# Lcomoment.coefficients: L-comoment Coefficient Matrix In wasquith/lmomco: L-Moments, Censored L-Moments, Trimmed L-Moments, L-Comoments, and Many Distributions

## Description

Compute the L-comoment coefficients from an L-comoment matrix of order k ≥ 2 and the k = 2 (2nd order) L-comoment matrix. However, if the first argument is 1st-order then the coefficients of L-covariation are computed. The function requires that each matrix has already been computed by the function Lcomoment.matrix.

## Usage

 1 Lcomoment.coefficients(Lk, L2) 

## Arguments

 Lk A k ≥ 2 L-comoment matrix from Lcomoment.matrix. L2 A k = 2 L-comoment matrix from Lcomoment.matrix(Dataframe,k=2).

## Details

The coefficient of L-variation is computed by Lcomoment.coefficients(L1,L2) where L1 is a 1st-order L-moment matrix and L2 is a k = 2 L-comoment matrix. Symbolically, the coefficient of L-covariation is

\hat{τ}_{[12]} = \frac{\hat{λ}_{2[12]}} {\hat{λ}_{1[12]}} \mbox{.}

The higher L-comoment coefficients (L-coskew, L-cokurtosis, ...) are computed by the function Lcomoment.coefficients(L3,L2) (k=3), Lcomoment.coefficients(L4,L2) (k=4), and so on. Symbolically, the higher L-comoment coefficients for k ≥ 3 are

\hat{τ}_{k[12]} = \frac{\hat{λ}_{k[12]}} {\hat{λ}_{2[12]}}\mbox{.}

Finally, the usual univariate L-moment ratios as seen from lmom.ub or lmoms are along the diagonal. The Lcomoment.coefficients function does not make use of lmom.ub or lmoms.

## Value

An R list is returned.

 type The type of L-comoment representation in the matrix: “Lcomoment.coefficients”. order The order of the coefficients. order=2 L-covariation, order=3 L-coskew, ... matrix A k ≥ 2 L-comoment coefficient matrix.

## Note

The function begins with a capital letter. This is intentionally done so that lower case namespace is preserved. By using a capital letter now, then lcomoment.coefficients remains an available name in future releases.

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.

Serfling, R., and Xiao, P., 2007, A contribution to multivariate L-moments—L-comoment matrices: Journal of Multivariate Analysis, v. 98, pp. 1765–1781.

Lcomoment.matrix, Lcomoment.coefficients
 1 2 3 4 5 6 D <- data.frame(X1=rnorm(30), X2=rnorm(30), X3=rnorm(30)) L1 <- Lcomoment.matrix(D,k=1) L2 <- Lcomoment.matrix(D,k=2) L3 <- Lcomoment.matrix(D,k=3) LkLCV <- Lcomoment.coefficients(L1,L2) LkTAU3 <- Lcomoment.coefficients(L3,L2)