Compute the tail concentration function (\(q_\mathbf{C}\)) of a copula \(\mathbf{C}(u,v)\) (COP) or diagnonal (diagCOP) of a copula \(\delta_\mathbf{C}(t) = \mathbf{C}(t,t)\) according to Durante and Semp (2015, p. 74):
$$q_\mathbf{C}(t) = \frac{\mathbf{C}(t,t)}{t} \cdot \mathbf{1}_{[0,0.5)} + \frac{1 - 2t + \mathbf{C}(t,t)}{1-t} \cdot \mathbf{1}_{[0.5, 1]}\mbox{\quad or}$$
$$q_\mathbf{C}(t) = \frac{\delta_\mathbf{C}(t)}{t} \cdot \mathbf{1}_{[0,0.5)} + \frac{1 - 2t + \delta_\mathbf{C}(t)}{1-t} \cdot \mathbf{1}_{[0.5, 1]}\mbox{,}$$
where \(t\) is a nonexceedance probability on the margins and \(\mathbf{1}(.)\) is an indicator function scoring 1 if condition is true otherwise zero on what interval \(t\) resides: \(t \in [0,0.5)\) or \(t \in [0.5,1]\). The \(q_\mathbf{C}(t; \mathbf{M}) = 1\) for all \(t\) for the M copula and \(q_\mathbf{C}(t; \mathbf{W}) = 0\) for all \(t\) for the W copula. Lastly, the function is related to the Blomqvist Beta (\(\beta_\mathbf{C}\); blomCOP) by
$$q_\mathbf{C}(0.5) = (1 + \beta_\mathbf{C})/2\mbox{,}$$
where \(\beta_\mathbf{C} = 4\mathbf{C}(0.5, 0.5) - 1\). Lastly, the \(q_\mathbf{C}(t)\) for \(0,1 = t\) is NaN and no provision for alternative return is made. Readers are asked to note some of the mathematical similarity in this function to Blomqvist Betas in blomCOPss in regards to tail dependency.
tailconCOP(t, cop=NULL, para=NULL, ...)Value(s) for \(q_\mathbf{C}\) are returned.
Nonexceedance probabilities \(t\);
A copula function;
Vector of parameters or other data structure, if needed, to pass to the copula; and
Additional arguments to pass to the copula function.
W.H. Asquith
Durante, F., and Sempi, C., 2015, Principles of copula theory: Boca Raton, CRC Press, 315 p.
taildepCOP, tailordCOP
tailconCOP(0.5, cop=PSP) == (1 + blomCOP(cop=PSP)) / 2 # TRUE
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