hypotheses testing

**rumlikehell** · January 5th 2009, 02:50 PM

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!

**Renkai** · March 24th 2009, 08:19 AM

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??

**matheagle** · March 24th 2009, 01:50 PM

Originally Posted by Renkai

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??

The usual technique is to say that if $S_y^2 / S_x^2$ is large, then $(1+S_y^2 / S_x^2)^m (1+S_x^2 / S_y^2)^n$ is large.
Let $A=S_y^2 / S_x^2$ , so you have $(1+A)^m \biggl(1+{1\over A}\biggr)^n$ , which increases as A increases.

I worked this out on page 2

**matheagle** · March 24th 2009, 03:43 PM

That t-test was for testing the equality of two means.
This post was about testing equality of variances.

**Renkai** · March 24th 2009, 06:43 PM

Thanks for the response. How about if we were testing the mean and variance at the same time

**matheagle** · March 24th 2009, 08:11 PM

Originally Posted by Renkai

Thanks for the response. How about if we were testing the mean and variance at the same time (i.e H0: equal mean & equal variance, H1:not equal mean and not equal variance). I can't seem to reduce it to a nice test statistic..any ideas?

one simple thing to do is a Bonferroni test.
Split the $\alpha$ 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.

**Renkai** · March 25th 2009, 02:11 AM

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.

**Renkai** · March 26th 2009, 03:08 PM

Yes, it should also also be nm/(n+m) as opposed to (1/n+1/m)/

I believe that Welch's test refers to when the variances are not known but not necessarily equal (for both H0 and H1).

**Renkai** · March 26th 2009, 05:59 PM

Originally Posted by chewwy

ah, ok.

why do you think the statistic should be that? surely it'll have to involve S_y^2 too??

Like I said I'm not sure :/ but I've seen it used in a few books for applied use with control populations. But what I'm saying could be wrong so any ideas would be great

**matheagle** · April 4th 2009, 09:19 PM

I worked it out just now. The estimators for either mean are the sample means
and the MLE of our common population variance under $H_0:\sigma^2_x=\sigma^2_y$ is

$\hat\sigma^2= { \sum_{k=1}^m(x_k-\bar x)^2 + \sum_{k=1}^n(y_k-\bar y)^2 \over m+n}$ .

While under the union of $H_0:\sigma^2_x=\sigma^2_y$ and $H_a:\sigma^2_x\ne\sigma^2_y$

we have $\hat\sigma^2_x={ \sum_{k=1}^m(x_k-\bar x)^2\over m}$ and $\hat\sigma^2_y={\sum_{k=1}^n(y_k-\bar y)^2 \over n}$ .

$\lambda ={ (\hat\sigma^2_x)^{m/2}(\hat\sigma^2_y)^{n/2}\over (\hat\sigma^2)^{(m+n)/2}}$ .

We reject $H_0:\sigma^2_x=\sigma^2_y$ when $\lambda$ is small

and this happens when the two sample variances differ since

$\hat\sigma^2={m\hat\sigma^2_x+n\hat\sigma^2_y\over m+n}$

It's the same argument that you use to show that the max of AB given A+B=1 occurs when A=B.

**matheagle** · April 5th 2009, 01:38 PM

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.

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