theorem press for linear models consider the linear model ( )where the nxp model matrix is of full rank given an outcome [ yn ]of ythe fitted values can be obtained as pywhere xxx(xx)- xis the projection matrix if the leverage value pi :pii for all nthen the predicted residual sum of squares can be written as press = ei pi ! where ei yi yi yi (xb ) is the -th residual proofit suffices to show that the -th predicted residual can be written as yi - ei /( pi let - denote the model matrix with the -th rowxi removedand define - similarly thenthe least-squares estimate for using all but the -th observation is - ( >- - )- >- - writing xx >- - xi > we have by the sherman-morrison formula (xx)- xi > (xx)- ( >- - )- (xx)- > (xx)- xi where > (xx)- xi pi - - xy xi yi combining all these identitieswe have - ( >- - )- >- - (xx)- xi > (xx)- (xy xi yi ( pi (xx)- xi > (xx)- xi pi yi = (xx)- xi yi pi pi - > ( xxi xi (xx)- xi yi = pi pi - > ( xxi (yi xi (xx)- xi ei = = pi pi - it follows that the predicted value for the -th observation is given by by- > - > > (xx)- xi ei pi ei = yi pi pi henceyi - ei pi ei /( pi ei /( pi example (polynomial regression (cont )we return to example where we estimated the generalization risk for various polynomial prediction functions using independent validation data insteadlet us estimate the expected generalization risk via crossvalidation (thus using only the training setand apply theorem to compute the press