1.4. Using the same notation as in Section 1.4.1 on studentized residuals, prove (1.38) by proceeding in the following s

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1 4 Using The Same Notation As In Section 1 4 1 On Studentized Residuals Prove 1 38 By Proceeding In The Following S 1 (242.16 KiB) Viewed 35 times

1.4. Using the same notation as in Section 1.4.1 on studentized residuals, prove (1.38) by proceeding in the following steps. (a) Let A be a p × p nonsingular matrix and U and V be p × 1 vectors. The matrix inversion lemma represents (A + UVÃ)−¹ in terms of A-¹ as follows: (A + UV¹)−¹ = A¯¹ − A¯¹U(I + Vª A¯¹U)¯¹V² A¯¹. Use the matrix inversion lemma to prove (x²-₁) X(-₁))-¹ = (XX)−¹ + (−i)- (X¹X)—¹x¡x/(X¹X)−¹ 1 - hii hii x² (X²_₁)X(-1))¯¹x; = 1 - hii (b) Show that, after the deletion of (xį, yi), we have -1 (X¹X)-¹xi B(-1) = 3 - ∙eir 1 - hii = Yi − x(-i) and e where ei = y₁ - x{ß. Letting ê(-i) ei/(s√1 – hii), derive the representations ei e'i n − p − (e;)² ². = ê(-i) - :S, 1 - hii √1 - hii n-p-1 (c) Show that Var(^(-i)) = s²_i)/(1 — hii), and hence prove (1.38). =