My $2E-2:
(1) There is something to be said for the mean-median definition of
skewness, as it uses only second powers and is hence more robust than
the third-moment definition. IIRC, the sampling distribution of sample
skewness (3rd power def'n) is rather wide.
I'm not sure what the _best_ measure is. One interesting possibility
would be, given one measure, to then determine the power transformation
that takes it to zero, and use *that* as the measurement. This has the
advantage that it frequently tells you something about the process
generating the data, and I would conjecture that it would be quite
robust against some forms of outlier - more so than the original
measure.
(2) Distinguishing between peakedness and heavy-tailedness is to some
extent a false dichotomy (compare: "I'm not overweight, I'm
undertall!"), as tails are heavy or light by comparison with the middle
of the distribution. For tails of a certain length to result in
leptokurtosis [which sounds like an occupational lung disease anyway]
there must be a tight peak in the middle to hold the variance down.
Of course, one can define heavy-tailedness in ways that explicitly
concentrate on the asymptotic behaviour, or on the sixth moment, or
(say) the top and bottom 5% of the distribution or sample. But the most
intuitive definition of heavy-tailedness is in contrast to the tightness
of a central peak.
(3) The high-moment measures concentrate on extreme tails if these
exist, even though they may contain vanishingly few data. This is not
always what one wants - consider the St. Petersburg Paradox.
This odd behaviour is not restricted to moment measures, though;
consider Tukey's "7-11" test, which is entirely based upon the extreme
values!
-Robert Dawson
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