I have this sample exam question and no model answer.
Let be a random sample from a distribution from a density function
where is an unknown parameter.
i. Show that the uniformly most powerful test (UMPT) for testing hypotheses
will reject Ho if and only if T>K for some constant K>0, where .
ii. Find the distribution of the test statistic T under Ho. [You may use the fact that the distribution with k degrees of freedom has moment generating function .
i. The likelihood function is
The likelihood ratio from Neyman-Person lemma is
If this ratio is higher than some number k (determined by required confidence level), then we will reject Ho:
and the required T>K is the above inequality.
does it make sense?
ii. Distribution of T - let me think it over a bit longer. I am thinking to find MGF of y^2 (with theta=1 the density is simplifed) and to show that it is a form of MGF of the chi squared.