Seismic risk management of insurance losses using extreme value theory and copula - Seismic Risk Analysis and Management of Civil Infrastructure Systems - page 771

100 100

80 80

60 60

40 40

20 20

0 0

0 0 0
L A - Group 2 (10 6 CAD)

60 80 100 0 0 0

L A - Group 2 (10 6 CAD)

60 80 100

Separation
distance

Separation
distance

: 1 km : 8 km

(a) (b)

28.6 Scatter plot and marginal distribution plots of joint POT data:
(a) Groups 2 and 3 (1 km separation), and (b) Groups 2 and 5 (8 km
separation).

The second step of the method is to investigate dependence characteris-
tics of the joint POT data. In this study, POT data points that exceed the
respective threshold values simultaneously are considered; this is the same
approach adopted by Dupuis and Jones (2006). The focus is deemed to be
reasonable, because the developed statistical model will be used to evaluate
risk measures related to very rare events, e.g. calculation of risk exposure
of a reinsurance policy where the retention level is generally much higher
than the adopted threshold level of the GP model. To inspect characteristics
of the seismic loss data for Groups 2 and 3 as well as Groups 2 and 5, scatter/
plots for the two cases are presented in Fig. 28.6. The average separation
distance between Groups 2 and 3 is 1 km, while that between Groups 2 and
5 is 8 km; therefore, based on physical features of ground motions, the cor-
relation of seismic loss data for Groups 2 and 3 is greater than that for
Groups 2 and 5. Inspection of Fig. 28.6 indicates that a tendency of upper
tail dependence for Groups 2 and 3 is evident (i.e. occurrence of a large
loss for one group is accompanied by a counterpart for the other group),
while the extent of upper tail dependence for Groups 2 and 5 is insignifi cant.
To examine the tail dependence characteristics evidenced in Fig. 28.6, the
POT data are transformed to the copula samples (i.e. C Emp ( u 1 , u 2 ) as in
Equation [28.6]), and scatter plots in terms of uniform marginals are shown
in Fig. 28.7. In Fig. 28.7, it is clearly seen that the right upper tail dependence
exists for Groups 2 and 3, while such dependence is not obvious for Groups
2 and 5.
To complete the analysis, copula fi tting is carried out by fi tting the four
parametric copula models mentioned in Section 28.2.2, i.e. normal, t ,
Gumbel, and HRT copulas, to the transformed copula samples (i.e. pairs of