(E.g., the trimean will have a better breakdown point but be less efficient than the mean; a worse breakdown point but more efficient than the median.)
What does "efficient" mean, in this context? Time to calculate would be my first guess, but the median should be faster to calculate than the trimean.
[EDITED to add:] Sorry, that's a bit rude; I should also give a brief explanation here.
Any estimator will be noisy. All else being equal, you would prefer one with less noise. There is a thing called the Cramér-Rao inequality that gives a lower bound on how noisy an estimator can be, as measured by its variance. (But see the note below.)
The efficiency of an estimator is the ratio between its variance and the bound given by Cramér-Rao. An estimator whose efficiency is 1 has as little variance as any estimator can have. (Such estimators need not exi...
In a previous post, I looked at some of the properties of using the median rather than the mean.
Inspired by Househalter's comment, it seems we might be able to take a compromise between median and mean. It seems to me that simply taking the mean of the lower quartile, median, and upper quartile would also have the nice features I described, and would likely be closer to the mean.
Furthermore, there's no reason to stop there. We can take the mean of the n-1 n-quantiles.
Two questions:
Note the unlike the median approach, for large enough n, this maximiser will pay Pascal's mugger.