finding UMVUE for $\theta_x$/$\theta_y$

Question

Let Xi ~ Exp($\theta_x$), Yj ~ Exp($\theta_y$), i = 1; ... ; n1, j = 1;...;n2. Find UMVUE of $\theta_x$/$\theta_y$.

Since $\bar{X}$ and $\bar{Y}$ are compelete sufficient statistic, by using Lehmann-scheffe theorem, UMVUE should be found by solving E($\bar{X}$/$\bar{Y}$), but due to inverse of exponential family, is it possible to find it?

Answer 1

Assuming independence between $X_i $ and $Y_j$ the calculation of the following expectation

$$\mathbb{E}\left[\frac{\overline{X}_n}{\overline{Y}_m} \right]$$

(I used $n,m$ instead of $n_1,n_2$ to simplify the notation...)

is very easy being

$$\mathbb{E}\left[\frac{\overline{X}_n}{\overline{Y}_m} \right]=\frac{m}{n}\mathbb{E}\left[\frac{\Sigma_iX_i}{\Sigma_iY_i} \right]=\frac{m}{n}\mathbb{E}\left[\Sigma_iX_i\cdot\frac{1}{\Sigma_iY_i} \right]=\frac{m}{n}\mathbb{E}\left[\Sigma_iX_i\right]\cdot\mathbb{E}\left[\frac{1}{\Sigma_iY_i} \right]$$

and as known

$$\frac{1}{\Sigma_iY_i}\sim \text{Inverse Gamma}$$