Compute the L-comoment (\(\lambda_{k[12]}\)) for a given pair of sample of \(n\) random variates \(\{(X_i^{(1)}, X_i^{(1)}), 1 \le i \le n \}\) from a joint distribution \(H(x^{(1)}, x^{(2)})\) with marginal distribution functions \(F_1\) and \(F_2\). When the \(X^{(2)}\) are sorted to form the sample order statistics \(X^{(2)}_{1:n} \le X^{(2)}_{2:n} \le \cdots \le X^{(2)}_{n:n}\), then the element of \(X^{(1)}\) of the unordered (at leasted expected to be) but shuffled set \(\{X^{(1)}_1, \ldots, X^{(1)}_n\}\) that is paired with \(X^{(2)}_{r:n}\) the concomitant \(X^{(12)}_{[r:n]}\) of \(X^{(2)}_{r:n}\). (The shuffling occurs by the sorting of \(X^{(2)}\).) The \(k \ge 1\)-order L-comoments are defined (Serfling and Xiao, 2007, eq. 26) as
$$\hat\lambda_{k[12]} = \frac{1}{n}\sum_{r=1}^n w^{(k)}_{r:n} X^{(12)}_{[r:n]}\mbox{,}$$
where \(w^{(k)}_{r:n}\) is defined under Lcomoment.Wk. (The author is aware that \(k \ge 1\) is \(k \ge 2\) in Serfling and Xiao (2007) but \(k=1\) returns sample means. This matters only in that the lmomco package returns matrices for \(k \ge 1\) by Lcomoment.matrix even though the off diagnonals are NAs.)
Lcomoment.Lk12(X1,X2,k=1)A single L-comoment.
A vector of random variables (a sample of random variable 1).
Another vector of random variables (a sample of random variable 2).
The order of the L-comoment to compute. The default is 1.
W.H. Asquith
Now directing explanation of L-comoments with some reference heading into R code. L-comoments of random variable X1 (a vector) are computed from the concomitants of X2 (another vector). That is, \(X2\) is sorted in ascending order to create the order statistics of X2. During the sorting process, X1 is reshuffled to the order of X2 to form the concomitants of X2 (denoted as X12). So the trailing 2 is the sorted variable and the leading 1 is the variable that is shuffled. The X12 in turn are used in a weighted summation and expectation calculation to compute the L-comoment of X1 with respect to X2 such as by Lk3.12 <- Lcomoment.Lk12(X1,X2,k=3). 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 (2007) and then Asquith (2011). The weights for the computation are derived from calls made by Lcomoment.Lk12 to the weight function Lcomoment.Wk. The L-comoments of X2 are computed from the concomitants of X1, and the X21 are formed by sorting X1 in ascending order and in turn shuffling X2 by the order of X1. The often asymmetrical L-comoment of X2 with respect to X1 is readily done (Lk3.21 <- Lcomoment.Lk12(X2,X1,k=3)) and is not necessarily equal to (Lk3.12 <- Lcomoment.Lk12(X1,X2,k=3)).
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.Wk
X1 <- rnorm(101); X2 <- rnorm(101) + X1
Lcoskew12 <- Lcomoment.Lk12(X1,X2, k=3)
Lcorr12 <- Lcomoment.Lk12(X1,X2,k=2)/Lcomoment.Lk12(X1,X1,k=2)
rhop12 <- cor(X1, X2, method="pearson")
print(Lcorr12 - rhop12) # smallish number
Run the code above in your browser using DataLab