# The UMVUE of ratio of parameters for two uniform distributions,

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.

• The title doesn't correctly represent what the question asks. – Glen_b Dec 17 '18 at 6:21

The sufficient statistic for a Uniform sample over $$(0,\theta)$$ is $$X_{(N)}$$ Hence a sufficient statistic for this problem is $$(X_{m:m},Y_{n:n})$$
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 $$\frac{m}{m+1}\frac{n-1}{n}X_{m:m}Y_{n:n}^{-1}$$