Problem 2. (15 points) Suppose Y~ Nn (µ, In). Let X € Rnxp be a fixed matrix with full column rank. In this exercise we consider estimators of the form û = Xß for some estimator 3. (i) Let μLS XLS be the corresponding estimator of the mean of Y based on the least squares estimator BLS = (XX)-¹XTY. Find the risk of this estimator directly in terms of Co= X(XTX)-¹XT, μ, n, and p only. (ii) Suppose μ = XB* for some ß* ERP. Show that R(μ, ûLS) = p. = = (iii) Consider the ridge estimator pridge = (X¹X+8Ip)-¹X¹Y and the corresponding matrix Cg = X(XTX + 8Ip)-¹X, where 820. Show that R(µ, pridge) = tr(C3) + || (In-Cs)μ||². (iv) Suppose that XTX = Ip and that μ = XB* for some ß*. Show that for every 3* 0 there exists 8 >0 such that R(XB*, pridge) < R(XB*, μLS).

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