= Solution
{c}
Put $A=-\sum_i\log x_i>0$, $B=-\sum_j\log y_j>0$, and $N=n+m$. The two independent <statistical samples> give the joint <likelihood function>
$$
L(\theta,\tau)=\theta^n\tau^m\exp\{-(\theta-1)A-(\tau-1)B\}.
$$
The unrestricted <maximum-likelihood estimators> are $\widehat\theta=n/A$ and $\widehat\tau=m/B$. Under the <null hypothesis>, the common <statistical parameter> $\eta$ has <log-likelihood function> $N\log\eta-(\eta-1)(A+B)$, maximized at $\widehat\eta=N/(A+B)$. With the <likelihood ratio> defined as the null maximum divided by the unrestricted maximum,
$$
\lambda=\frac{(N/(A+B))^Ne^{-N+A+B}}
{(n/A)^n(m/B)^me^{-N+A+B}}
=\frac{N^N}{n^nm^m}\frac{A^nB^m}{(A+B)^N}.
$$
The statistic in the paper is $T=A/(A+B)$, since both original sums of <natural logarithms> are negative. Hence
$$
\boxed{\lambda=c(n,m)T^n(1-T)^m,\qquad
c(n,m)=\frac{(n+m)^{n+m}}{n^nm^m}}.
$$
It has maximum one at $T=n/N$, corresponding to equal fitted rates. This is the <likelihood-ratio test for equality of beta shape parameters>.
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