assocMeasures | R Documentation |
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.
tau (copula, ...)
rho (copula, ...)
lambda(copula, ...)
iTau (copula, tau, ...)
iRho (copula, rho, ...)
copula |
an R object of class |
tau |
a numerical value of Kendall's tau in [-1, 1]. |
rho |
a numerical value of Spearman's rho in [-1, 1]. |
... |
currently nothing. |
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.
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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.insmatheco.2010.03.008")}
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.
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])
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