/m11 answer: Make your answer as near to closed-form as possible. That is, simplify the 2 log lambda(x) expression as much as possible. Make your answer as analogous as you can to the formula for the F-test statistic, to make comparisons easier.

For problem 3a) are you looking for a close formed solution for a threshold to reject/accept the hypothesis or simply describe a procedure for a ratio test for a particular X and y and the steps that we would have to undertake?

/m9 answer: You can use either the F test or the LRT test--just be clear which!

In Problem 4bc), should we use the F-statistic or LRT statistic to compare the regressions?

Thanks,
I oversaw that... I understood H1 to have the only constraint that P(z0)> 1/2, but even then I was wrong(only MLE(z0) would have been 1, 1/2 for the other values).

/m5 answer: Yes, we observe one and only one of the N+1 possible values.

"If this is the case the MLE under H1 should just be one for any of the possible values" No; look again at the constraints that are part of H1.

Hi,

In problem 1:

Do you mean by a single observation that we observe one and only one of the N+1 possible values?
If this is the case the MLE under H1 should just be one for any of the possible values, so the likelihood ratio would actually be 2 for x_0, and 2N for x_n , n > 0.
Am I understanding the problem right?

Thanks,

Samory

/m4 answer: You have the essence of the answer. Remember that lambda(x) involves maximizing under each hypothesis, so here, lambda(x) = 2 if x = z_i for i > 0.

Let k be the threshold. The size of the test is P(reject H0|H0) = P(lambda(x) > k|H0) = 0.5 if k = 2.

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.