Originally Posted by

**theodds** Okay, it may be more instructive if I just do it. Also, ignore my initial post in this thread; for some reason I thought we were making a LRT, but everything after that is fine. Consider testing H: lambda = lambda_0 against H': lambda = lambda_1 for lambda_1 > lambda_0. The MP test of this hypothesis is to reject when

$\displaystyle

R(X) = \left(\frac {\lambda_1}{\lambda_0}\right) ^ {\sum X_i} e^{-n(\lambda_1 - \lambda_0)} > C_\alpha

$

where we choose C to get the desired size (NP Lemma). Now, R(X) is monotonically increasing in $\displaystyle \sum X_i$ due to the fact that $\displaystyle \lambda_1 > \lambda_0$. Equivalently, we reject H when $\displaystyle \sum X_i > k$ where k is chosen to give the desired size. Now, this test does not depend on the choice of lambda_1 because in choosing k we only make use of the distribution of $\displaystyle \sum X_i$ under H. Thus, reject when $\displaystyle \sum X_i > k$ is UMP for the given hypothesis (it is most powerful for all lambda > lambda_0).