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Numerically determine the global property of the positively quadrant dependency (PQD) characteristic of a copula as described by Nelsen (2006, p. 188). The random variables P
).
Nelsen (2006) shows that a copula is PQD when
blomCOP
, giniCOP
, rhoCOP
, and tauCOP
.
The concept of negatively quadrant dependency (NQD) is the reverse:
Conceptually, PQD is related to the probability that two random variables are simultaneously small (or simultaneously large) is at least as great as it would be if they were independent. The graph of a PQD copula lies on or above the copulatic surface of the independence copula
Albeit a “global” property of a copula, there can be “local” variations in the PQD/NQD state. Points in wolfCOP
because as those examples involve the copulatic difference from independence
isCOP.PQD(cop=NULL, para=NULL, uv=NULL, empirical=FALSE, verbose=TRUE, ...)
A copula function;
Vector of parameters or other data structure, if needed, to pass to the copula;
An optional R data.frame
of
A logical that will use sample versions for Gini Gamma, Spearman Rho, and Kendall Tau. This feature is only applicable if the copula is empirical and therefore the para
argument is the data.frame
of
A logical that will report the four concordance measures; and
Additional arguments to pass, which are then passed to subordinate functions.
If uv=NULL
then a logical for the global property of PQD is returned but if argument uv
is a data.frame
, then an R list
is returned, and that list holds the global condition in global.PQD
and local condition assessments in local.PQD
and local.NQD
.
Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.
# NOT RUN {
isCOP.PQD(cop=PSP) # TRUE
# }
# NOT RUN {
# }
# NOT RUN {
# Example concerning Empirical Bivariate Copula and sample versions for comparison.
set.seed(10); n <- 1000
para <- list(cop1=PLACKETTcop, cop2=PLACKETTcop, para1=0.145, para2=1.45,
alpha=0.81, beta=0.8)
D <- simCOP(n=n, cop=composite2COP, para=para, cex=0.5, col=rgb(0,0,0,0.2), pch=16)
#tauCOP(cop=EMPIRcop, para=D) # ??? but == 1
cor(D$U, D$V, method="kendall") # -0.3224705
blomCOP(cop=EMPIRcop, para=D) # -0.332
giniCOP(cop=EMPIRcop, para=D) # -0.3692037
GINI <- sum(abs(rank(D$U)+rank(D$V)-n-1)) - sum(abs(rank(D$U)-rank(D$V)))
print(GINI/as.integer(n^2/2)) # -0.369996
rhoCOP(cop=EMPIRcop, para=D) # ??? but fails
cor(D$U, D$V, method="spearman") # -0.456694
lmomco::lcomoms2(D)$T2 # 1.0000000 -0.4568357
# -0.4567859 1.0000000
# }
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