-----LKCDF (LET)--------------------------------
 
LKCDF
 
Name:
    LKCDF (LET)
 
Type:
    Library Function
 
Purpose:
    Compute the Lagrange Katz cumulative distribution
    function.
 
Description:
    The Lagrange Katz distribution has the following probability
    mass function:

       p(x;a,b,beta)=(a/beta)/((a/beta) + (x*b/beta) + x)*
                     ((a/beta)+x*b/beta+x  x)*
                     beta**x*(1-beta)**((a/beta)+x*b/beta)
                     x = 0, 1, 2, 3, ,... ;
                     a > 0; b > -beta; beta < 1
    The cumulative distribution is computed using the following
    recurrence relation given on page 243 of Consul and Famoye
    (see Reference section below)

        p(x+1) = {(a+b*(x+1)+beta*x)/(x+1)}*(1-beta)**(b/beta)*
                 PROD[i=1 to x-1][(1 + b/(a+b*x+beta*i)]*p(x)

    where

        P(0) = (1-beta)**(a/beta)
        P(1) = a*(1-beta)**(b/beta)*P(0)

Syntax:
    LET <y> = LKCDF(<x>,<a>,<b>,<b>) 
              <SUBSET/EXCEPT/FOR qualification>
    where <x> is a non-negative integer variable, number, or
               parameter;
          <a> is a number, parameter, or variable that specifies
               the first shape parameter;
          <b> is a number, parameter, or variable that
               specifies the second shape parameter;
          <beta> is a number, parameter, or variable that
               specifies the third shape parameter;
          <y> is a variable or a parameter (depending on what <x>
               is) where the computed Lagrange Katz cdf value is
               stored;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.
 
Examples:
    LET A = LKCDF(0,2,0.2,0.5)
    LET Y = LKCDF(X,A,B,BETA)
    PLOT LKCDF(X,2,0.2,0.5) FOR X = 0  1  20

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:
    LKPDF                    = Compute the Lagrange Katz probability
                               mass function.
    LKPPF                    = Compute the Lagrange Katz percent
                               point function.
    GNBPDF                   = Compute the generalized negative
                               binomial probability mass function.
    GLSPDF                   = Compute the generalized logarithmic
                               series probability mass function.
    BNBPDF                   = Compute the beta-negative binomial
                               probability mass function.
    NBPDF                    = Compute the negative binomial
                               probability mass function.
    BINPDF                   = Compute the binomial probability
                               mass function.
    LPOPDF                   = Compute the Lagrange Poisson
                               probability mass function.
 
Reference:
    Consul and Famoye (2006), "Lagrangian Probability
    Distribution", Birkhauser, chapter 10.

    Consul and Famoye (1996), "The Lagrangian Katz Family of
    Distributions", Communications in Statistics--Theory
    and Methods", 25, pp. 415-434.

Applications:
    Distributional Modeling
 
Implementation Date:
    2006/8
 
Program:
    title size 3
    tic label size 3
    label size 3
    legend size 3
    height 3
    x1label displacement 12
    y1label displacement 15
    .
    multiplot corner coordinates 0 0 100 95
    multiplot scale factor 2
    label case asis
    title case asis
    case asis
    tic offset units screen
    tic offset 3 3
    title displacement 2
    y1label Probability Mass
    x1label X
    .
    ylimits 0 1
    major ytic mark number 6
    minor ytic mark number 3
    xlimits 0 20
    line blank
    spike on
    .
    multiplot 2 2
    .
    title Theta = 0.3, Beta = 1.8, M =2
    plot lkcdf(x,0.3,1.8,2) for x = 1 1 20
    .
    title Theta = 0.5, Beta = 1.5, M = 2
    plot lkcdf(x,0.5,1.5,2) for x = 1 1 20
    .
    title Theta = 0.7, Beta = 1.2, M = 2
    plot lkcdf(x,0.7,1.2,2) for x = 1 1 20
    .
    title Theta = 0.9, Beta = 1.1, M = 2
    plot lkcdf(x,0.9,1.1,2) for x = 1 1 20
    .
    end of multiplot
    .
    justification center
    move 50 97
    text Lagrange Katz Cumulative Distribution Functions

