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:
- As n increases, does this quantity tend to the mean if it exists? (I suspect yes).
- For some distributions (eg Cauchy distribution) this quantity will tend to a limit as n increases, even if there is no mean. Is this an effective way of extending means to distributions that don't possess them?
Note the unlike the median approach, for large enough n, this maximiser will pay Pascal's mugger.
The wheels in this case come from robust statistics.
One example of a good robust estimator for the center is the [truncated mean]https://en.wikipedia.org/wiki/Truncated_mean). To put it simply: throw away the lowest x% and the highest x% of the samples, and take the mean of the rest. If x=0 you get the regular mean, if x=50% then you get the median.
Descriptive statistics, also, probably.
"Please do my literature search for me" is not a reasonable request, though.