# TLmom: A Sample Trimmed L-moment In wasquith/lmomco: L-Moments, Censored L-Moments, Trimmed L-Moments, L-Comoments, and Many Distributions

## Description

A sample trimmed L-moment (TL-moment) is computed for a vector. The r ≥ 1 order of the L-moment is specified as well as the level of symmetrical trimming. A trimmed TL-moment \hat{λ}^{(t_1,t_2)}_r is

\hat{λ}^{(t_1,t_2)}_r = \frac{1}{r}∑^{n-t_2}_{i=t_1+1} ≤ft[ \frac{∑\limits^{r-1}_{k=0}{ (-1)^k {r-1 \choose k} {i-1 \choose r+t_1-1-k} {n-i \choose t_2+k} }}{{n \choose r+t_1+t_2}} \right] x_{i:n} \mbox{,}

where t_a represents the trimming level of the t_2-largest or t_1-smallest values, r represents the order of the L-moment, n represents the sample size, and x_{i:n} represents the ith sample order statistic (x_{1:n} ≤ x_{2:n} ≤ … ≤ x_{n:n}).

## Usage

 1 TLmom(x, order, trim=NULL, leftrim=NULL, rightrim=NULL, sortdata=TRUE) 

## Arguments

 x A vector of data values. order L-moment order to use in the computations. Default is 1 (the mean). trim Level of symmetrical trimming to use in the computations. Although NULL is in the argument list, the default is 0—the usual L-moment is returned. leftrim Level of trimming of the left-tail of the sample, which should be left to NULL if no or symmetrical trimming is used. rightrim Level of trimming of the right-tail of the sample, which should be left to NULL if no or symmetrical trimming is used. sortdata A logical switch on whether the data should be sorted. The default is TRUE.

## Value

An R list is returned.

 lambda The TL-moment of order=order, \hat{λ}^{(t_1,t_2)}_r where r is the moment order, t_1 is left-tail trimming, and t_2 is right-tail trimming. order L-moment order computed. Default is 1 (the mean). trim Level of symmetrical trimming used in the computation. leftrim Level of left-tail trimming used in the computation, which will equal trim if symmetrical trimming was used. rightrim Level of right-tail trimming used in the computation, which will equal trim if symmetrical trimming was used.

## Note

The presence of the sortdata switch can be dangerous. L-moment computation requires that the data be sorted into the “order statistics”. Thus the default behavior of sortdata=TRUE is required when the function is called on its own. In practice, this function would almost certainly not be used on its own because multiple trimmed L-moments would be needed. Multiple trimmed L-moments are best computed by TLmoms, which calls TLmom multiple times. The function TLmoms takes over the sort operation on the data and passes sortdata=FALSE to TLmom for efficiency. (The point of this discussion is that CPU time is not wasted sorting the data more than once.)

W.H. Asquith

## References

Elamir, E.A.H., and Seheult, A.H., 2003, Trimmed L-moments: Computational statistics and data analysis, v. 43, pp. 299–314.

TLmoms
 1 2 X1 <- rcauchy(30) TL <- TLmom(X1,order=2,trim=1)