pGHGBeta: 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 pGHGBeta(p,n,a,b,c) 

Arguments

 p vector of probabilities 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 pGHGBeta gives the cumulative density values 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)