Studentized Residuals When we compute the studentized residuals, Ti 32 V (I _ PX)i,iSSres/(n _ p _1)1 as long as n p 1 is large, then we claim that...

Hi, could you please help me answer three STAT question, thank you.

1. Studentized Residuals When we compute the studentized residuals,Ti 32 V (I _ PX)i,iSSres/(n _ p _1)1 as long as n — p — 1 is large, then we claim that 5.5 has a t(n — p — 1)distribution which is close to a N(U, 1). Show that actually 3%- can’thave a t (n — p — 1) distribution by showing that Isi] 3 V1), —p— 1.Hint: write $87.33 as 211:1 1:52 and use the fact that (I — PX)”- E (0, 1]for all i. 2. The Outlier Efi'ectConsider the linear regression setting where we have n sample pointswith s N N(0, of) and m sample points with 8 ~ N(0, (73).1 (a) Show that for some 6 6 [0,1], n+m Z Var (sz) = 603 + (1 — (5)03. i=1 1n+m (b) Assuming that (I? < 03, show thatSS2 res 2E —~ .01 < (n+m—p— 1) <023. Variance Stabilization: For the standard linear regression model,Y=fi0+fil$l+---+fip$p+eilet ,u = EY. Compute the variance stabilizing transformation when. . .(80 Val" (Y) = H2(b) Var (Y) = if for It > 2(c) Var (Y) = e“((0 WHY) = V2 1 Here, the m points could represent a few outliers in the dataset which will effect thehypothesis tests and confidence intervals.