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GLDCDFName:
The original parameterization, referred to as the RS generalized Tukey-Lambda 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 Tukey-Lambda 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. The generalized Tukey-Lambda cumulative distribution function 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 Tukey-Lambda cdf value is stored; and where the <SUBSET/EXCEPT/FOR qualification> is optional. Note that the location and scale parameters are optional.
LET Y = GLDCDF(X,0.5,0.2) PLOT GLDCDF(X,0.5,0.2) FOR X = -2 0.01 5
Ramberg and Schmeiser (1974), "An Approximate Method for Generating Asymmetric Random Variables", Communications of the Association for Computing Machinery, 17, pp. 78-82. Ozturk and Dale (1985), "Least Squares Estimation of the Parameters of the Generalized Lambda Distribution", Technometrics, Vol. 27, No. 1, pp. 81-84. Friemer, Mudholkar, Kollia, and Lin (1988), "A Study of the Generalized Lambda Family", Communications in Statistics-Theory and Methods, 17, pp. 3547-3567. King and MacGillivray (1999), "A Starship Estimation Method for the Generalized Lambda Distributions", Australia and New Zealand Journal of Statistics, 41(3), pp. 353-374. 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. 408-424.
MULTIPLOT 4 4 MULTIPLOT SCALE FACTOR 2.5 MULTIPLOT CORNER COORDINATES 0 0 100 95 LABEL CASE ASIS X1LABEL X Y1LABEL Probability 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 GLDCDF(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 Tukey-Lambda Distributions
Date created: 4/14/2006 |