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The r largest order statistics approach is widely used in extreme valueanalysis because it may use more information from the data than just the blockmaxima. In practice, the choice of r is critical. If r is too large, bias canoccur; if too small, the variance of the estimator can be high. The limitingdistribution of the r largest order statistics, denoted by GEVr, extends thatof the block maxima. Two specification tests are proposed to select rsequentially. The first is a score test for the GEVr distribution. Due to thespecial characteristics of the GEVr distribution, the classical chi-squareasymptotics cannot be used. The simplest approach is to use the parametricbootstrap, which is straightforward to implement but computationally expensive.An alternative fast weighted bootstrap or multiplier procedure is developed forcomputational efficiency. The second test uses the difference in estimatedentropy between the GEVr and GEV(r-1) models, applied to the r largest orderstatistics and the r-1 largest order statistics, respectively. The asymptoticdistribution of the difference statistic is derived. In a large scalesimulation study, both tests held their size and had substantial power todetect various misspecification schemes. A new approach to address the issue ofmultiple, sequential hypotheses testing is adapted to this setting to controlthe false discovery rate or familywise error rate. The utility of theprocedures is demonstrated with extreme sea level and precipitation data.

Automated Selection of r for the r Largest Order Statistics Approach with Adjustment for Sequential Testing