Though with the knowledge that an ancillary statistic is a statistic has distribution that is independent of the parameter, I feel like I still don't know very well for verifying a statistic is an ancillary statistic. :(
It's a problem that I got no clue to start....
Let $((X_1,Y_1),(X_2,Y_2),\ldots, (X_n,Y_n))$ be a sample from $$f_{X,Y}(x,y;\theta )=e^{-(x\theta +y/\theta )}$$
Show: $\bar{X}_n\bar{Y}_n$ is an ancillary statistic
Could anyone give me some clue to work with it? Thanks a lot!!
Hint: For every $\theta$, with respect to the distribution $f(\ ;\ ;\theta)$, $(X_k,Y_k)$ is distributed as $(\theta^{-1}U_k,\theta V_k)$ where $(U_k,V_k)$ is i.i.d. standard exponential. Thus, $\bar X_n\bar Y_n$ is distributed as $\bar U_n\bar V_n$.