I am attempting to show that the smallest order statistic T is minimally sufficient for the mean of a distribution when the variance is known. In particular, iid random variables $X_1,\ldots,X_n$ have a pdf that is given by $f(x;θ)=(1/\sigma)e^{(-(x-\theta)/\sigma)}$.
I know that the two necessary conditions for minimal sufficiency are: i.) sufficiency; and ii.) that the statistic cannot be further reduced to another sufficient statistic. After having shown that T is sufficient, I failed to prove the second condition.
Since the statistic is clearly minimally sufficient (I wouldn't have been told to prove it, otherwise), it must be true that $T$ cannot be reduced to another arbitrarily defined, sufficient statistic $T'$. In other words, $T$ must be a defined function of $T'$. However, if $T' = \sum_i X_i$, for example, i.e. $T'$ is sufficient, then $T$ would only be a defined function of $T'$ if $n=1$. Hence, the second condition of minimal sufficiency would not hold. I must be missing something...