Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 42 / 1 / i / Solution

For the standard normal distribution, $m=0$ and $f(0)=1/\sqrt{2\pi }$, so the sample median has asymptotic variance $\pi /2$ under scaling by $\sqrt{n}$. The central limit theorem gives $\sqrt{n}{\bar{X}}_n\stackrel{d}{\to }N(0,1)$ for the sample mean. Thus the sample median has $\pi /2$ times the asymptotic variance of the sample mean, and its asymptotic relative efficiency against the sample mean is