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

## Description

These functions provide the ability for generating probability density values, cumulative probability density values and moment about zero values for the Gaussian Hypergeometric Generalized Beta distribution bounded between [0,1]

## Usage

 1 mazGHGBeta(r,n,a,b,c) 

## Arguments

 r vector of moments n single value for no of binomial trials a single value for shape parameter alpha representing as a b single value for shape parameter beta representing as b c single value for shape parameter lambda representing as c

## Details

The probability density function and cumulative density function of a unit bounded Gaussian Hypergeometric Generalized Beta Distribution with random variable P are given by

g_{P}(p)= \frac{1}{B(a,b)}\frac{2F1(-n,a;-b-n+1;1)}{2F1(-n,a;-b-n+1;c)} p^{a-1}(1-p)^{b-1} \frac{c^{b+n}}{(c+(1-c)p)^{a+b+n}}

; 0 ≤ p ≤ 1

G_{P}(p)= \int^p_0 \frac{1}{B(a,b)}\frac{2F1(-n,a;-b-n+1;1)}{2F1(-n,a;-b-n+1;c)} t^{a-1}(1-t)^{b-1}\frac{c^{b+n}}{(c+(1-c)t)^{a+b+n}} \,dt

; 0 ≤ p ≤ 1

a,b,c > 0

n = 1,2,3,...

The mean and the variance are denoted by

E[P]= \int^1_0 \frac{p}{B(a,b)}\frac{2F1(-n,a;-b-n+1;1)}{2F1(-n,a;-b-n+1;c)} p^{a-1}(1-p)^{b-1}\frac{c^{b+n}}{(c+(1-c)p)^{a+b+n}} \,dp

var[P]= \int^1_0 \frac{p^2}{B(a,b)}\frac{2F1(-n,a;-b-n+1;1)}{2F1(-n,a;-b-n+1;c)} p^{a-1}(1-p)^{b-1}\frac{c^{b+n}}{(c+(1-c)p)^{a+b+n}} \,dp - (E[p])^2

The moments about zero is denoted as

E[P^r]= \int^1_0 \frac{p^r}{B(a,b)}\frac{2F1(-n,a;-b-n+1;1)}{2F1(-n,a;-b-n+1;c)} p^{a-1}(1-p)^{b-1}\frac{c^{b+n}}{(c+(1-c)p)^{a+b+n}} \,dp

r = 1,2,3,...

Defined as B(a,b) as the beta function Defined as 2F1(a,b;c;d) as 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 mazGHGBeta give the moments about zero in vector form.

## 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 28 29 30 #plotting the random variables and probability values col<-rainbow(5) a<-c(.1,.2,.3,1.5,2.15) plot(0,0,main="Probability density graph",xlab="Random variable",ylab="Probability density values", xlim = c(0,1),ylim = c(0,10)) for (i in 1:5) { lines(seq(0,1,by=0.001),dGHGBeta(seq(0,1,by=0.001),7,1+a[i],0.3,1+a[i])$pdf,col = col[i]) } dGHGBeta(seq(0,1,by=0.01),7,1.6312,0.3913,0.6659)$pdf #extracting the pdf values dGHGBeta(seq(0,1,by=0.01),7,1.6312,0.3913,0.6659)$mean #extracting the mean dGHGBeta(seq(0,1,by=0.01),7,1.6312,0.3913,0.6659)$var #extracting the variance #plotting the random variables and cumulative probability values col<-rainbow(6) a<-c(.1,.2,.3,1.5,2.1,3) plot(0,0,main="Cumulative density graph",xlab="Random variable",ylab="Cumulative density values", xlim = c(0,1),ylim = c(0,1)) for (i in 1:6) { lines(seq(0.01,1,by=0.001),pGHGBeta(seq(0.01,1,by=0.001),7,1+a[i],0.3,1+a[i]),col=col[i]) } pGHGBeta(seq(0,1,by=0.01),7,1.6312,0.3913,0.6659) #acquiring the cumulative probability values mazGHGBeta(1.4,7,1.6312,0.3913,0.6659) #acquiring the moment about zero values #acquiring the variance for a=1.6312,b=0.3913,c=0.6659 mazGHGBeta(2,7,1.6312,0.3913,0.6659)-mazGHGBeta(1,7,1.6312,0.3913,0.6659)^2 #only the integer value of moments is taken here because moments cannot be decimal mazGHGBeta(1.9,15,5,6,1)