quality:corruption, information:extraction
Intervals

These asymptotic distributions can be used for prediction, testing hypotheses, constructing other estimators, etc.. As an example consider the problem of prediction. Suppose x 0 {\displaystyle x_{0}} is some point within the domain of distribution of the regressors, and one wants to know what the response variable would have been at that point. The mean response is the quantity y 0 = x 0 T β {\displaystyle y_{0}=x_{0}^{\mathrm {T} }\beta }, whereas the predicted response is y ^ 0 = x 0 T β ^ {\displaystyle {\hat {y}}_{0}=x_{0}^{\mathrm {T} }{\hat {\beta }}}. Clearly the predicted response is a random variable, its distribution can be derived from that of β ^ {\displaystyle {\hat {\beta }}}: