TLmom: A Sample Trimmed L-moment

In wasquith/lmomco: L-Moments, Censored L-Moments, Trimmed L-Moments, L-Comoments, and Many Distributions

A Sample Trimmed L-moment

Description

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

\hat{\lambda}^{(t_1,t_2)}_r = \frac{1}{r}\sum^{n-t_2}_{i=t_1+1} \left[ \frac{\sum\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} \le x_{2:n} \le \dots \le x_{n:n}).

Usage

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{\lambda}^{(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.)

Author(s)

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.

See Also

TLmoms

Examples

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

wasquith/lmomco documentation built on Nov. 13, 2024, 4:53 p.m.