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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)?

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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.

Presumably your teacher proved this under some additional sufficient conditions, in which case it would be a good idea to check the required conditions.

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