The beta-geometric distribution has the following probability density function:
with , , 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:
Converting this to the parameterization above yields
Dataplot computes the percent point function by summing the cumulative distribution function until the specified probability is obtained.
where <p> is a number, parameter, or variable in the interval (0,1);
<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 <p> is) where the computed beta-geometric ppf value is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
LET A = BGEPPF(P,2.1,4)
PLOT BGEPPF(P,ALPHA,BETA) FOR P = 0 0.01 0.99
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
To reset the unshifted parameterization (i.e., starting at x = 1), enter the command
This distribution is also sometimes given with and reversed. In this case, the probability mass functions become
Irwin developed the Waring distribution based on the Waring expansion. The probability mass function for the Waring distribution is
The Waring distribution can be computed with the shifted form of the beta-geometric distribution with the following change in parameters:
= c - a
If a = 1, then the Waring distribution reduces to the Yule distribution.
You can compute the Waring (and Yule) percent point functions using the BGEPPF routine with the above re-parameterization or you can use the WARPPF or YULPPF routines directly (enter HELP WARPDF or HELP YULPDF for details).
XLIMITS 0 1 XTIC OFFSET 0.5 0.5 . TITLE CASE ASIS LABEL CASE ASIS Y1LABEL Number of Successes X1LABEL Probability TITLE DISPLACEMENT 2 Y1LABEL DISPLACEMENT 15 X1LABEL DISPLACEMENT 12 . MULTIPLOT 2 2 MULTIPLOT CORNER COORDINATES 0 0 100 95 MULTIPLOT SCALE FACTOR 2 . TITLE Alpha = 0.5, Beta = 0.5 PLOT BGEPPF(P,0.5,0.5) FOR P = 0 0.01 0.99 . TITLE Alpha = 3, Beta = 0.5 PLOT BGEPPF(P,3.0,0.5) FOR P = 0 0.01 0.99 . TITLE Alpha = 0.5, Beta = 3 PLOT BGEPPF(P,0.5,3.0) FOR P = 0 0.01 0.99 . TITLE Alpha = 3, Beta = 3 PLOT BGEPPF(P,3.0,3.0) FOR P = 0 0.01 0.99 . END OF MULTIPLOT . CASE ASIS JUSTIFICATION CENTER MOVE 50 97 TEXT Beta-Geometric Percent Point Functions
Date created: 8/23/2006