Measures of Deviation (and Dependence) for Heavy-Tailed Distributions and their Estimation under Interval and Fuzzy Uncertainty

Traditionally, in science and engineering, most statistical techniques are based on the assumption that the random variables are normally distributed. For such distributions, a natural characteristic of the “average” value is the mean, and a natural characteristic of the deviation from the average is the variance. However, in many practical situations, e.g., in economics and finance, we encounter probability distributions for which the variance is infinite; such distributions are called heavy-tailed. For such distributions, we describe which characteristics can be used to describe the average and the deviation from the average, and how to estimate these characteristics under interval and fuzzy uncertainty. We also discuss what are the reasonable analogues of correlation for such heavy-tailed distributions.