9.4 Effect of Outliers, 9.5 Robustness of the F-Test to Nonnormality: 9.5.1 Effect of the Regressor Variables, 9.5.2 Quadratically Balanced F-Tests

Fill in the details of the proof of Theorem 9.2.

Show that the theory of Section 9.5.2 can also be applied to the case $H: \mathbf{A} \mathbf{\beta } = \mathbf{c}$, where $\mathrm{c} \neq \mathbf{0}$.

Consider the full-rank regression model

where the $\varepsilon_{i}$ are independently and identically distributed as $N\left(0, \sigma^{2}\right)$, and suppose that we wish to test $H: \beta_{1} = \beta_{2} = \cdots = \beta_{p-1} = 0$.

Assuming that $H$ is true, find an approximate expression for $\mathbb {E}\left[Z\right]$, where $Z = \frac{1}{2} \log F$, in terms of the diagonal elements of $\mathbf{X}\left(\mathbf{X}^{T} \mathbf{X}\right)^{-1} \mathbf{X}^{T}$.