-----TNPPDF (LET)--------------------------------
 
TNPPDF
 
Name:
    TNPPDF (LET)
 
Type:
    Library Function
 
Purpose:
    Compute the truncated 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) = gammaa*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> = TNPPDF(<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 = TNPPDF(3,1.5)
    LET A = TNPPDF(3,1.5,2)
    LET Y = TNPPDF(X,GAMMA,A,LOC,SCALE)
    PLOT TNPPDF(X,GAMMA,A,LOC,SCALE) FOR X = XSTART  0.01  XSTOP
 
Note:
    The Pareto distribution can be extended with location and
    scale parameters by using the relationship

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

    Since a defines the lower bound, the location parameter is
    typically set to zero.

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 PROBABILITY PLOT Y X
       PARETO PROBABILITY PLOT Y XLOW XHIGH
       PARETO KOLMOGOROV SMIRNOV GOODNESS OF FIT Y
       PARETO CHI-SQUARE GOODNESS OF FIT Y X
       PARETO CHI-SQUARE GOODNESS OF FIT Y XLOW XHIGH

    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 PPCC PLOT Y X
       PARETO PPCC PLOT Y XLOW XHIGH
       PARETO KS PLOT Y
       PARETO KS PLOT Y X
       PARETO KS PLOT Y XLOW XHIGH

    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

Note:
    Library functions are distinguished from let subcommands
    in the following ways.
    1) Functions enclose the input value in parenthesis.  Let
       subcommands use spaces.
    2) Functions can accept (and return) either parameters
       (i.e., single values) or variables (i.e., an array of
       values) while let subcommands are specific in which they
       accept as input and what they return as output.
    3) Functions can accept expressions while let subcommands
       do not.  For example, the following is legal:
           LET Y2 = ABS(Y1-INT(Y1))
       For let subcommands, you typically have to do something
       like the following:
           LET YTEMP = Y**2 + 8
           LET A = SUM YTEMP
 
Default:
    None
 
Synonyms:
    None
 
Related Commands:
    PARCDF = Compute the Pareto cumulative distribution 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 TNPPDF(X,1) FOR X = 1 0.1 10
    TITLE Gamma = 2
    PLOT TNPPDF(X,2) FOR X = 1 0.1 10
    TITLE Gamma = 5
    PLOT TNPPDF(X,5) FOR X = 1 0.1 10
    TITLE Gamma = 0.5
    PLOT TNPPDF(X,0.5) FOR X = 1 0.1 10
    END OF MULTIPLOT
    .
    MOVE 50 97
    JUSTIFICATION CENTER
    TEXT Pareto PDF Functions
 
