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Dataplot Vol 1 Auxiliary Chapter

PARPDF

Name:
    PARPDF (LET)
Type:
    Library Function
Purpose:
    Compute the Pareto probability density function with shape parameters gamma and a.
Description:
    The standard form of the Pareto probability density function is:

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

    with gamma and a denoting the tail length shape parameter and the lower bound parameter, respectively. The default value of a is 1.

    Note that although the a parameter is typically called a location parameter (and it is in the sense that it defines the lower bound), it is not a location parameter in the technical sense that the following relation does not hold:

      f(x;gamma,a) = f((x-a);gamma,0)

    For this reason, Dataplot treats a as a shape parameter. In Dataplot, the a shape parameter is optional with a default value of 1.

Syntax:
    LET <y> = PARPDF(<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 pdf value is saved;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

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

Examples:
    LET A = PARPDF(3,1.5)
    LET A = PARPDF(3,1.5,2)
    LET Y = PARPDF(X,GAMMA,A,LOC,SCALE)
Note:
    The Pareto distribution can be extended with location and scale parameters using the relationship

      f(x;gamma,a,loc,scale) = (1/scale)*f(x;gamma,a,0,1)

    Most applications of the Pareto distribution use the standard form (i.e., location = 0 and scale = 1).

Note:
    Pareto random numbers, probability plots, and goodness of fit tests can be generated with the commands:

      LET GAMMA = <value>
      LET A = <value>
      LET Y = PARETO RANDOM NUMBERS FOR I = 1 1 N
      PARETO PROBABILITY PLOT Y
      PARETO KOLMOGOROV SMIRNOV GOODNESS OF FIT Y
      PARETO CHI-SQUARE GOODNESS OF FIT Y

    The following commands can be used to estimate the shape parameters for the Pareto distribution:

      LET GAMMA1 = <value>
      LET GAMMA2 = <value>
      LET A = <value>
      PARETO PPCC PLOT Y
      PARETO KS PLOT Y

    The default values for gamma1 and gamma2 are 0.2 and 10, respectively. Note that only the gamma parameter is estimated for these plots. The default value of A is 1. If the value of A is greater than the data minimum, then it is automatically set to the data minimum.

    You can generate maximum likelihood estimates for the Pareto distribution with the command

      PARETO MAXIMUM LIKELIHOOD Y

    The maximum likelihood estimate of the lower bound parameter is:

      ahat = minimum X(i)

    This estimate is used in the following equation to find the maximum likelihood estimate of the tail length parameter:

      alphahat = n/SUM[i=1 to n][LOG(x(i)/khat)]
Default:
    None
Synonyms:
    None
Related Commands:
    PARCDF = Compute the Pareto cumulative distribution function.
    PARCHAZ = Compute the Pareto cumulative hazard function.
    PARHAZ = Compute the Pareto hazard function.
    PARPPF = Compute the Pareto percent point 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.
    EXPPDF = Compute the exponential probability density function.
Reference:
    "Continuous Univariate Distributions: Volume 1", Second Edition, Johnson, Kotz, and Balakrishnan, Wiley, 1994, chapter 19.
Applications:
    Distributional Modeling
Implementation Date:
    1994/4
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 Density
    X1LABEL X
    .
    TITLE Gamma = 1
    PLOT PARPDF(X,1) FOR X = 1 0.1 10
    TITLE Gamma = 2
    PLOT PARPDF(X,2) FOR X = 1 0.1 10
    TITLE Gamma = 5
    PLOT PARPDF(X,5) FOR X = 1 0.1 10
    TITLE Gamma = 0.5
    PLOT PARPDF(X,0.5) FOR X = 1 0.1 10
    END OF MULTIPLOT
    .
    MOVE 50 97
    JUSTIFICATION CENTER
    TEXT Pareto PDF 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.