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The fitting method is designed only for the tail. Below u (where typically data are abundant) one could use a smooth version of the empirical df. From the latter plot, quantile estimates can be deduced. Figure 10 contains as an example the estimate for the 99.9% quantile x0.999 together with the profile likelihood. The latter can be used to find confidence intervals for x0.999. The 95% and 99% intervals are given. Figure 11 contains the same picture, but the (symmetric) confidence intervals are calculated using the Wald statistic. Finally, the 99.9% quantile estimates across a whole range of models (depending on the threshold value, or number of exceedances used) are given in Figure 12. Though the estimate of x0.999 settles between 1400 and 1500, the 95% Wald intervals are rather wide, ranging from 500 to about 2200. The above analysis yields a summary about the high quantiles of the fire insurance data based on the information on extremes available in the data. The analysis can be used as a tool in the final pricing of risks corresponding to high layers (catastrophic, rare events). All the methods used are based on extremes and are fairly standard. 4.2 An ARCH Example To further illustrate some of the available techniques, we simulated an ARCH(1) time series of length 99,000. The time series, called testarch, has the form Name /8042/03 04/21/99 09:19AM Plate # 0 pg 37 # 8 EXTREME VALUE THEORY AS A RISK MANAGEMENT TOOL 37 NAAJ (SOA) Figure 11 Estimate of x0.999 with 95% Wald-Statistic Confidence Interval Figure 12 Estimates of the Quantile x0.999 as a Function of the Threshold u Figure 8 Maximum Likelihood Fit of the Mean-Excess Tail Fu Based on Exceedances above u 5 100 Figure 9 Tail Fit for Based on a Threshold Value of F u 5 100, Doubly Logarithmic Scale Figure 10 Tail Fit with an Estimate for x0.999 and the Corresponding Profile Likelihood 2 1/2 j 5 X (b 1 lj ) , n $ 1, (11) nn n21 where {Xn} are iid N(0, 1) random variables. In our simulation, we took b 5 1, l 5 0.5. From known results of Kesten (1973) (see also EKM, Theorem 8.4.12; Goldie 1991; Vervaat 1979) 22k P(j . x) , c(x ), x → `, (12) 1 and we get from Table 3.2 of de Haan et al. (1989) that k 5 2.365 (see also Hooghiemstra and Meester 1995). There are several reasons why we choose to simulate an ARCH process: