I was also wandering what the answer to this question was. I can reduce the Likelihood ratio to:
(1+Sy^2 / Sx^2)^m x (1+Sx^2 / Sy^2)^n >C, but how can I arrange this to obtain the F-test??
hello,
can anyone get me going on this problem:
"Find the likelihood ratio test for testing equality of variances of two independent, normally distributed variables X and Y, their random samples being respectively X_1, ..., X_m and Y_1, ..., Y_n. Both expectation and variance of each variable are unknown."
my question is, how do I apply LR-method to this problem? i've tried, but failed to get the test statistic that i should be getting: Sx^2 / Sy^2
(Sx^2 and Sy^2 being the sample variances).
thanks!
one simple thing to do is a Bonferroni test.
Split the in half and test them separately.
That's not such a bad idea when you have only 2 or 3 parameters you're testing. It's a bad idea if you have a lot of tests you want to perform.
In that case you wnat to do a Tukey/Scheffe test on your parameters.
Ok, but if I'm asked to come up with one test statistic using the LR method, I reduce the equation partially however I don't know how to reduce it further to a obtain a proper test statistic from which I can use tables to find the value of critical region.
I worked it out just now. The estimators for either mean are the sample means
and the MLE of our common population variance under is
.
While under the union of and
we have and .
.
We reject when is small
and this happens when the two sample variances differ since
It's the same argument that you use to show that the max of AB given A+B=1 occurs when A=B.
I do NOT see where anyone is assuming that the means are equal.
MAYBE if you actually stated BOTH hypotheses someone could answer your question.
BUT from....
"Find the likelihood ratio test for testing equality of variances of two independent, normally distributed variables X and Y, their random samples being respectively X_1, ..., X_m and Y_1, ..., Y_n. Both expectation and variance of each variable are unknown."
I found the correct MLEs and the likelihood ratio test.