I have this sample exam question and no model answer.

Question.

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 .

Answer.

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:

take logs

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.