tauCOP: The Kendall Tau and Concordance Function of a Copula

Description

Compute the measure of association known as the Kendall Tau ($\tau_\mathbf{C}$) of a copula ($\tau_\mathbf{C}$) according to Nelsen (2006, sec. 5.1.1 and p. 161) by $$\tau_\mathbf{C} = \mathcal{Q}(\mathbf{C}, \mathbf{C}) = 4\int\!\!\int_{\mathcal{I}^2} \mathbf{C}(u,v)\,\mathrm{d}\mathbf{C}(u,v) - 1\mbox{,}$$ where $\mathcal{Q}(\mathbf{C}, \mathbf{C})$ is a concordance function (concordCOP) of a copula with itself. Nelsen (2006, p. 164) reports however that this form is often not amenable to computation when there is a singular component to the copula and that the expression $$\tau_\mathbf{C} = 1 - 4\int\!\!\int_{\mathcal{I}^2} \frac{\delta\mathbf{C}(u,v)}{\delta u} \frac{\delta\mathbf{C}(u,v)}{\delta v}\, \mathrm{d}u\mathrm{d}v$$ is to be preferred. Such an expression hence relies on the partial numerical derivatives of the copula provided by derCOP and derCOP2. The Nelsen (2006) preferred expression is used by the tauCOP function. Nelsen (2006, pp. 175--176) reports that the relation between $\tau_\mathbf{C}$ and $\rho_\mathbf{C}$ (rhoCOP) is $-1 \le 3\tau - 2\rho \le 1$ (see rhoCOP for more details).

Nelsen (2006, pp. 160--161) lists some special identities involving $\mathcal{Q}(\mathbf{C}_1,\mathbf{C}_2)$: $$\mathcal{Q}(\mathbf{M}, \mathbf{M}) = 4\int_0^1 u\,\mathrm{d}u - 1 = 1\mbox{,}$$ $$\mathcal{Q}(\mathbf{M}, \mathbf{\Pi}) = 4\int_0^1 u^2\,\mathrm{d}u - 1 = 1/3\mbox{,}$$ $$\mathcal{Q}(\mathbf{M}, \mathbf{W}) = 4\int_{1/2}^1 (2u-1)\,\mathrm{d}u - 1 = 0\mbox{,}$$ $$\mathcal{Q}(\mathbf{W}, \mathbf{\Pi}) = 4\int_0^1 u(1-u)\,\mathrm{d}u - 1 = -1/3\mbox{,}$$ $$\mathcal{Q}(\mathbf{W}, \mathbf{W}) = 4\int_0^1 0\,\mathrm{d}u - 1 = -1\mbox{, and}$$ $$\mathcal{Q}(\mathbf{\Pi}, \mathbf{\Pi}) = 4\int\!\!\int_{\mathcal{I}^2} uv\,\mathrm{d}u\mathrm{d}v - 1 = 0\mbox{.}$$

Kendall Tau also can be expressed in terms of the Kendall Function ($F_K(z)$; kfuncCOP): $$\tau_\mathbf{C} = 3 - 4\int_0^1 F_K(t)\,\mathrm{d}t\mbox{,}$$ which is readily verified by code shown in Examples. This definition might be useful if integration errors are encountered for some arbitrary copula and arbitrary parameter set. In fact, should two attempts (see source code) at dual integration of the partial derivatives occur, the implementation switches over to integration of the Kendall Function (e.g. tauCOP(cop=N4212cop, para=2)). Note, Durante and Sempi have erroneously dropped the multiplication by “$4$” as shown above in their definition of $\tau_\mathbf{C}$ as a function of $F_K(t)$ (Durante and Sempi, 2015, eq. 3.9.4, p. 121).

Usage

tauCOP( cop=NULL,  para=NULL,
       cop2=NULL, para2=NULL, as.sample=FALSE, brute=FALSE, delta=0.002, ...)
concordCOP(cop=NULL,  para=NULL, cop2=NULL, para2=NULL, ...)

Value

The value for $\tau_\mathbf{C}$ is returned.

Arguments

cop: A copula function;
para: Vector of parameters or other data structure, if needed, to pass to the copula;
cop2: A second copula function;
para2: Vector of parameters or other data structure, if needed, to pass to the second copula;
as.sample: A logical controlling whether an optional R data.frame in para is used to compute the $\hat\tau$ by dispatch to cor() function in R with method = "kendall";
brute: Should brute force be used instead of two nested integrate() functions in R to perform the double integration;
delta: The $\mathrm{d}u$ and $\mathrm{d}v$ for the brute force integration using brute; and
...: Additional arguments to pass on to derCOP and derCOP2.

Author

W.H. Asquith

References

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

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