Variance of Function of Sample standard deviation

Question

I'm trying to calculate Var(s_x/s_y), where X is distributed N(μ_x, θ_x) and Y is distributed N(μ_y, θ_y).

Note: s_x, s_y are the sample standard deviations of X and Y.

Any help is greatly appreciated!

Thanks

Answer 1

Depending on how accessible this is to you, this might be more of a long comment than an answer.

If the sample size is $n$, famously $s_x/\sigma_x,\,s_y/\sigma_y$ are $\chi_{n-1}$-distributed, where $X$ has standard deviation $\sigma_x$ etc.; I'm not sure from your notation whether you intend $\theta_x=\sigma_x$ or $\theta_x=\sigma_x^2$. I say "famously"; this is covered if you're taught how to derive the Student's $t$-distribution. In that problem, we show a mean-$0$ normally distributed offset from a sample mean, divided by a $\chi_n$-distributed sample standard deviation, is $t$-distributed.

Although your problem is different, they both require the same technique (if I expand your problem to one of finding the distribution of $s_x/s_y$), as each obtains the distribution of the ratio of two independent variables. (Even when variables aren't independent, if their joint distribution is continuous the resulting PDF is this integral.)

In your case, the final result is proportional to the square root of a variable with an $F$-distribution; as explained here, $F$-distributions are by definition the ratios of independent $\chi^2$ variables, each scaled to have mean $1$. The square root of an $F$-distribution will have a folded $t$-distribution. Correction: that last part is only true in a special case.

As for finding the variance you seek, its evaluation will require you to be au fait with the Beta function.