Studentized Residual

Definition

A studentized residual is a technique used to detect outliers

Internally Studentized Residual

$r_i=\frac{e_i}{s\sqrt{1-h_{ii}}}$ where $s=\frac{1}{n-p}\sum _{i=1}^n{\widehat{ϵ}}_i^2$

$\frac{r_i^2}{n-p}\sim \mathrm{Beta}(\frac{1}{2},\frac{n-p-1}{2})$

Externally Studentized Residual

$r_i^{\ast }=\frac{e_i}{s_{(i)}\sqrt{1-h_{ii}}}\sim (n-p-1)$ where $s_{(i)}^2=\frac{1}{n-p}\sum _{j=1}^n{\widehat{ϵ}}_j^2\cdot 1(j\ne i)$ is the unbiased estimator of ${\sigma }^2$ based on $n-1$ observations after deleting the $i$-th observation

Facts

$s_{(i)}^2=s^2\frac{n-p-r_i^2}{n-p-1}$ $r_i^{\ast }=r_i\sqrt{\frac{n-p-1}{n-p-r_i^2}}$