isCOP.RTI: Is a Copula Right-Tail Increasing

Description

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

The RTI 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, $\mathrm{RTI}(V{\mid}U)$ (or $\mathrm{RTI}(V{\mid}U)$) means that the probability $P[Y > y \mid X > x]$ (or $P[X > x \mid Y > y]$) is a nondecreasing function of $x$ (or $y$) for all $y$ (or $x$).

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

Usage

isCOP.RTI(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.