# pMcGBB: McDonald Generalized Beta Binomial 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 function values and cumulative probability function values for the McDonald Generalized Beta Binomial Distribution.

## Usage

 1 pMcGBB(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 representing as a b single value for shape parameter beta representing as b c single value for shape parameter gamma representing as c

## Details

Mixing Generalized Beta Type-1 Distribution with binomial distribution the probability function value and cumulative probability function can be constructed and are denoted below.

The cumulative probability function is the summation of probability function values

P_{McGBB}(x)= {n \choose x} \frac{1}{B(a,b)} (∑_{j=0}^{n-x} (-1)^j {n-x \choose j} B(\frac{x}{c}+a+\frac{j}{c},b) )

a,b,c > 0

The mean, variance and over dispersion are denoted as

E_{McGBB}[x]= n\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})}

Var_{McGBB}[x]= n^2(\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) +n(\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})}-\frac{B(a+b,\frac{2}{c})}{B(a,\frac{2}{c})})

over dispersion= \frac{\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}{\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})}-(\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})})^2}

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

n = 1,2,3,...

## Value

The output of pMcGBB gives cumulative probability function 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 19 20 21 22 23 24 25 26 #plotting the random variables and probability values col<-rainbow(5) a<-c(1,2,5,10,0.6) plot(0,0,main="Mcdonald generalized beta-binomial probability function graph", xlab="Binomial random variable",ylab="Probability function values",xlim = c(0,10),ylim = c(0,0.5)) for (i in 1:5) { lines(0:10,dMcGBB(0:10,10,a[i],2.5,a[i])$pdf,col = col[i],lwd=2.85) points(0:10,dMcGBB(0:10,10,a[i],2.5,a[i])$pdf,col = col[i],pch=16) } dMcGBB(0:10,10,4,2,1)$pdf #extracting the pdf values dMcGBB(0:10,10,4,2,1)$mean #extracting the mean dMcGBB(0:10,10,4,2,1)$var #extracting the variance dMcGBB(0:10,10,4,2,1)$over.dis.para #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(0,10),ylim = c(0,1)) for (i in 1:4) { lines(0:10,pMcGBB(0:10,10,a[i],a[i],2),col = col[i]) points(0:10,pMcGBB(0:10,10,a[i],a[i],2),col = col[i]) } pMcGBB(0:10,10,4,2,1) #acquiring the cumulative probability values 

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