Best way to measure the tail weight of a distribution
I am trying to solve this problem and after blindly testing a dozen different distributions. I realised that I am more interested in the extreme values rather than mod/median/mean, therefore I wanted to check its tail weight and try to find a distribution fitting large values of $x$ rather than all values of $x$.
I checked the literature and it feels like although kurtosis is used a lot for tail weight, there is no consensus.
A few works (or tutorials), describe tail weight as skewness. One would expect the more skewed a distribution is, the more likely it is to find a value far from $\mu+\sigma$ (or $\mu+2\sigma$), which makes sense.
I also read that it might be measured as kurtosis. If it is more than 3, the chance of finding a value far from $\mu+\sigma$ (or $\mu+2\sigma$) is lower. Yet here, more descriptive than any paper or tutorial I read, it is stated that kurtosis is more of a measure of shoulders than tails.
I guess I am quite confused at this point, so what I have written so far might not make sense.
So, is there an accepted (universal) measure of tail weight that somehow alluded my google search? If not, what is the best way to measure the tail weight? Is there a list of tail weights for distributions depending on their parameters?
Thanks in advance!
Solution 1:
I don't believe there is a universally accepted measure of "tail weights". In particular higher moments starting with skew and kurtosis may not exist (or be unbounded/infinite if one prefers to think in those terms).
As a practical matter one is sometimes asked to recommend a statistical test that distinguishes a sampled population from proposed "thin-tailed" exponentially distributed models (or normal distributions in the two-tailed case), i.e. to test for the likelihood of a fat-tailed distribution. Such tests are notoriously unsatisfactory without an abundance of sampling data. An introductory discussion is given by John D. Cook's blog post. It is an active topic in the research literature.