r-th Largest Order Statistic Model

 It is also possible to extend the block maxima methods to other order statistics. The simplest case is to look at minima, where one needs only take the negative of the data and then use the regular maximum methods (see, for example, sectionRelating the Point Process Model to the Poisson-GP Example 3). It is also possible to model other order statistics more generally. One such method is referred to as the r-th largest order statistic model. This model has essentially been replaced by the threshold exceedance methods (see Generalized Pareto Distribution (GPD) and Peaks Over Threshold(POT)/Point Process (PP) Approach chapters) in practice, but extRemes does facilitate r-th largest model fitting as it is often desired for pedagogical reasons. For help on using the r-th largest model, see Coles (2001) (b) and Coles (2001) (a).

Although limited in scope, it is possible to perform an r-th largest order statistics model fit using extRemes. The (common format) dataset Ozone4H.dat is included in the data directory. Data for fitting this model must be in a much different form than data used for all the other model fits with extRemes. Instead of one response column, there needs to be as many columns as r. That is, if interest is in the fourth-highest value, then there must be at least four columns of data giving the maxima, second-, third- and fourth-highest values, respectively; missing values are allowed. In the case of Ozone4H.dat, there are five columns: the first (obs) is simply an index from 1 to 513, the second (r1) are maxima, followed by r2, r3 and r4. Here, all of the data come from 1997, but from 513 different monitoring stations in the eastern United States. The order statistics represent the maximum, second-, third- and fourth-highest daily maximum 8-hour average ozone for 1997 (see Fuentes (2003) or Gilleland and Nychka (2005) for more about these data). After loading Ozone4H.dat, saved in R as Ozone4H, the r-th largest order statistic model can be applied in the following manner.

7Multiple panels of plots will be plotted. The user must hit return at the R session window to view each plot. This may interrupt seeing fit results until all plots are viewed. See Coles (2001) (b) for an explanation of these plots.

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