An interesting side item from a X validated question about calculating the Fisher information for the Normal median (as an estimator of the mean). While this information is not available in closed form, it has a “nice” expression
![1+n\mathbb E[Z_{n/2:n}\varphi(Z_{n/2:n})]-n\mathbb E[Z_{n/2:n-1}\varphi(Z_{n/2:n-1})]+](https://s0.wp.com/latex.php?latex=1%2Bn%5Cmathbb+E%5BZ_%7Bn%2F2%3An%7D%5Cvarphi%28Z_%7Bn%2F2%3An%7D%29%5D-n%5Cmathbb+E%5BZ_%7Bn%2F2%3An-1%7D%5Cvarphi%28Z_%7Bn%2F2%3An-1%7D%29%5D%2B&bg=000000&fg=B0B0B0&s=0&c=20201002)
which can easily be approximated by simulation (much faster than by estimating the variance of said median). This shows that the median is about 1.57 less informative than the empirical mean. Bonus points for computing the information brought by the MAD statistic! (The information loss against the MLE is 2.69, since the Monte Carlo ratio of their variances is 0.37.)
