3. In this question we consider the derivation of the 0,, statistic for model selection discussed in lectures. Consider the general linear model and write if. for the tted value at 5:.- and MSE(E] for its mean squared error. Recall that if the error variance :12 is known, then an estimate of Emss] : EVE-dis) + Eli-3135*??? .1, _ H [wereas [,2 m where a is the estimate of the residual error variance. Substituting 3%. (the estimated error variance based on a model including all predictors} for s2 in this expression gives Mallow's Cpl You now have to provide a justication for (1) In what follows, we let Xd be the design matrix for the proposed model, and X. be the design matrix for the true model; Let d denote the vector of parameters in the proposed model and {i be the vector of parameters for the true modeL We thus have for the true model 1 = XLIB' + E with corresponding hat matrix H1. = XIlXITXt]_l-XE- (a) Writing if for the vector of tted values [using Xd, not the unknown XI} and observing that if: XleiX-irth! = Hay show that 21mm} = cased}. i Deduee that 'ml'l = p. (h) Now consider the bias part. Using the fact that Hg is idempotent (11'de 2 Hi.) and symmetric, show that 23mm) = we! Hem. {c} Now consider the expectation of the dierence between the estimated error variance 32 and the true error variance 0"!) Show that (11 132'E[3'2 - 0\"} = ElyTU Ham] - {n p10\"- Henee, deduce that (n p)E[fr\"2 r12] = x?\" Home