In multiple linear regression, what statistical property should studentized residuals adhere to?

In multiple linear regression, studentized residuals are calculated to help assess the fit of the regression model and identify potential outliers. The correct characteristic of studentized residuals is that they should follow a t-distribution, particularly when the sample size is small. This is because studentized residuals are derived from the ordinary residuals, which are scaled by an estimate of their standard deviation.

When the assumptions of regression are met, including homoscedasticity and normality of the residuals, the distribution of studentized residuals converges to a t-distribution. This property is particularly important in hypothesis testing and constructing confidence intervals, as the t-distribution accounts for the increased variability that can occur when estimating standard errors from sample data.

The normal distribution is not appropriate for studentized residuals because they account for the variability in the residual values and leverage points in the regression model. Additionally, it is not required that all studentized residuals be positive, as they can take on both negative and positive values indicating how far off each data point is from the predicted values. While normally distributed residuals are a desired property of the underlying errors in the regression model, specifically studentized residuals should follow a t-distribution.