giniCOP: The Gini's Gamma of a Copula

Description

Compute the measure of association known as Gini's 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's Gamma measures a concordance relation of distance between $\mathbf{C}(u,v)$ and monotone dependence, as represented by the Fréchet{Frechet}-Hoeffding lower bound and Fréchet{Frechet}-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 the single unnested integration are readily deployed (and fast) and accommodated using the integrate() function in R. 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 occassionally 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 Rdata.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.

encoding

utf8

References

Genest, C., Nešlehová{Neslehova}, 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., Rodríguez-Lallena{Rodriguez-Lallena}, J.A., Úbeda-Flores{Ubeda-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

giniCOP(cop=PSP)                                 #                   0.3819757
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

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

# 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)
hoefCOP(para=D[sample(1:n, n/5),],
                      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
wolfCOP(cop=composite2COP, para=para)         # 0.5257662 (theoretical)
#wolfCOP(para=D, as.sample=TRUE) # major CPU hog, n too big
wolfCOP(para=D[sample(1:n, n/5),],
                      as.sample=TRUE)         # 0.5338009 (re-sample)

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