Copula Selection

Neptune fits 6 copulas (Gaussian, Student's t, Gumbel, Clayton, Frank, and Joe) to each copula scenario. The Akaike Information Criterion (AIC) is computed and tabulated for each copula model. The Akaike Information Criterion is computed using the following equation (Burnham and Anderson 1998):

$//<![CDATA[ \begin{array}{l}AIC_c = \frac{2Nk}{N-k-1} - 2ln(\hat{L})\end{array} //]]>$

where $//<![CDATA[ \begin{array}{l}k\end{array} //]]>$ is the number of parameters estimated by the probability model, $//<![CDATA[ \begin{array}{l}N\end{array} //]]>$ is the number of data points, and $//<![CDATA[ \begin{array}{l}\hat{L}\end{array} //]]>$ is the maximized value of the likelihood function of the probability model i.e. $//<![CDATA[ \begin{array}{l}\hat{L} = p(x|\hat{\theta}, M)\end{array} //]]>$ where $//<![CDATA[ \begin{array}{l}\hat{\theta}\end{array} //]]>$ are the parameter values that maximize the likelihood function and $//<![CDATA[ \begin{array}{l}M\end{array} //]]>$ is the probability model. 

The above form of AIC is used for small samples sizes ($//<![CDATA[ \begin{array}{l}N/k\end{array} //]]>$ < 40). As the sample size increases with respect to the number of parameters, this equation converges to the standard form of the AIC:

$//<![CDATA[ \begin{array}{l}AIC = 2k - 2ln(\hat{L})\end{array} //]]>$

If the user entered two copula scenarios on the Copula Scenarios tab, the Next and Previous chips in the upper right of the tab (boxed in red) can be used to toggle between the copula model fits for each scenario. The name of the copula scenario is shown in the upper left.

In addition, the samples and probability isolines are plotted for each copula model. The user must select a copula model for each copula scenario. By default, HEC-Neptune populates the Selected Copula with the copula with the lowest AIC. The user can change this selection.

The dependence characteristics of the copula models is described in the table below.