The question of whether the distributions of weather or climate variables and their impacts have heavy (i.e., Pareto-like) tails is important for several reasons. For one thing, its neglect can result in a substantial underestimation of the likelihood of extreme events (e.g., in terms of "return periods" or "design values" used in engineering design). For another, the question of whether extreme weather or climate events and their impacts are increasing has been addressed through trend analyses using conventional statistics (in effect, assuming medium-tailed distributions). If these distributions were heavy-tailed, then such trend analyses might well be suspect. In particular, the standard errors associated with the estimated slopes of trend lines would be underestimated.

The evidence of heavy tails in weather or climate variables and their impacts is reviewed. In examples, the generalized Pareto distribution (GP) is fitted to the extreme upper tail or the generalized extreme value (GEV) distribution to the maximum of a sequence by the method of maximum likelihood. The evidence for a heavy tail is evaluated by the likelihood-ratio test for the shape parameter of the GP or GEV being nonzero (i.e., test of GP vs. exponential or GEV vs. Gumbel).

Among weather or climate variables, precipitation amount has the strongest evidence for a heavy tail. This evidence is not overwhelming when individual sites are analyzed separately, but becomes stronger when either relatively long records are available or when regional analyses are performed (e.g., assuming common shape parameter within the region). For impact variables related to the weather or climate, the evidence of heavy tails is quite a bit stronger (e.g., economic damage from extreme events such as hurricanes), a result that is of special interest to the insurance industry. The question is raised as to whether the source of the heavy tail in impact variables is attributable to the underlying weather or climate variable or rather to an inherent tendency of variables related to income or wealth.