isCOP.LTD: Is a Copula Left-Tail Decreasing

Description

Numerically set a logical whether a copula is left-tail decreasing (LTD) as described by Nelsen (2006, pp. 192--193) and Salvadori et al. (2007, p. 222). A copula $\mathbf{C}(u,v)$ is left-tail decreasing for $\mathrm{LTD}(V{\mid}U)$ if and only if any $v \in [0,1]$, $$\frac{\delta \mathbf{C}(u,v)}{\delta u} \le \frac{\mathbf{C}(u,v)}{u}$$ for almost all $u \in [0,1]$. Similarly, a copula $\mathbf{C}(u,v)$ is left-tail decreasing for $\mathrm{LTD}(U{\mid}V)$ if and only if any $u \in [0,1]$, $$\frac{\delta \mathbf{C}(u,v)}{\delta v} \le \frac{\mathbf{C}(u,v)}{v}$$ for almost all $v \in [0,1]$ where the later definition is controlled by the wrtV=TRUE argument.

The LTD concept is associated with the concept of tail monotonicity (Nelsen, 2006, p. 191). Specifically, but reference to Nelsen (2006) definitions and geometric interpretations is recommended, $\mathrm{LTD}(V{\mid}U)$ (or $\mathrm{LTD}(V{\mid}U)$) means that the probability $P[Y \le y|X \le x]$ (or $P[X \le x|Y \le y]$) is a nonincreasing function of $x$ (or $y$) for all $y$ (or $x$).

A positive LTD of either $\mathrm{LTD}(V{\mid}U)$ or $\mathrm{LTD}(U{\mid}V)$ implies positively quadrant dependency (PQD, isCOP.PQD) but the condition of PQD does not imply LTD. Finally, the accuracy of the numerical assessment of the returned logical by isCOP.LTD is dependent on the the smallness of the delta argument passed into the function.

Usage

isCOP.LTD(cop=NULL, para=NULL, wrtV=FALSE, delta=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);

delta

The increment of ${u,v} \mapsto [0+\Delta\delta, 1-\Delta\delta, \Delta\delta]$ set by wrtV; and

...

Additional arguments to pass to the copula or derivative of a copula function.

Value

A logical TRUE or FALSE is returned.

References

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

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.LTD(cop=P, delta=0.01) # independence should be FALSE
# Positive association
isCOP.LTD(cop=PSP)                               # TRUE
# Negative association Plackett
isCOP.LTD(cop=PLACKETTcop, para=0.15)            # FALSE
# Positive association Plackett
isCOP.LTD(cop=PLACKETTcop, para=15)              # TRUE
# Negative association Plackett
isCOP.LTD(cop=PLACKETTcop, wrtv=TRUE, para=0.15) # FALSE
# Positive association Plackett
isCOP.LTD(cop=PLACKETTcop, wrtV=TRUE, para=15)   # TRUE