In this question we consider derivation of the C, statistic for model selection discussed in lectures....

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In this question we consider derivation of the C, statistic for model selection discussed in lectures. Consider fitting the proposed general linear model and write ; for the fitted value at r; and MSE() for its mean squared error. Recall that if the error variance o? is known, then an estimate of E MSE() Jp EVar() E; Bias () is (n - p)( - o²) (1) where ở is the estimate of the residual error variance for the proposed model and p is the number of parameters. Substituting o (the estimated error variance based on a model including all predictors) for o in this expression gives Mallow's Cp. You now have to provide a justification for (1). In what follows, we let X1 be the design matrix for the proposed model, and X2 be the design matrix for the true model. Let B(1) denote the vector of parameters in the proposed model and 3(2) be the vector of parameters for the true model. Write b(1) for the least squares estimator of 8(1) and b2) for the least squares estimator of g(2) (a) Writing for the vector of fitted values for the proposed model and observing that j = X;(X X1)Xy = Hy, show that EVar() = o'tr(H1), where tr(A) denotes the trace of A and H1 = X1(X X1)X denotes the hat matrix corresponding to the proposed model. By using the rules given in lectures about matrix traces, deduce that ΣVar() = p. In this question we consider derivation of the C, statistic for model selection discussed in lectures. Consider fitting the proposed general linear model and write ; for the fitted value at r; and MSE() for its mean squared error. Recall that if the error variance o? is known, then an estimate of E MSE() Jp EVar() E; Bias () is (n - p)( - o²) (1) where ở is the estimate of the residual error variance for the proposed model and p is the number of parameters. Substituting o (the estimated error variance based on a model including all predictors) for o in this expression gives Mallow's Cp. You now have to provide a justification for (1). In what follows, we let X1 be the design matrix for the proposed model, and X2 be the design matrix for the true model. Let B(1) denote the vector of parameters in the proposed model and 3(2) be the vector of parameters for the true model. Write b(1) for the least squares estimator of 8(1) and b2) for the least squares estimator of g(2) (a) Writing for the vector of fitted values for the proposed model and observing that j = X;(X X1)Xy = Hy, show that EVar() = o'tr(H1), where tr(A) denotes the trace of A and H1 = X1(X X1)X denotes the hat matrix corresponding to the proposed model. By using the rules given in lectures about matrix traces, deduce that ΣVar() = p.