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-tailednessmeant
that the underlying distributionwas capable of producing anomalous observations
in the sense that they were ‘too far’ from themain 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.