R: Dependence Measures for Bivariate Copulas

assocMeasures {copula} | R Documentation

## Dependence Measures for Bivariate Copulas

### Description

These functions compute Kendall's tau, Spearman's rho, and the tail dependence index for bivariate copulas. `iTau` and `iRho`, sometimes called “calibration” functions are the inverses: they determine (“calibrate”) the copula parameter (which must be one-dimensional!) given the value of Kendall's tau or Spearman's rho.

### Usage

    tau (copula, ...)
    rho (copula, ...)
    lambda(copula, ...)
    iTau (copula, tau, ...)
    iRho (copula, rho, ...)

### Arguments

`copula` | an R object of class `"copula"` (or also `"acopula"` or `"nacopula"`; note however that some methods may not be available for some copula families).
`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 functions `iTau()` and `iRho()` in fact return 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.

Iwan Kojadinovic and Jun 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 `tau()`, and the two tail indices `lambda()`, and slot `iTau` for `iTau()`, see the examples and `copGumbel`, etc.

### Examples

    gumbel.cop <- gumbelCopula(3)
    tau(gumbel.cop)
    rho(gumbel.cop)
    lambda(gumbel.cop)
    iTau(joeCopula(), 0.5)

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

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

              all.equal(iTau (gumbel.cop, 0.681),
                        copGumbel@iTau(0.681))
    )

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

[Package copula version 1.0-1 Index]