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Dataplot Vol 1 Auxiliary Chapter

PA2CDF

Name:
    PA2CDF (LET)
Type:
    Library Function
Purpose:
    Compute the Pareto cumulative distribution of the second kind with shape parameters gamma and a.
Description:
    The standard form of the Pareto cumulative distribution function of the second kind is:

      F(x;gamma,a) = 1 - a**gamma/(x+a)**gamma    x, gamma, a > 0

    with gamma and a denoting the shape parameters. The a parameter is optional and has a default value of 1.

    The Pareto distribution of the second kind is sometimes referred to as the Lomax distribution.

Syntax:
    LET <y> = PA2CDF(<x>,<gamma>,<a>,<loc>,<scale>)
                            <SUBSET/EXCEPT/FOR qualification>
    where <x> is a variable, a number, or a parameter;
                <gamma> is a number or parameter that specifies the tail length shape parameter;
                <a> is a number or parameter that specifies the optional lower bound shape parameter;
                <loc> is a number or parameter that specifies the optional location parameter;
                <scale> is a number or parameter that specifies the optional scale parameter;
                <y> is a variable or a parameter (depending on what <x> is) where the computed Pareto cdf value is saved;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    The a, loc, and scale parameters are all optional.

Examples:
    LET A = PA2CDF(3,1.5)
    LET X2 = PA2CDF(X1,GAMMA)
    LET X2 = PA2CDF(X1,GAMMA,A)
    PLOT PA2CDF(X,GAMMA,A) FOR X = 0 0.01 10
Note:
    The Pareto cumulative distribution of the second kind can be extended with location and scale parameters by using the relationship

      F(x;gamma,a,loc,scale) = F((x-loc)/scale;gamma,a,0,1)
Note:
    Johnson, Kotz, and Balakrishnan (see Reference section below) define Pareto distributions of the first, second, third, and fourth kinds. Dataplot supports Pareto distributions of the first and second kinds.
Default:
    None
Synonyms:
    None
Related Commands:
    PA2PDF = Compute the Pareto second kind probability density function.
    PA2PPF = Compute the Pareto second kind percent point function.
    PARPDF = Compute the Pareto probability density function.
    GEPPDF = Compute the generalized Pareto probability density function.
    EV1PDF = Compute the extreme value type I probability density function.
    WEIPDF = Compute the Weibull probability density function.
    NORPDF = Compute the normal probability density function.
    LGNPDF = Compute the lognormal probability density function.
    EXPPDF = Compute the exponential probability density function.
Reference:
    "Continuous Univariate Distributions: Volume 1", 2nd. Ed., Johnson, Kotz, and Balakrishnan, John Wiley, 1994, (chapter 20).
Applications:
    Distributional Modeling, Income Distributions
Implementation Date:
    1995/10
Program:
     
    MULTIPLOT 2 2
    MULTIPLOT CORNER COORDINATES 0 0 100 95
    MULTIPLOT SCALE FACTOR 2
    .
    CASE ASIS
    TITLE CASE ASIS
    LABEL CASE ASIS
    TITLE DISPLACEMENT 2
    Y1LABEL DISPLACEMENT 15
    X1LABEL DISPLACEMENT 12
    Y1LABEL Probability
    X1LABEL X
    .
    LET A = 1
    TITLE Gamma = 0.1
    PLOT PA2CDF(X,0.1,A) FOR X = 0.01  0.01  5
    TITLE Gamma = 1
    PLOT PA2CDF(X,1,A) FOR X = 0.01  0.01  5
    TITLE Gamma = 2
    PLOT PA2CDF(X,2,A) FOR X = 0.01  0.01  5
    TITLE Gamma = 5
    PLOT PA2CDF(X,5,A) FOR X = 0.01  0.01  5
    END OF MULTIPLOT
    .
    MOVE 50 97
    JUSTIFICATION CENTER
    TEXT Pareto Second Kind CDF Functions
        
    plot generated by sample program

Date created: 8/23/2006
Last updated: 8/23/2006
Please email comments on this WWW page to alan.heckert@nist.gov.