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I attempted this via a proof of contradiction:

Assume via contradiction that there exists a $\delta(X)$ such that $E[\delta(X)]=\sigma^2$ $\forall \mu \in \Theta$.

I know that for a fixed $\sigma$ that $X$ is a complete sufficient statistic for $\mu$. My goal is to find a function of $X$ whose expected value is $0$ for a fixed $\sigma^2$ and which is not equal to $0$ a.e.

Fix $\sigma_0$ then $X$ is a complete sufficient static of $\mu$ . By assumption there exists a $\delta(X)$ which is an unbiased estimator for $\sigma_0^2$, $\forall \mu$.

By completeness there exists a unique $\delta^{*}(X)$ such that $E[\delta^{*}(X)]=g(\mu)$, $\forall \sigma^2$, where $\delta^{*}(X)$ cannot be a constant. Then consider $\delta^{'}(X)=\delta(X)-\delta^{*}(X)$ whose $E[\delta^{'}(X)=\delta(X)-\delta^{*}(X)]=g(u)-\sigma_0^2$. If we set $g(u)=\sigma_0^2$ then we have a contradiction since $\delta^{'}$ is non zero and is an estimator of 0.

I know my argument is not correct, but I am just lost as to how to prove this. Any help?

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