kfuncCOP: The Kendall (Distribution) Function of a Copula

Description

To begin, there are at least three terms in the literature for what appear as the same function supported by the kfuncCOP function. The Kendall Function also is known as Kendall Distribution Function (Nelsen, 2006, p. 163) and Kendall Measure (Salvadori et al., 2007, p. 148). Each of these is dealt with in sequel to set the manner of the rather lengthy documentation for this function.

KENDALL FUNCTION---The Kendall Function ($F_K$) (Joe, 2014, pp. 419--422) is the cumulative distribution function (CDF) of the vector $\mathbf{U} = (U_1, U_2, \ldots)$ or $\mathbf{U} = (u,v)$ (bivariate) where $\mathbf{U}$ is distributed as the copula: $\mathbf{U} \sim \mathbf{C}(u,v)$. Letting $Z$ be the random variable for $\mathbf{C}(u,v): Z = \mathbf{C}(u,v)$, the Kendall Function is defined as

$$F_K(z; \mathbf{C}) = \mathrm{Pr}[Z \le z; \mathbf{U} \sim \mathbf{C}(u,v)]\mbox{,}$$

where $F_K$ is the nonexceedance probability of the joint probability $z$ stemming from the $\mathbf{C}$. Note, unlike its univariate counterpart, $F_K(z)$ is rarely uniformly distributed (Nelsen et al., 2001, p. 278). The inverse $F_K^{(-1)}(z)$ is implemented by the kfuncCOPinv function, which could be used for simulation of the correct joint probability using a single unformly distributed $\sim$ U(0,1) random variable. A reminder is needed that $Z$ is the joint probability and $F_K(z)$ is the Kendall Function.

Joe (2014) and others as cited list various special cases of $F_K(z)$, inequalities, and some useful identities suitable for validation study:
$\mbox{}\quad\bullet\quad\mbox{}$For $\mathbf{M}(u,v)$ (see M): $F_K(z) = z$ for all $0 < z < 1$ for all $d \ge 2$ dimensions;
$\mbox{}\quad\bullet\quad\mbox{}$For $\mathbf{W}(u,v)$ (see W): $F_K(z) = 1$ for all $0 < z < 1$ for $d = 2$ (bivariate only);
$\mbox{}\quad\bullet\quad\mbox{}$For $\mathbf{\Pi}(u,v)$ (see P): $F_K(z) = z - z \log z$ for $0 < z < 1$ for $d = 2$ (bivariate only);
$\mbox{}\quad\bullet\quad\mbox{}$For any $\mathbf{C}$: $z \le F_K(z)$ for $0 < z < 1$; and
$\mbox{}\quad\bullet\quad\mbox{}$For any $\mathbf{C}$: $\mathrm{E}[Z] = 1 - \int_0^1 F_K(t)\,\mathrm{d}t \ge z$ (Nelsen, 2001, p. 281) --- Z expectation, not $F_K$!
$\mbox{}\quad\bullet\quad\mbox{}$For any $\mathbf{C}$: $\tau_\mathbf{C} = 3 - 4\int_0^1 F_K(t)\,\mathrm{d}t$ (Nelsen, 2006, p. 163; see tauCOP [Examples]).
$\mbox{}\quad\bullet\quad\mbox{}$For any $\mathbf{C}$: $F_K(t)$ does not uniquely determine the copula.

The last item is from Durante and Sempi (2015, p. 118), and later discussion herein will concern an example of theirs. By coincidence within a few days before receipt of the Durante and Sempi book, experiments using kfuncCOP suggested that numerically the Galambos (GLcop), Gumbel--Hougaard (GHcop), and Hüsler--Reiss (HRcop) extreme value copulas for the same Kendall Tau ($\tau_\mathbf{C}$) all have the same $F_K(t)$. Therefore, do all EV-copulas have the same Kendall Function? Well in fact, they do and Durante and Sempi (2015, p. 207) show that $F_K(z) = z - (1 - \tau_\mathbf{C})z \log(z)$ for an EV-copula.

