Abstract

https://ssrn.com/abstract=2834165
 


 



Stochastic Tail Exponent for Asymmetric Power Laws


Nassim Nicholas Taleb


NYU-Tandon School of Engineering

September 2, 2016


Abstract:     
We examine random variables in the power law/slowly varying class with stochastic tail exponent, the exponent α having its own distribution. We show the effect of stochasticity of α on the expectation and higher moments of the random variable. For instance, the moments of a right-tailed or right-asymmetric variable, when finite, increase with the variance of α; those of a left-asymmetric one decreases. The same applies to conditional shortfall (CVar), or mean-excess functions.

We prove the general case and examine the specific situation of lognormally distributed α ∈ [b,∞), b>1.

The stochasticity of the exponent induces a significant bias in the estimation of the mean and higher moments in the presence of data uncertainty. This has consequences on sampling error as uncertainty about α translates into a higher expected mean.

The bias is conserved under summation, even upon large enough a number of summands to warrant convergence to the stable distribution. We establish inequalities related to the asymmetry.

We also consider the situation of capped power laws (i.e. with compact support), and apply it to the study of violence by Cirillo and Taleb (2016). We show that uncertainty concerning the historical data increases the true mean.

Number of Pages in PDF File: 4

Keywords: Risk, Power Laws, Statistics, Tail Risk


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Date posted: September 6, 2016  

Suggested Citation

Taleb, Nassim Nicholas, Stochastic Tail Exponent for Asymmetric Power Laws (September 2, 2016). Available at SSRN: https://ssrn.com/abstract=2834165 or http://dx.doi.org/10.2139/ssrn.2834165

Contact Information

Nassim Nicholas Taleb (Contact Author)
NYU-Tandon School of Engineering ( email )
Bobst Library, E-resource Acquisitions
20 Cooper Square 3rd Floor
New York, NY 10003-711
United States
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