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copBasic (version 2.2.7)

rhoCOP: The Spearman Rho of a Copula

Description

Compute the measure of association known as the Spearman Rho ρC of a copula according to Nelsen (2006, pp. 167--170, 189, 208) by ρC=12I2C(u,v)dudv3, or ρC=12I2[C(u,v)uv]dudv, where the later equation is implemented by rhoCOP as the default method (method="default"). This equation, here having p=1 and kp(1)=12, is generalized under hoefCOP. The absence of the 12 in the above equation makes it equal to the covariance of a copula defined by the Hoeffding Identity (Joe, 2014, p. 54): cov(U,V)=I2[C(u,v)uv]dudv or cov(U,V)=I2[C^(u,v)uv]dudv, which\ is cov(U,V)=I2[u+v1+C(1u,1v)uv]dudv.

Depending on copula family (Joe, 2014, pp. 56 and 267), the alternative formulation for ρC could be used ρC=312I2uδC(u,v)δududv=312I2vδC(u,v)δvdudv, where the first integral form corresponds to Joe (2014, eq. 248, p. 56) and is the method="joe21", and the second integral form is the method="joe12".

The integral I2C(u,v)dudv, represents the “volume under the graph of the copula and over the unit square” (Nelsen, 2006, p. 170) and therefore ρC is simple a rescaled volume under the copula. The second equation for ρC expresses the “average distance” between the joint distribution and statistical independence Π=uv. Nelsen (2006, pp. 175--176) shows that the following relation between ρC and τC (tauCOP) exists 13τ2ρ1.

Usage

rhoCOP(cop=NULL, para=NULL, method=c("default", "joe21", "joe12"),
                            as.sample=FALSE, brute=FALSE, delta=0.002, ...)

Value

The value for ρC is returned.

Arguments

cop

A copula function;

para

Vector of parameters or other data structure, if needed, to pass to the copula;

method

The form of integration used to compute (see above);

as.sample

A logical controlling whether an optional R data.frame in para is used to compute the ρ^ by dispatch to cor() function in R with method = "spearman";

brute

Should brute force be used instead of two nested integrate() functions in R to perform the double integration;

delta

The du and dv for the brute force integration using brute; and

...

Additional arguments to pass.

Author

W.H. Asquith

References

Joe, H., 2014, Dependence modeling with copulas: Boca Raton, CRC Press, 462 p.

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

See Also

blomCOP, footCOP, giniCOP, hoefCOP, tauCOP, wolfCOP, joeskewCOP, uvlmoms