Editing Studentized residual (section)

==Internal and external studentization==

The usual estimate of ''σ''<sup>2</sup> is  the ''internally studentized'' residual

:<math>\widehat{\sigma}^2={1 \over n-m}\sum_{j=1}^n \widehat{\varepsilon\,}_j^{\,2}.</math>

where ''m'' is the number of parameters in the model (2 in our example).

But if the ''i''&nbsp;th case is suspected of being improbably large, then it would also not be normally distributed. Hence it is prudent to exclude the ''i''&nbsp;th observation from the process of estimating the variance when one is considering whether the ''i''&nbsp;th case may be an outlier, and instead use the ''externally studentized'' residual, which is

:<math>\widehat{\sigma}_{(i)}^2={1 \over n-m-1}\sum_{\begin{smallmatrix}j = 1\\j \ne i\end{smallmatrix}}^n \widehat{\varepsilon\,}_j^{\,2},</math>

based on all the residuals ''except'' the suspect ''i''&nbsp;th residual. Here is to emphasize that <math>\widehat{\varepsilon\,}_j^{\,2} (j \ne i)</math> for suspect ''i'' are computed with ''i''&nbsp;th case excluded. 

If the estimate ''σ''<sup>2</sup> ''includes'' the ''i''&nbsp;th case, then it is called the '''''internally studentized''''' residual, <math>t_i</math> (also known as the ''standardized residual'' <ref>[https://stat.ethz.ch/R-manual/R-devel/library/stats/html/influence.measures.html Regression Deletion Diagnostics] R docs</ref>).
If the estimate <math>\widehat{\sigma}_{(i)}^2</math> is used instead, ''excluding'' the ''i''&nbsp;th case, then it is called the '''''externally studentized''''', <math>t_{i(i)}</math>.