  The r largest order statistics approach is widely used in extreme value
analysis because it may use more information from the data than just the block
maxima. In practice, the choice of r is critical. If r is too large, bias can
occur; if too small, the variance of the estimator can be high. The limiting
distribution of the r largest order statistics, denoted by GEVr, extends that
of the block maxima. Two specification tests are proposed to select r
sequentially. The first is a score test for the GEVr distribution. Due to the
special characteristics of the GEVr distribution, the classical chi-square
asymptotics cannot be used. The simplest approach is to use the parametric
bootstrap, which is straightforward to implement but computationally expensive.
An alternative fast weighted bootstrap or multiplier procedure is developed for
computational efficiency. The second test uses the difference in estimated
entropy between the GEVr and GEV(r-1) models, applied to the r largest order
statistics and the r-1 largest order statistics, respectively. The asymptotic
distribution of the difference statistic is derived. In a large scale
simulation study, both tests held their size and had substantial power to
detect various misspecification schemes. A new approach to address the issue of
multiple, sequential hypotheses testing is adapted to this setting to control
the false discovery rate or familywise error rate. The utility of the
procedures is demonstrated with extreme sea level and precipitation data.
