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Suppose we have two estimators $\hat{\theta}_1$ and $\hat{\theta}_2$ of $\theta$, both with the same bias.

If we have

$$ \begin{align} &\hat{\theta}_1 \xrightarrow{a.s.}\ \theta \\ &\hat{\theta}_2 \nrightarrow_{a.s}\ \theta \text{ but }\hat{\theta}_2 \xrightarrow{p}\ \theta \end{align} $$

Question: Do we necessarily have

$$ Avar(\hat{\theta}_1) \le Avar(\hat{\theta}_2)?$$

Here, $Avar$ denotes the asymptotic variance.

That is, is a strongly consistent always (asymptotically) preferable to one that is known not to be strongly consistent?

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