The bounded inverse Weibull distribution: An extreme value alternative for application to environmental maxima?

Abstract

There has long been interest in making inferences about future low-probability natural events that have magnitudes greater than any in the past record. Given a stationary time series, the unbounded Type 1 and Type 2 asymptotic extreme value distributions are often invoked as giving theoretical justification for extrapolating to large magnitudes and long return periods for hydrological variables such as rainfall and river discharge. However, there is a problem in that environmental extremes are bounded above by the bounded nature of their causal variables. Extrapolation using unbounded asymptotic models therefore cannot be justified from extreme value theory and at some point there will be over-prediction of future magnitudes. This creates the apparent contradiction, for example, of annual rainfall maxima being well approximated by Type 2 extreme value distributions despite the bounded nature of rainfall magnitudes. An alternative asymptotic extreme value approach is suggested for further investigation, with the model being the asymptotic distribution of minima (Weibull distribution) applied to block maxima reciprocals. Two examples are presented where data that are well matched by Type 1 or Type 2 extreme value distributions give reciprocals suggestive of lower bounds (upper bound γ to the original data). The asymptotic model here is a 3-parameter Weibull distribution for the reciprocals, with positive location parameter γ⁻¹. When this situation is demonstrated from data, parameter estimation can be carried out with respect to the distribution of reciprocals of 3-parameter Weibull random variables. This distribution is referenced here as the bounded inverse Weibull distribution. A maximum likelihood parameter estimation methodology is presented, together with a parametric bootstrap approach for obtaining one-sided upper confidence limits to γ. When data permits estimation of γ, the bounded inverse Weibull distribution is suggested as an improved alternative to Type 1 or Type 2 extreme value distributions because the upper bound reality is recognised. However, extensive application to many data sets is required to evaluate the practical utility of the bounded approach for extrapolating beyond the largest recorded event.