lmomco (version 0.88)

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, L4/L2) are computed. In conventional nomenclature, the L-moments are

$$\hat{\lambda}_1 = \mbox{L1} = \mbox{mean, }$$ $$\hat{\lambda}_2 = \mbox{L2} = \mbox{L-scale, }$$ $$\hat{\lambda}_3 = \mbox{L3} = \mbox{third L-moment, }$$ $$\hat{\lambda}_4 = \mbox{L4} = \mbox{fourth L-moment, }$$ $$\hat{\lambda}_5 = \mbox{L5} = \mbox{fifth L-moment, }$$ $$\hat{\tau} = \mbox{LCV} = \lambda_2/\lambda_1 = \mbox{coefficient of L-variation, }$$ $$\hat{\tau}_3 = \mbox{TAU3} = \lambda_3/\lambda_2 = \mbox{L-skew, }$$ $$\hat{\tau}_4 = \mbox{TAU4} = \lambda_4/\lambda_2 = \mbox{L-kurtosis, and}$$ $$\hat{\tau}_5 = \mbox{TAU5} = \lambda_5/\lambda_2 = \mbox{not named.}$$

Usage

lmom.ub(x)

Arguments

x

A vector of data values.

Value

• An R list is returned.
• L1
  Arithmetic mean.
• L2
  L-scale---analogous to standard deviation.
• 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.

source

The Perl code base of 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. This implementation of L-moment calculation requires a minimum of five data points.

References

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, vol. 52, p. 105--124.

Hosking, J.R.M., 1996, FORTRAN routines for use with the method of L-moments: Version 3, IBM Research Report RC20525, T.J. Watson Research Center, Yorktown Heights, New York.

Hosking, J.R.M. and Wallis, J.R., 1997, Regional frequency analysis---An approach based on L-moments: Cambridge University Press.

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

See Also

lmom2pwm, pwm.ub, pwm2lmom, lmoms, and lmorph

Examples

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