isCOP.RTI: Is a Copula Right-Tail Increasing

Description

Numerically set a logical whether a copula is right-tail increasing. A copula $\mathbf{C}(u,v)$ is right-tail decreasing for $RTI(X \mid Y)$ or $RTI(V \mid U)$ if, and only if, for every $v \in [0,1]$, $$\frac{\delta \mathbf{C}(u,v)}{\delta u} \le \frac{v - \mathbf{C}(u,v)}{1 - u}\mbox{.}$$

The $RTI$ test is associated with the concept of tail monotonicity. $RTI(V \mid U)$ means that the random variable $Y$ is more likely to acquire larger values as random variable $X$ increases. Finally, the accuracy of the numerical determination of the logical is dependent on the the smallness of the delt argument passed into the function.

The opposite with-respect-to $RTI(X \mid Y)$ is right-tail decreasing ($RTI(U \mid V)$) if, and only if, for every $u \in [0,1]$, $$\frac{\delta \mathbf{C}(u,v)}{\delta v} \le \frac{u - \mathbf{C}(u,v)}{1- v}\mbox{,}$$ which is controlled by the wrtV=TRUE argument.

Usage

isCOP.RTI(cop=NULL, para=NULL, wrtV=FALSE, verbose=FALSE,
          delt=0.005, ...)

Arguments

cop

A copula function;

para

Vector of parameters, if needed, to pass to the copula;

wrtV

A logical to toggle between with respect to $v$ or $u$ (default);

verbose

A logical to show the looping index on the outer loop;

delt

The increment of axis set by wrtV; and

...

Additional arguments to pass.

Value

A logical TRUE or FALSE is returned.

References

Salvadori, G., De Michele, C., Kottegoda, N.T., and Rosso, R., 2007, Extremes in nature---An approach using copulas: Dordrecht, Netherlands, Springer, Water Science and Technology Library 56, 292 p.

Examples

Run this code

isCOP.RTI(cop=P, delt=.01) # independency should be FALSE
# but function returns TRUE, note that same logic for isCOP.LTD returns
# FALSE.

# Positive association
isCOP.RTI(cop=PSP) # compute fast, has lower tail dependency
# TRUE

# Negative association Plackett
isCOP.RTI(cop=PLACKETTcop, para=c(.15))
# FALSE

# Positive association Plackett
isCOP.RTI(cop=PLACKETTcop, para=c(15))
# TRUE



# Negative association Plackett
isCOP.RTI(cop=PLACKETTcop, wrtv=TRUE, para=c(.15))
# FALSE

# Positive association Plackett
isCOP.RTI(cop=PLACKETTcop, wrtV=TRUE, para=c(15))
# TRUE