Description Usage Arguments Details Value Note Author(s) References See Also Examples

Compute the L-comoment (*λ_{k[12]}*) for a given pair of sample of *n* random variates *\{(X_i^{(1)}, X_i^{(1)}), 1 ≤ i ≤ 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} ≤ X^{(2)}_{2:n} ≤ \cdots ≤ X^{(2)}_{n:n}*, then the element of *X^{(1)}* of the unordered (at leasted expected to be) but shuffled set *\{X^{(1)}_1, …, 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 ≥ 1*-order L-comoments are defined (Serfling and Xiao, 2007, eq. 26) as

*\hatλ_{k[12]} = \frac{1}{n}∑_{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 ≥ 1* is *k ≥ 2* in Serfling and Xiao (2007) but *k=1* returns sample means. This matters only in that the lmomco package returns matrices for *k ≥ 1* by `Lcomoment.matrix`

even though the off diagnonals are `NAs`

.)

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

`X1` |
A 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. |

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)`

).

A single L-comoment.

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.Lk12`

or similar remains an available name in future releases.

W.H. Asquith

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`

1 2 3 4 5 | ```
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
``` |

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