For the first problem, I can see that the likelihood ratio test is :

\lambda (x) = 2(1-1/N) for x = Z_0
            = 2Nb_i for x = Z_i, when i = 1..N

where b_i is the undefined probability under H_1.

Since b_i < 1/N, I can see that when x=Z_0, \lambda(x) < 2 and \lambda(x) > 2 otherwise.

I think this is what we are trying to show, but I'm unsure how to show that k=2 is our threshold and how that is related to the size 1/2 or how to get the power function of this test.

Is this a correct way to approach the problem and do you have any suggestions on how get more intuition on finding the power function?

Thanks

For H0, yes, p(z0)=1/2 & p(z1)=p(z2)...=p(zN)=1/2N.

H1 is a composite hypothesis: there are many different parameter values allowed under H1. The only constraint is that p(z0) = 1-1/N.

Edited by author 02-16-2005 06:50 PM
Does the first question mean p(z0)=1/2 & p(z1)=p(z2)...=p(zN)=1/2N for H0 and p(z0)=1-1/N & p(z1)=p(z2)=...=p(zN)=undefined for H1?

Please ask questions here about the fourth assignment, due Tuesday March 1.