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Dataplot Vol 2 Vol 1

GLSCDF

Name:
    GLSCDF (LET)
Type:
    Library Function
Purpose:
    Compute the generalized logarithmic series cumulative distribution function.
Description:
    The generalized logarithmic series distribution has the following probability mass function:

      p(x;theta,beta) = (1/(beta*x))*(beta*x x)theta**x*(1-theta)**(beta*x-x)/ (-LOG(1-theta)) x = 1, 2, ...; 0 < theta < 1; 1 <= beta < -1/theta

    with theta and beta denoting the shape parameters and Gamma denoting the gamma function (enter HELP GAMMA for details).

    The cumulative distribution function is computed using the following recurrence relation given on pages 228-229 of Consul and Famoye:

      p(x+1;theta,beta) = (beta - x/(x+1))*theta*(1-theta)^(beta-1)* PROD[j=1 to x-1][(1+beta/(beta*x - j))*p(x;theta,beta)

    with

      p(1;theta,beta) = theta*(1-theta)**(beta-1)/(-LOG(1-theta))

      p(2;theta,beta) = (beta-(1/2))*theta*(1-theta)^(beta-1)*p(1;theta,beta)

Syntax:
    LET <y> = GLSCDF(<x>,<theta>,<beta>)
                            <SUBSET/EXCEPT/FOR qualification>
    where <x> is a positive integer variable, number, or parameter;
                <theta> is a number, parameter, or variable in the range (0,1) that specifies the first shape parameter;
                <beta> is a number, parameter, or variable that specifies the second shape parameter;
                <y> is a variable or a parameter (depending on what <x> is) where the computed generalized logarithmic series cdf value is stored;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
    LET A = GLSCDF(3,0.5,1.4)
    LET Y = GLSCDF(X,0.3,1.6)
    PLOT GLSCDF(X,0.3,1.6) FOR X = 1 1 20

Default:
    None
Synonyms:
    None
Related Commands:
    GLSPDF = Compute the generalized logarithmic series probability mass function.
    GLSPPF = Compute the generalized logarithmic series percent point function.
    DLGPDF = Compute the logarithmic series probability mass function.
    YULPDF = Compute the Yule probability mass function.
    ZETPDF = Compute the Zeta probability mass function.
    BGEPDF = Compute the beta geometric probability mass function.
    POIPDF = Compute the Poisson probability mass function.
    BINPDF = Compute the binomial probability mass function.
Reference:
    Consul and Famoye (2006), "Lagrangian Probability Distribution", Birkhauser, chapter 11.
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
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
plot glscdf(x,0.3,1.8) for x = 1 1 20
.
title Theta = 0.5, Beta = 1.5
plot glscdf(x,0.5,1.5) for x = 1 1 20
.
title Theta = 0.7, Beta = 1.2
plot glscdf(x,0.7,1.2) for x = 1 1 20
.
title Theta = 0.9, Beta = 1.1
plot glscdf(x,0.9,1.1) for x = 1 1 20
.
end of multiplot
.
justification center
move 50 97
text Cumulative Distribution for Generalized Logarithmic Series
    plot generated by sample program

Date created: 8/23/2006
Last updated: 8/23/2006
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