In practice, when analysing a sample of extremes {tout(j), xout(j)}m=1 obtained from a climate time series {t(i), x(i)}™=1, the questions often regard the tails of the distribution of Xout. It is here at the upper values, where the "climate risk" is located, where tail probabilities p, return periods 1/p (in time units) and return levels Xp are often associated with events of high socioeconomical relevance (Fig. 6.2). For example, authorities dealing with flood protection may be interested in HQ1ooo, the 1000-year return level of runoff at a certain river station.

The requirement of accurate methods of tail estimation was also noticed in the Earth Sciences literature, for example, by Dargahi-Noubary

(1989). He argued in favour of the POT-GP and against the block extremes-GEV approach because of information wastage caused by the latter. Notably, he also remarked that methods suited for estimating distribution parameters need not be optimal for estimating tail probabilities. An example in the Earth Sciences literature, where accurate tail estimation was an objective, was given by Caers et al. (1999b), who applied parametric bootstrap resampling to earthquake, diamond (an extreme with a "positive" connotation) and impact crater size data.

Smith (1987) proposed to estimate the tail probability of a GP distribution (Fig. 6.2) as follows:

Herein, £ is the estimate of the shape parameter and a the estimate of the transformed scale parameter (Eq. 6.13). This tail probability estimator is defined for xout > u (if £ > 0) or 0 < (xout — u) < — a/£ (if £ < 0). Smith (1987) showed by means of theoretical studies of its asymptotic properties that this estimator has "often" a better performance than a previous tail estimator suggested by Hill (1975).

In Eq. (6.22), the expression m n-1 is an estimate of the time-constant rate of occurrence of an extreme event within a time unit. The term within square brackets is the tail probability conditional on that an extreme occurred. Eq. (6.22) can therefore be seen (Kallache M 2009, personal communication) as a manifestation of the hybrid Poisson-extreme value distribution approach (Section 6.3.3) in the stationary setting, as a counterpart to Eq. (6.42).

Smith's (1987) estimator (Eq. 6.22) applies to "within-sample" thresholds. If u > max({xout(j)}m=1) for positive extremes, then m = 0 and pT =0, which is not a helpful estimation. When confronted with the task to estimate such "out-of-sample" probabilities or quantiles, other methods (Hall and Weissman 1997; El-Aroui and Diebolt 2002; Ferreira et al. 2003) can be tried. These methods are based on estimating a quantile "within" (< max({xout(j)}m=1) for positive extremes) and transforming it to "outside" (> max({xout(j)}m=1)). A related task is to make long-range predictions of extremes, based on extrapolation; for that purpose Hall et al. (2002) found good coverage performance of calibrated bootstrap CIs (Section 3.8). For the analysis of climate risk, however, long-range predictions based on "out-of-sample" estimations bear the danger of considerable errors caused by nonstationarities. The assumption made so far, namely that the distribution of X(i) does not change with time, is rather strong. It may, for example, be questionable to estimate an HQ1000 based on 150 years of data.

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