need help in c,d,e

Applied Statistics: Sheet 2 MT 2021 4. When the k'th observation (yk, Xk) is removed from a normal linear model Y = XB + E, the MLE parameter estimate B_k based on the reduced data is related to the MLE B = (XTX)-1XTy computed from the full data by B_ K = B - (XTX ) Xk1 - hkk ek where ek = yk - yk, hkk = xK(XTX)-1xx is the k'th leverage component, and yx = X, B. [You may assume this, you are not being asked to show it.] (a) Show that yx - XEB_k = ex/ (1 - hkk). (b) Show that var(yx - XEB_k) = 02/(1 -hkk). (c) Let s_k be the residual standard error in the analysis with (yk, Xk) omitted. Define the studentised residuals rk, for k = 1, ..., n, and show they are given by rk = (1 - hkk) 1/2 ( yk - X / B-k) S _ k (d) Show that rk ~ t(n -p -1) (assume without proof that B and s are independent in the primary fit). We often check normal qqplots for r'. Why do we compare r' to the order statistics of the N(0, 1) distribution? (e) We plot r' against y and look for evidence of correlation. We have seen that e = y - y and y are independent. Show that under the normal linear model r' and y are independent. = yo - yR - han - ex- hikk T-hik ek = ek (+ hik ) = ek. I hick