Moments of undersampled distributions: Application to the size of epidemics

The total number of fatalities of an epidemic outbreak is a dramatic but extremely informative quantity. Knowledge of the statistics of this quantity allows the calculation of the mean total number of fatalities conditioned to the fact that the outbreak has surpassed a given number of fatalities, which is very relevant for risk assessment. However, the fact that the total number of fatalities seems to be characterized by a power-law tailed distribution with exponent (for the complementary cumulative distribution function) smaller than one poses an important theoretical difficulty, due to the non-existence of a mean value for such distributions. Cirillo and Taleb (2020) propose a transformation from a so-called dual variable, which displays a power-law tail, to the total number of fatalities, which becomes bounded by the total world population. Here, we (i) show that such a transformation is ad hoc and unphysical; (ii) propose alternative transformations and distributions (also ad hoc); (iii) argue that the right framework for this problem is statistical physics, through finite-size scaling; and (iv) demonstrate that the real problem is not the non-existence of the mean value for power-law tailed distributions but the fact that the tail of the different theoretical distributions (which is what distinguishes one model from the other) is far from being well sampled with the available number of empirical data. Our results are also valid for many other hazards displaying (apparent) power-law tails in their size.