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Dataplot Vol 2 Vol 1

CONCDF

Name:
    CONCDF (LET)
Type:
    Library Function
Purpose:
    Compute the Consul probability mass function.
Description:
    The Consul distribution has the following probability mass function:

      p(x;theta,m) = (1/x)*(m*x x-1)*theta**(x-1)*(1-theta)**(m*x-x+1) x = 1, ...; 0 < theta < 1; 1 <= m <= 1/theta

    with theta and m denoting the shape parameters. The case where m = 1 reduces to the geometric distribution. For this reason, the Consul distribution is also referred to as the generalized geometric distribution.

    The mean and variance of the Consul distribution are:

      mu = 1/(1 - theta*m)

      sigma2 = m*theta*(1 - theta)/(1 - m*theta)**3

    The Consul distribution is also parameterized in terms of its mean, mu, which results in the following probability mass function:

      p(x;mu,m) = (1/x)*(m*x x-1)* ((mu-1)/(m*mu))**(x-1)*(1-(mu-1)/(m*mu))**(m*x-x+1) x = 1, ...; mu > 1; m >= 1

    theta can be expressed in terms of mu:

      theta = (mu-1)/(m*mu)

    For this parameterization, the mean and variance are:

      mu

      sigma2 = mu*(mu-1)*(m*mu-mu+1)/m

    Dataplot supports both parameterizations (see the Note section below).

    Dataplot computes the cumulative distribution function using the following recurrence relation given on page 320 of Consul and Famoye:

      P(x) = {(m-1)*(x-1) + 1)/(x-1)}*theta*(1-theta)**(m-1)* PROD[i=1 to x-2][1 + m/(m*x-m-i)]*P(x-1)

    where P(1) = (1-theta)**m. If the mu parameterization is used, it is converted to the theta parameterization and the above recurrence relation is applied.

Syntax:
    LET <y> = CONCDF(<x>,<shape>,<m>)
                            <SUBSET/EXCEPT/FOR qualification>
    where <x> is a positive integer variable, number, or parameter;
                <shape> is a number, parameter, or variable that specifies the valuie of theta (or mu);
                <m> 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 Consul cdf value is stored;
                and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
    LET A = CONCDF(3,0.5,1.4)
    LET Y = CONCDF(X,0.3,1.6)
    PLOT CONCDF(X,0.3,1.6) FOR X = 1 1 20
Note:
    To use the MU parameterization, enter the command

      SET CONSUL DEFINITION MU

    To restore the THETA parameterization, enter the command

      SET CONSUL DEFINITION THETA
Default:
    None
Synonyms:
    None
Related Commands:
    CONCDF = Compute the Consul cumulative distribution function.
    CONPDF = Compute the Consul probability mass function.
    GEOPDF = Compute the geometric probability mass function.
    GETPDF = Compute the Geeta probability mass function.
    GLSPDF = Compute the generalized logarithmic series probability mass 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 8.

    Famoye (1997), "Generalized Geometric and Some of its Applications", Journal of Mathematical Sciences, 8, pp. 1-13.

    Consul (1990), "New Class of Location-Parameter Discrete Probability Distributions and Their Chaacterizations", Communications in Statistics--Theory and Methods, 19, pp. 4653-4666.

Applications:
    Distributional Modeling
Implementation Date:
    2006/8
Program: 
     
set consul definition theta
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, M = 1.8
plot concdf(x,0.3,1.8) for x = 1 1 20
.
title Theta = 0.5, M = 1.5
plot concdf(x,0.5,1.5) for x = 1 1 20
.
title Theta = 0.7, M = 1.2
plot concdf(x,0.7,1.2) for x = 1 1 20
.
title Theta = 0.9, M = 1.1
plot concdf(x,0.9,1.1) for x = 1 1 20
.
end of multiplot
.
justification center
move 50 97
text Cumulative Distribution Functions for Consul
    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.