taildepCOP: The Upper and Lower Tail Dependency Parameters of a Copula

Description

Compute the upper and lower tail dependency parameters (if they exist), respectively, of a copula according to Nelson (2006). The parameters are expressions of conditional probability. The upper tail parameter $\lambda^U_\mathbf{C}$ is defined as the probability that $Y$ is greater than the $100t$-th percentile defined by quantile function $y(t)$ given that $X$ also is greater than the $100t$-th percentile defined by quantile function $x(t)$ or $$\lambda^U_\mathbf{C} = \lim_{t{\rightarrow 1^{-}}} \mathrm{Pr}[Y > y(t)\mid X > x(t)]\mbox{.}$$ Similarly, the lower tail dependency parameter $\lambda^L_\mathbf{C}$ is the same with reversed inequalities or $$\lambda^L_\mathbf{C} = \lim_{t{\rightarrow 0^{+}}} \mathrm{Pr}[Y \le y(t)\mid X \le x(t)]\mbox{.}$$

Nelson (2006) notes that $\lambda^U_\mathbf{C}$ and $\lambda^L_\mathbf{C}$ are nonparametric and depend only on the copula of $X$ and $Y$ and shows that each can be computed as follows: $$\lambda^U_\mathbf{C} = 2 - \lim_{t{\rightarrow 1^{-}}} \frac{1 - \mathbf{C}(t,t)}{1-t}\mbox{\ and}$$ $$\lambda^L_\mathbf{C} = \lim_{t{\rightarrow 0^{+}}} \frac{\mathbf{C}(t,t)}{t}\mbox{.}$$

Usage

taildepCOP(cop=NULL, para=NULL, plotem=FALSE,
           verbose=FALSE, tol=1e-6, ...)

Arguments

cop

A copula function;

para

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

plotem

Plot the parameters by nonexceedance probability with indicated limited values that are the ;

verbose

Show incremental progress;

tol

Tolerance on convergence; and

...

Additional arguments to pass.

Value

A list with $\lambda^U_\mathbf{C}$ and $\lambda^L_\mathbf{C}$ is returned.

References

Nelson, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.

Examples

Run this code

# Plot the tail dependencies by nonexceedance probability
# for a positive association Plackett copula
TD <- taildepCOP(cop=PLACKETTcop,para=c(3), plotem=TRUE)
print(TD)


# Plot the tail dependencies by nonexceedance probability
# for the PSP copula
TD <- taildepCOP(cop=PSP, para=c(3), plotem=TRUE)
print(TD)

# How about some composited Plackett-Plackett copulas?
# Each has upper and lower tail dependence parameters equal to zero.
para <- list(cop1=PLACKETTcop,  cop2=PLACKETTcop,
             para1=c(0.00395),  para2=c(4.67),
             alpha=0.9392,      beta=0.5699)
taildepCOP(cop=composite2COP, para=para, plotem=TRUE, verbose=TRUE)

para <- list(cop1=PLACKETTcop,  cop2=PLACKETTcop,
             para1=c(0.14147),  para2=c(20.96),
             alpha=0.0411,      beta=0.6873)
taildepCOP(cop=composite2COP,para=para)

para <- list(cop1=PLACKETTcop,  cop2=PLACKETTcop,
             para1=c(0.10137),  para2=c(4492.87),
             alpha=0.0063,      beta=0.0167)
taildepCOP(cop=composite2COP,para=para)