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.

中文翻译：

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欠采样分布的时刻：在流行病规模上的应用

流行病爆发的死亡总数是一个巨大但信息量极大的数字。了解该数量的统计数据可以计算平均死亡人数，条件是疫情已经超过给定的死亡人数，这与风险评估非常相关。然而，由于不存在平均值，死亡总数似乎具有指数（对于互补累积分布函数）小于 1 的幂律尾部分布这一事实，造成了重要的理论困难。对于这样的分布。Cirillo 和 Taleb（2020）提出从所谓的双变量（显示幂律尾部）到死亡总数的转换，该变量受到世界总人口的限制。在这里，我们（i）表明这种转变是临时的且非物理的；(ii) 提出替代转换和分配（也是临时的）；(iii) 认为解决这个问题的正确框架是通过有限尺寸缩放的统计物理学；(iv) 证明真正的问题不是幂律尾部分布的平均值不存在，而是不同理论分布的尾部（这是一个模型与另一个模型的区别）远不存在的事实利用现有的经验数据进行了很好的抽样。我们的结果对于许多其他在其大小上表现出（明显）幂律尾部的危险也有效。