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Pi,, and the ordinary residuals are given next. 6. For i = 1. , n , (a) ifpii = 1 or 0, then pij = 0. 2. 2 0. iJ Proof. We prove part (d) and leave the proofs of parts (a)-(c) as exercises for the reader. Define Z = (X : Y), PX = X(X’X)-*X’’, and Pz = Z(ZTZ)-IZT. 12) and since the diagonal elements of Pz are less than or equal to 1, part (d) follows. 6 indicate that if pii is large (near 1) or small (near 0), then pij is small for all j f i. Part (d) indicates that the larger pii, the smaller the ith ordinary residual, ek As we shall see in Chapter 4, observations with large pii tend to have small residuals, and thus may go undetected in the usual plots of residuals.

6. For i = 1. , n , (a) ifpii = 1 or 0, then pij = 0. 2. 2 0. iJ Proof. We prove part (d) and leave the proofs of parts (a)-(c) as exercises for the reader. Define Z = (X : Y), PX = X(X’X)-*X’’, and Pz = Z(ZTZ)-IZT. 12) and since the diagonal elements of Pz are less than or equal to 1, part (d) follows. 6 indicate that if pii is large (near 1) or small (near 0), then pij is small for all j f i. Part (d) indicates that the larger pii, the smaller the ith ordinary residual, ek As we shall see in Chapter 4, observations with large pii tend to have small residuals, and thus may go undetected in the usual plots of residuals.

If A is a k x k nonsingular matrix, then det(A - BCT)= det(A) det(1- CTA-*B). 3CT). Hence det(A BCT) = det(A) det(1- CTA-'B). 13. Let P/ be an m X m minor of P given by the intersection of the rows and columns of P indexed by I. If hl I h2 5 ... , m. ). (c) rank(X(,)) < k iff hm = 1. Proof. Let P22 be an ~tl x m minor of P. Without loss of generality, let these be the last m rows and columns of P. 22) where A is a diagonal mamx with the eigenvalues of P22 in the diagonal and V is a mamx containing the corresponding normalized eigenvectors as columns.