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BNBPDFName:
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.
<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; <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 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 re-parameterized beta-negative 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 CHI-SQUARE GOODNESS OF FIT Y
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 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 Beta-Negative 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 pre-erase off line color blue . plot bnbpdf(x,alpha,beta,k) for x = 0 1 20 limits pre-erase on . beta negative binomial chi-square goodness of fit y3 xlow xhigh . y1label Theoretical x1label Data char x line blank beta negative binomial probability plot y3 xlow xhigh CHI-SQUARED GOODNESS-OF-FIT 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 NON-EMPTY CELLS = 16 NUMBER OF PARAMETERS USED = 3 TEST: CHI-SQUARED TEST STATISTIC = 17.00824 DEGREES OF FREEDOM = 12 CHI-SQUARED 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 |