I cannot find a consistent definition of the studentized residual and the RMSEP, because I've noticed that various websites, lecture notes and software packages mix up 1 or 2 definitions along the way to the point that a "compound" definition ends up very different between one reference source and another!(adsbygoogle = window.adsbygoogle || []).push({});

So I'm going to write all of my definitions from the ground up. Would someone be so kind as to confirm to me if my definitions 4, 5, 7 and 8 are correct?

> Regarding (4) and (5), should I divide my PRESS by the sample size [itex]n[/itex] or should I divide it by the degrees of freedom, as I would calculate the RMSE?

> Regarding (7) and (8), am I correct to use the jackknifed residual in the numerator and the RMSEP (instead of the RMSE) in the denominator? Is there an intuitive explanation as to why I should prefer the jackknifed residual over the internally studentized residual?

DEFINITION 1.My raw residuals are [itex]\hat{e}_{i}=Y_{i}-\hat{Y}_{i}[/itex] where [itex]Y_{i}[/itex]'s are the actual values and [itex]\hat{Y}_{i}[/itex] are the values predicted by the regression equation.

DEFINITION 2.The hat matrix is defined as [itex]H[/itex] such that the vector of values predicted by the regression equation [itex]\hat{Y}=HY[/itex], where [itex]Y[/itex] is the vector of actual values.

DEFINITION 3.The jackknifed residuals are defined as [itex]\hat{e}_{i,-i}=Y_{i}-\hat{Y}_{i,-i}[/itex] where [itex]\hat{Y}_{i,-i}[/itex] are the values predicted by the regression equation estimated while excluding [itex]Y_{i}[/itex]

DEFINITION 4.Given a sample size of [itex]n[/itex] data points and [itex]k[/itex] predictor variables, the RMSE is simply the SSE divided by the degrees of freedom, [itex]\sqrt{\dfrac{SSE}{n-k-1}}[/itex].

DEFINITION 5.Given a sample size of [itex]n[/itex] data points, the predicted residual sums of squares (PRESS) is [itex]PRESS=\sum_{i=1}^{n}\hat{e}_{i,-i}=\sum_{i=1}^{n}\left(y_{i}-\hat{y}_{i,-i}\right)^{2}[/itex] so the root mean squared error of prediction (RMSEP) is [itex]RMSEP=\sqrt{\dfrac{PRESS}{n}}[/itex]

DEFINITION 6.The standardized residual is the raw residual divided by its RMSE, i.e. [itex]\dfrac{\hat{e}_{i}}{RMSE}[/itex].

DEFINITION 7.The internally studentized residual is [itex]\dfrac{\hat{e}_{i}}{RMSE\sqrt{1-h_{ii}}}[/itex] where the leverage [itex]h_{ii}\in\left[0,1\right][/itex] is the [itex]i[/itex]th diagonal entry of the hat matrix .

DEFINITION 8.The studentized deleted residual is calculated using the jackknifed residuals, so it is computed as [itex]\dfrac{\hat{e}_{i,-i}}{RMSEP\sqrt{1-h_{ii}}}[/itex].

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# Why are definitions for the studentized residual so confusing?

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