# pGBeta1: Generalized Beta Type-1 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 Generalized Beta Type-1 Distribution bounded between [0,1]

## Usage

 1 pGBeta1(p,a,b,c) 

## Arguments

 p vector of probabilities a single value for shape parameter alpha representing as a b single value for shape parameter beta representing as a c single value for shape parameter gamma representing as c

## Details

The probability density function and cumulative density function of a unit bounded Generalized Beta Type-1 Distribution with random variable P are given by

g_{P}(p)= \frac{c}{B(a,b)} p^{ac-1} (1-p^c)^{b-1}

; 0 ≤ p ≤ 1

G_{P}(p)= \frac{p^{ac}}{aB(a,b)} 2F1(a,1-b;p^c;a+1)

0 ≤ p ≤ 1

a,b,c > 0

The mean and the variance are denoted by

E[P]= \frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})}

var[P]= \frac{B(a+b,\frac{2}{c})}{B(a,\frac{2}{c})}-(\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})})^2

The moments about zero is denoted as

E[P^r]= \frac{B(a+b,\frac{r}{c})}{B(a,\frac{r}{c})}

r = 1,2,3,....

Defined as B(a,b) is beta function Defined as 2F1(a,b;c;d) is 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 pGBeta1 gives the cumulative density values in vector form.

## References

Manoj, C., Wijekoon, P. & Yapa, R.D., 2013. The McDonald Generalized Beta-Binomial Distribution: A New Binomial Mixture Distribution and Simulation Based Comparison with Its Nested Distributions in Handling Overdispersion. International Journal of Statistics and Probability, 2(2), pp.24-41.

Janiffer, N.M., Islam, A. & Luke, O., 2014. Estimating Equations for Estimation of Mcdonald Generalized Beta - Binomial Parameters. , (October), pp.702-709.

Roozegar, R., Tahmasebi, S. & Jafari, A.A., 2015. The McDonald Gompertz Distribution: Properties and Applications. Communications in Statistics - Simulation and Computation, (May), pp.0-0.

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 #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),dGBeta1(seq(0,1,by=0.001),a[i],1,2*a[i])$pdf,col = col[i]) } dGBeta1(seq(0,1,by=0.01),2,3,1)$pdf #extracting the pdf values dGBeta1(seq(0,1,by=0.01),2,3,1)$mean #extracting the mean dGBeta1(seq(0,1,by=0.01),2,3,1)$var #extracting the variance pGBeta1(0.04,2,3,4) #acquiring the cdf values for a=2,b=3,c=4 mazGBeta1(1.4,3,2,2) #acquiring the moment about zero values mazGBeta1(2,3,2,2)-mazGBeta1(1,3,2,2)^2 #acquiring the variance for a=3,b=2,c=2 #only the integer value of moments is taken here because moments cannot be decimal mazGBeta1(3.2,3,2,2) 

Amalan-ConStat/R-fitODBOD documentation built on Oct. 1, 2018, 7:13 p.m.