
GLDPDFName:
The original parameterization, referred to as the RS generalized TukeyLambda distribution, given by Ramberg and Schmeiser has the percent point function
with , , , and denoting the location, the scale, and the two shape parameters, respectively. One drawback of this parameterization is that it does not define a valid probability distribution for certain values of the parameters. Futhermore, the regions that do not define a valid probability distribution are not simple. For this reason, Friemer, Mudholkar, Kollia, and Lin developed an alternative parameterization, referred to as the FMLKL generalized TukeyLambda distribution, that has the percent point function
with , , , and denoting the location, the scale, and the two shape parameters, respectively. Note that = 0 or = 0 results in division by zero in the above formula. If = 0, then
Likewise, if = 0, then
The advantage of the FMKL parameterization is that it defines a valid probability distribution for all real values of and . For this reason, Dataplot uses the FMKL parameterization. Although the probability density function does not exist in simple closed form, it can be computed from the following relationship:
with f, F, and G' denoting the probability density function, the cumulative distribution function, and the derivative of the percent point function, respectively. The derivative of the percent point function is also known as the sparsity function. Using the above, the probability density function for the standard generalized TukeyLambda distribution (FMKL parameterization) is
with F(x) denoting the generalized TukeyLambda cumulative distribution function. F(x) is computed by numerically inverting the percent point function. A few relevant properties for this distribution are:
<SUBSET/EXCEPT/FOR qualification> where <x> is a variable, number, or parameter; <l3> is a number or parameter that specifies the first shape parameter; <l4> is a number or parameter that specifies the second shape parameter; <loc> is a number or parameter that specifies the location parameter; <scale> is a number or parameter that specifies the scale parameter; <y> is a variable or a parameter (depending on what <x> is) where the computed generalized TukeyLambda pdf value is stored; and where the <SUBSET/EXCEPT/FOR qualification> is optional. Note that the location and scale parameters are optional.
LET Y = GLDPDF(X,0.5,0.2) PLOT GLDPDF(X,0.5,0.2) FOR X = 2 0.01 5
LET LAMBDA4 = <value> LET Y = GENERALIZED TUKEY LAMBDA RANDOM NUMBERS ... FOR I = 1 1 N GENERALIZED TUKEY LAMBDA PROBABILITY PLOT Y GENERALIZED TUKEY LAMBDA KOLMOGOROV SMIRNOV GOODNESS OF FIT Y GENERALIZED TUKEY LAMBDA CHISQUARE GOODNESS OF FIT Y The following commands can be used to estimate the shape parameters for the generalized TukeyLambda distribution:
LET LAMBDA32 = <value> LET LAMBDA41 = <value> LET LAMBDA42 = <value> GENERALIZED TUKEYLAMBDA PPCC PLOT Y The default values for LAMBDA31, LAMBDA32, LAMBDA41, and LAMBDA42 are 1, 5, 1 and 5, respectively. We are still resolving some issues with the GENERALIZED TUKEY LAMBDA KS PLOT command, so we recommend that you not use this for now. Alternatively, you can perform a least squares regression fit using the following commands:
LET N = SIZE Y LET PIN = UNIFORM ORDER STATISTIC MEDIANS FOR I = 1 1 N FIT Y = GLDPPF(PIN,L3,L4,ALOC,ASCALE)
Ramberg and Schmeiser (1974), "An Approximate Method for Generating Asymmetric Random Variables", Communications of the Association for Computing Machinery, 17, pp. 7882. Ozturk and Dale (1985), "Least Squares Estimation of the Parameters of the Generalized Lambda Distribution", Technometrics, Vol. 27, No. 1, pp. 8184. Friemer, Mudholkar, Kollia, and Lin (1988), "A Study of the Generalized Lambda Family", Communications in StatisticsTheory and Methods, 17, pp. 35473567. King and MacGillivray (1999), "A Starship Estimation Method for the Generalized Lambda Distributions", Australia and New Zealand Journal of Statistics, 41(3), pp. 353374. Karian and Dudewicz (2000), Fitting Statistical Distributions: The Generalized Bootstrap Methods, New York, Chapman & Hall. Su (2005), "A Discretized Approach to Flexibly Fit Generalized Lambda Distributions to Data", Journal of Modern Applied Statistical Methods, Vol. 4, No. 2, pp. 408424.
MULTIPLOT 4 4 MULTIPLOT SCALE FACTOR 2.5 MULTIPLOT CORNER COORDINATES 0 0 100 95 LABEL CASE ASIS X1LABEL X Y1LABEL Probability Density X1LABEL DISPLACEMENT 16 Y1LABEL DISPLACEMENT 18 TITLE DISPLACEMENT 2 XLIMITS 10 10 LET LAMBDA3 = DATA 0.5 0 0.5 2 LET LAMBDA4 = DATA 0.5 0 0.5 2 LOOP FOR K = 1 1 4 LET L3 = LAMBDA3(K) LET XLOW = 10 IF L3 > 0 LET XLOW = 1/L3 END OF IF LOOP FOR L = 1 1 4 LET L4 = LAMBDA4(L) LET XUPP = 10 IF L4 > 0 LET XUPP = 1/L4 END OF IF TITLE L3 = ^L3, L4 = ^L4 PLOT GLDPDF(X,L3,L4) FOR X = XLOW 0.01 XUPP END OF LOOP END OF LOOP END OF MULTIPLOT MOVE 50 97 JUSTIFICATION CENTER CASE ASIS TEXT Generalized TukeyLambda Distributions
Date created: 4/14/2006 