be decided by means of hypotheses tests this is the classical approach to model selectionand will be discussed in section as consequence of the central limit theoremone can use the same approach when the error terms are not necessarily normalprovided their variance is finite and the sample size is large finallywhen using bayesian approachcomparison of two models can be achieved by computing their so-called bayes factor (see section all of the above strategies can be thought of as specifications of simple rule formulated by william of occamwhich can be interpreted aswhen presented with competing modelschoose the simplest one that explains the data this age-old principleknown as occam' razoris mirrored in famous quote of einsteinoccam' razor everything should be made as simple as possiblebut not simpler in linear regressionthe number of parameters or predictors is usually reasonable measure of the simplicity of the model cross-validation and predictive residual sum of squares we start by considering the -fold cross-validationalso called leave-one-out crossvalidationfor the linear model ( we partition the data into data setsleaving out precisely one observation per data setwhich we then predict based on the remaining observationssee section for the general case let - denote the prediction for the -th observation using all the data except yi the error in the predictionyi - is called predicted residual -in contrast to an ordinary residualei yi - yi which is the difference between an observation and its fitted value yi gt (xi obtained using the whole sample in this waywe obtain the collection of predicted residuals {yi - }ni= and summarize them through the predicted residual sum of squares (press)press = leave-one-out cross-validation predicted residual press (yi - ) dividing the press by gives an estimate of the expected generalization risk in generalcomputing the press is computationally intensive as it involves training and predicting separate times for linear modelshoweverthe predicted residuals can be calculated quickly using only the ordinary residuals and the projection matrix xxonto the linear space spanned by the columns of the model matrix (see ( )the -th diagonal element pii of the projection matrix is called the -th leverageand it can be shown that pii (see exercise leverage