Generalized Gumbel model for r-largest order statistics, with an application to peak streamflow

The effective use of available information in extreme value analysis is critical because extreme values are scarce. Thus, applying the r-largest order statistics (rLOS) instead of the block maxima is encouraged in general. The Gumbel distribution for rLOS (rGD) has been employed for modeling the r-largest data. However, the rGD is not flexible enough to capture the variability of the r-largest data because only two parameters are used. This study extends the rGD to the generalized Gumbel distribution for rLOS (rGGD) to address this problem, which incorporates three parameters including a shape parameter. We derive some probability functions of the rGGD. The maximum likelihood estimation, delta method, entropy difference test for r selection, and cross-validated likelihood are considered for inference. The usefulness and practical effectiveness of the rGGD are illustrated by Monte Carlo simulation and an application to the peak streamflow data at Oykel Bridge in the UK. This new r-largest model should be helpful for the design of engineering structures to prevent severe damage by extreme events.