copBasic (version 2.0.1)

tauCOP: The Kendall's Tau and Concordance Function of a Copula

Description

Compute the measure of association known as Kendall's 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. However, Nelsen (2006, p. 164) reports 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. Nelsen's preferred expression is used by the tauCOP function. Nelsen (2006, pp. 175--176) reports that the following relation between $\tau_\mathbf{C}$ and $\rho_\mathbf{C}$ (rhoCOP) exists $-1 \le 3\tau - 2\rho \le 1$ (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's 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 that 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, ...)

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 Rdata.frame in para is used to compute the $\hat\tau$ by dispatch to cor() function in Rwith method = "kendall";

brute

Should brute force be used instead of two nested integrate() functions in Rto 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.

Value

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

encoding

utf8

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.

Nelsen, R.B., Quesada-Molina, J.J., Rodríguez-Lallena{Rodriguez-Lallena}, J.A., Úbeda-Flores{Ubeda-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.

See Also

blomCOP, footCOP, giniCOP, hoefCOP, rhoCOP, wolfCOP, joeskewCOP, uvlmoms, derCOP, derCOP2, kfuncCOP

Examples

tauCOP(cop=PSP) # 1/3
# Now compute Kendall Tau via integration of the Kendall Function.
# 3 - 4*integrate(function(t) kfuncCOP(t, cop=PSP), 0, 1)$value # 0.3333314

tauCOP(cop=PSP, brute=TRUE) # 0.3306625
# CPU heavy example showing that the dual-integration (fast) results in
# a Kendall's Tau that matches a sample version
dotau <- function(n) {
   uv <- simCOP(n=n, cop=PSP, ploton=FALSE, points=FALSE)
   return(cor(uv$U, uv$V, method="kendall"))
}
taus <- replicate(100, dotau(1000))
tau.sample <- mean(taus); print(tau.sample) # 0.3333763
# Nelsen (2006, pp. 160-161, numeric results shown thereine)
# The rational values or integers may be derived analytically.
tauCOP(cop=M, cop2=M) #   1, correct
tauCOP(cop=M, cop2=P) # 1/3, correct
tauCOP(cop=P, cop2=M) # 1/3, correct
tauCOP(cop=M, cop2=W) #   0, correct
tauCOP(cop=W, cop2=M) # throws warning, swaps copulas, == tauCOP(M,W)
tauCOP(cop=W, cop2=P) # throws warning, swaps copulas, approx. -1/3
tauCOP(cop=P, cop2=W) # -1/3, correct
tauCOP(cop=P, cop2=P) #    0, correct
tauCOP(cop=M, cop2=W, brute=TRUE) #    0, correct

para <- list(cop1=PLACKETTcop,  cop2=PLACKETTcop,
             para1=0.00395, para2=4.67, alpha=0.9392, beta=0.5699)
tauCOP(cop=composite2COP, para=para)

para <- list(cop1=PLACKETTcop,  cop2=PLACKETTcop,
             para1=0.14147, para2=20.96, alpha=0.0411, beta=0.6873)
tauCOP(cop=composite2COP, para=para)

para <- list(cop1=PLACKETTcop,  cop2=PLACKETTcop,
             para1=0.10137, para2=4492.87, alpha=0.0063, beta=0.0167)
# Theoretical attempt fails because para2 is large and thus a singularity
# is emerging and internal copula swapping does not help.
tauCOP(cop=composite2COP, para=para) # fails
tauCOP(cop=composite2COP, para=para, brute=TRUE) # about 0.95