P: The Product (Independence) Copula

In wasquith/copBasic: General Bivariate Copula Theory and Many Utility Functions

The Product (Independence) Copula

Description

Compute the product copula (Nelsen, 2006, p. 12), which is defined as

\mathbf{\Pi}(u,v) = uv\mbox{.}

This is the copula of statistical independence between U and V and is sometimes referred to as the independence copula. The two extreme antithesis copulas are the Fréchet–Hoeffding upper-bound (M) and Fréchet–Hoeffding lower-bound (W) copulas.

Usage

P(u, v, ...)

Arguments

u	Nonexceedance probability u in the X direction;
v	Nonexceedance probability v in the Y direction; and
...	Additional arguments to pass.

Value

Value(s) for the copula are returned.

Author(s)

W.H. Asquith

References

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

See Also

M, W, rhoCOP

Examples

P(c(0.4, 0, 1), c(0, 0.6, 1))

## Not run: 
n <- 100000 # giant sample size, L-comoments are zero
# PERFECT INDEPENDENCE
UV <- simCOP(n=n, cop=P, graphics=FALSE)
lmomco::lcomoms2(UV, nmom=4)
# The following are Taus_r^{12} and Taus_r^{21}
# L-corr:        0.00265 and  0.00264 ---> ZERO
# L-coskew:     -0.00121 and  0.00359 ---> ZERO
# L-cokurtosis:  0.00123 and  0.00262 ---> ZERO

# MODEST POSITIVE CORRELATION
rho <- 0.6; # Spearman Rho
theta <- PLACKETTpar(rho=rho) # Theta = 5.115658
UV <- simCOP(n=n, cop=PLACKETTcop, para=theta, graphics=FALSE)
lmomco::lcomoms2(UV, nmom=4)
# The following are Taus_r^{12} and Taus_r^{21}
# L-corr        0.50136 and  0.50138 ---> Pearson R == Spearman Rho
# L-coskews    -0.00641 and -0.00347 ---> ZERO
# L-cokurtosis -0.00153 and  0.00046 ---> ZERO 
## End(Not run)

wasquith/copBasic documentation built on Dec. 13, 2024, 6:39 p.m.