lmomco (version 2.4.14)

pwm: 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)

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”.

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.

Author

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