tailconCOP {copBasic}

The Tail Concentration Function of a Copula

Description

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.

Usage

tailconCOP(t, cop=NULL, para=NULL, ...)

Arguments

t

Nonexceedance probabilities t;

cop

A copula function;

para

Vector of parameters or other data structure, if needed, to pass to the copula; and

...

Additional arguments to pass to the copula function.

Value

Value(s) for q_\mathbf{C} are returned.

Author(s)

W.H. Asquith

References

Durante, F., and Sempi, C., 2015, Principles of copula theory: Boca Raton, CRC Press, 315 p.

See Also

taildepCOP, tailordCOP

Examples

tailconCOP(0.5, cop=PSP) == (1 + blomCOP(cop=PSP)) / 2 # TRUE