Need for most accurate discrete approximations explains effectiveness of statistical methods based on heavy-tailed distributions

Abstract

© Springer International Publishing AG 2016. In many practical situations, it is effective to use statistical methods based on Gaussian distributions, and, more generally, distribution for which tails are light – in the sense that as the value increases, the corresponding probability density tends to 0 very fast. There are many theoretical explanations for this effectiveness. On the other hand, in many other cases, it is effective to use statistical methods based on heavy-tailed distributions, in which the probability density is asymptotically described, e.g., by a power law. In contrast to the light-tailed distributions, there is no convincing theoretical explanation for the effectiveness of the heavy-tail-based statistical methods. In this paper, we provide such a theoretical explanation. This explanation is based on the fact that in many applications, we approximate a continuous distribution by a discrete one. From this viewpoint, it is desirable, among all possible distributions which are consistent with our knowledge, to select a distribution for which such an approximation is the most accurate. It turns out that under reasonable conditions, this requirement (of allowing the most accurate discrete approximation) indeed leads to the statistical methods based on the power-law heavy-tailed distributions.