pwm: Unbiased Sample Probability-Weighted Moments

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

Unbiased Sample Probability-Weighted Moments

Description

Unbiased sample probability-weighted moments (PWMs) are computed from a sample. The \beta_r's are computed using

\beta_r = n^{-1}\sum^n_{j=1} {j-1 \choose r} x_{j:n}\mbox{.}

Usage

pwm(x, nmom=5, sort=TRUE)

Arguments

x	A vector of data values.
nmom	Number of PWMs to return (r = nmom - 1).
sort	Do the data need sorting? The computations require sorted data. This option is provided to optimize processing speed if presorted data already exists.

Value

An R list is returned.

betas	The PWMs. Note that convention is the have a \beta_0, but this is placed in the first index i=1 of the betas vector.
source	Source of the PWMs: “pwm”.

Author(s)

W.H. Asquith

References

Greenwood, J.A., Landwehr, J.M., Matalas, N.C., and Wallis, J.R., 1979, Probability weighted moments—Definition and relation to parameters of several distributions expressable in inverse form: Water Resources Research, v. 15, pp. 1,049–1,054.

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.

See Also

lmoms, pwm2lmom, pwm

Examples

# Data listed in Hosking (1995, table 29.2, p. 551)
H <- c(3,4,5,6,6,7,8,8,9,9,9,10,10,11,11,11,13,13,13,13,13,
       17,19,19,25,29,33,42,42,51.9999,52,52,52)
# 51.9999 was really 52, but a real non censored data point.
z <-  pwmRC(H,52,checkbetas=TRUE)
str(z)
# Hosking(1995) reports that A-type L-moments for this sample are
# lamA1=15.7 and lamAL-CV=.389, and lamAL-skew=.393
pwm2lmom(z$Abetas)
# WHA gets 15.666, 0.3959, and 0.4030

# See p. 553 of Hosking (1995)
# Data listed in Hosking (1995, table 29.3, p. 553)
D <- c(-2.982, -2.849, -2.546, -2.350, -1.983, -1.492, -1.443,
       -1.394, -1.386, -1.269, -1.195, -1.174, -0.854, -0.620,
       -0.576, -0.548, -0.247, -0.195, -0.056, -0.013,  0.006,
        0.033,  0.037,  0.046,  0.084,  0.221, 0.245, 0.296)
D <- c(D,rep(.2960001,40-28)) # 28 values, but Hosking mentions
                              # 40 values in total
z <-  pwmRC(D,.2960001)
# Hosking reports B-type L-moments for this sample are
# lamB1 = -.516 and lamB2 = 0.523
pwm2lmom(z$Bbetas)
# WHA gets -.5162 and 0.5218

lmomco documentation built on May 29, 2024, 10:06 a.m.