lmomco (version 1.7.3)

pwm.pp: Plotting-Position Sample Probability-Weighted Moments

Description

The sample Probability-Weighted Moments (PWMs) are computed from the plotting positions of the data. The first five $\beta_r$'s are computed by default. The plotting-position formula for a sample size of $n$ is

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

where $pp_i$ is the nonexceedance probability $F$ of the $i$th ascending data values. An alternative form of the plotting position equation is

$$p_i = \frac{i + a}{n + 1 - 2a}\mbox{,}$$

where $a$ is a single plotting position coefficient. Having $a$ provides $A$ and $B$, therefore the parameters $A$ and $B$ together specify the plotting-position type. 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.pp(x, pp=NULL, A=NULL, B=NULL, a=0, nmom=5, sort=TRUE)

Arguments

x

A vector of data values;

pp

An optional vector of nonexceedance probabilities. If present then A and B or a are ignored;

A

A value for the plotting-position formula. If A and B are both zero then the unbiased PWMs are computed through pwm.ub;

B

Another value for the plotting-position formula. If A and B are both zero then the unbiased PWMs are computed through pwm.ub;

a

A single plotting position coefficient from which, if not NULL, $A$ and $B$ will be internally computed;

nmom

Number of PWMs to return;

sort

Does 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.pp

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.ub, pwm.gev, pwm2lmom

Examples

pwm <- pwm.pp(rnorm(20), A=-0.35, B=0)

X <- rnorm(20)
pwm <- pwm.pp(X, pp=pp(X)) # weibull plotting positions