The Prevalence and Impact of Heavy Tails on Hydrologic Extremes and Other Statistics

The prevalence of heavy tailed (HT) populations in hydrology is becoming increasingly commonplace due in part to the increasing need and use of high frequency and high-resolution data.   In addition to the impact of HT on extremes, HT populations can have a profound impact on a wide range of other hydrologic statistics and methods associated with planning,  management and design for  extremes.   We review the known impacts of HT populations on the instability and bias in a wide range of commonly used hydrologic statistics. Experiments reveal that HT distributions result in the degradation of many commonly used statistical methods including the bootstrap, probability plots, the central limit theorem, and the law of large numbers.     We document the gross instability of perhaps the best-behaved statistic of all, the sample mean (SM) when computed from HT distributions.  The SM is ubiquitous because it is a component of and related to a myriad of statistical methods, thus its unstable behavior provides a window into future challenges faced by the hydrologic community.  We outline many challenges associated with HT data, for example, upper product moments are often infinite for HT populations, yet upper L-moment always exist, so that the theory of L-moments is uniquely suited to HT distributions and data.  We introduce a magnification factor for evaluating the impact of HT distributions on the behavior of extreme quantiles