Studentized Residual - Motivation

Motivation

See also: Errors and residuals in statistics

The key reason for studentizing is that, in regression analysis of a multivariate distribution, the variances of the residuals at different input variable values may differ, even if the variances of the errors at these different input variable values are equal. The issue is the difference between errors and residuals in statistics, particularly the behavior of residuals in regressions.

Consider the simple linear regression model

Given a random sample (X_i, Y_i), i = 1, ..., n, each pair (X_i, Y_i) satisfies

where the errors ε_i, 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

and

(Here ε_i is the ith error, and is the ith residual.)

Moreover, and most importantly, the residuals, unlike the errors, do not all have the same variance: the variance decreases as the corresponding x-value gets farther from the average x-value. This is a feature of the regression better fitting values at the ends of the domain, not the data itself, and is also reflected in the influence functions of various data points on the regression coefficients: endpoints have more influence. This can also be seen because the residuals at endpoints depend greatly on the slope of a fitted line, while the residuals at the middle are relatively insensitive to the slope. 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.

It is not simply a matter of the population parameters (mean and standard deviation) being unknown – it is that regressions yield different residual distributions at different data points, unlike point estimators of univariate distributions, which share a common distribution for residuals.