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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.

Write $S^2=Q/(n-1)$ and $S=\sqrt{S^2}$, with $n\ge 2$. Two exact facts about a normal sample areand independence of the normal sample mean and sample variance holds. A chi-squared distribution with $n-1$ statistical degrees of freedom is a sum of $n-1$ independent ${\chi }_1^2$ variables, each with mean one and variance two. The central limit theorem therefore givesFor $g(x)=\sqrt{x}$, the derivative at ${\mu }^2$ is $1/(2\mu )$ because $\mu >0$. The delta method gives the asymptotic normality of a normal sample standard deviation:The exact independence of $\overline{X}$ and $S$ makes their limiting normal distributions independent. Consequently,ThusThis has the same leading asymptotic variance as $\hat{\mu }$. The weights $1/3$ and $2/3$ are those in the optimal asymptotic combination of a normal sample mean and standard deviation. This does not require $S$ to be an exactly unbiased estimator of $\mu$.