tailconCOP: 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.

Usage

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

Arguments

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.

References

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

Examples

Run this code

# NOT RUN {
tailconCOP(0.5, cop=PSP) == (1+blomCOP(cop=PSP))/2 # TRUE
# }