I'm looking for an example for a statistical model for which Maximum Likelihood Estimator $\hat{\theta}(\vec{Y})$ for $\theta$ exists, and satisfies the following conditions:
- It is unique. Equivalently, the maximal value of the likelihood is obtained only for one point $\theta$, for every sample vector $\vec{y}$. Note that this condition implies that $S(\vec{Y})$ is a function of any sufficient statistic (this is a simple result).
- $\hat{\theta}(\vec{Y})$ itself is not sufficient.
I've been looking for examples here, but I only found examples for MLEs that aren't functions of sufficient statistics (as the first condition doesn't hold). I also tried playing with examples but it didn't quite work. I'd appreciate your help finding one.