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 \(i\)th 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 \(j\)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 \(0, 1, 2, \dots\) for the PWM order.
pwm.pp(x, pp=NULL, A=NULL, B=NULL, a=0, nmom=5, sort=TRUE)An R
list is returned.
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 of the PWMs: “pwm.pp”.
A vector of data values.
An optional vector of nonexceedance probabilities. If present then A and B or a are ignored.
A value for the plotting-position formula. If A and B are both zero then the unbiased PWMs are computed through pwm.ub.
Another value for the plotting-position formula. If A and B are both zero then the unbiased PWMs are computed through pwm.ub.
A single plotting position coefficient from which, if not NULL, \(A\) and \(B\) will be internally computed;
Number of PWMs to return.
Do the data need sorting? The computations require sorted data. This option is provided to optimize processing speed if presorted data already exists.
W.H. Asquith
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.
pwm.ub, pwm.gev, pwm2lmom
pwm <- pwm.pp(rnorm(20), A=-0.35, B=0)
X <- rnorm(20)
pwm <- pwm.pp(X, pp=pp(X)) # weibull plotting positions
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