lcomCOP: L-comoments and Bivariate L-moments of a Copula

Description

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 the lcomoms2() function 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.

Usage

lcomCOP(cop=NULL, para=NULL, as.bilmoms=FALSE, orders=2:5, ...)

Value

An R

list of the L-comoments or bivariate L-moments is returned depending on as.bilmoms setting.

bilmomUV: 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;
bilmomVU: 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;
lcomUV: 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
lcomVU: 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.

Arguments

cop: A copula function;
para: Vector of parameters or other data structure, if needed, to pass to the copula;
as.bilmoms: A logical to trigger return of the $\delta_k$ and the return vectors will be named differently;
orders: 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.

Author

W.H. Asquith

References

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.