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.
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Notes
- 1.
Note that treating Y as belonging to the Fréchet class is a mistake. If a random variable has a finite upper bound, it cannot belong to the Fréchet class, but rather to the Weibull class [2].
- 2.
Note that we require that the upper bound H be both finite and known –as with the world’s population or the capital of a bank. In case H is not known, we fail to see the difference with an unbounded random variable.
- 3.
Note that the use of logarithmic transformation is quite natural in the context of utility.
- 4.
There are alternative methods to face finite (or concave) upper bounds, i.e., the use of tempered power laws (with exponential dampening) [7] or stretched exponentials [8]; while being of the same nature as our exercise, these methods do not allow for immediate applications of extreme value theory or similar methods for parametrization.
- 5.
We call it “shadow”, as it is not immediately visible from the data.
- 6.
Remember that for a GPD random variable Z, \(E\left[ Z^ p\right] <\infty \) iff \(\xi < 1/p\).
- 7.
Because of the similarities between \(1-F(y)\) and \(1-G(z)\), at least up until M, the GPD approximation will give two statistically undistinguishable estimates of \(\xi \) for both tails [3].
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Taleb, N.N., Cirillo, P. (2018). On the Shadow Moments of Apparently Infinite-Mean Phenomena. In: Morales, A., Gershenson, C., Braha, D., Minai, A., Bar-Yam, Y. (eds) Unifying Themes in Complex Systems IX. ICCS 2018. Springer Proceedings in Complexity. Springer, Cham. https://doi.org/10.1007/978-3-319-96661-8_16
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