lmomco (version 1.3.3)

Lcomoment.Lk12: Compute a Single Sample L-comoment

Description

Compute the $k \ge 1$ order L-comoment ($\lambda_{k[r:n]}$) for a given pair of random variables. The order of the L-comoments is specified.

Usage

Lcomoment.Lk12(X1,X2,k=1)

Arguments

X1

An vector of random variables (a sample of random variable 1).

X2

Another vector of random variables (a sample of random variable 2).

k

The order of the L-comoment to compute. The default is 1.

Value

• A single L-comoment.

Details

L-comoments of random variable $X1$ are computed from the concomitants of $X2$. That is, $X2$ is sorted in ascending order to create the order statistics of $X2$. $X1$ is in turn reshuffled to the order of $X2$ for form the concomitants of $X2$ (denoted as $X^{(12)}$). The concomitants inturn are used in a weighted summation and expectation calculation to compute the L-comoment of $X1$ with respect to $X2$. The inverse can also be done (Lcomoment.Lk12(X2,X1,k=1)) and is not necessarily equal to (Lcomoment.Lk12(X1,X2,k=1)). The notation of Lk12 is to read Lambda for kth order L-comoment, where the 12 portion of the notation reflects that of Serfling and Xiao (2006). The weights for the computation are derived from calls by Lcomoment.Lk12 to Lcomoment.Wk.

$$\hat{\lambda}_{k[12]} = \frac{1}{n}\sum_{r=1}^{n} w^{(k)}_{r:n} x^{(12)}_{[r:n]}$$

The L-comoments of $X2$ are computed from the concomitants of $X1$ ($X^{(21)}$) are formed by sorting $X1$ in ascending order and in turn shuffling $X2$ by the order of $X1$. The sample concomitants are thus formed ($x^{(12)}_{[r:n]}$). By symmetry the L-comoment is

$$\hat{\lambda}_{k[21]} = \frac{1}{n}\sum_{r=1}^{n} w^{(k)}_{r:n} x^{(21)}_{[r:n]}$$

References

Hosking, J.R.M. and Wallis, J.R., 1997, Regional frequency analysis---An approach based on L-moments: Cambridge University Press.

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

See Also

Lcomoment.matrix, Lcomoment.Wk

Examples

X1   <- rnorm(20)
X2   <- rnorm(20)
Lk12 <- Lcomoment.Lk12(X1,X2,k=1)