
BNBPDFName:
The formula for the betanegative 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 betanegative binomial and the negative binomial distributions. It also demonstrates the relation between the betanegative binomial and the betabinomial and betageometric distributions. It also provides a computationally convenient formula since the betanegative binomial can be computed as the sums and differences of log gamma functions.
<SUBSET/EXCEPT/FOR qualification> where <x> is a number, parameter, or variable containing nonnegative 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 betanegative binomial pdf value is stored; and where the <SUBSET/EXCEPT/FOR qualification> is optional.
LET A = BNBPDF(X,2.1,4,2.5) PLOT BNBPDF(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 reparameterized betanegative binomial distribution. Irwin's uses the parameterization
= c  a k = k
LET ALPHA = <value> LET BETA = <value> LET Y = BETA NEGATIVE BINOMIAL RANDOM NUMBERS ... FOR I = 1 1 N
BETA NEGATIVE BINOMIAL PROBABILITY PLOT Y
BETA NEGATIVE BINOMIAL CHISQUARE GOODNESS OF FIT Y
J. O. Irwin (1975), "The Generalized Waring Distribution Part 1", Journal of the Royal Statistical Society, Series A, 138, pp. 1831. J. O. Irwin (1975), "The Generalized Waring Distribution Part 2", Journal of the Royal Statistical Society, Series A, 138, pp. 204227. J. O. Irwin (1975), "The Generalized Waring Distribution Part 3", Journal of the Royal Statistical Society, Series A, 138, pp. 374378. 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. 197216.
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 Mass 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 BNBPDF(X,0.5,0.5,K) FOR X = 0 1 50 . TITLE Alpha = 3, Beta = 0.5, K = 3 PLOT BNBPDF(X,3.0,0.5,K) FOR X = 0 1 50 . TITLE Alpha = 0.5, Beta = 3, K = 3 PLOT BNBPDF(X,0.5,3.0,K) FOR X = 0 1 50 . TITLE Alpha = 3, Beta = 3, K = 3 PLOT BNBPDF(X,3.0,3.0,K) FOR X = 0 1 50 . END OF MULTIPLOT . CASE ASIS JUSTIFICATION CENTER MOVE 50 97 TEXT BetaNegative Binomial Probability Mass Functions Program 2: let alpha = 1.5 let beta = 3 let k = 4 . let y = beta negative binomial random numbers for i = 1 1 500 let amax = maximum y let amax2 = amax + 0.5 class lower 0.5 class upper amax2 class width 1 let y2 x2 = binned y let y3 xlow xhigh = integer frequency table y . tic offset units screen tic offset 3 3 . relative histogram y2 x2 limits freeze preerase off line color blue . plot bnbpdf(x,alpha,beta,k) for x = 0 1 20 limits preerase on . beta negative binomial chisquare goodness of fit y3 xlow xhigh . y1label Theoretical x1label Data char x line blank beta negative binomial probability plot y3 xlow xhigh CHISQUARED GOODNESSOFFIT TEST NULL HYPOTHESIS H0: DISTRIBUTION FITS THE DATA ALTERNATE HYPOTHESIS HA: DISTRIBUTION DOES NOT FIT THE DATA DISTRIBUTION: BETA NEGATIVE BINOMIAL SAMPLE: NUMBER OF OBSERVATIONS = 500 NUMBER OF NONEMPTY CELLS = 16 NUMBER OF PARAMETERS USED = 3 TEST: CHISQUARED TEST STATISTIC = 17.00824 DEGREES OF FREEDOM = 12 CHISQUARED CDF VALUE = 0.850712 ALPHA LEVEL CUTOFF CONCLUSION 10% 18.54935 ACCEPT H0 5% 21.02607 ACCEPT H0 1% 26.21697 ACCEPT H0
Date created: 8/23/2006 