The UMVUE of ratio of parameters for two uniform distributions,

Question

Let $X_1,\ldots,X_m$ be i.i.d. having the uniform distribution $U(0, \theta_x)$ and $Y_1,\ldots, Y_n$ be i.i.d. having the uniform distribution $U(0, \theta_y)$. Suppose that $X_i$’s and $Y_j$’s are independent and that $\theta_x > 0$ and $\theta_y > 0$. Find the UMVUE of $\theta_x/\theta_y$ when $n > 1$.

I don't know how to proceed with this exercise and I'd like some help.

I do know that the UMVUE of a unfiorm distirbution is $(n+1)X_{(n)}/n$ but I think that this isn't the way to do this exercise.

Thanks.

Answer 1

First, one only need look at sufficient statistics:

The sufficient statistic for a Uniform sample over $(0,\theta)$ is $X_{(N)}$ Hence a sufficient statistic for this problem is $(X_{m:m},Y_{n:n})$

Second, one need find an unbiased estimator based on a sufficient statistic:

Since$$\mathbb{E}_{\theta_x}[X_{m:m}]=\frac{m}{m+1}\theta_x\ \text{ and }\ \mathbb{E}_{\theta_y}[Y_{n:n}^{-1}]=\frac{n}{n-1}\theta_y^{-1}$$ an unbiased estimator of $\theta_x\theta_y^{-1}$ is $$\frac{m}{m+1}\frac{n-1}{n}X_{m:m}Y_{n:n}^{-1}$$

Last, one can call for a completeness argument.