= Optimal asymptotic combination of a normal sample mean and standard deviation
{title2=$\widetilde\mu=(\overline X+2S)/3$}
In the <normal mean proportional to standard deviation model>, both $\overline X$ and $S$ estimate the positive <statistical parameter> $\mu$. Their independent limiting <normal distributions> have rescaled <variances> $\mu^2$ and $\mu^2/2$, respectively, by <asymptotic normality of a normal sample standard deviation>. Thus for fixed $w\in\mathbb R$,
$$
\sqrt n\{w\overline X+(1-w)S-\mu\}\xrightarrow dN\!\left(0,\mu^2\left[w^2+\frac{(1-w)^2}{2}\right]\right).
$$
The bracket is $3(w-1/3)^2/2+1/3$, minimized uniquely at $w=1/3$. Hence $(\overline X+2S)/3$ has leading sampling <variance> $\mu^2/(3n)$, the inverse expected <Fisher information>. The result is first-order optimality within these fixed-weight combinations and does not assert that $S$ is an exactly <unbiased estimator>.
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