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Dataplot Vol 2 Vol 1

GL5PDF

Name:
    GL5PDF (LET)
Type:
    Library Function
Purpose:
    Compute the type 5 generalized logistic probability density function with shape parameter alpha
Description:
    Johnson, Kotz, and Balakrishnan (see Reference Section below) define five different versions of the generalized logistic distribution. The first four are referred to as type 1, type 2, type 3, and type 4. The fifth is a version used by Hosking. Although they do not give this version a specific name, Dataplot refers to it as either the generalized logistic type 5 or the Hosking generalized logistic distribution.

    The standard form of the Hosking generalized logistic distribution has the probability density function:

      f(x;alpha) = (1 - alpha*x)**((1/alpha)-1)/
[1 + (1 - alpha*x)**(1/alpha)]**2   
x <= 1/alpha  when alpha > 0    x >= 1/alpha  when alpha < 0

    The general form of the Hosking generalized logistic probability density function can be obtained by replacing x in the above formula with (x-loc)/scale.

Syntax:
    LET <y> = GL5PDF(<x>,<alpha>,<loc>,<scale>)
                            <SUBSET/EXCEPT/FOR qualification>
    where <x> is a variable, number or parameter;
                <alpha> is a number or parameter that specifies the value of the first shape parameter;
                <loc> is a number or parameter that specifies the value of the location parameter;
                <scale> is a number or parameter that specifies the value of the scale parameter;
                <y> is a variable or a parameter (depending on what <x> is) where the computed generalized logistic type 5 pdf value is stored;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    The location and scale parameters are optional.

Examples:
    LET A = GL5PDF(-3,2)
    LET X2 = GL5PDF(X1,ALPHA)
    PLOT GL5PDF(X,2) FOR X = -5 0.01 0.5
Note:
    For this parameterization, alpha = 0 is equivalent to the logistic distribution.
Note:
    Hosking generalized logistic random numbers, probability plots, and goodness of fit tests can be generated with the commands:

      LET ALPHA = <value>
      LET Y = HOSKING GENERALIZED LOGISTIC RANDOM NUMBERS ...
                  FOR I = 1 1 N
      HOSKING GENERALIZED LOGISTIC PROBABILITY PLOT Y
      HOSKING GENERALIZED LOGISTIC KOLMOGOROV SMIRNOV ...
                  GOODNESS OF FIT Y
      HOSKING GENERALIZED LOGISTIC CHI-SQUARE ...
                  GOODNESS OF FIT Y

    The following commands can be used to estimate the shape parameter for the Hosking generalized logistic distribution:

      LET ALPHA1 = <value>
      LET ALPHA2 = <value>
      HOSKING GENERALIZED LOGISTIC PPCC PLOT Y
      HOSKING GENERALIZED LOGISTIC KS PLOT Y

    The default values for ALPHA1 and ALPHA2 are -2 and 2, respectively.

    Bootstrap samples for these plots can be obtained with the commands

      LET ALPHA1 = <value>
      LET ALPHA2 = <value>
      BOOTSTRAP GENERALIZED LOGISTIC TYPE 2 PLOT Y
      BOOTSTRAP GENERALIZED LOGISTIC TYPE 2 KS PLOT Y

    Alternatively, L-moment based estimates can be obtained with the command

      GENERALIZED LOGISTIC MLE Y

    Note that fitting becomes more problematic the further that the absolute value of alpha is from zero. The greater the absolute value of alpha, the greater the occurence of extreme values which distort the fitting procedures.

    Some informal simulations showed good performance for an absolute value less than 1 (excellent for |alpha| ≤ 0.5). However, the performance rapidly declines as |alpha| gets larger than 1. For that reason, be sure to apply fitting diagnostics (probability plots, goodness of fit tests) when fitting this distribution.

Default:
    None
Synonyms:
    TYPE 5 can be used as a synonym for HOSKING. For example, the following commands are equivalent:

      LET Y = HOSKING GENERALIZED LOGISTIC RANDOM NUMBERS ...
                  FOR I = 1 1 100
      LET Y = GENERALIZED LOGISTIC HOSKING RANDOM NUMBERS ...
                  FOR I = 1 1 100
      LET Y = TYPE 2 GENERALIZED LOGISTIC RANDOM NUMBERS ...
                  FOR I = 1 1 100
      LET Y = TYPE II GENERALIZED LOGISTIC RANDOM NUMBERS ...
                  FOR I = 1 1 100
      LET Y = GENERALIZED LOGISTIC TYPE 2 RANDOM NUMBERS ...
                  FOR I = 1 1 100
      LET Y = GENERALIZED LOGISTIC TYPE II RANDOM NUMBERS ...
                  FOR I = 1 1 100
Related Commands:
    GL5CDF = Compute the generalized logistic type 5 (Hosking) cumulative distribution function.
    GL5PPF = Compute the generalized logistic type 5 (Hosking) percent point function.
    GLOPDF = Compute the generalized logistic type 1 probability density function.
    GL2PDF = Compute the generalized logistic type 2 probability density function.
    GL43DF = Compute the generalized logistic type 3 probability density function.
    GL4PDF = Compute the generalized logistic type 4 probability density function.
    LOGPDF LOGPDF = Compute the logistic probability density function.
    NORPDF = Compute the normal probability density function.
    LGNPDF = Compute the logmormal probability density function.
Reference:
    "Continuous Univariate Distributions - 2", 2nd. Ed., Johnson, Kotz, and Balakrishnan, John Wiley, 1994 (pp. 140-147).
Applications:
    Distributunial Modeling
Implementation Date:
    2006/3
Program:
    LET A = DATA -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1
    MULTIPLOT 3 3
    MULTIPLOT CORNER COORDINATES 0 0 100 95
    MULTIPLOT SCALE FACTOR 3
    LABEL CASE ASIS
    TITLE CASE ASIS
    TITLE DISPLACEMENT 2
    X1LABEL X
    Y1LABEL Probability Density
    X1LABEL DISPLACEMENT 12
    Y1LABEL DISPLACEMENT 15
    .
    LOOP FOR K = 1 1 9
       LET ALPHA = A(K)
       LET XSTART = -5
       LET XSTOP = 5
       IF ALPHA > 0
          LET XSTOP = 1/ALPHA
       END OF IF
       IF ALPHA < 0
          LET XSTART = 1/ALPHA
       END OF IF
       TITLE Alpha = ^ALPHA
       PLOT GL5PDF(X,ALPHA) FOR X = XSTART  0.01  XSTOP
    END OF LOOP
    END OF MULTIPLOT
    CASE ASIS
    MOVE 50 97
    JUSTIFICATION CENTER
    TEXT Generalized Logistic Type 5 PDF's
        
    plot generated by sample program

Date created: 3/27/2006
Last updated: 3/27/2006
Please email comments on this WWW page to alan.heckert@nist.gov.