Seismic risk management of insurance losses using extreme value theory and copula - Seismic Risk Analysis and Management of Civil Infrastructure Systems - page 772

1. 0 1. 0

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0.0 0.0

0.0 0.2 0.4

U 1 - Group 2

0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

U 1 - Group 2

Separation
distance

Separation
distance

: 1 km : 8 km

(a) (b)

28.7 Scatter/plot and marginal distribution plots of transformed
copula samples: (a) Groups 2 and 3 (1 km separation), and (b) Groups
2 and 5 (8 km separation).

u 1 and u 2 shown in Fig. 28.7). The model fi tness evaluated based on the AIC
value indicates that the HRT copula is superior to other copulas consis-
tently (but the Gumbel copula achieved a similar AIC value). This is in
agreement with the fi nding by Goda and Ren (2010). To visually inspect the
copula fi tting, bivariate copula density plots of the empirical copula samples
and the three fi tted normal, Gumbel, and HRT copulas for the joint POT
data for Groups 2 and 3 (i.e. Fig. 28.7a) are compared in Fig. 28.8. (A plot
for the t copula is not shown because it is very similar to the one for the
normal copula.) It is observed that the empirical copula has strong upper
tail dependence; the normal copula has the symmetrical moderate upper
and lower tail dependence; the Gumbel copula has strong upper tail depen-
dence and weak lower tail dependence; and the HRT copula has strong
upper tail dependence only. These plots confi rm the adequacy of the Gumbel
and HRT copulas for characterising seismic loss dependence for Groups 2
and 3, which is highly correlated in the upper tail. The above steps complete
the statistical modelling of joint seismic loss data based on extreme value
theory and copula. The fi tted model can be used to simulate seismic loss
samples.
Moreover, to investigate seismic loss dependence in terms of separation
distance, the preceding analysis is repeated for other pairs of groups with
different separation distances (see Fig. 28.3). As a measure of seismic loss
dependence, the Kendall's

coeffi cient is useful, because it is invariant
under strictly increasing transformations of random variables (unlike the
Pearson's linear correlation coeffi cient) and can be calculated from the
(estimated) HRT copula parameter as

τ

τ = θ /( θ + 2) (see Equation [28.10]).

Figure 28.9 shows the estimated Kendall's τ coeffi cient as a function of