UMVU estimator for iid observations from uniform distribution

Question

Let $X_1, \cdots, X_m$ and $Y_1,\cdots,Y_n$ be independent i.i.d observations from the uniform distributions $U(0,\theta_1)$ and $U(0,\theta_2)$ , respectively. Find the UMVU estimator of $\frac{\theta_1}{\theta_2}$ .

I know how to find UMVU estimators for $\theta$ or $\theta_2$ by themselves but I am not sure what would be procedure to find UMVU of $\frac{\theta_1}{\theta_2}$.

Answer 1

Start in the following way

$$\mathbb{E}\left[\frac{X_{(m)}}{Y_{(n)}}\right]=\mathbb{E}\left[X_{(m)}\cdot\frac{1}{Y_{(n)}}\right]=\mathbb{E}[X_{(m)}]\cdot\mathbb{E}\left[\frac{1}{Y_{(n)}}\right]$$

You know the distribution of Max thus you can go on in your calculations...