pwm.pp {lmomco}

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

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

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

pp_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}pp_i^r \times x_{j:n} \mbox{,}

where x_{j:n} is the jth 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 for the PWM order.

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

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.pp”.

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

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