studentized residual all data except the -th observation this gives rise to the studentized residual defined as ei : - pii where - is an estimate of obtained by fitting all the observations except the -th and ei yi yi is the -th (randomresidual exercise shows that we can takefor example ky - -ib - ( - where - is the model matrix with the -th row removedis an unbiased estimator of efficientlyusing in ( )as the latter will typically be we wish to compute - available once we have fitted the linear model to this enddefine ui as the -th unit vector [ ]and let - ( : (yi - )ui ei ui pii where we have used the fact that yi - ei /( pii )as derived in the proof of theorem now apply exercise to prove that - cook' distance ( ps ei /( pii np- using the notation from exercises - cook' distance for observation is defined as (ikb - di :ps it measures the change in the fitted values when the -th observation is removedrelative to the residual variance of the model (estimated via by using similar arguments as those in exercise show that di pii ei ( pii ) it follows that there is no need to "omit and refitthe linear model in order to compute cook' distance for the -th response prove that if we add an additional feature to the general linear modelthen the coefficient of determinationis necessarily non-decreasing in value and hence cannot be used to compare models with different numbers of predictors let :[ xn ]and :[ un ]in the fundamental theorem we use the fact that if xi (ui ) are independentthen kxk has (per definitiona noncentral kh distribution show that kxk has moment generating function etkuk /( - ( ) / / and so the distribution of kxk depends on only through the norm kuk