Using the notation of Section 9.2, assume that the means \(\mu_{j}\) satisfy a
linear function of \(j\), nanely \(\mu_{j}=c+d[j-(b+1) / 2] .\) Let independent
random samples of size \(a\) be taken from the \(b\) normal distributions having
means \(\mu_{1}, \mu_{2}, \ldots, \mu_{b}\), respectively, and common unknown
variance \(\sigma^{2}\).
(a) Show that the maximum likelihood estimators of \(c\) and \(d\) are,
respectively, \(\hat{c}=\bar{X}_{. .}\) and
$$
\hat{d}=\frac{\sum_{j=1}^{b}[j-(b-1) / 2]\left(\bar{X}_{. j}-\bar{X}_{.
.}\right)}{\sum_{j=1}^{b}[j-(b+1) / 2]^{2}}
$$
(b) Show that
$$
\begin{aligned}
\sum_{i=1}^{a} \sum_{j=1}^{b}\left(X_{i j}-\bar{X}_{.
.}\right)^{2}=\sum_{i=1}^{a} \sum_{j=1}^{b}\left[X_{i j}-\bar{X}_{.
.}-\hat{d}\left(j-\frac{b+1}{2}\right)\right]^{2} \\
&+\hat{d}^{2} \sum_{j=1}^{b} a\left(j-\frac{b+1}{2}\right)^{2}
\end{aligned}
$$
(c) Argue that the two terms in the right-hand member of Part (b), once
divided by \(\sigma^{2}\), are independent random variables with \(\chi^{2}\)
distributions provided that \(d=0\)
(d) What \(F\) -statistic would be used to test the equality of the means, that
is, \(H_{0}: d=0 ?\)
