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/m15 answer: For example, you could create a dataset with two predictors, where one has true coefficient 1 and the other has true coefficient b where, b ranges from -1 to +1. When b is zero, that is H0. When b =/= 0, that is H1. Make a plot with b on the horizontal axis, and P(reject H0) on the vertical axis.

Let N be the number of data points. Show curves for several different N, e.g. N = 2, 5, 10, 20, 50. Ideally, each curve will dip down to power = alpha for b = 0, and stay at power = 1.0 for H1. The curves should approach this ideal as N gets larger.

For a given N, the test that is better is the one whose power curve is closer to the ideal.

I'm still confused about "investigate the power function". could you give an example of appropriate H0 and H1. I guess I'm not sure about what would be an appropriate synthetic dataset would be. Thanks

/m13 answer: Yes, the F test assumes an unknown variance. Explain carefully whether or not the LRT test can have the same flexibility.

"Investigate the power function" means compute P(reject|theta) for many values of theta. Some of these values should be inside H0, and some should be outside H0.

For values of theta far from the boundary of H0 (and inside) the power should tend to zero, while for values outside and far from the boundary the power should tend to 1.

Specifically, you should compute the power function for more than just one value of theta inside H0, and one value inside H1.

Hi,
For 3a) should we assume an unknown variance for the sake of comparison with the F test,since it seems the F test assumes an unknown variance?

For problem 3b)
I'm a little confused as to what is meant by "investigate the power function". The power function is defined for all possible parameters, so should we just fix a true parameter under H0, a true parameter under H1, compute the power function for both and draw conclusions?

Thanks,

Samory

/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