Let $X\_1, ..., X\_n$ be independent and uniformly distributed on $(\theta\_1-\theta\_2,\theta\_1+\theta\_2)$ for $\theta\_1 \in \mathbb{R}$, $\theta\_2>0$.

Find UMVUEs for

a) $\theta\_1$ and $\theta\_2$,

b) $\theta\_1/\theta\_2$.

Naturally, I would like to use the Lehmann-Scheffé theorem that says:

If $V$ is a complete, sufficient statistic for $\theta$ and $\mathbb{E}\_\theta[g(V)]=h(\theta)$ holds. Then $g(V)$ is an UMVUE for $h(\theta)$.

So first, I have to find such a statistic for any of my $\theta$s.

In the lecture we learned that $(X\_{(1)}, X\_{(n)})$ sufficient for $(a,b)$ if the $X\_i$ are uniform on $(a,b)$ (where $X\_{(1)}<...<X\_{(n)}$ is the order statistic for $X\_1, ..., X\_n$).

So it follows that in my setting $(X\_{(1)}, X\_{(n)})$ is sufficient for $(\theta\_1-\theta\_2,\theta\_1+\theta\_2)$? But looking at the proof, $(X\_{(1)}, X\_{(n)})$ is already a sufficient statistic for $(\theta\_1, \theta\_2)$, too, right?

So I have to check for completeness and thus show that for all functions $g$ mapping from the range of $V$ to $\mathbb{R}$ from $\mathbb{E}\_\theta[g(X\_{(1)}, X\_{(n)})]=0$ for all $\theta=(\theta\_1, \theta\_2)$ follows $\mathbb{P}\_\theta(g(X\_{(1)}, X\_{(n)})=1)=1$ $\mathbb{P}\_\theta$-almost-surely.

Here I'm stuck: How can I show that?