pMcGBB function

McDonald Generalized Beta Binomial Distribution

McDonald Generalized Beta Binomial Distribution

These functions provide the ability for generating probability function values and cumulative probability function values for the McDonald Generalized Beta Binomial Distribution.

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.

Returns

The output of pMcGBB gives cumulative probability function values in vector form.

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.

PMcGBB(x)=(nx)1B(a,b)(∑j=0n−x(−1)j(n−xj)B(xc+a+jc,b)) P_{McGBB}(x)= {n \choose x} \frac{1}{B(a,b)} (\sum_{j=0}^{n-x} (-1)^j {n-x \choose j} B(\frac{x}{c}+a+\frac{j}{c},b) ) a,b,c>0 a,b,c > 0

The mean, variance and over dispersion are denoted as

EMcGBB[x]=nB(a+b,1c)B(a,1c) E_{McGBB}[x]= n\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})} VarMcGBB[x]=n2(B(a+b,2c)B(a,2c)−(B(a+b,1c)B(a,1c))2)+n(B(a+b,1c)B(a,1c)−B(a+b,2c)B(a,2c)) 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})}) overdispersion=B(a+b,2c)B(a,2c)−(B(a+b,1c)B(a,1c))2B(a+b,1c)B(a,1c)−(B(a+b,1c)B(a,1c))2 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 x = 0,1,2,...n n=1,2,3,... n = 1,2,3,...

Examples

#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

References

\insertRef manoj2013mcdonaldfitODBOD

\insertRef janiffer2014estimatingfitODBOD

\insertRef roozegar2017mcdonaldfitODBOD

fitODBOD packageRead PDF manual
  • Maintainer: Amalan Mahendran
  • License: MIT + file LICENSE
  • Last published: 2024-11-20

Useful links