The formula for the beta-negative binomial probability mass function is
with , , and k denoting the shape parameters and denoting the gamma function.
Note that there are a number of different parameterizations and formulations of this distribution in the literature. We use the above formulation because it makes clear the relation between the beta-negative binomial and the negative binomial distributions. It also demonstrates the relation between the beta-negative binomial and the beta-binomial and beta-geometric distributions. It also provides a computationally convenient formula since the beta-negative binomial can be computed as the sums and differences of log gamma functions. Dataplot compumtes the cumulative distribution function using the following recurrence relation derived by Hesselager:
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;
<k> 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 beta-negative binomial cdf value is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
LET A = BNBCDF(X,2.1,4,2.5)
PLOT BNBCDF(X,ALPHA,BETA,K) FOR X = 0 1 20
Irwin developed the generalized Waring distribution based on a generalization of the Waring expansion. The generalized Waring distribution is a re-parameterized beta-negative binomial distribution. Irwin's uses the parameterization
= c - a
k = k
J. O. Irwin (1975), "The Generalized Waring Distribution Part 1", Journal of the Royal Statistical Society, Series A, 138, pp. 18-31.
J. O. Irwin (1975), "The Generalized Waring Distribution Part 2", Journal of the Royal Statistical Society, Series A, 138, pp. 204-227.
J. O. Irwin (1975), "The Generalized Waring Distribution Part 3", Journal of the Royal Statistical Society, Series A, 138, pp. 374-378.
Johnson, Kotz, and Kemp (1992), "Univariate Discrete Distributions", Second Edition, Wiley, chapter 6.
Luc Devroye (1992), "Random Variate Generation for the Digamma and Trigamma Distributions", Journal of Statistical Computation and Simulation", Vol. 43, pp. 197-216.
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 TITLE DISPLACEMENT 2 Y1LABEL DISPLACEMENT 15 X1LABEL DISPLACEMENT 12 . MULTIPLOT 2 2 MULTIPLOT CORNER COORDINATES 0 0 100 95 MULTIPLOT SCALE FACTOR 2 . LET K = 3 TITLE Alpha = 0.5, Beta = 0.5, K = 3 PLOT BNBCDF(X,0.5,0.5,K) FOR X = 0 1 50 . TITLE Alpha = 3, Beta = 0.5, K = 3 PLOT BNBCDF(X,3.0,0.5,K) FOR X = 0 1 50 . TITLE Alpha = 0.5, Beta = 3, K = 3 PLOT BNBCDF(X,0.5,3.0,K) FOR X = 0 1 50 . TITLE Alpha = 3, Beta = 3, K = 3 PLOT BNBCDF(X,3.0,3.0,K) FOR X = 0 1 50 . END OF MULTIPLOT . CASE ASIS JUSTIFICATION CENTER MOVE 50 97 TEXT Beta-Negative Binomial Cumulative Distribution Functions
Date created: 8/23/2006