# Fat Tail distribution

The concept of a “fat tail” distribution can be a bit tricky to define, as different sources may have varying interpretations. In fact, the term “tail” itself can be a bit slippery. But what really adds to the confusion is that the term has been co-opted by non-mathematicians to mean any probability distribution with a thicker-than-average tail. For those of us who aren’t math whizzes, this might seem like a more intuitive way to describe things. But it’s important to remember that it’s not always mathematically precise. So while “fat tail” might be a catchy and descriptive term, it’s worth digging a bit deeper to really grasp what it means in the context of probability distributions.

In this article, we’ll take a look at three common meanings of the term fat tail distribution:

1. A synonym for a heavy tailed distribution.
2. A subset of heavy tailed distribution.
3. A synonym for Leptokutric.

## Fat tail distribution definitions

1. A synonym for heavy tailed distribution.

Imagine a graph where the a two-tailed a fat cat has tails that are fatter than a two tailed skinny cat. That’s what a heavy tailed distribution looks like. Although a heavy tailed distribution tends to have heavier tails than the distribution, it can also be described as a distribution that has heavier tails than an exponential distribution ; in other words, there is a higher probability of very large values as the distribution it approaches zero at a lower rate.

In trading and finance, this means there’s a higher chance of extreme events happening, making it more important to consider these possibilities. One example is the likelihood of values being 4-5 standard deviations away from the mean, which is rare in a normal distribution but has a 4% chance in a “fat” or heavy tail distribution, or once every 25 events . So while it may be rare, it’s more possible than you might think.

2. A subclass of heavy tailed distributions.

Some authors use the term “fat tail” to mean a subclass of heavy tailed distributions that show power law decay behavior and infinite variance. A power law in statistics is like a magical formula that connects two things together, where changing one thing results in a change in the other thing proportional to a power of the change. It’s like watching a superhero movie where powers grow exponentially! For example, if you take a square and increase the length of its side, the area doesn’t just get a little bigger but multiplies by a whopping four. It’s like watching a tiny acorn turn into a mighty oak tree.

As a more formal example of this type of super-power behavior, Taylor  defines a distribution X with a fat right tail as one with a positive exponent α (called the tail index) so that
P(X > x) ∼ x−α
As x → ∞
With Taylor’s definition, every fat-tailed distribution is heavy tailed, but not every heavy tailed distribution has a fat tail. For example, the Weibull distribution is heavy-tailed but not fat-tailed.

3. A synonym for Leptokutric.

A leptokurtic distribution is a fancy way of saying that a certain set of data has a wider or flatter shape with fatter tails than usual. This leads to a greater chance of extreme positive or negative events. In fact, a leptokurtic distribution has fatter than the normal distribution — hence the synonym “fat tail distribution.” Plot of several probability distributions. The (N)ormal distribution in in black. The Laplace (D)ouble Exponential {red), (L)ogistic (green) and hyperbolic (S)ecant (orange) have higher kurtosis and fatter tails. The other three distributions have thinner tails: raised (C)osine (aqua), (W)igner semicircle (blue) and (U)niform (magenta).

This type of distribution falls under one of three categories in kurtosis analysis, the other two being mesokurtic and platykurtic. Mesokurtic has no kurtosis and is associated with the normal distribution, while platykurtic has thinner tails and less kurtosis; If you come across some data that falls under the leptokurtic category, be prepared for some wild outliers.

## References

 Bryson, M. (1974). Heavy Tailed Distributions: Properties and Tests. Technometrics 16(1):61-68 (February 1974).

 Neto, J. (2014). Power Laws and Heavy Tailed Distributions. Retrieved December 10, 2017 from: http://www.di.fc.ul.pt/~jpn/r/powerlaw/powerlaw.html

 Taylor, J. (2016). Heavy-tailed Distributions. Retrieved December 10, 2017 from: https://math.la.asu.edu/~jtaylor/teaching/Spring2016/STP421/lectures/stable.pdf

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