PROBABILITY WEIGHTED MOMENTS

PROBABILITY WEIGHTED MOMENTS (LET)

Let Subcommand

Compute the probability weighted moments of a variable.

Given a random variable X with a cumulative distribution function F, the probability weighted moments are defined to be:
\( M(p,r,s) = E[ X^{p} \{F(X)\}^{r} \{1 - F(X)\}^{s} ] \)

Two special cases are

\( \alpha_{r} = M(1,0,r) = E[ X \{1 - F(X)\}^{r} ] \)

\( \beta_{r} = M(1,r,0) = E[ X \{F(X)\}^{r} ] \)

For an ordered sample x_1:n <= x_2:n <= ... <= x_n:n, unbiased estimators of

\( a_{r} = \frac{1}{n} \sum_{j=1}^{n} { \frac{(n-j)(n-j-1) \ldots (n-j+1)} {(n-1)(n-2) \ldots (n-r)} } x_{j:n} \)
\( b_{r} = \frac{1}{n} \sum_{j=1}^{n} { \frac{(j-1)(j-2) \ldots (j-r)} {(n-1)(n-2) \ldots (n-r)} } x_{j:n} \)

The primary use of probability weighted moments (and the related L-moments) is in the estimation of parameters for a probability distribution. For a more detailed description of probability weighted moments and L-moments, 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 moment-based 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 L-moment based estimates for the maximum likelihood estimates for the generalized Pareto and the generalized extreme value distributions).

LET <y> = PROBABILITY WEIGHTED MEANS <x> <nmom>
            <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.

LET <y> = BETA PROBABILITY WEIGHTED MEANS <x> <nmom>
            <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 5
LET PWM = PROBABILITY WEIGHTED MEANS Y 4 SUBSET Y > 0

Dataplot computes probability weighted moments using the SAMPWM routine written by Hoskings and documented in "Research Report: Fortran Routines for use with the Method of L-Moments" (see the Reference section below).

This routine is available from the statlib archive at the URL

http://lib.stat.cmu.edu/general/lmoments

L-moments are linear combinations of probability weighted moments and are frequently used in the application of probability weighted moments.

If the parameter is omitted from the command, the first four sample probability weighted moments are computed.

By default, Dataplot computes the alpha probability weighted moments.

None

L MOMENTS	= Compute the sample L-moments of a variable.
MAXIMUM LIKELIHOOD	= Compute maximum likelihood estimates for a probability distribution.

"Research Report: Fortran Routines for use with the Method of L-Moments", J. R. M. Hosking, IBM Research Division, T. J. Watson Research Center, Yorktown Heights, NY 10598, 6/2000.

"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, 1049-1054.

"Estimation of the Generalized Extreme Value Distribution by the Method of Probability-Weighte Moments", Hosking, Wallis, and Wood, Technometrics, 27, 1985, 251-261.

"Probability Weighted Moments Compared with Some Traditional Techniques in Estimating Gumbel Parameters and Quantiles", Landwehr, Matalas, Wallis, Water Resources Research, 15, (1979a), 1055-1064.

Distributional Modeling

2005/12

 
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