Minimal sufficient statistic for normal distribution with known variance Let $X_1, ..., X_n$ be a random sample from the $N(\theta,1)$ distribution. Find a minimal sufficient statistic for $\theta$.
Now, I can find a sufficient statistic using the factorisation theorem ($\sum X_i$), but I don't think that this statistic is in fact minimal sufficient.
The question seems to imply that there exists a minimal sufficient statistic, but I'm not even sure that there is one.
MY QUESTION: How would I go about proving that there is no minimal sufficient statistic, or if there is one, what is it!?
Any hints greatly appreciated!

By the factorization criterion
$$
\mathcal{L}(\theta)=\frac{1}{(2\pi)^{n/2}}\exp\{-\sum_{i=1}^nX_i^2/2 +\bar{X}_n \theta -n\theta^2/2\} 
$$
$$
  \qquad = \exp\{\bar{X}_n\theta-n\theta^2/2\}\times(2\pi)^{-n/2}\exp\{-\sum X_i^2/2\}.
$$
So $\bar{X}_n$ is sufficient statistic.
