Fat-tail distributions

#statistics, Identifying power law data

Under at least one definition, a fat-tail distribution is one whose tail is fatter than that of the exponential distribution.

Examples

• Cauchy
• Pareto
• Zipf
• Weibull with low k parameter

All exist in Stan.

Pareto and Zipf are both simple power laws with a negative exponent, scaled so that their cumulative distributions equal 1. The difference is that Zipf is discrete.

The "80-20 law", according to which 20% of all people receive 80% of all income, and 20% of the most affluent 20% receive 80% of that 80%, and so on, holds precisely when the Pareto index is α = log₄(5) = log(5)/log(4), approximately 1.161.

80-20 also implies 64/4 and approx. 50/1 (51.2/0.8)

Cauchy has no well-defined mean.