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Consider observing $X \mid \sigma \sim N(\mu, \sigma^2)$ and $\sigma \sim F$ for some known distribution $F$ supported on the positive reals. We observe a single draw $(X, \sigma)$. An estimator $T(X, \sigma)$ is unbiased for $\mu$ if $E[T(X,\sigma)] = \mu$.

$T(X, \sigma) = X$ is one such choice. Another choice is $T(X,\sigma) = w(\sigma) X$ for any $w(\sigma)$ where $E[w(\sigma)] = 1$. Similarly, one could construct some unbiased estimator that uses $X^2, \ldots$. There may be other more exotic choices as well.

Among these estimators, what is the minimum variance estimator?

The Cramer-Rao bound for $\mu$ with likelihood $f(x,\sigma) = f(\sigma) \frac{1}{\sigma} \varphi((x-\mu)/ \sigma)$ is $1/E[1/\sigma^2]$. This is a lower bound on the minimum variance. Is there an estimator that achieves it?

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