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Dataplot Vol 1 Auxiliary Chapter

GLDPDF

Name:
    GLDPDF (LET)
Type:
    Library Function
Purpose:
    Compute the generalized Tukey-Lambda probability density function.
Description:
    The generalized Tukey-Lambda distribution is defined in terms of its percent point function. Note that there are two parameterizations of this distribution in the literature.

    The original parameterization, referred to as the RS generalized Tukey-Lambda distribution, given by Ramberg and Schmeiser has the percent point function

      G(p;lambda1,lambda2,lambda3,lambda4) = 
lambda1 + (p^lambda3 - (1-p)^lambda4)/lambda2

    with lambda1, lambda2, lambda3, and lambda4 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

      G(p;lambda1,lambda2,lambda3,lambda4) = 
lambda1 + (1/lambda2)*
[(p^lambda3-1)/lambda3 - ((1-p)^lambda4-1)/lambda4]   
lambda2 > 0

    with lambda1, lambda2, lambda3, and lambda4 denoting the location, the scale, and the two shape parameters, respectively.

    Note that lambda3 = 0 or lambda4 = 0 results in division by zero in the above formula.

    If lambda3 = 0, then

      (p^lambda3 - 1)/lambda3 = log(p)

    Likewise, if lambda4 = 0, then

      ((1-p)**lambda4 - 1)/lambda4 = log(1-p)

    The advantage of the FMKL parameterization is that it defines a valid probability distribution for all real values of lambda3 and lambda4. 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:

      f(x) = 1/G'(F(x))

    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 Tukey-Lambda distribution (FMKL parameterization) is

      f(x;lambda3,lambda4) = 1/[F(x)**(lambda3-1) + (1-F(x))**(lambda4-1)]

    with F(x) denoting the generalized Tukey-Lambda cumulative distribution function. F(x) is computed by numerically inverting the percent point function.

    A few relevant properties for this distribution are:

    1. If lambda3 = lambda4, then the generalized Tukey-Lambda distribution reduces to the symmetric Tukey-Lambda distribution.

    2. The lambda3 parameter controls the behavior of the lower tail. If lambda3 > 0, then the distribution is bounded below at -1/lambda3. If lambda3 ≤ 0, then the distribution is unbounded below.

    3. The lambda4 parameter controls the behavior of the upper tail. If lambda4 > 0, then the distribution is bounded above at 1/lambda4. If lambda4 ≤ 0, then the distribution is unbounded above.

    4. The kth moment is finite only if min(lambda3, lambda4) > -1/k. So this distribution has finite mean only if min(lambda3, lambda4) > -1 and finite variance only if min(lambda3, lambda4) > -0.5.
Syntax:
    LET <y> = GLDPDF(<x>,<l3>,<l4>,<loc>,<scale>)
                            <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 pdf value is stored;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    Note that the location and scale parameters are optional.

Examples:
    LET A = GLDPDF(0.9,0.5,0.2)
    LET Y = GLDPDF(X,0.5,0.2)
    PLOT GLDPDF(X,0.5,0.2) FOR X = -2 0.01 5
Note:
    Generalized Tukey-Lambda random numbers, probability plots, and goodness of fit tests can be generated with the commands:

      LET LAMBDA3 = <value>
      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 CHI-SQUARE GOODNESS OF FIT Y

    The following commands can be used to estimate the shape parameters for the generalized Tukey-Lambda distribution:

      LET LAMBDA31 = <value>
      LET LAMBDA32 = <value>
      LET LAMBDA41 = <value>
      LET LAMBDA42 = <value>
      GENERALIZED TUKEY-LAMBDA 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 Y = SORT Y
      LET N = SIZE Y
      LET PIN = UNIFORM ORDER STATISTIC MEDIANS FOR I = 1 1 N
      FIT Y = GLDPPF(PIN,L3,L4,ALOC,ASCALE)
Default:
    None
Synonyms:
    None
Related Commands:
    GLDCDF = Compute the generalized Tukey-Lambda cumulative distribution function.
    GLDPDF = Compute the generalized Tukey-Lambda percent point function.
    LAMPPF = Compute the Tukey-Lambda probability density function.
    GHPPF = Compute the g-and-h probability density function.
    JSUPPF = Compute the Johnson SU probability density function.
    JSBPPF = Compute the Johnson SB probability density function.
    PPCC PLOT = Generate a ppcc plot.
    PROBABILITY PLOT = Generate a probability plot.
Reference:
    Ramberg and Schmeiser (1972), "An Approximate Method for Generating Symmetric Random Variables", Communications of the Association for Computing Machinery, 15, pp. 987-990.

    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.

Applications:
    Distributional Modeling
Implementation Date:
    2006/3
Program:
     
    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 Tukey-Lambda Distributions
        
    plot generated by sample program

Date created: 4/14/2006
Last updated: 4/14/2006
Please email comments on this WWW page to alan.heckert@nist.gov.