Studentized residual and it variance

Question

I am recently reading regression diagnose and came across with studentized residual. Everything went smoothly in understanding but I am stuck at a formula to calculate sample variance after the jth observation is removed.

So the studentized residual is defined as $$r_j = \frac{\hat{e_j}}{s_{(j)} \sqrt{1-h_j}}$$

There is no problem understanding above. The question comes from the formula $${s_{(j)}}^2 = \frac{s^2(n-p)-\hat{e_j}^2/(1-h_j) }{n-p-1}$$

I don't understand how this was derived. I tried to break it down by calculating the sample variance after removing the jth observation, and also tried to look at it through the difference between the squared residual sum of the full model and the reduced model. However, none of the above brought me this formula.

So if any one could please help or provide some hint, that would be greatly appreciated.

Answer 1

• Note that $s^2$ is defined as $SSE/(n-p)$.
• Note that $s_{(j)}^2$ is defined as $SSE_{(j)}/(n-p-1)$, where $SSE_{(j)}$ denotes the sum of squared residuals when performing regression after removing the $j$th observation.

Thus, you are asking to prove $SSE_{(j)} = SSE - \hat{e}_j^2/(1-h_j)$. This is proved in Proposition 13.5.5 of Christensen's "Plane Answers to Complex Questions" [PDF]. It relies on a couple of intermediate results preceding it in the same section, including a special case of the Woodbury matrix identity (Proposition 13.5.1 in the book).