# If best unbiased estimator exists then it's maximum likelihood estimator?

Our teacher proved in class that if the best unbiased estimator exists, then it is an MLE using a theorem that if $$\hat{\theta}-\theta$$ is proportional to the score of $$\theta$$ with probability $$1$$, then $$\hat{\theta}$$ is the best unbiased estimator since it attains the CR bound.

In general, I know that MLE attains the CR bound asymptotically. So I'm in doubt whether the statement in the title holds for a finite sample. Could anyone provide some insight about relationship between the best unbiased estimator and MLE in finite sample case (and proof)?

This cannot be true as a general theorem, since there are situations where the MLE is not unbiased, but other unbiased estimators exist. For example, this occurs when estimating the parameter $\theta$ in the model $X_1, ..., X_n \sim \text{IID U}[0, \theta]$. In these situations the best unbiased estimator must be in the class of unbiased estimators (which is non-empty) and so it cannot be the MLE.