Joe (2014, p. 420) also indicates that strength of lower-tail dependence (taildepCOP) affects $F_K(z)$ as $z \rightarrow 0^{+}$, whereas strength of upper-tail dependence affects $F_K(z)$ as $z \rightarrow 1^{-}$. (A demonstration of tail dependence dependence is made in section Note.) Also compared to comonotonicity copula [$\mathbf{M}$] there are no countermonotonicity copula ($\mathbf{W}_{d > 2}$) for dimensions greater the bivariate (Joe, 2014, p. 214)

Joe (2014) does not explicitly list an expression of $F_K(z)$ that is computable directly for any $\mathbf{C}(u,v)$, and Nelsen (2006, p. 163) only lists a form (see later in documentation) for Archimedean copulas. Salvadori et al. (2007, eq. 3.47, p. 147) also list the Archimedean form; however, Salvadori et al. (2007, eq. 3.49, p. 148) also list a form computable directly for any $\mathbf{C}(u,v)$. Considerable numerical experiments and derivations involving the $\mathbf{\Pi}(u,v)$ copula and results for $K_\mathbf{C}(z)$ shown later, indicate that the correct Kendall form for any $\mathbf{C}(u,v)$ is

$$F_K(z) \equiv z + \int_z^1 \frac{\delta\mathbf{C}(u,t)}{\delta u}\,\mathrm{d}u\mbox{,}$$

where $t = \mathbf{C}^{(-1)}(u,z)$ for $0 \le z \le 1$, $t$ can be computed by the COPinv function, and the partial derivative $\delta\mathbf{C}(u,t)/\delta u$ can be computed by the derCOP function. It is a curiosity that this form is not in Joe (2014), Nelsen et al. (2001, 2003), or Nelsen (2006), but actually in Salvadori et al. (2007).

KENDALL MEASURE---The actual expression for any $\mathbf{C}(u,v)$ by Salvadori et al. (2007, eq. 3.49, p. 148) is for Kendall Measure ($K_\mathbf{C}$) of a copula: $$K_\mathbf{C}(z) = z - \int_z^1 \frac{\delta\mathbf{C}(u,t)}{\delta u}\,\mathrm{d}u\mbox{,}$$ where $t = \mathbf{C}^{(-1)}(u,z)$ for $0 \le z \le 1$. Those authors report that $K_\mathbf{C}(z)$ is the CDF of a random variable $Z$ whose distribution is $\mathbf{C}(u,v)$. This is clearly appears to be the same meaning as Joe (2014) and Nelsen (2006). The minus “$-$” in the above equation is very important.

Salvadori et al. (2007, p. 148) report that “the function $K_\mathbf{C}(z)$ represents a fundamental tool for calculating the return period of extreme events.” The complement of $K_\mathbf{C}(z)$ is $\overline{K}_\mathbf{C}(z) = 1 - K_\mathbf{C}(z)$, and the $\overline{K}_\mathbf{C}(z)$ inverse $$\frac{1}{1 - K_\mathbf{C}(z)} = \frac{1}{\overline{K}_\mathbf{C}(z)} = T_{\mathrm{KC}}$$ is referred to as a secondary return period (Salvadori et al., 2007, pp. 161--170).

KENDALL DISTRIBUTION FUNCTION---Nelsen (2006, p. 163) defines the Kendall Distribution Function (say $K^\star_\mathbf{C}(t)$) as

$$K^\star_\mathbf{C}(t) = t - \frac{\phi(t)}{\phi'(t^{+})}\mbox{,}$$

where $\phi(t)$ is a generator function of an Archimedean copula and $\phi'(t^{+})$ is a one-sided derivative (Nelsen, 2006, p. 125), and $\phi(t)$ is $\phi(\mathbf{C}(u,v)) = \phi(u) + \phi(v)$. This same form is listed by Salvadori et al. (2007, eq. 3.47).

Nelsen (2006) does not seem to list a more general definition for any $\mathbf{C}(u,v)$. Because there is considerable support for Archimedean copulas in R, copBasic has deliberately been kept from being yet another Archimedean-based package. This is made for more fundamental theory and pedogogic reasons without the algorithmic efficiency relative to the many convenient properties of Archimedean copulas.

