Discussion
A sample of size \(n\) is obtained from a Poisson distribution with parameter \(m\). The most powerful (MP) test of less than size \(\alpha\) to test \(H_{0} : m \le m_{0}\) against \(H_{1} : m \gt m_{0}\) is given as :
where \(T = \sum_{1}^{n} X_{i}\)
Answer: A
The keyed test is \(\phi(x) = 1\) if \(T \gt t_{0}\), 0 otherwise.
The Poisson family has monotone likelihood ratio in \(T = \sum X_{i}\): for \(m_{1} \gt m_{0}\), \(f_{m_{1}}/f_{m_{0}} = e^{-n(m_{1} - m_{0})}(m_{1}/m_{0})^{T}\) increases with T. So the best test of \(m \le m_{0}\) against \(m \gt m_{0}\) rejects for large T, which rules out the two options rejecting for small T.
T is discrete, so a test of exact size \(\alpha\) generally needs randomisation at \(T = t_{0}\) (option B). The question asks for the most powerful test of size less than \(\alpha\), that is the non-randomised one: reject when \(T \gt t_{0}\), with \(t_{0}\) the smallest value keeping \(P_{m_{0}}(T \gt t_{0}) \le \alpha\). That is option A.
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