This is going to be a short blog because, although I am interested in heavy tailed distributions, it is not my area of expertise and I don’t want to pretend to any special competence or insights. However, I thought you might share my interest .. so, read on.

Ok, heavy tails .. simply means that more of the probability mass is concentrated in the tails of the distributions than in the normal/gaussian distribution.

So, its a relative term, and means that we are more likely to be “surprised” than we think we are (based on our adsorbed gaussian expectations).

Sort of surprise about surprises.

Or shocked about shocks .. we can be very unpleasantly surprised (“shocked”) if something happens (the “shock”) if we dramatically underestimated the probability (even just in our heads, in unspoken and unexplicated assumptions about the shape of the likelihood) of some extreme event – and it happens.

If there is a lot of money attached to this, then it can be a very shocking event indeed. (The question of whether we can, in reality, insure against such shocks we will leave for another time).

So, I was idly perusing a document entitled “Stable Distributions: Models for Heavy Tailed Data” by John Nolan .. it is somewhat heavy going and a definite work in progress, you can googlefind it on the net.

Take out of it what you will, but I was struck by a few things ..

• firstly, that there are sometimes good theoretical reasons for expecting non Gaussian Stable model (Stable distributions being a class of probability distributions that allow skew ness and heavy tails), as in the hitting times for a Brownian motion yielding a Levy distribution.
• secondly, just how different even the simplest of heavy tailed distributions and the Gaussian really really are. This REALLY REALLY matters in practice. A Gaussian (normal distribution) has little probability mass above 3 (standard deviations). Remember the 1.96 sigma rule (2.5% beyond those tails), then 2.58 sigma ..we are down to nothing. But in a Cauchy distribution which looks like a normal distribution (ie is symmetric) we have something like 100 times the probability mass in the tails as we do in the normal. In a sample of data from a Cauchy distribution and a sample of data from a Gaussian distribution, there will be MORE THAN ONE HUNDRED TIMES AS MANY VALUES ABOVE 3 (standard deviations) in the former compared to the latter. VERY NASTY SURPRISES are MUCH MORE COMMON THAN YOU THINK.
• The Levy distribution is a skewed to the right (or left) distribution with all the probability mass concentrated above (below) zero, and even heavier tails than the Cauchy.
• Stable distributions are a generalization of the above (Gaussian/Normal, Cauchy, Levy) allowing for varying degrees of skewness and tail heaviness but they are analytically intractable : this is not such a big deal these days, plenty of grunt power available

OK, if I was heavily into decision making in fields that involved mega bucks and I wanted to avoid nasty surprises, I would be watching this like a hawk. And try learning how to do some sensible implementations.

Finance is the obvious candidate, but get this : “Outliers are Prevalent in Social Variables”. Nicholas Taleb argues this in “The Black Swan: Why Don’t We Learn that We Don’t Learn?” (again, you can googlefind this, or better yet/as well buy his book “Fooled By Randomness”).

I am not going to quote him, I would rather you read him direct – his works are very accessible and very stimulating. Just one point : some of the notions underlying Stable distributions go back a long way to Pareto and thence forward to Mandelbrot (he of the fractals and chaos fame).

I guess it depends on your sets of responsibility and interests as to how much this is relevant to you. There are of course many more immediate and accessible problems to be studied, but to me there is a flavour of an “edge” in this.. if I can study it and do something with it, yes I think it will be profitable (and not just in the sense of insurance against large losses).

fwiw

some implementations (in R) at Jim Lindsey’s site http://popgen.unimaas.nl/~jlindsey/rcode.html

Probability functions and generalized regression models for stable distributions http://popgen.unimaas.nl/~jlindsey/rcode/stable.zip