lmomco (version 0.6)

pwm.ub: Unbiased Sample Probability-Weighted Moments

Description

Unbiased sample Probability-Weighted Moments (PWMs) are computed from a sample. The first five $\beta_r$'s are computed. The unbiased PWMs are computed by the the plotting-position formula by a call to pwm.pp{data,A=0,B=0}. The plotting-position formula is

$$p_i = \frac{i+A}{n+B} \mbox{,}$$

where $p_i$ is the nonexceedance probability $F$ of the $i$th ascending data values. The parameters $A$ and $B$ together specify the plotting position type, and $n$ is the sample size. The PWMs are computed by

$$\beta_r = n^{-1}\sum_{i=1}^{n}p_i^r \times x_{j:n} \mbox{,}$$

where $x_{j:n}$ is the $j$th order statistic $x_{1:n} \le x_{2:n} \le x_{j:n} \dots \le x_{n:n}$ of random variable X, and $r$ is $0, 1, 2, \dots$.

Usage

pwm.ub(x)

Arguments

x

A vector of data values.

Value

• An R list is returned.
• BETA0
  The first PWM---equal to the arithmetic mean.
• BETA1
  The second PWM.
• BETA2
  The third PWM.
• BETA3
  The fourth PWM.
• BETA4
  The fifth PWM.

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, vol. 15, p. 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, 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.

See Also

pwm.pp, pwm.gev, pwm2lmom

Examples

pwm <- pwm.ub(rnorm(20))