Wed Least Squares. Suppose we have the usual linear model 33 Y: = :flfz'jj +623 1 S i E 5'1: i=1 but, instead of the 63' having common variance 02, they are of the form Var(eg-) : 031-02 with 02 possibly unknown but a1, ...., an known. The other assumptions, that the 61- are uncorrelated and have mean 0, are the same as usual. In this case, the usual procedure is to modify the standard least squares procedure so that we minimize Z 05:10? Zi'z'jjl2- (3-58) 1' j The purpose of this exercise is to derive this result from two different points of view. (a) Suppose on; = M/Ni for constant AI and integers N1, ..., NR. By considering the case in which there are N1 observations at 11:11, ..., 5313), N2 observations at 3:91, ...,x2p, etc., show that the re sult (3.58) is valid in this case, and hence argue that it is valid for general ag's. (b) For the simple linear regression Yz'Zo-liiJrEa line with the above assumptions on 63', nd the BLUE of 61, ie deter mine constants ci such that the estimator 61 = Z ciy has mini- mum variance among all unbiased estimators of this form. Show that the answer is consistent with the result of ( a)