Abstract
It is suggested here that in many environmental and other contexts the severity of an extreme event might usefully be represented by the sum of the excesses of a measured variable over a high threshold. The general form of the limiting distributions of such sums for a wide class of models has been derived by Anderson and Dancy, and has suggested methods for the statistical analysis of data concerning extreme severity. This work is reviewed here, and some extensions to the distributional theory are presented. An application of the methods to atmospheric ozone levels, which calls for the extension of the approach to take account of covariate information is reported.