REGRESSION

Question 1 Consider the general linear regression model, Y = X3px1 + , where Y is an n x 1 response vector, X is an n x p full rank predictor matrix of real numbers, Box1 is a p x 1 vector of parameters and & is an n x 1 with mean 0 and variance-covariance matrix ) = 21. The maximum likelihood (ML) estimator of Box | is Box1= (X'X) X'Y and the hat (prediction) matrix is H = X(X'X) X'. (a) Let A and D be nonsingular matrices of orders & and m, respectively, B be k x m, and C be k x m. Then, provided that the inverses exists (A+BDC) =A -A B(D-+CA 'B) C'A-1. The inverse of the scatter matrix can be expressed as (X'X) = (X, Xon) + xix, ) , where X( denotes the design matrix with ith observation deleted. Denote the predicted leverage as him) = x'(X'")X())"x; and deduce that the relationship between the leverage values hu = x'(X'X) x; and the predicted leverage hi( is monotonic, i.e. express h, as a function of hi()