lmom.ub: Unbiased Sample L-moments by Direct Sample Estimators

Description

Unbiased sample L-moments are computed for a vector using the direct sample estimation method as opposed to the use of sample probability-weighted moments. The L-moments are the ordinary L-moments and not the trimmed L-moments (see TLmoms). The mean, L-scale, coefficient of L-variation ($\tau$, LCV, L-scale/mean), L-skew ($\tau_3$, TAU3, L3/L2), L-kurtosis ($\tau_4$, TAU4, L4/L2), and $\tau_5$ (TAU5, L5/L2) are computed. In conventional nomenclature, the L-moments are $$ \hat{\lambda}_1 = \mbox{L1} = \mbox{mean, }$$ $$ \hat{\lambda}_2 = \mbox{L2} = \mbox{L-scale, }$$ $$ \hat{\lambda}_3 = \mbox{L3} = \mbox{third L-moment, }$$ $$ \hat{\lambda}_4 = \mbox{L4} = \mbox{fourth L-moment, and }$$ $$ \hat{\lambda}_5 = \mbox{L5} = \mbox{fifth L-moment. }$$

The L-moment ratios are $$ \hat{\tau} = \mbox{LCV} = \lambda_2/\lambda_1 = \mbox{coefficient of L-variation, }$$ $$ \hat{\tau}_3 = \mbox{TAU3} = \lambda_3/\lambda_2 = \mbox{L-skew, }$$ $$ \hat{\tau}_4 = \mbox{TAU4} = \lambda_4/\lambda_2 = \mbox{L-kurtosis, and}$$ $$ \hat{\tau}_5 = \mbox{TAU5} = \lambda_5/\lambda_2 = \mbox{not named.}$$

It is common amongst practitioners to lump the L-moment ratios into the general term “L-moments” and remain inclusive of the L-moment ratios. For example, L-skew then is referred to as the 3rd L-moment when it technically is the 3rd L-moment ratio. The first L-moment ratio has no definition; the lmoms function uses the NA of R in its vector representation of the ratios.

The mathematical expression for sample L-moment computation is shown under TLmoms. The formula jointly handles sample L-moment computation and sample TL-moment computation.

Usage

lmom.ub(x)

Value

An R

list is returned.

L1: Arithmetic mean.
L2: L-scale---analogous to standard deviation (see also gini.mean.diff.
LCV: coefficient of L-variation---analogous to coe. of variation.
TAU3: The third L-moment ratio or L-skew---analogous to skew.
TAU4: The fourth L-moment ratio or L-kurtosis---analogous to kurtosis.
TAU5: The fifth L-moment ratio.
L3: The third L-moment.
L4: The fourth L-moment.
L5: The fifth L-moment.
source: An attribute identifying the computational source of the L-moments: “lmom.ub”.

Arguments

x: A vector of data values.

Author

W.H. Asquith

Details

The L-moment ratios ($\tau$, $\tau_3$, $\tau_4$, and $\tau_5$) are the primary higher L-moments for application, such as for distribution parameter estimation. However, the actual L-moments ($\lambda_3$, $\lambda_4$, and $\lambda_5$) are also reported. The implementation of lmom.ub requires a minimum of five data points. If more or fewer L-moments are needed then use the function lmoms.

References

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.

Hosking, J.R.M., 1990, L-moments---Analysis and estimation of distributions using linear combinations of order statistics: Journal of the Royal Statistical Society, Series B, v. 52, pp. 105--124.

Wang, Q.J., 1996, Direct sample estimators of L-moments: Water Resources Research, v. 32, no. 12., pp. 3617--3619.

Examples

Run this code

lmr <- lmom.ub(c(123,34,4,654,37,78))
lmorph(lmr)
lmom.ub(rnorm(100))

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