Let $X_1, . . . , X_n$ be a random sample from $N(µ_X, σ^2_X)$ and let $Y_1, \ldots, Y_n$ be a random sample from $N(\mu_Y , \sigma^2_Y$), where $\mu_X \in \mathbb R$, $\mu_Y \in \mathbb R$, $\sigma^2_X > 0$ and $\sigma^2_Y > 0$. Assume that $X_i$’s and $Y_j$’s are independent.
(1) Find the UMVUE of $\sigma^2_X/\sigma^2_Y$
(2) Find the UMVUE of $\sigma_X/\sigma_Y$
My attempt:
A complete and sufficient statisitc for $(\sigma^2_X, \sigma^2_Y)$ is $(S^2_X, S^2_Y)$. Also since we know that the sample variance is a biased estimator for the variance, does this mean that the UMVUE is just $S^2_X/S^2_Y$? And similarly for $(2)$ just $S_X/S_Y$? Or have I got this completely wrong?