# dBetaCorrBin: Beta-Correlated 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 Beta-Correlated Binomial Distribution.

## Usage

 1 dBetaCorrBin(x,n,cov,a,b) 

## Arguments

 x vector of binomial random variables n single value for no of binomial trials cov single value for covariance a single value for alpha parameter b single value for beta parameter

## Details

The probability function and cumulative function can be constructed and are denoted below

The cumulative probability function is the summation of probability function values

x = 0,1,2,3,...n

n = 1,2,3,...

0 < a,b

-∞ < cov < +∞

0 < p < 1

p=\frac{a}{a+b}

Θ=\frac{1}{a+b}

The Correlation is in between

\frac{-2}{n(n-1)} min(\frac{p}{1-p},\frac{1-p}{p}) ≤ correlation ≤ \frac{2p(1-p)}{(n-1)p(1-p)+0.25-fo}

where fo=min [(x-(n-1)p-0.5)^2]

The mean and the variance are denoted as

E_{BetaCorrBin}[x]= np

Var_{BetaCorrBin}[x]= np(1-p)(nΘ+1)(1+Θ)^{-1}+n(n-1)cov

Corr_{BetaCorrBin}[x]=\frac{cov}{p(1-p)}

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

## Value

The output of dBetaCorrBin gives a list format consisting

pdf probability function values in vector form

mean mean of Beta-Correlated Binomial Distribution

var variance of Beta-Correlated Binomial Distribution

corr correlation of Beta-Correlated Binomial Distribution

mincorr minimum correlation value possible

maxcorr maximum correlation value possible

## References

Paul, S.R., 1985. A three-parameter generalization of the binomial distribution. Communications in Statistics - Theory and Methods, 14(6), pp.1497-1506.

## 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 27 28 29 30 #plotting the random variables and probability values col<-rainbow(5) a<-c(9.0,10,11,12,13) b<-c(8.0,8.1,8.2,8.3,8.4) plot(0,0,main="Beta-Correlated 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,dBetaCorrBin(0:10,10,0.001,a[i],b[i])$pdf,col = col[i],lwd=2.85) points(0:10,dBetaCorrBin(0:10,10,0.001,a[i],b[i])$pdf,col = col[i],pch=16) } dBetaCorrBin(0:10,10,0.001,10,13)$pdf #extracting the pdf values dBetaCorrBin(0:10,10,0.001,10,13)$mean #extracting the mean dBetaCorrBin(0:10,10,0.001,10,13)$var #extracting the variance dBetaCorrBin(0:10,10,0.001,10,13)$corr #extracting the correlation dBetaCorrBin(0:10,10,0.001,10,13)$mincorr #extracting the minimum correlation value dBetaCorrBin(0:10,10,0.001,10,13)$maxcorr #extracting the maximum correlation value #plotting the random variables and cumulative probability values col<-rainbow(5) a<-c(9.0,10,11,12,13) b<-c(8.0,8.1,8.2,8.3,8.4) plot(0,0,main="Beta-Correlated binomial probability function graph",xlab="Binomial random variable", ylab="Probability function values",xlim = c(0,10),ylim = c(0,1)) for (i in 1:5) { lines(0:10,pBetaCorrBin(0:10,10,0.001,a[i],b[i]),col = col[i],lwd=2.85) points(0:10,pBetaCorrBin(0:10,10,0.001,a[i],b[i]),col = col[i],pch=16) } pBetaCorrBin(0:10,10,0.001,10,13) #acquiring the cumulative probability values 

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