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I've been reading about the HL estimator, and a question came to mind. I could fairly easily create a mean-estimator where I trim or clip 29% of the data on either side and have a statistic with a similar breakdown point, but how would this statistic compare to the HL in terms of efficiency, other robustness measurements, or anything else?

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  • $\begingroup$ efficiency at what (ie. are we comparing at the normal? the logistic? the Cauchy? Some contaminated distribution? a skewed distribution?)? which robust measurements do you mean? What are the circumstances in which you want to compare them? $\endgroup$ – Glen_b -Reinstate Monica Sep 17 '16 at 2:02
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The issue with comparing two estimators is we need to specify the circumstances we're comparing and the measures by which we want to measure. To answer a question like this in all circumstances would be very broad. We can do some exploration but we could fill volumes with it.

I made some arbitratry choices and performed an initial comparison of the center of symmetry for a set of symmetric continuous distributions at n=28 of Hodges-Lehmann and trimmed mean estimators (that can each tolerate 8 arbitrary values but not 9, so the trimmed mean will trim 8 observations from each end, and average the remaining 12).

These choices may not reflect what you're interested in but allow us to get the beginning of an idea of how they compare.

The distributions I looked at (I believe in increasing order of heavy-tailedness, though this ordering is not necessarily the most suitable way to order them considering the comparisons being made; ordering on some robust measure of kurtosis might come closer to making sense here):

  • uniform

  • normal

  • 90-10 mixture of $N(\mu,\sigma^2)$ and $N(\mu,(10\sigma)^2)$

  • average of two logistic random variables

  • logistic

  • Laplace (double exponential)

  • t-distribution with 5 df

  • Cauchy

Here are the simulated relative efficiencies I got:

Distribution RE(trim:HL)
Uniform      0.580
Normal       0.860
N.Mix        0.863
Logis2       0.905
Logis        0.940
Laplace      1.124
t5           0.965
Cauchy       1.429

For the lighter tailed distributions the Hodges-Lehmann was more efficient but among heavier-tailed distributions the comparison indicates that the trimmed mean does relatively well, and at the Cauchy does considerably better.

However, relative performance depends on what distributions you choose to consider and may vary somewhat with the particular sample sizes you look at. We also didn't consider performance on skewed distributions (though the two would no longer be consistent estimates of the same population location, making comparison difficult)

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