Describe the generalised likelihood ratio test and the type of statistical question for which it is useful.

Suppose that $X_{1}, \ldots, X_{n}$ are independent and identically distributed random variables with the Gamma $(2, \lambda)$ distribution, having density function $\lambda^{2} x \exp (-\lambda x), x \geqslant 0$. Similarly, $Y_{1}, \ldots, Y_{n}$ are independent and identically distributed with the Gamma $(2, \mu)$ distribution. It is desired to test the hypothesis $H_{0}: \lambda=\mu$ against $H_{1}: \lambda \neq \mu$. Derive the generalised likelihood ratio test and express it in terms of $R=\sum_{i} X_{i} / \sum_{i} Y_{i}$.

Let $F_{\nu_{1}, \nu_{2}}^{(1-\alpha)}$ denote the value that a random variable having the $F_{\nu_{1}, \nu_{2}}$ distribution exceeds with probability $\alpha$. Explain how to decide the outcome of a size $0.05$ test when $n=5$ by knowing only the value of $R$ and the value $F_{\nu_{1}, \nu_{2}}^{(1-\alpha)}$, for some $\nu_{1}, \nu_{2}$ and $\alpha$, which you should specify.

[You may use the fact that the $\chi_{k}^{2}$ distribution is equivalent to the $\operatorname{Gamma}(k / 2,1 / 2)$ distribution.]