 Optimal asymptotic combination of a normal sample mean and standard deviation

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 $w\in \mathbb{R}$,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.