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)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.
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”.
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.
# NOT RUN {
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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