Lcomoment.Lk12 {lmomco} | R Documentation |
Compute the k >= 1 order L-comoment (λ_{k[r:n]}) for a given pair of random variables. The order of the L-comoments is specified.
Lcomoment.Lk12(X1,X2,k=1)
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. |
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{λ}_{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{λ}_{k[21]} = frac{1}{n}sum_{r=1}^{n} w^{(k)}_{r:n} x^{(21)}_{[r:n]}
A single L-comoment.
The function begins with a capital letter. This is intentionally done so that lower case
namespace is preserved. L-comoments are new in the literature and experimental in this package.
By using a capital letter now, then lcomoment.Lk12
or similar remains an available name in future
releases.
W.H. Asquith
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.
Lcomoment.matrix
, Lcomoment.Wk
X1 <- rnorm(20) X2 <- rnorm(20) Lk12 <- Lcomoment.Lk12(X1,X2,k=1)