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

/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.

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

/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.

zeldar

AT6GqQ comment2 ,