copBasic (version 2.1.5)

semicorCOP: The Lower and Upper Semi-Correlations of a Copula

Description

Compute the lower semi-correlations $${\rho }_{\mathbf{C}}^{N-}(u,v;a)={\rho }_N^{-}(a)\text{ and}$$ compute the upper semi-correlations $${\rho }_{\mathbf{C}}^{N+}(u,v;a)={\rho }_N^{+}(a)$$ of a copula $\mathbf{C}(u,v)$ (Joe, 2014, p. 73) using numerical simulation. The semi-correlations are defined as $${\rho }_N^{-}(a)=\mathrm{cor}[Z_1,Z_2|Z_1<-a,Z_2<-a]\text{,}$$ $${\rho }_N^{+}(a)=\mathrm{cor}[Z_1,Z_2|Z_1>+a,Z_2>+a]\text{, and}$$ $${\rho }_N(a>-\mathrm{\infty })=\mathrm{cor}[Z_1,Z_2]\text{,}$$ where $\mathrm{cor}[{\mathrm{z}}_1,{\mathrm{z}}_2]$ is the familiar Pearson correlation function, which is in R the syntax cor(..., method="pearson"), parameter $a\ge 0$ is a truncation point that identifies truncated tail regions (Joe, 2014, p. 73), and lastly $(Z_1,Z_2)\sim \mathbf{C}(\mathrm{\Phi },\mathrm{\Phi })$ and thus from the standard normal distribution $(Z_1,Z_2)=({\mathrm{\Phi }}^{-1}(u),{\mathrm{\Phi }}^{-1}(v))$ where the random variables $(U,V)\sim \mathbf{C}$.

Usage

semicorCOP(cop=NULL, para=NULL, truncation=0, n=0, as.sample=FALSE, ...)

Arguments

cop

A copula function;

para

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

truncation

The truncation value for $a$, which is in standard normal variates;

n

The sample size $n$ for simulation estimates of the ${\rho }_N$;

as.sample

A logical controlling whether an optional data.frame in para is used to compute the ${\hat{\rho }}_N$ (see Note); and

...

Additional arguments to pass to the copula.

Value

The value(s) for $rh{o}_N$, ${\rho }_N^{-}$, ${\rho }_N^{+}$ are returned.

References

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

See Also

giniCOP, rhoCOP, tauCOP

Examples

# NOT RUN {
# Gumbel-Hougaard copula with Pearson rhoN = 0.4 (by definition)
run <- sapply(1:20, function(i) semicorCOP(cop=GHcop, para=1.350, n=600))
mean(unlist(run[1,])) # cor.normal.scores
mean(unlist(run[2,])) # minus.semicor
mean(unlist(run[3,])) # plus.semicor
sd(  unlist(run[1,])) # cor.normal.scores (These are our sampling variations
sd(  unlist(run[2,])) # minus.semicor      for the n=600 used as a Monte
sd(  unlist(run[3,])) # plus.semicor       Carlo simulation.)
# The function returns:    rhoN = 0.3945714, rhoN-= 0.1312561, rhoN+= 0.4108908
#  standard deviations           (0.0378331)       (0.0744049)       (0.0684766)
# Joe (2014, p. 72) shows: rhoN = 0.4,       rhoN-= 0.132,     rhoN+= 0.415
#  standard deviations           (not avail)       (0.08)            (0.07)
# We see alignment with Joe's results with his n=600. #
# }