In this post, we explore finite-sample and asymptotic Gaussian efficiency values of the sample median and the Harrell-Davis median.
I have conducted a numerical simulation which enumerates various sample sizes (2..100); generates 1,000,000 samples from the normal distribution; estimates the mean, the sample median, and the Harrell-Davis median for these samples; calculates the finite-sample relative efficiency of the sample median and the Harrell-Davis median to the mean (the Gaussian efficiency). Here are the results:
As we can see, on small samples the Harrell-Davis median estimator is noticeably more efficient than the classic sample median estimator.
Asymptotically, the Gaussian efficiency of both median estimator is $\approx 64\%$.