Generalized Extreme Value plotting-position probability-weighted moments (PWMs) are computed from a sample. The first five \(\beta_r\)'s are computed by default. The plotting-position formula for the Generalized Extreme Value distribution is
$$pp_i = \frac{i-0.35}{n} \mbox{,}$$
where \(pp_i\) is the nonexceedance probability \(F\) of the \(i\)th ascending values of the sample of size \(n\). 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 \(0, 1, 2, \dots\). Finally, pwm.gev dispatches to pwm.pp(data,A=-0.35,B=0) and does not have its own logic.
pwm.gev(x, 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.gev”.
A vector of data values.
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.pp, pwm2lmom