On the Shadow Moments of Apparently Infinite-Mean Phenomena

Book chapter (2018)

Authors

Nassim Nicholas Taleb New York University
P. Cirillo Applied Probability
Research Group
Applied Probability
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Published Date

2018

Language

English

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Research Group

Applied Probability

Abstract

We propose an approach to compute the conditional moments of fat-tailed phenomena that, only looking at data, could be mistakenly considered as having infinite mean. This type of problems manifests itself when a random variable Y has a heavy-tailed distribution with an extremely wide yet bounded support. We introduce the concept of dual distribution, by means of a logarithmic transformation that smoothly removes the upper bound. The tail of the dual distribution can then be studied using extreme value theory, without making excessive parametric assumptions, and the estimates one obtains can be used to study the original distribution and compute its moments by reverting the transformation. The central difference between our approach and a simple truncation is in the smoothness of the transformation between the original and the dual distribution, allowing use of extreme value theory.