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Is unbiasedness a necessary condition for an estimator to be efficient?

For example, if $\hat {\theta}= \frac{\sum_i^n X_i}{3}$, I assume $\hat {\theta}$ can't be efficient in a Cramer-Rao lower bound context because $E[\hat {\theta}]= \frac {\theta}{3}$.

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Clearly not.

A possible way to compare two estimators is to use Mean Squared Error : $\begin{align*} MSE = Bias^2 + Variance \end{align*}$.

There are some biased estimators with very good variances, this being better choices than some other unbiased estimators with awfullly high variances.

See this blog post for an illustration in Python.

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