Hybrid estimation using L-moments for the r-largest extreme value model with hydrological applications

Incorporating more data using the r-largest order statistics (rLOS), rather than relying solely on a single set of maxima from each block, can enhance the precision of extreme value distribution estimation. The maximum likelihood estimation (MLE) method is commonly employed for parameter estimation of the rLOS model. However, the MLE tends to overestimate high quantiles, particularly for small sample and for distributions with heavy right tails. To mitigate this issue, we explored an L-moment-based estimation method tailored to rLOS, which retains some of the advantages of the L-moment estimator (LME). We proposed a weighted LME (WelMET), which transforms certain order statistics to resemble block maxima, and we introduced a hybrid estimator combining WelMET with MLE. Our simulation study demonstrated that while WelMET tends to underestimate high quantiles, the hybrid estimator is nearly unbiased due to compensating for the biases in both directions. Furthermore, the hybrid estimator exhibited a lower mean squared error than other methods, including MLE and WelMET. The optimal weight for the hybrid estimator was determined through Monte Carlo simulation. We applied this approach to annual 5-largest daily rainfall data from Maha Sarakham, Thailand.