Heavy-tailed densities

The concept of heavy- or long-tailed densities (or distributions) has attracted much well-deserved attention in the literature. A quick search in Google using the keywords long-tailed statistics retrieves almost 12 million items. The concept has become a pillar of the theory of extremes, and through its connection with outlier-prone distributions, long-tailed distributions also play a central role in the theory of robustness. The concept of tail heaviness is by now ubiquitous, appearing in a diverse set of disciplines that includes: economics, communications, atmospheric sciences, climate modeling, social sciences, physics, modeling of complex systems, etc. Nevertheless, the precise meaning of ‘long-’ or ‘heavy tails’ remains somewhat elusive. Thus, in a substantial portion of the early literature, long-tailedness meant that the underlying distribution was capable of producing anomalous observations in the sense that they were ‘too far’ from the main body of observations. Implicit in these informal definitions was the notion that any distribution that behaved that way had to do so because its tails were longer than those of the normal distribution. This paper discusses tail orderings and several approaches for the classification of probability distributions according to tail heaviness. It is concluded that an approach based on the limiting behavior of the residual life function, and its corresponding characterizations based on functions of regular variation and asymptotic distribution of extreme spacings, provides the more natural and illuminating concepts of tail behavior. WIREs Comput Stat 2013, 5:30–40. doi: 10.1002/wics.1236