Minimal sufficient statistics for uniform distribution on $(-\theta, \theta)$

Solution 1:

so I realised there was a mistake in the assignment.

The statistic $(X_{(1)},X_{(n)})$ is NOT minimal sufficient for $\theta$. The minimal statistic is $\max\{ -X_{(1)}, X_{(n)} \}$ which follows easily from the fact that the density of $X_1,\dots,X_n$ can be expressed as

$$ \frac{1}{(2\theta)^n} \mathbb{1}_{[\max\{ -X_{(1)}, X_{(n)} \} < \theta]} .$$

As for the first question, I found a reference - Theorem 2.29, Mark J. Schervish, Theory of Statistics, 1995. One needs to check if one density is a multiple of the other and the multiplicative constant does not depend on $\theta$.