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For an i.i.d sequence of Random Variables $X_1, \dots, X_n$, each with mean $\mu = \mathbb E[X]$, the goal is to estimate some continuous function $f$ evaluated at the mean, $f[\mathbb E[X]]$.

If there is some unbiased estimator, $L$, so that $\mathbb E L = f[\mathbb E[X]]$, then can this unbiased estimator always be turned into a consistent series of estimators, where $L' = \frac{1}{n}\sum_1^n L_i$ for $n$ independent samplings of $L$.

It looks like $L'$ should converge in probability to $f[\mathbb E[X]]$ by the law of large numbers, and therefore be consistent.

Is my reasoning sound here? I tried this out with a few examples and it seems to work but I have little experience so I can't tell, even though it is a basic question.

Thanks for the help!

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  • $\begingroup$ The subject line is ambiguous. Does it mean "Is there any unbiased estimator that can be changed...?" or "Is it the case that any unbiased estimator, no matter which one, can be changed...?"? In the former cases, "any" means "some", and in the latter case, it means "every". Just changing it to "every" would disambiguate it, if that's what was meant. $\endgroup$ May 9, 2019 at 18:06

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Your reasoning is sound. Forget about the $X$'s, and write $\theta:=f(\mu)$. Your question boils down to:

If $L_1,L_2,\ldots,L_n$ are iid with mean $\theta$, does $\frac1n\sum_1^n L_i$ converge in probability to $\theta$?

The answer is yes. Indeed, by the strong law of large numbers, the convergence is almost sure.

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