giniCOP: The Gini Gamma of a Copula

Description

Compute the measure of association known as the Gini Gamma $\gamma_\mathbf{C}$ (Nelsen, 2006, pp. 180--182), which is defined as $$\gamma_\mathbf{C} = \mathcal{Q}(\mathbf{C},\mathbf{M}) + \mathcal{Q}(\mathbf{C},\mathbf{W})\mbox{,}$$ where $\mathbf{C}(u,v)$ is the copula, $\mathbf{M}(u,v)$ is the M function, and $\mathbf{W}(u,v)$ is the W function. The function $\mathcal{Q}(a,b)$ (concordCOP) is a concordance function (Nelsen, 2006, p. 158). Nelsen also reports that “Gini Gamma measures a concordance relation of “distance” between $\mathbf{C}(u,v)$ and monotone dependence, as represented by the Fr<U+00E9>chet--Hoeffding lower bound and Fr<U+00E9>chet--Hoeffding upper bound copulas [$\mathbf{M}(u,v)$, M and $\mathbf{W}(u,v)$, W respectively]”

A simpler method of computation and the default for giniCOP is to compute $\gamma_\mathbf{C}$ by

$$\gamma_\mathbf{C} = 4\biggl[\int_\mathcal{I} \mathbf{C}(u,u)\,\mathrm{d}u + \int_\mathcal{I} \mathbf{C}(u,1-u)\,\mathrm{d}u\biggr] - 2\mbox{,}$$ or in terms of the primary diagonal $\delta(t)$ and secondary diagonal $\delta^\star(t)$ (see diagCOP) by $$\gamma_\mathbf{C} = 4\biggl[\int_\mathcal{I} \mathbf{\delta}(t)\,\mathrm{d}t + \int_\mathcal{I} \mathbf{\delta^\star }(t)\,\mathrm{d}t\biggr] - 2\mbox{.}$$

The simpler method is more readily implemented because single integration is fast. Lastly, Nelsen et al. (2001, p. 281) show that $\gamma_\mathbf{C}$ also is computable by $$\gamma_\mathbf{C} = 2\,\mathcal{Q}(\mathbf{C},\mathbf{A})\mbox{,}$$ where $\mathbf{A}$ is a convex combination (convex2COP, using $\alpha = 1/2$) of the copulas $\mathbf{M}$ and $\mathbf{W}$ or $\mathbf{A} = (\mathbf{M}+\mathbf{W})/2$. However, integral convergence errors seem to trigger occasionally, and the first definition by summation $\mathcal{Q}(\mathbf{C},\mathbf{M}) + \mathcal{Q}(\mathbf{C},\mathbf{W})$ thus is used. The convex combination is demonstrated in the Examples section.

Usage

giniCOP(cop=NULL, para=NULL, by.concordance=FALSE, as.sample=FALSE, ...)

Arguments

cop

A copula function;

para

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

by.concordance

Instead of using the single integrals (Nelsen, 2006, pp. 181--182) to compute $\gamma_\mathbf{C}$, use the concordance function method implemented through concordCOP;

as.sample

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

...

Additional arguments to pass, which are dispatched to the copula function cop and possibly concordCOP if by.concordance=TRUE, such as delta used by that function.

Value

The value for $\gamma_\mathbf{C}$ is returned.

References

Genest, C., Ne<U+0161>lehov<U+00E1>, J., and Ghorbal, N.B., 2010, Spearman's footrule and Gini's gamma---A review with complements: Journal of Nonparametric Statistics, v. 22, no. 8, pp. 937--954.

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

Nelsen, R.B., Quesada-Molina, J.J., Rodr<U+00ED>guez-Lallena, J.A., <U+00DA>beda-Flores, M., 2001, Distribution functions of copulas---A class of bivariate probability integral transforms: Statistics and Probability Letters, v. 54, no. 3, pp. 277--282.

Examples

Run this code

# NOT RUN {
giniCOP(cop=PSP)                                 #                 = 0.3819757
# }
# NOT RUN {
giniCOP( cop=PSP, by.concordance=TRUE)           # Q(C,M) + Q(C,W) = 0.3820045
# use convex combination ---triggers integration warning but returns anyway
cxpara <- list(alpha=1/2, cop1=M, cop2=W) # parameters for convex2COP()
2*tauCOP(cop=PSP, cop2=convex2COP, para2=cxpara) #    2*Q(C,A)     = 0.3819807
# where the later issued warnings on the integration
# }
# NOT RUN {
# }
# NOT RUN {
n <- 2000; UV <- simCOP(n=n, cop=N4212cop, para=9.3, graphics=FALSE)
giniCOP(para=UV, as.sample=TRUE)                     # 0.9475900 (sample version)
giniCOP(cop=N4212cop, para=9.3)                      # 0.9479528 (copula integration)
giniCOP(cop=N4212cop, para=9.3, by.concordance=TRUE) # 0.9480267 (concordance function)
# where the later issued warnings on the integration
# }
# NOT RUN {
# }
# NOT RUN {
# The canoncial example of theoretical and sample estimators of bivariate
# association for the package: Blomqvist Beta, Spearman Footrule, Gini Gamma,
# Hoeffding Phi, Kendall Tau, Spearman Rho, and Schweizer-Wolff Sigma
# and comparison to L-correlation via lmomco::lcomoms2().
n <- 9000; set.seed(56)
para <- list(cop1=PLACKETTcop, cop2=PLACKETTcop, para1=1.45, para2=21.9,
             alpha=0.41, beta=0.08)
D <- simCOP(n=n, cop=composite2COP, para=para, cex=0.5, col=rgb(0,0,0,0.2), pch=16)
blomCOP(cop=composite2COP, para=para)         # 0.4037908 (theoretical)
blomCOP(para=D, as.sample=TRUE)               # 0.4008889 (sample)
footCOP(cop=composite2COP, para=para)         # 0.3721555 (theoretical)
footCOP(para=D, as.sample=TRUE)               # 0.3703623 (sample)
giniCOP(cop=composite2COP, para=para)         # 0.4334687 (theoretical)
giniCOP(para=D, as.sample=TRUE)               # 0.4311698 (sample)
tauCOP(cop=composite2COP, para=para)          # 0.3806909 (theoretical)
tauCOP(para=D,  as.sample=TRUE)               # 0.3788139 (sample)
rhoCOP(cop=composite2COP, para=para)          # 0.5257662 (theoretical)
rhoCOP(para=D,  as.sample=TRUE)               # 0.5242380 (sample)
lmomco::lcomoms2(D)$T2      # 1               # 0.5242388 (sample matrix)
                            # 0.5245154 1
hoefCOP(cop=composite2COP, para=para)         # 0.5082776 (theoretical)
subsample <- D[sample(1:n, n/5),] # subsampling for speed
hoefCOP(para=subsample, as.sample=TRUE)       # 0.5033842 (re-sample)
#hoefCOP(para=D, as.sample=TRUE) # major CPU hog, n too big
# because the Ds are already "probabilities" just resample as shown above
wolfCOP(cop=composite2COP, para=para)         # 0.5257662 (theoretical)
#wolfCOP(para=D, as.sample=TRUE) # major CPU hog, n too big
wolfCOP(para=subsample, as.sample=TRUE)       # 0.5338009 (re-sample)
# }

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