copBasic (version 1.7.1)

tauCOP: The Kendall's Tau of a Copula

Description

Compute the measure of association known as Kendall's Tau ($\tau_K$) of a copula ($\tau_\mathbf{C}$) according to Nelsen (2006, sec. 5.1.1 and p. 161) by $$\tau_\mathbf{C} = Q(\mathbf{C}, \mathbf{C}) = 4\int_0^1 \int_0^1 \mathbf{C}(u,v)\,\mathrm{d}\mathbf{C}(u,v) - 1\mbox{,}$$ where $Q(\mathbf{C}, \mathbf{C})$ is a concordance function of a copula with itself. However, Nelsen (2006, p. 164) reports that this form is often not amenable to computation when a copula 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 the concordance function $Q(a,b)$: $$Q(\mathbf{M}, \mathbf{M}) = 4\int_0^1 u\,\mathrm{d}u - 1 = 1\mbox{,}$$ $$Q(\mathbf{M}, \mathbf{\Pi}) = 4\int_0^1 u^2\,\mathrm{d}u - 1 = 1/3\mbox{,}$$ $$Q(\mathbf{M}, \mathbf{W}) = 4\int_{1/2}^1 (2u-1)\,\mathrm{d}u - 1 = 0\mbox{,}$$ $$Q(\mathbf{W}, \mathbf{\Pi}) = 4\int_0^1 u(1-u)\,\mathrm{d}u - 1 = -1/3\mbox{,}$$ $$Q(\mathbf{W}, \mathbf{W}) = 4\int_0^1 0\,\mathrm{d}u - 1 = -1\mbox{, and}$$ $$Q(\mathbf{\Pi}, \mathbf{\Pi}) = 4\int_0^1\int_0^1 uv\,\mathrm{d}u\mathrm{d}v - 1 = 0\mbox{.}$$

Usage

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

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;

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.

References

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

See Also

blomCOP, giniCOP, rhoCOP, derCOP, derCOP2

Examples

tauCOP(cop=PSP) # 1/3
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