Problem 2.8. Show that \(H=X\left (X^tX\right )^{-1}X^t\) is symmetric and idempotent, and that \(tr(H)=k\) when \(X\) is \(n\times k\) of full column rank. Where to start: the cyclic property of the trace.

Problem 2.9. Use Theorem 2.5.1 to show that \(E\left (SSE\right )=\left (n-k\right )\sigma ^2\) in the general linear model, whether or not the fitted model is the true one. Where to start: \(SSE=\underline {Y}^t(I-H)\underline {Y}\) and \((I-H)X=0\).

Problem 2.10. Verify by Theorem 2.5.4 that \(SSTr\) and \(SSE\) of Example 2.5.7 are independent, by showing that the product of their matrices is the zero matrix.

Problem 2.11. Let \(A\) be symmetric and idempotent of rank \(r\). Prove that \(I-A\) is symmetric and idempotent of rank \(n-r\), and that \(A(I-A)=0\). Explain what each of these three facts contributes to the analysis of variance table. Where to start: Theorem 2.3.6(iv) has the first part.

Problem 2.12. A student proposes to test \(H_0\) by comparing \(SSTr/SST\) with a chi-square table. Explain, using Theorem 2.5.3, why this is wrong on two separate counts.

Problem 2.13. Suppose \(\underline {Y}\sim N_n\left (\underline {\mu },\sigma ^2I\right )\) and \(A\) is symmetric with eigenvalues \(2,1,1,0,\dots ,0\). What is the distribution of \(\underline {Y}^tA\underline {Y}/\sigma ^2\)? Is it chi-square? Justify your answer from the proof of Theorem 2.5.3.