copBasic (version 2.1.4)

P: 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 Fr<U+00E9>chet--Hoeffding upper bound (M) and Fr<U+00E9>chet--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.

References

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

See Also

M, W

Examples

# NOT RUN {
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 <- 0.5; # Spearman's Rho
theta <- PLACKETTpar(rho=0.5) # 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
# }