
PROBABILITY WEIGHTED MOMENTSName:
Two special cases are
\( \beta_{r} = M(1,r,0) = E[ X \{F(X)\}^{r} ] \) For an ordered sample x_{1:n} <= x_{2:n} <= ... <= x_{n:n}, unbiased estimators of
The primary use of probability weighted moments (and the related Lmoments) is in the estimation of parameters for a probability distribution. For a more detailed description of probability weighted moments and Lmoments, see the papers listed in the Reference section below (in particular, the papers by Hoskings). Estimates based on probability weighted moments are often considered to be superior to standard momentbased estimates. They are sometimes used when maximum likelihood estimates are unavailable or difficult to compute. They may also be used as starting values for maximum likelihood estimates. Estimation methods based on probability weighted moments are discussed in the papers listed in the Reference section below (Dataplot generates Lmoment based estimates for the maximum likelihood estimates for the generalized Pareto and the generalized extreme value distributions).
<SUBSET/EXCEPT/FOR qualification> where <x> is the response variable; <nmom> is the number of probability weighted moments that will be generated; <y> is a variable where the computed probability weighted moments are saved; and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax computes the alpha probability weighted moments.
<SUBSET/EXCEPT/FOR qualification> where <x> is the response variable; <nmom> is the number of probability weighted moments that will be generated; <y> is a variable where the computed probability weighted moments are saved; and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax computes the beta probability weighted moments.
LET PWM = PROBABILITY WEIGHTED MEANS Y 4 SUBSET Y > 0
This routine is available from the statlib archive at the URL
By default, Dataplot computes the alpha probability weighted moments.
"Probability Weighted Moments: Definition and Relation to Parameters of Several Distributions Expressable in Inverse Form", Greenwood, Landwehr, Matalas, and Wallis, Water Resources Research, 15, 1079, 10491054. "Estimation of the Generalized Extreme Value Distribution by the Method of ProbabilityWeighte Moments", Hosking, Wallis, and Wood, Technometrics, 27, 1985, 251261. "Probability Weighted Moments Compared with Some Traditional Techniques in Estimating Gumbel Parameters and Quantiles", Landwehr, Matalas, Wallis, Water Resources Research, 15, (1979a), 10551064.
LET GAMMA = 0.3 LET Y = GENERALIZED PARETO RANDOM NUMBERS FOR I = 1 1 100 LET Y = 5*Y + 2 LET PROBMOME = PROBABILITY WEIGHTED MOMENTS Y  
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Date created: 12/02/2005 