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Dataplot Vol 2 Vol 1

GHPPF

Name:
    GHPPF (LET)
Type:
    Library Function
Purpose:
    Compute the g-and-h percent point function.
Description:
    The percent point function of the g-and-h distribution is defined as follows:

      G(p,g,h) = (EXP(g*Zp) - 1)*EXP(h*Zp**2/2)/g

    with Zp denoting the normal percent point function of p. When g = 0 and h = 0, the g-and-h distribution reduces to a standard normal distribution.

    The value of g controls the degree of skewness. For g = 0, the distribution is symmetric. As the absolute value of g increases, the amount of the skewness increases. The sign of g controls the direction of the skewness (but not the amount). Positive values of g skew the distribution to the right tail while negative values of g skew the distribution to the left tail. Values for g are typically in the range (-1,1).

    The value of h controls the elongation, or how heavy the tails are, of the distribution. For h = 0, the elongation is equivalent to that of a normal distribution. For h = 1, the elongation is equivalent to that of a Cauchy distribution. Values of h are typically in the range (0,1).

    Specifying values for both g and h gives this distribution great flexibility in modeling data.

    The input value is a real number between 0 and 1 (since it corresponds to a probability).

    The g-and-h distribution can be generalized with location and scale parameters in the usual way.

Syntax:
    LET <y> = GHPPF(<p>,<g>,<h>,<loc>,<scale>)
                            <SUBSET/EXCEPT/FOR qualification>
    where <p> is a variable, number, or parameter in the range 0 to 1;
                <g> is a number or parameter that specifies the skewness shape parameter;
                <h> is a number or parameter that specifies the elongation 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 <p> is) where the computed g-and-h ppf value is stored;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    Note that the location and scale parameters are optional.

Examples:
    LET A = GHPPF(0.95,0.5,0.2)
    LET Y = GHPPF(P,0.5,0.2)
    PLOT GHCDF(P,0.5,0.2) FOR P = 0.01 0.01 0.01
Default:
    None
Synonyms:
    None
Related Commands:
    GHCDF = Compute the g-and-h cumulative distribution function.
    GHPDF = Compute the g-and-h probability density function.
    LAMPDF = Compute the Tukey-Lambda probability density function.
    NORPDF = Compute the standard normal probability density function.
    LOGPDF = Compute the logistic probability density function.
    JSUPDF = Compute the Johnson SU probability density function.
Reference:
    "Summarizing Shape Numerically: The g-and-h Distributions", David C. Hoaglin, chapter 11 in "Exploring Data Tables, Trends, and Shapes", Eds. Hoaglin, Mosteller, and Tukey, Wiley, 1985.
Applications:
    Distributional Modeling
Implementation Date:
    2004/5
Program:
     
    MULTIPLOT 2 2
    MULTIPLOT SCALE FACTOR 2
    MULTIPLOT CORNER COORDINATES 0 0 100 95
    Y1LABEL X
    X1LABEL PROBABILITY
    X1LABEL DISPLACEMENT 12
    Y1LABEL DISPLACEMENT 12
    TITLE G = 0.2, H = 0.2
    PLOT GHPPF(P,0.2,0.2) FOR P = 0.01 .01 0.99
    TITLE G = 0.5, H = 0.2
    PLOT GHPPF(P,0.5,0.2) FOR P = 0.01 .01 0.99
    TITLE G = 0.2, H = 0.5
    PLOT GHPPF(P,0.2,0.5) FOR P = 0.01 .01 0.99
    TITLE G = 0.5, H = 0.5
    PLOT GHPPF(P,0.5,0.5) FOR P = 0.01 .01 0.99
    END OF MULTIPLOT
    MOVE 50 97
    JUSTIFICATION CENTER
    TEXT G-AND-H DISTRIBUTIONS
        
    plot generated by sample program

Date created: 7/7/2004
Last updated: 7/7/2004
Please email comments on this WWW page to alan.heckert@nist.gov.