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Dataplot Vol 2 Vol 1

BGECDF

Name:
    BGECDF (LET)
Type:
    Library Function
Purpose:
    Compute the beta-geometric cumulative distribution function with shape parameters alpha and beta.
Description:
    If the probability of success parameter, p, of a geometric distribution has a Beta distribution with shape parameters alpha and beta, the resulting distribution is referred to as a beta-geometric distribution. For a standard geometric distribution, p is assumed to be fixed for successive trials. For the beta-geometric distribution, the value of p changes for each trial.

    The beta-geometric distribution has the following probability density function:

      P(x;alpha,beta) = B(alpha+1,x+beta-1)/B(alpha,beta) x = 1, 2, ...; alpha, beta > 0

    with alpha, beta, and B denoting the two shape parameters and the complete beta function, respectively. See the documentation for the BETA command for a description of the complete beta function.

    Dataplot computes the cumulative distribution function using a recurrence relation given by Hesselager. Hesselager gives the recurrence relation as:

      p(x;alpha,beta) = [(x+alpha-1)/(x+alpha+beta)]*p(x-1;alpha,beta)

    Converting this to the parameterization above yields

      p(x;alpha,beta) = [(x+beta-2)/(x+alpha+beta-1)]*p(x-1;alpha,beta)
Syntax:
    LET <y> = BGECDF(<x>,<alpha>,<beta>)
                            <SUBSET/EXCEPT/FOR qualification>
    where <x> is a number, parameter, or variable containing non-negative integer values;
                <alpha> is a number, parameter, or variable 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 beta-geometric cdf value is stored;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
    LET A = BGECDF(4,0.5,0.9)
    LET A = BGECDF(X,2.1,4)
    PLOT BGECDF(X,ALPHA,BETA) FOR X = 1 1 20
Note:
    Some sources shift this distribution to start at x = 0. In this case, the probability mass function is

      P(x;alpha,beta) = B(alpha+1,x+beta)/B(alpha,beta) x = 0, 1, 2, ...; alpha, beta > 0

    We will refer to the first parameterization as the unshifted parameterization and the second parameterization as the shifted parameterization.

    To specify the shifted parameterization (i.e., starting at x = 0), enter the command

      SET BETA GEOMETRIC DEFINITION SHIFTED

    To reset the unshifted parameterization (i.e., starting at x = 1), enter the command

      SET BETA GEOMETRIC DEFINITION UNSHIFTED

    This distribution is also sometimes given with alpha and beta reversed. In this case, the probability mass functions become

      P(x;alpha,beta) = B(alpha+1,x+beta-1)/B(alpha,beta) x = 1, 2, ...; alpha, beta > 0

      and

      P(x;alpha,beta) = B(alpha+1,x+beta)/B(alpha,beta) x = 0, 1, 2, ...; alpha, beta > 0

    To use this parameterization, simply interchange the order in which you give the alpha and beta arguments to the BGECDF command.
Note:
    The beta-geometric as given above is derived as a beta mixture of geometric random variables.

    Irwin developed the Waring distribution based on the Waring expansion. The probability mass function for the Waring distribution is

      P(x;c,a) = (c-a)*(a+x-1)!*c!/[c*(a-1)!*(c+x)!] x = 0, 1, 2, ...; a > 0; c > a

    The Waring distribution can be computed with the shifted form of the beta-geometric distribution with the following change in parameters:

      beta = a
      alpha = c - a

    If a = 1, then the Waring distribution reduces to the Yule distribution.

    You can compute the Waring (and Yule) distributions using the BGECDF routine with the above re-parameterization or you can use the WARCDF or YULCDF routines directly (enter HELP WARCDF or HELP YULCDF for details).

Default:
    None
Synonyms:
    None
Related Commands:
    BGEPDF = Compute the beta-geometric probability mass function.
    BGECDF Compute the beta-geometric cumulative distribution function.
    BETPDF = Compute the beta probability density function.
    GEOPDF = Compute the geometric probability mass function.
    WARPDF = Compute the Waring probability mass function.
    YULPDF = Compute the Yule probability mass function.
    BBNPDF = Compute the beta-binomial probability mass function.
    BNBPDF = Compute the beta-negative binomial (generalized Waring) probability mass function.
Reference:
    Ole Hesselager (1994), "A Recursive Procedure for Calculations of Some Compound Distributions", Astin Bulliten, Vol. 24, No. 1, pp. 19-32.

    Sudhir R. Paul (2004), "Applications of the Beta Distribution" in "Handbook of the Beta Distribution", edited by Gupta and Nadarajah, Marcel-Dekker, pp. 431-436.

    J. O. Irwin (1963), "The Place of Mathematics in Medical and Biological Statistics", Journal of the Royal Statistical Society, Series A, 126, pp. 1-44.

    Johnson, Kotz, and Kemp (1992), "Univariate Discrete Distributions", Second Edition, Wiley, chapter 6.

Applications:
    Distributional Modeling
Implementation Date:
    2006/7
Program: 
     
XLIMITS 0 50
XTIC OFFSET 0.5 0.5
LINE BLANK
SPIKE ON
SPIKE THICKNESS 0.3
.
TITLE CASE ASIS
LABEL CASE ASIS
X1LABEL Number of Successes
Y1LABEL Probability
.
MULTIPLOT 2 2
MULTIPLOT CORNER COORDINATES 0 0 100 95
MULTIPLOT SCALE FACTOR 2
.
TITLE Alpha = 0.5, Beta = 0.5
PLOT BGECDF(X,0.5,0.5) FOR X = 0 1 50
.
TITLE Alpha = 3, Beta = 0.5
PLOT BGECDF(X,3.0,0.5) FOR X = 0 1 50
.
TITLE Alpha = 0.5, Beta = 3
PLOT BGECDF(X,0.5,3.0) FOR X = 0 1 50
.
TITLE Alpha = 3, Beta = 3
PLOT BGECDF(X,3.0,3.0) FOR X = 0 1 50
.
END OF MULTIPLOT
.
CASE ASIS
JUSTIFICATION CENTER
MOVE 50 97
TEXT Beta-Geometric Cumulative Distribution Functions
    plot generated by sample program

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