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Dataplot Vol 2 Vol 1

QBICDF

Name:
    QBICDF (LET)
Type:
    Library Function
Purpose:
    Compute the quasi-binomial type I cumulative distribution function.
Description:
    The quasi-binomial type I distribution has the following probability mass function:

      p(x;p,phi) = (m x)*p*(p + x*phi)**(x-1)*(1-p-x*phi)**(m-x) x = 0, 1, 2, ..., m; 0 <= p <= 1; -p/m < phi < (1-p)/m

    with p, phi, and m denoting the shape parameters.

    The quasi-binomial type I distribution is used to model Bernoulli trials. The parameter p denotes the initial probability of success, m denotes the number of Bernoulli trials, and phi denotes how the probability of success increases or decreases with the number of successes. Specificially, when phi = 0, the quasi-binomial type I distribution reduces to the binomial distribution. When phi ≠ 0, the probability of success in the xth trial becomes

      p + x*phi

    The cumulative distribution function is computed using the following recurrence relation given by Consul and Famoye:

      p(x+1) = {(m-x)*(p+x*phi)/((x+1)*(1-p-x*phi))}* (1 + phi/(p+x*phi))**x*(1 - phi/(1-p-x*phi))**(m-x-1)*p(x)
Syntax:
    LET <y> = QBICDF(<x>,<p>,<theta>,<beta>)
                            <SUBSET/EXCEPT/FOR qualification>
    where <x> is a positive integer variable, number, or parameter;
                <p> is a number, parameter, or variable in the range (0,1) that specifies the first shape parameter;
                <phi> is a number, parameter, or variable that specifies the second shape parameter;
                <m> 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 quasi binomial type I cdf value is stored;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
    LET A = QBICDF(10,0.5,0.005,20)
    LET Y = QBICDF(X,0.7,0.01,20)
    PLOT QBICDF(X,0.3,0.005,20) FOR X = 0 1 20
Default:
    None
Synonyms:
    None
Related Commands:
    QBIPDF = Compute the quasi-binomial type I probability mass function.
    QBIPPF = Compute the quasi-binomial type I percent point function.
    BINPDF = Compute the binomial probability mass function.
    BBNPDF = Compute the beta-binomial probability mass function.
    NBPDF = Compute the negative binomial probability mass function.
Reference:
    Consul and Famoye (2006), "Lagrangian Probability Distribution", Birkhauser, pp. 70-80.
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 P = 0.3, Phi = 0.01, M = 20
plot qbicdf(x,0.3,0.01,20) for x = 1 1 20
.
title P = 0.3, Phi = -0.01, M = 20
let phi = -0.01
plot qbicdf(x,0.3,phi,20) for x = 1 1 20
.
title P = 0.7, Phi = 0.01, M = 20
plot qbicdf(x,0.7,0.01,20) for x = 1 1 20
.
title P = 0.7, Phi = -0.01, M = 20
let phi = -0.01
plot qbicdf(x,0.7,phi,20) for x = 1 1 20
.
end of multiplot
.
justification center
move 50 97
text Cumulative Distribution Functions for Quasi Binomial Type I
    plot generated by sample program

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