Two special cases are
The L-moments are defined as
with denoting the rth shifted Legendre ploynomials. The L-moment estimators are linear combinations of order statistics.
L-moments are related to probability weighted moments by the equation
with denoting the binomal coefficient.
The L-moment ratios are defined to be
This command returns the L-moment estimates for orders 1 and 2 and L-moment ratios for higher orders.
For an ordered sample x1:n <= x2:n <= ... <= xn:n, unbiased estimators of
The primary use of probability weighted moments and L-moments is in the estimation of parameters for a probability distribution. Estimates based on probability weighted moments and L-moments are generally superior to standard moment-based estimates. The L-moment estimators have some desirable properties for parameter estimation. In particular, they are robust with respect to outliers and their small sample bias tends to be small. L-moment estimators can often be used when the maximum likelihood estimates are unavailable, difficult to compute, or have undesirable properties. They may also be used as starting values for maximum likelihood estimates.
Estimation methods based on L-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).
where <x> is the response variable;
<nmom> is the number of L-moments that will be generated;
<y> is a variable where the computed L-moments are saved;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
LET XMOM = L MOMENTS Y 4 SUBSET Y > 0
Hoskings software provides L-moment estimators for 11 different distributions. Dataplot currently implements the L-moment estimates for the generalized Pareto and generalized extreme value distributions. To obtain these estimates, enter the commands
GENERALIZED EXTREME VALUE MAXIMUM LIKELIOHOOD Y
We plan to implement the L-moment estimators for additional distributions in future releases of Dataplot.
"L-moments: Analysis and Estimation of Distribution using Linear Combinations of Order Statistics", Hoskings, Journal of the Royal Statistical Society, Series B, 52, 1990, pp. 105-124.
"The Estimation of Extreme Quantiles of Wind Velocity Using L-Moments in the Peaks-Over-Threshold Approach", Pandey, Van Gelder, and Vrijling, Structural Safety, 23, 2001, pp. 179-192.
"Probability Weighted Moments: Definition and Relation to Parameters of Several Distributions Expressable in Inverse Form", Greenwood, Landwehr, Matalas, and Wallis, Water Resources Research, 15, 1979, 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.
"Extreme Value and Related Models with Applications in Engineering and Science", Castillo, Hadi, Balakrishnan, and Sarabia, Wiley, 2005, pp. 117-119.
LET GAMMA = -0.3 LET Y = GENERALIZED PARETO RANDOM NUMBERS FOR I = 1 1 100 LET Y = 5*Y LET NMOM = 3 LET XMOM = L MOMENTS Y NMOM PRINT XMOM GENERALIZED PARETO MLE YThe following output is generated
VARIABLES--XMOM 0.4160070E+01 -0.3669064E-01 -0.4920776E+00 GENERALIZED PARETO PARAMETER ESTIMATION: SUMMARY STATISTICS: NUMBER OF OBSERVATIONS = 100 SAMPLE MEAN (ALL DATA) = 4.160070 SAMPLE VARIANCE (ALL DATA) = 9.603471 SAMPLE MINIMUM = 0.7464492E-02 SAMPLE MAXIMUM = 12.82312 USER-SPECIFIED THRESHOLD = 0.000000 METHOD OF MOMENTS: (VALID IF SHAPE PARAMETER < 1) SAMPLE MEAN (AFTER SUBTRACT LOCATION) = 4.152607 SAMPLE VARIANCE (AFTER SUBTRACT LOCATION) = 9.603471 ESTIMATE OF LOCATION = 0.7463745E-02 ESTIMATE OF SCALE = 5.804547 ESTIMATE OF GAMMA = -0.3978080 VARIANCE OF GAMMA = 0.1448942E-01 VARIANCE OF SCALE = 0.6819228 COVARIANCE OF GAMMA AND SCALE = 0.8959930E-01 L-MOMENTS: (L-MOMENT ESTIMATES WORK BEST FOR VALUES OF SHAPE PARAMETER IN (-0.5,0.5)) FIRST SAMPLE L-MOMENT = 4.160070 SECOND SAMPLE L-MOMENT = 1.718239 THIRD SAMPLE L-MOMENT = 0.2119006 ESTIMATE OF LOCATION = 0.2070879 ESTIMATE OF SCALE = 5.141252 ESTIMATE OF SHAPE = -0.3006007 ELEMENTAL PERCENTILE METHOD: ESTIMATE OF LOCATION = 0.7463745E-02 ESTIMATE OF SCALE = 5.473905 ESTIMATE OF GAMMA = -0.4396029 MAXIMUM LIKELIHOOD: (MAXIMUM LIKELIHOOD ESTIMATES DO NOT EXIST IF SHAPE PARAMTER < -1 AND MAY PERFORM POORLY IF < -0.5) ESTIMATE OF LOCATION = 0.7463745E-02 ESTIMATE OF SCALE = 3.383862 ESTIMATE OF GAMMA = -0.2713313 VARIANCE OF GAMMA = 0.1616283E-01 VARIANCE OF SCALE = 0.2911481 COVARIANCE OF GAMMA AND SCALE = 0.4302010E-01 NOTE: FOR THE GENERALIZED PARETO DISTRIBUTION, LARGE SAMPLE SIZES MAY BE REQUIRED FOR THE NORMAL APPROXIMATIONS TO BE ACCURATE (E.G., > 500). CONFIDENCE INTERVAL FOR SCALE PARAMETER NORMAL APPROXIMATION CONFIDENCE LOWER UPPER VALUE (%) LIMIT LIMIT ------------------------------------------- 50.000 3.01992 3.74780 75.000 2.76315 4.00457 90.000 2.49633 4.27139 95.000 2.32630 4.44142 99.000 1.99399 4.77373 99.900 1.60836 5.15936 CONFIDENCE INTERVAL FOR SHAPE PARAMETER NORMAL APPROXIMATION CONFIDENCE LOWER UPPER VALUE (%) LIMIT LIMIT ------------------------------------------- 50.000 -.357081 -.185581 75.000 -.417579 -.125084 90.000 -.480447 -.622159E-01 95.000 -.520508 -.221549E-01 99.000 -.598805 0.561420E-01 99.900 -.689665 0.147002
Date created: 1/6/2006