On the shadow moments of apparently infinite-mean phenomena

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 log-transformation that 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 log- 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. War casualties, operational risk, environment blight, complex networks and many other econophysics phenomena are possible fields of application.