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 for any $v \in [0,1]$ that the following holds $$\frac{\delta \mathbf{C}(u,v)}{\delta u} \le \frac{\mathbf{C}(u,v)}{u}$$ 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 for any $u \in [0,1]$ that the following holds $$\frac{\delta \mathbf{C}(u,v)}{\delta v} \le \frac{\mathbf{C}(u,v)}{v}$$ 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 $P[Y \le y \mid X \le x]$ (or $P[X \le x \mid Y \le y]$) 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, ...)

Value

A logical TRUE or FALSE is returned.

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.

Author

W.H. Asquith

References

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