inference for normal linear models so far we have not assumed any distribution for the random vector of errors [ en ]in linear model xb when the error terms {ei are assumed to be normally distributed (that is{ei ~iid ( ))whole new avenues open up for inference on linear models in section we already saw that for such normal linear modelsestimation of and can be carried out via maximum likelihood methodsyielding the same estimators from ( and ( the following theorem lists the properties of these estimators in particularit shows /( pare independent and unbiased estimators of and respectively that and theorem properties of the estimators for normal linear model consider the linear model xb ewith ( in )where is pdimensional vector of parameters and dispersion parameter the following results hold are independent the maximum likelihood estimators and (bs (xx) / kh where rank( - proofusing the pseudo-inverse (definition )we can write the random vector as ywhich is linear transformation of normal random vector consequentlyb has multivariate normal distributionsee theorem the mean vector and covariance matrix follow from the same theoremeb xey xx and cov( bxs in ( ) (xx) are independentdefine ( xb to show that and note that / has (uin distributionwith expectation vector xb/ direct application of theorem now shows that ( ( )/ is independent of ( / since xxb xy ( and ky ( /nit follows that is independent of finallyby the same theorem( the random variable ky / has khn- distributionas ( has the same expectation vector as as corollarywe see that each estimator bi of bi has normal distribution with expect ation bi and variance ui ( ui kui where ui [ ]is the -th unit vectorin other wordsthe variance is [(xx)]ii it is of interest to test whether certain regression parameters bi are or notsince if bi the -th explanatory variable has no direct effect on the expected response and so could be removed from the model standard procedure is to conduct hypothesis test (see section for review of hypothesis testingto test the null hypothesis bi