dAddBin: Additive Binomial Distribution

In fitODBOD: Modeling Over Dispersed Binomial Outcome Data Using BMD and ABD

Additive Binomial Distribution

Description

These functions provide the ability for generating probability function values and cumulative probability function values for the Additive Binomial Distribution.

Usage

dAddBin(x,n,p,alpha)

Arguments

x	vector of binomial random variables.
n	single value for no of binomial trials.
p	single value for probability of success
alpha	single value for alpha 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.

P_{AddBin}(x)= {n \choose x} p^x (1-p)^{n-x}(\frac{alpha}{2}(\frac{x(x-1)}{p}+\frac{(n-x)(n-x-1)}{(1-p)}-\frac{alpha(n-1)n}{2})+1)

The alpha is in between

\frac{-2}{n(n-1)}min(\frac{p}{1-p},\frac{1-p}{p}) \le alpha \le (\frac{n+(2p-1)^2}{4p(1-p)})^{-1}

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

n = 1,2,3,...

0 < p < 1

-1 < alpha < 1

The mean and the variance are denoted as

E_{Addbin}[x]=np

Var_{Addbin}[x]=np(1-p)(1+(n-1)alpha)

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

Value

The output of dAddBin gives a list format consisting

pdf probability function values in vector form.

mean mean of Additive Binomial Distribution.

var variance of Additive Binomial Distribution.

References

\insertRef

johnson2005univariatefitODBOD \insertRefkupper1978usefitODBOD \insertRefpaul1985threefitODBOD \insertRefmorel2012overdispersionfitODBOD

Examples

#plotting the random variables and probability values
col <- rainbow(5)
a <- c(0.58,0.59,0.6,0.61,0.62)
b <- c(0.022,0.023,0.024,0.025,0.026)
plot(0,0,main="Additive 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,dAddBin(0:10,10,a[i],b[i])$pdf,col = col[i],lwd=2.85)
  points(0:10,dAddBin(0:10,10,a[i],b[i])$pdf,col = col[i],pch=16)
}

dAddBin(0:10,10,0.58,0.022)$pdf     #extracting the probability values
dAddBin(0:10,10,0.58,0.022)$mean    #extracting the mean
dAddBin(0:10,10,0.58,0.022)$var     #extracting the variance

#plotting the random variables and cumulative probability values
col <- rainbow(5)
a <- c(0.58,0.59,0.6,0.61,0.62)
b <- c(0.022,0.023,0.024,0.025,0.026)
plot(0,0,main="Additive 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,pAddBin(0:10,10,a[i],b[i]),col = col[i],lwd=2.85)
points(0:10,pAddBin(0:10,10,a[i],b[i]),col = col[i],pch=16)
}

pAddBin(0:10,10,0.58,0.022)       #acquiring the cumulative probability values

fitODBOD documentation built on Oct. 10, 2024, 5:07 p.m.