The similarity of $F_K(z)$, $K_\mathbf{C}(z)$, and $K^\star_\mathbf{C}(t)$, however, is obvious---research shows that there are no syntatic differences between $F_K(z)$ and $K_\mathbf{C}(t)$ and $K^\star_\mathbf{C}(z)$---they all are the CDF of the joint probability $Z$ of the copula. Consider now that Salvadori et al. show $K_\mathbf{C}$ having the form $a - b$ and not a form $a + b$ as previously shown for $F_K(z)$. Which form is thus correct? The greater bulk of this documentation seeks to answer that question, and it must be concluded that Salvadori et al. (2007, eq. 3.49) definition for $K_\mathbf{C}(z)$ has a typesetting error.

Usage

kfuncCOP(z, cop=NULL, para=NULL, wrtV=FALSE, as.sample=FALSE,
            verbose=FALSE, subdivisions=100L,
            rel.tol=.Machine$double.eps^0.25, abs.tol=rel.tol, ...)
kmeasCOP(z, cop=NULL, para=NULL, wrtV=FALSE, as.sample=FALSE,
            verbose=FALSE, subdivisions=100L,
            rel.tol=.Machine$double.eps^0.25, abs.tol=rel.tol, ...)

Value

The value(s) for $F_K(z)$ is returned.

Arguments

z: The values for $z$;
cop: A copula function;
para: Vector of parameters or other data structure, if needed, to pass to the copula;
wrtV: A logical to toggle between with respect to $v$ or $u$ (default);
as.sample: A control on whether an optional R data.frame in para is used to compute the empirical $\hat{F}_K(z)$. Let vector length of para be denoted $m$ and $i = (1,\ldots,m)$, if as.sample=TRUE, then for each of the probability pairs in para, the empirical copula (EMPIRcop) function with additional arguments ... is used to generate a vector $(0, F^\sharp_{K,m}, 1)$ of length $m+2$, then a vector of $(0, z^\sharp_m, 1)$ again of length $m+2$ where $z^\sharp = (i-0.5)/m$ is created and linear interpolation by the approx() function in R for each of the $z$ values in z is used to estimate $\hat{F}_K(z)$ (see source code). If as.sample="genest", then the $\hat{F}_K(z)$ is estimated without interpolation using a simple empirical copula basis and “pseudo-observations” after Genest et al. (2006, p. 339). The default $\hat{F}_K$ (as.sample=TRUE) is the linear interpolation based on the default call of Weibull form of the empirical copula, but that can be confirmed by ctype="weibull" (see Note);
verbose: A logical supressing warnings from integrate() in R that are usually related to “integral divergence” for $z \rightarrow 0^{+}$. The constructed behavior of kfuncCOP is to return $F_K(z \rightarrow 0^{+}) = 0$ if numerical integration returns NULL, and such a construction is made to avoid “end points not of opposite sign” during $F_K(z)$ inversion by kfuncCOPinv;
subdivisions: Argument of same name passed to integrate(),
rel.tol: Argument of same name passed to integrate(),
abs.tol: Argument of same name passed to integrate(), and
...: Additional arguments to pass.

Author

W.H. Asquith

References

Durante, F., and Sempi, C., 2015, Principles of copula theory: Boca Raton, CRC Press, 315 p.

Genest, C., Quessy, J.F., Rémillard, B., 2006, Goodness-of-fit procedures for copula models based on the probability integral transformation: Scandinavian Journal of Statistics, v. 33, no. 2, pp. 337--366.

Joe, H., 2014, Dependence modeling with copulas: Boca Raton, CRC Press, 462 p.

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

Nelsen, R.B., Quesada-Molina, J.J., Rodríguez-Lallena, J.A., Úbeda-Flores, M., 2001, Distribution functions of copulas---A class of bivariate probability integral transforms: Statistics and Probability Letters, v. 54, no. 3, pp. 277--282.

Nelsen, R.B., Quesada-Molina, J.J., Rodríguez-Lallena, J.A., Úbeda-Flores, M., 2003, Kendall distribution functions: Statistics and Probability Letters, v. 65, no. 3, pp. 263--268.