 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.
