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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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Efficiency can be looked at in two different ways. Many times efficiency is reserved for estimators that are unbiased to begin with. In that case, if it achieves the Cramer-Rao lower bound, it is considered efficient.

The other interpretation of efficiency is from a relative perspective. That is, the relative efficiency of two estimators can be compared. In that situation, it is not reserved only for unbiased estimators. Both bias and variance are taken into account. The Mean Squared Error (MSE) is one risk function of many used to compare the relative efficiency of two estimators.

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