Compute the L-comoments (Serfling and Xiao, 2007; Asquith, 2011) through the bivariate L-moments (ratios) (\(\delta^{[\ldots]}_{k;\mathbf{C}}\)) of a copula \(\mathbf{C}(u,v; \Theta)\) The L-comoments include L-correlation (Spearman Rho), L-coskew, and L-cokurtosis. As described by Brahimi et al. (2015), the first four bivariate L-moments \(\delta^{[12]}_k\) for random variable \(X^{(1)}\) or \(U\) with respect to (wrt) random variable \(X^{(2)}\) or \(V\) are defined as
$$\delta^{[12]}_{1;\mathbf{C}} = 2\int\!\!\int_{\mathcal{I}^2}
\mathbf{C}(u,v)\,\mathrm{d}u\mathrm{d}v - \frac{1}{2}\mbox{,}$$
$$\delta^{[12]}_{2;\mathbf{C}} = \int\!\!\int_{\mathcal{I}^2}
(12v - 6)
\mathbf{C}(u,v)\,\mathrm{d}u\mathrm{d}v - \frac{1}{2}\mbox{,}$$
$$\delta^{[12]}_{3;\mathbf{C}} = \int\!\!\int_{\mathcal{I}^2}
(60v^2 - 60v + 12)
\mathbf{C}(u,v)\,\mathrm{d}u\mathrm{d}v - \frac{1}{2}\mbox{, and}$$
$$\delta^{[12]}_{4;\mathbf{C}} = \int\!\!\int_{\mathcal{I}^2}
(280v^3 - 420v^2 + 180v - 20)
\mathbf{C}(u,v)\,\mathrm{d}u\mathrm{d}v - \frac{1}{2}\mbox{,}$$
where the bivariate L-moments are related to the L-comoment ratios by
$$6\delta^{[12]}_k = \tau^{[12]}_{k+1}\mbox{\quad and \quad}6\delta^{[21]}_k = \tau^{[21]}_{k+1}\mbox{,}$$
where in otherwords, “the third bivariate L-moment \(\delta^{[12]}_3\) is one sixth the L-cokurtosis \(\tau^{[12]}_4\).” The first four bivariate L-moments yield the first five L-comoments. The terms and nomenclature are not easy and also the English grammar adjective “ratios” is not always consistent in the literature. The \(\delta^{[\ldots]}_{k;\mathbf{C}}\) are ratios. The sample L-comoments are supported by the lmomco package, and in particular for the bivariate case, they are supported by lcomoms2 of that package.
Similarly, the \(\delta^{[21]}_k\) are computed by switching \(u \rightarrow v\) in the polynomials within the above integrals multiplied to the copula in the system of equations with \(u\). In general, \(\delta^{[12]}_k \not= \delta^{[21]}_k\) for \(k > 1\) unless in the case of permutation symmetric (isCOP.permsym) copulas. By theory, \(\delta^{[12]}_1 = \delta^{[21]}_1 = \rho_\mathbf{C}/6\) where \(\rho_\mathbf{C}\) is the Spearman Rho rhoCOP.
The integral for \(\delta^{[12]}_{4;\mathbf{C}}\) does not appear in Brahimi et al. (2015) but this and the other forms are verified in the Examples and discussion in Note. The four \(k \in (1,2,3,4)\) for \(U\) wrt \(V\) and \(V\) wrt \(U\) comprise a full spectrum of system of seven (not eight) equations. One equation is lost because \(\delta^{[12]}_1 = \delta^{[21]}_1\).
Chine and Benatia (2017) describe trimmed L-comoments as the multivariate extensions of the univariate trimmed L-moments (Elamir and Seheult, 2003) that are implemented in lmomco. These are not yet implemented in copBasic.
lcomCOP(cop=NULL, para=NULL, as.bilmoms=FALSE, orders=2:5, ...)A copula function;
Vector of parameters or other data structure, if needed, to pass to the copula;
A logical to trigger return of the \(\delta_k\) and the return vectors will be named differently;
The orders of the L-comoments to return, which is internally adjusted if the argument as.bilmoms is set. There is no first order L-comoment and the first index on returned values is set to NA to remain index consistent with the lmomco package. An order greater than 5 is not supported; and
Additional arguments to pass to the densityCOP function.
An R list of the L-comoments or bivariate L-moments is returned depending on as.bilmoms setting.
The bivariate L-moments \(\delta^{[12]}_k\) of \(U\) with respect to \(V\) for \(k \in [1,2,3,4]\) if orders is 2:5 and there is no NA index as for the L-comoments;
The bivariate L-moments \(\delta^{[21]}_k\) of \(V\) with respect to \(U\) for \(k \in [1,2,3,4]\) if orders is 2:5 and there is no NA index as for the L-comoments;
The L-comoments \(\tau^{[12]}_k\) of \(V\) with respect to \(U\) for \(k \in [2,3,4,5]\) if orders is 2:5 and index 1 is NA; and
The L-comoments \(\tau^{[21]}_k\) of \(V\) with respect to \(U\) for \(k \in [2,3,4,5]\) if orders is 2:5 and index 1 is NA.
Asquith, W.H., 2011, Distributional analysis with L-moment statistics using the R environment for statistical computing: Createspace Independent Publishing Platform, ISBN 978--146350841--8.
Brahimi, B., Chebana, F., and Necir, A., 2015, Copula representation of bivariate L-moments---A new estimation method for multiparameter two-dimensional copula models: Statistics, v. 49, no. 3, pp. 497--521.
Chine, Amel, and Benatia, Fatah, 2017, Bivariate copulas parameters estimation using the trimmed L-moments methods: Afrika Statistika, v. 12, no. 1, pp. 1185--1197.
Elamir, E.A.H, and Seheult, A.H., 2003, Trimmed L-moments: Computational Statistics and Data Analysis, v. 43, p. 299--314.
Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.
Serfling, R., and Xiao, P., 2007, A contribution to multivariate L-moments---L-comoment matrices: Journal of Multivariate Analysis, v. 98, pp. 1765--1781.
# NOT RUN {
para <- list(alpha=0.5, beta=0.93, para1=4.5, cop1=GLcop, cop2=PSP)
copBasic:::lcomCOP(cop=composite2COP, para=para)$lcomUV[3]
# Lcomom:T3[12]= +0.156
copBasic:::lcomCOP(cop=composite2COP, para=para)$lcomVU[3]
# Lcomom:T3[21]= -0.0668
bilmoms(cop=composite2COP, n=10000, para=para, sobol=TRUE)$bilcomoms$T3
# Tau3[12]=+0.1566, Tau3[21]=-0.0655
# The numerical default Monte Carlo integration of bilmoms()
# matches the numerical integration of lcomCOP albeit with a
# substantially slower and less elegant means in bilmoms().
# }
Run the code above in your browser using DataLab