2.3 Examples

In this section we show possible applications for the Gumbel-Hougaard copula, i.e. for $ \tt copula=4$. First we try to visualize $ C_4(u,v)$ in Figure 2.1.


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Figure 2.1: Plot of $ C_4(u,v)$ for $ \theta =3$
\includegraphics[width=0.99\defpicwidth]{plot1.ps}

In the next Figure 2.2 we show an example of copula sampling for fixed parameters $ \sigma _1=1$, $ \sigma _2=1$, $ \theta =3$ for copulas numbered 4, 5, 6, and 12, see Table 2.1.



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Figure 2.2: 10000-sample output for $ \sigma _1=1$, $ \sigma _2=1$, $ \theta =3$
\includegraphics[width=1.2\defpicwidth]{plot2.ps}

In order to investigate the connection between the Gaussian and Copula based dependency structure we plot $ \theta $ against correlation $ \rho $ in Figure 2.3. We assume that tmin and tmax hold the minimum respectively maximum possible $ \theta $ values. Those can also be obtained by tmin= 6701 VaRcopula (0,0,0,8,copula) and tmax= 6704 VaRcopula (0,0,0,9,copula). Care has to be taken that the values are finite, so we have set the maximum absolute $ \theta $ bound to $ 10$.



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Figure 2.3: Plot of $ \theta $ against correlation $ \rho $ for $ C_4$.
\includegraphics[width=0.9\defpicwidth]{plot3.ps}