SED navigation bar go to SED home page go to Dataplot home page go to NIST home page SED Home Page SED Staff SED Projects SED Products and Publications Search SED Pages
Dataplot Vol 2 Vol 1

PAPCDF

Name:
    PAPCDF (LET)
Type:
    Library Function
Purpose:
    Compute the Polya-Aeppli cumulative distribution function.
Description:
    The formula for the Polya-Aeppli probability mass function is

      p(x;theta,p) = EXP(-theta) x = 0; = EXP(-theta)*p**x*SUM[j=1 to x][(x-1 j-1)*(theta*(1-p)/p)**j/j!] x = 1, ...; 0 < p < 1; theta > 0

    with theta and p denoting the shape parameters.

    The cumulative distribution function is computed using the following recurrence relation (from page 379 of Johnson, Kemp, and Kotz)

      p(x+1;theta,p) = (theta*(1-p)/(x+1))*SUM[j=0 TO x][(x+1-j)*p**(x-j)* p(x;theta,p)]
Syntax:
    LET <y> = PAPCDF(<x>,<theta>,<p>)
                            <SUBSET/EXCEPT/FOR qualification>
    where <x> is a non-negative integer variable, number, or parameter;
                <theta> is a positive number or parameter that specifies the first shape parameter;
                <p> is a positive number or parameter that specifies the second shape parameter;
                <y> is a variable or a parameter where the computed Polya-Aeppli cdf value is stored;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
    LET A = PAPCDF(3,3,0.5)
    LET Y = PAPCDF(X1,2,0.3)
    PLOT PAPCDF(X,2,0.3) FOR X = 0 1 20
Default:
    None
Synonyms:
    None
Related Commands:
    PAPPDF = Compute the Polya-Aeppli probability mass function.
    PAPPPF = Compute the Polya-Aeppli percent point function.
    LPOPDF = Compute the Lagrange-Poisson percent point function.
    BTAPDF = Compute the Borel-Tanner probability mass function.
    LOSPDF = Compute the lost games probability mass function.
    POIPDF = Compute the Poisson probability mass function.
    HERPDF = Compute the Hermite probability mass function.
    BINPDF = Compute the binomial probability mass function.
    NBPDF = Compute the negative binomial probability mass function.
    GEOPDF = Compute the geometric probability mass function.
    INTEGER FREQUENCY TABLE = Generate a frequency table at integer values with unequal bins.
    COMBINE FREQUENCY TABLE = Convert an equal width frequency table to an unequal width frequency table.
    KS PLOT = Generate a minimum chi-square plot.
    MAXIMUM LIKELIHOOD = Perform maximum likelihood estimation for a distribution.
References:
    Douglas (1980), "Analysis with Standard Contagious Distributions", International Co-operative Publishing House, Fairland, MD.

    Evans (1953), "Experimental Evidence Concerning Contagious Distributions in Ecology", Biometrika, 40, pp. 186-211.

    Johnson, Kotz, and Kemp (1992), "Univariate Discrete Distributions", Second Edition, Wiley, pp. 378-382.

Applications:
    Distributional Modeling
Implementation Date:
    2006/6
Program: 
     
title size 3
tic label size 3
label size 3
legend size 3
height 3
multiplot scale factor 1.5
x1label displacement 12
y1label displacement 17
.
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 = 0.5, P = 0.5
plot papcdf(x,0.5,0.5) for x = 1 1 20
.
title Theta = 1, P = 0.5
plot papcdf(x,1,0.5) for x = 1 1 20
.
title Theta = 2.5, P = 0.5
plot papcdf(x,2.5,0.5) for x = 1 1 20
.
title Theta = 5, P = 0.5
plot papcdf(x,5,0.5) for x = 1 1 20
.
end of multiplot
.
justification center
move 50 97
text Cumulative Distribution for Polya-Aeppli
    plot generated by sample program

Date created: 6/20/2006
Last updated: 6/20/2006
Please email comments on this WWW page to [email protected].