4. (Constrained linear regression) Consider the linear regression model Y = X* +, where Y = Rn, X = Rnxd, * is in a closed convex set KC Rd, and ~ the constrained least squares estimator defined by Bargmin ||Y X|| 2. _ BEK N(0,02 In). Let be (a) Using the fact ||Y X|| ||Y X*||2, show that *) 2 || X (3 - 8*)||2 = 4 (exx (8-8)) (b) Let r be the rank of X. Show that . ||X (B *)||2 / E || X X*|| 40. n n (Hint: Use Part (a) and take a supremum over all possible values of the vector v = X(3-3).) ||X (B-B*)||2