24.9 Fitting Copulas to Data

\(J(z)\) and \(\chi(z)\) are the empirical statistics we’re interested in that are related to \(\tau\) and \(R\)

We graph them empirically

24.9.1 \(J(z)\)

\[J(z) = - z^2 + \dfrac{4 \int_0^z \int_0^z C(u,v) \cdot c(u,v)dudv}{C(z,z)}\]

Recall continuous formula for \(\tau\) (24.4)

Since \(C(1,1) =1\)

We have

\[\lim \limits_{z \rightarrow 1} J(z) = \tau\]

\(J(z)\) is the build up of \(\tau\) from 0 (when no data points are considered), up to \(\tau\) when all the data points are considered

• Maybe??

Graph the empirical \(J(z)\) and compare to the theoretical \(J(z)\) for each copulas

Note:

• 3.3.14 is incorrect

• Look at 3.3.16 we can tell that t-copula is the best fit and MM2 is the next closest, MM1 is the worst

24.9.2 \(\chi(z)\)

Don’t have much on an intuitive explanation for what the graph represents

\(z \rightarrow 1\) it approaches \(R\)

\[\chi(z) = 2 - \dfrac{\mathrm \ln C(z,z)}{\ln(z)}\]

\[R = \lim \limits_{z \rightarrow 1} R(z) = \lim \limits_{z \rightarrow 1} \dfrac{1 -2z + C(z,z)}{1-z}\]

Compare the empirical graph of \(\chi(z)\) with the theoretical copulas to determine which is best fit