pGrassiaIIBin: Grassia-II-Binomial Distribution

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

Grassia-II-Binomial Distribution

Description

These functions provide the ability for generating probability function values and cumulative probability function values for the Grassia-II-Binomial Distribution.

Usage

pGrassiaIIBin(x,n,a,b)

Arguments

x	vector of binomial random variables.
n	single value for no of binomial trials.
a	single value for shape parameter a.
b	single value for shape parameter b.

Details

Mixing Gamma distribution with Binomial distribution will create the the Grassia-II-Binomial distribution, only when (1-p)=e^(-lambda) of the Binomial distribution. The probability function and cumulative probability function can be constructed and are denoted below.

The cumulative probability function is the summation of probability function values.

P_{GrassiaIIBin}[x]= {n \choose x} ∑_{j=0}^{x} {x \choose j} (-1)^{x-j} (1+b(n-j))^{-a}

a,b > 0

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

n = 1,2,3,...

The mean, variance and over dispersion are denoted as

E_{GrassiaIIBin}[x] = (\frac{b}{b+1})^a

Var_{GrassiaIIBin}[x] = n^2[(\frac{b}{b+2})^a - (\frac{b}{b+1})^{2a}] + n(\frac{b}{b+1})^a{1-(\frac{b+1}{b+2})^a}

over dispersion= \frac{(\frac{b}{b+2})^a - (\frac{b}{b+1})^{2a}}{(\frac{b}{b+1})^a[1-(\frac{b}{b+1})^a]}

Value

The output of pGrassiaIIBin gives cumulative probability values in vector form.

References

Grassia, A., 1977. On a family of distributions with argument between 0 and 1 obtained by transformation of the gamma and derived compound distributions. Australian Journal of Statistics, 19(2), pp.108-114.

Examples

#plotting the random variables and probability values
col <- rainbow(5)
a <- c(0.3,0.4,0.5,0.6,0.8)
plot(0,0,main="Grassia II 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,dGrassiaIIBin(0:10,10,2*a[i],a[i])$pdf,col = col[i],lwd=2.85)
points(0:10,dGrassiaIIBin(0:10,10,2*a[i],a[i])$pdf,col = col[i],pch=16)
}

dGrassiaIIBin(0:10,10,4,.2)$pdf    #extracting the pdf values
dGrassiaIIBin(0:10,10,4,.2)$mean   #extracting the mean
dGrassiaIIBin(0:10,10,4,.2)$var    #extracting the variance
dGrassiaIIBin(0:10,10,4,.2)$over.dis.para  #extracting the over dispersion value

#plotting the random variables and cumulative probability values
col <- rainbow(4)
a <- c(0.3,0.4,0.5,0.6)
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,pGrassiaIIBin(0:10,10,2*a[i],a[i]),col = col[i])
points(0:10,pGrassiaIIBin(0:10,10,2*a[i],a[i]),col = col[i])
}

pGrassiaIIBin(0:10,10,4,.2)   #acquiring the cumulative probability values

fitODBOD documentation built on Jan. 15, 2023, 5:11 p.m.