Studentized residual

In statistics, a Studentized residual, named in honor of William Sealey Gosset, who wrote under the pseudonym Student, is a residual adjusted by dividing it by an estimate of its standard deviation. Studentization of residuals is an important technique in the detection of outliers.

Errors versus residuals

It is very important to understand the difference between errors and residuals in statistics. Consider simple linear regression model

Related Topics:
Errors and residuals in statistics - Linear regression

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:Y_i=lpha_0+lpha_1 x_i+arepsilon_i,

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

where the errors εi, i = 1, ..., n, are independent and all have the same variance σ2. The residuals are not the true, and unobservable, errors, but rather are estimates, based on the observable data, of the errors. When the method of least squares is used to estimate α0 and α1, then the residuals, unlike the errors, cannot be independent since they satisfy the two constraints

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:sum_{i=1}^n widehat{arepsilon}_i=0

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

and

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:sum_{i=1}^n widehat{arepsilon}_i x_i=0.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

(Here arepsilon_i is the ith error, and widehat{arepsilon}_i is the ith residual.) Moreover, the residuals, unlike the errors, do not all have the same variance: the variance increases as the corresponding x-value gets farther from the average x-value. The fact that the variances of the residuals differ, even though the variances of the true errors are all equal to each other, is the principal reason for the need for Studentization.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~