# dGHGBB: Gaussian Hypergeometric Generalized Beta Binomial... In Amalan-ConStat/R-fitODBOD: Modeling Over Dispersed Binomial Outcome Data Using BMD and ABD

## Description

These functions provide the ability for generating probability function values and cumulative probability function values for the Gaussian Hypergeometric Generalized Beta Binomial distribution.

## Usage

 1 dGHGBB(x,n,a,b,c) 

## Arguments

 x vector of binomial random variables n single value for no of binomial trials a single value for shape parameter alpha value representing a b single value for shape parameter beta value representing b c single value for shape parameter lambda value representing c

## Details

Mixing Gaussian Hypergeometric Generalized beta distribution with binomial distribution will create the Gaussian Hypergeometric Generalized Beta Binomial distribution. The probability function and cumulative probability function can be constructed and are denoted below.

The cumulative probability function is the summation of probability function values

P_{GHGBB}(x)=\frac{1}{2F1(-n,a;-b-n+1;c)} {n \choose x} \frac{B(x+a,n-x+b)}{B(a,b+n)}(c^x)

a,b,c > 0

x = 0,1,2,...n

n = 1,2,3,...

The mean, variance and over dispersion are denoted as

E_{GHGBB}[x]= nE_{GHGBeta}

Var_{GHGBB}[x]= nE_{GHGBeta}(1-E_{GHGBeta})+ n(n-1)Var_{GHGBeta}

over dispersion= \frac{var_{GHGBeta}}{E_{GHGBeta}(1-E_{GHGBeta})}

Defined as B(a,b) is the beta function. Defined as 2F1(a,b;c;d) is the Gaussian Hypergeometric function

NOTE : If input parameters are not in given domain conditions necessary error messages will be provided to go further.

## Value

The output of dGHGBB gives a list format consisting

pdf probability function values in vector form

mean mean of Gaussian Hypergeometric Generalized Beta Binomial Distribution

var variance of Gaussian Hypergeometric Generalized Beta Binomial Distribution

over.dis.para over dispersion value of Gaussian Hypergeometric Generalized Beta Binomial Distribution

## References

Rodriguez-Avi, J., Conde-Sanchez, A., Saez-Castillo, A. J., & Olmo-Jimenez, M. J. (2007). A generalization of the beta-binomial distribution. Journal of the Royal Statistical Society. Series C (Applied Statistics), 56(1), 51-61.

Available at : http://dx.doi.org/10.1111/j.1467-9876.2007.00564.x

Pearson, J., 2009. Computation of Hypergeometric Functions. Transformation, (September), p.1–123.

hypergeo_powerseries
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 #plotting the random variables and probability values col<-rainbow(6) a<-c(.1,.2,.3,1.5,2.1,3) plot(0,0,main="GHGBB probability function graph",xlab="Binomial random variable", ylab="Probability function values",xlim = c(0,7),ylim = c(0,0.9)) for (i in 1:6) { lines(0:7,dGHGBB(0:7,7,1+a[i],0.3,1+a[i])$pdf,col = col[i],lwd=2.85) points(0:7,dGHGBB(0:7,7,1+a[i],0.3,1+a[i])$pdf,col = col[i],pch=16) } dGHGBB(0:7,7,1.3,0.3,1.3)$pdf #extracting the pdf values dGHGBB(0:7,7,1.3,0.3,1.3)$mean #extracting the mean dGHGBB(0:7,7,1.3,0.3,1.3)$var #extracting the variance dGHGBB(0:7,7,1.3,0.3,1.3)$over.dis.par #extracting the over dispersion value #plotting the random variables and cumulative probability values col<-rainbow(4) a<-c(1,2,5,10) plot(0,0,main="Cumulative probability function graph",xlab="Binomial random variable", ylab="Cumulative probability function values",xlim = c(0,7),ylim = c(0,1)) for (i in 1:4) { lines(0:7,pGHGBB(0:7,7,1+a[i],0.3,1+a[i]),col = col[i]) points(0:7,pGHGBB(0:7,7,1+a[i],0.3,1+a[i]),col = col[i]) } pGHGBB(0:7,7,1.3,0.3,1.3) #acquiring the cumulative probability values