In the previous post, we obtained the finite-sample Gaussian efficiency values of the sample median and the Harrell-Davis median. In this post, we extended these results and get the finite-sample Gaussian efficiency values of the trimmed Harrell-Davis median estimator based on the highest density interval of the width \(1/\sqrt{n}\).

Similarly to the previous experiment,
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 (`SM`

),
the Harrell-Davis median for these samples (`HD`

),
and the trimmed Harrell-Davis median based on the highest density interval of size \(1/\sqrt{n}\) (`THD-SQRT`

);
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, `THD-SQRT`

is less efficient than `HD`

(which is the price of robustness),
but it is still more efficient than `SM`

.