# Can the F test be generalized for a null hyp where variances are unequal?

Sorry for the question, maybe this appears in a book a little more advanced than the ones I have seen.

Suppose we have two samples from normal distributions with sample variances $$S_{n,1}^2$$ and $$S_{m,2}^2$$, and population variances $$\sigma_{n,1}^2$$ and $$\sigma_{m,2}^2$$

F test for equality of variances says that if $$\sigma_1=\sigma_2$$, then

$$F:=\frac{S_{n,1}^2}{S_{m,2}^2}\sim F(n-1,m-1)$$,

As far as I understand, this comes from the facts that:

1. $$\frac{S_n^2}{\sigma^2}\sim\chi^2_{n-1}$$, and
2. $$X\sim\chi^2_\nu, Y\sim\chi^2_{\xi}\rightarrow\frac{\xi}{\nu}\cdot\frac{X}{Y}\sim F(\nu,\xi)$$

But in that case, one could set $$H_0: \frac{\sigma_2}{\sigma_1}=k$$, and then the statistic:

$$F:=\frac{\frac{S_{n,1}^2}{\sigma_1^2}}{\frac{S_{m,2}^2}{\sigma_2^2}}=\frac{\sigma_2^2}{\sigma_1^2}\frac{S_{n,1}^2}{S_{m,2}^2}=k^2\frac{S_{n,1}^2}{S_{m,2}^2}$$, should also be distributed according to $$F(n-1,m-1)$$.

Am I right?, did I make some wrong assumption I didn't notice?

If I'm right, what is this test called?

• I don't know that it has a specific name beyond being a test of variances, but consider that you're just testing the equality of variance of one variable and a new variable that is $k$ times your other variable, so it's still just an equality of variance test / variance ratio test with an extra step (defining a new variable). I'd name it in much the same way I'd name a two sample t-test that hypothesized $\mu_2 = \mu_1+\delta$; that's still a t-test and still a test of means) Commented May 17, 2022 at 22:54
• Thanks, @Glen_b Commented May 17, 2022 at 22:58