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Stochastic Tail Exponent For Asymmetric Power Laws

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  • Nassim Nicholas Taleb

Abstract

We examine random variables in the power law/regularly varying class with stochastic tail exponent, the exponent $\alpha$ having its own distribution. We show the effect of stochasticity of $\alpha$ 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 $\alpha$; 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 $\alpha \in [b,\infty), b>1$. 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 $\alpha$ 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.

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  • Nassim Nicholas Taleb, 2016. "Stochastic Tail Exponent For Asymmetric Power Laws," Papers 1609.02369, arXiv.org, revised Apr 2017.
  • Handle: RePEc:arx:papers:1609.02369
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    1. Pasquale Cirillo & Nassim Nicholas Taleb, 2016. "Expected shortfall estimation for apparently infinite-mean models of operational risk," Quantitative Finance, Taylor & Francis Journals, vol. 16(10), pages 1485-1494, October.
    2. Cirillo, Pasquale & Taleb, Nassim Nicholas, 2016. "On the statistical properties and tail risk of violent conflicts," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 452(C), pages 29-45.
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