# Comparing the variance of paired observations

I have $N$ paired observations ($X_i$, $Y_i$) drawn from a common unknown distribution, which has finite first and second moments, and is symmetric around the mean.

Let $\sigma_X$ the standard deviation of $X$ (unconditional on $Y$), and $\sigma_Y$ the same for Y. I would like to test the hypothesis

$H_0$: $\sigma_X = \sigma_Y$

$H_1$: $\sigma_X \neq \sigma_Y$

Does anyone know of such a test? I can assume in first analysis that the distribution is normal, although the general case is more interesting. I am looking for a closed-form solution. Bootstrap is always a last resort.

You could use the fact that the distribution of the sample variance is a chi square distribution centered at the true variance. Under your null hypothesis, your test statistic would be the difference of two chi squared random variates centered at the same unknown true variance. I do not know whether the difference of two chi-squared random variates is an identifiable distribution but the above may help you to some extent.