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 manner of the 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 that 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 correct joint probability using a single unformly distributed $\sim$ U(0,1) random variable.
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).
$\mbox{}\quad\bullet\quad\mbox{}$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.
This last item is from Durante and Sempi (2015, p. 118) and see later discussion herein concerning 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 HHRcop) extreme value copulas for the same Kendall Tau ($\tau_\mathbf{C}$) all have the same $F_K(t)$. Therefore, do all EVs 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 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 and Salvadori et al. (2007, eq. 3.47, p. 147) also do the same. 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 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 important to note that this form is not in Joe (2014), Nelsen et al. (2001, 2003), Nelsen (2006), or Salvadori et al. (2007).
KENDALL MEASURE---The expression for any $\mathbf{C}(u,v)$ by Salvadori et al. (2007, eq. 3.49, p. 148) is for Kendall's 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
Salvadori et al. (2007, p. 148) report that
$$K^\star_\mathbf{C}(t) = t - \frac{\phi(t)}{\phi'(t^{+})}\mbox{,}$$
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 $\phi(\mathbf{C}(u,v)) = \phi(u) + \phi(v)$.
Nelsen (2006) does not seem to list a more general definition for any $\mathbf{C}$. Because there is considerable support for Archimedean copulas in R,
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 from $a + b$ as previously shown for $F_K(z)$. Which form is thus correct? The greater bulk of this documentation now deals with that question, and it must be concluded that Salvadori et al. (2007, pp. 161--165) definition for $K_\mathbf{C}(z)$ has a typesetting error.
kfuncCOP(z, cop=NULL, para=NULL, wrtV=FALSE, as.sample=FALSE,verbose=FALSE,...)
kmeasCOP(z, cop=NULL, para=NULL, wrtV=FALSE, as.sample=FALSE,verbose=FALSE,...)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 integrate() in Rthat are usually related to kfuncCOP is to return $F_K(z \rightarrow 0^{+}) = 0$ if nuThe $K_\mathbf{C}(z)$ as shown herein simply can not reproduce $F_K(z; \mathbf{\Pi}) = z - z\log z$ for the $\mathbf{\Pi}$ copula unless the kmeasCOP was further complicated by fact that neither Durante and Sempi (2015), Joe (2014), Nelsen (2006), and others do not actually present a general equation for $F_K(z)$ computation for any $\mathbf{C}$. Prior demonstrations numerically and graphically show that the implementation of kfuncCOP is valid.
Lastly and again because of the subtle differences evidently between
Genest, C., Quessy, J.F.,
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.,
Nelsen, R.B., Quesada-Molina, J.J.,
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.
kfuncCOPinv, tauCOP, derCOP, derCOP2, derCOPinv, derCOPinv2# Salvadori et al. (2007, p. 148, fig. 3.5 [right])
zs <- seq(0,1, by=.01)
plot(zs, kmeasCOP(zs, cop=GHcop, para=5), log="y", type="l", lwd=4,
xlab="Z <= z", ylab="Kendall Function", xlim=c(0,1), ylim=c(0.001,1))Run the code above in your browser using DataLab