copula (version 0.99-4)

AssocMeasures: Dependence measures for copulas

Description

These functions compute Kendall's Tau, Spearman's Rho, and the tail dependence index for bivariate copulas. Calibration is the inverse function: it calibrates the copula parameter given the value of Kendall's Tau or Spearman's Rho.

Usage

kendallsTau (copula)
spearmansRho(copula)
tailIndex(copula, ...)
calibKendallsTau (copula, tau)
calibSpearmansRho(copula, rho)

Arguments

copula

an Robject of class "copula".

tau

a numerical value of Kendall's Tau in [-1, 1].

rho

a numerical value of Spearman's Rho in [-1, 1].

...

currently nothing.

Details

The calibration function in fact returns a moment estimate of the parameter for one-parameter copulas.

When there are no closed-form expressions for Kendall's tau or Spearman's rho, the calibration functions use numerical approximation techniques (see the last reference). For closed-form expressions, see Frees and Valdez (1998). For the t copula, the calibration function based on Spearman's rho uses the corresponding expression for the normal copula as an approximation.

References

E.W. Frees and E.A. Valdez (1998) Understanding relationships using copulas. North American Actuarial Journal 2, 1--25.

I. Kojadinovic and J. Yan (2010). Comparison of three semiparametric methods for estimating dependence parameters in copula models. Insurance: Mathematics and Economics 47, 52--63.

See Also

The acopula class objects have slots, tau, lambdaL, and lambdaU providing functions for kendallsTau(), and the two tail indices tailIndex(), and slot tauInv for calibKendallsTau(), see the examples, and copGumbel, etc.

Examples

gumbel.cop <- gumbelCopula(3)
kendallsTau(gumbel.cop)
spearmansRho(gumbel.cop)
tailIndex(gumbel.cop)

stopifnot(all.equal(kendallsTau(gumbel.cop), copGumbel@tau(3)),

          all.equal(tailIndex(gumbel.cop),
                    c(copGumbel@lambdaL(3), copGumbel@lambdaU(3)),
                    check.attr=FALSE),

          all.equal(calibKendallsTau (gumbel.cop, 0.681),
                    copGumbel@tauInv(0.681))
)


## let us compute the sample versions
x <- rcopula(gumbel.cop, 200)
cor(x, method = "kendall")
cor(x, method = "spearman")
## compare with the true parameter value 3
calibKendallsTau (gumbel.cop, cor(x, method="kendall")[1,2])
calibSpearmansRho(gumbel.cop, cor(x, method="spearman")[1,2])