Internal and External Studentization
The usual estimate of σ2 is
where m is the number of parameters in the model (2 in our example). But it is desirable to exclude the ith observation from the process of estimating the variance when one is considering whether the ith case may be an outlier. Consequently one may use the estimate
based on all but the ith case. If the latter estimate is used, excluding the ith case, then the residual is said to be externally studentized; if the former is used, including the ith case, then it is internally studentized.
If the errors are independent and normally distributed with expected value 0 and variance σ2, then the probability distribution of the ith externally studentized residual is a Student's t-distribution with n − m − 1 degrees of freedom, and can range from to .
On the other hand, the internally studentized residuals are in the range, where r.d.f. is the number of residual degrees of freedom, namely n − m. If "i.s.r." represents the internally studentized residual, and again assuming that the errors are independent identically distributed Gaussian variables, then
where t is a random variable distributed as Student's t-distribution with r.d.f. − 1 degrees of freedom. In fact, this implies that i.s.r.2/r.d.f. follows the beta distribution B(1/2,(r.d.f. − 1)/2). When r.d.f. = 3, the internally studentized residuals are uniformly distributed between and .
If there is only one residual degree of freedom, the above formula for the distribution of internally studentized residuals doesn't apply. In this case, the i.s.r.'s are all either +1 or −1, with 50% chance for each.
The standard deviation of the distribution of internally studentized residuals is always 1, but this does not imply that the standard deviation of all the i.s.r.'s of a particular experiment is 1. For instance, the internally studentized residuals when fitting a straight line going through (0, 0) to the points (1, 4), (2, −1), (2, −1) are, and the standard deviation of these is not 1.
Read more about this topic: Studentized Residual
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