need help to construct UMPT (uniformly most powerful test)
Let be a random sample from a Poisson distribution with mean unknown. Construct the uniformly most powerful test for testing hypotheses
Suppose n=10, sample mean is 2.42, and =1.5. Will you reject Ho at a significance level of 5%?
I started doing by the book and got stuck. I know it is a trivial problem, statisticians do this all the time, I need to do it once only, to understand )))
Start by writing down the likelihood function
By Neyman-Pearson lemma,
(my book uses the likelihood over the total sample space in the numerator)
So I find the ratio (I replaced with
Now then, I will reject Ho if this ratio is large. This ratio will be larger the larger is. But how large it should be (for say 95% confidence level?)
Here is where I am stuck. I don't know distribution of this ratio. How do I decide on the critical region? (which will be a most powerful region, I reckon). Do I need to know a distribution?... Do I use the assymptotic distribution of the 2 log of the likelihood ratio? do I use Poisson table? When it comes to a most powerful test, my head is a mess.
thank you !