For the balanced block design, show that the log determinant is

\[
\log \operatorname{det}(\Sigma)=n \log \left(\sigma^{2}ight)+m \log (1+b \theta)
\]

Show that the ML estimate satisfies \(1+b \hat{\theta}=(m-1) F / m\). Hence deduce that the ordinary \(\log\) likelihood ratio statistic for testing \(\theta=0\) is

\[
\log \operatorname{det} \hat{\Sigma}_{0}-\log \operatorname{det} \hat{\Sigma}_{1}=n \log \left(\frac{n-m+(m-1) F}{n}ight)-m \log \left(\frac{(m-1) F}{m}ight)
\]

while the REML statistic is

\[
(n-1) \log \left(1+\frac{(m-1)(F-1)}{n-1}ight)-(m-1) \log F
\]

What does this expression tell you about the null distribution of the REML likelihood-ratio statistic?