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

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

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)

Arguments

x	A vector of data values.

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.

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”.

Note

The lmom.ub function was among the first functions written for lmomco and actually written before lmomco was initiated. The ub was to be contrasted with plotting-position-based estimation methods: pwm.pp \rightarrow pwm2lmom. Further, at the time of development the radical expansion of lmomco beyond the Hosking (1996) FORTRAN libraries was not anticipated. The author now exclusively uses lmoms but the numerical results should be identical. The direct sample estimator algorithm by Wang (1996) is used in lmom.ub and a more generalized algorithm is associated with lmoms.

Author(s)

W.H. Asquith

Source

The Perl code base of W.H. Asquith

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.

See Also

lmom2pwm, pwm.ub, pwm2lmom, lmoms, lmorph

Examples

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

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