pGHGBB {fitODBOD}

Gaussian Hypergeometric Generalized Beta Binomial Distribution

Description

These functions provide the ability for generating probability function values and cumulative probability function values for the Gaussian Hypergeometric Generalized Beta Binomial distribution.

Usage

pGHGBB(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 value representing a.

b

single value for shape parameter beta value representing b.

c

single value for shape parameter lambda value representing c.

Details

Mixing Gaussian Hypergeometric Generalized Beta distribution with Binomial distribution will create the Gaussian Hypergeometric Generalized Beta 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_{GHGBB}(x)=\frac{1}{2F1(-n,a;-b-n+1;c)}{n \choose x} \frac{B(x+a,n-x+b)}{B(a,b+n)}(c^x)

a,b,c > 0

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

n = 1,2,3,...

The mean, variance and over dispersion are denoted as

E_{GHGBB}[x]= nE_{GHGBeta}

Var_{GHGBB}[x]= nE_{GHGBeta}(1-E_{GHGBeta})+ n(n-1)Var_{GHGBeta}

over dispersion= \frac{var_{GHGBeta}}{E_{GHGBeta}(1-E_{GHGBeta})}

Defined as B(a,b) is the beta function. Defined as 2F1(a,b;c;d) is the Gaussian Hypergeometric function.

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

Value

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

References

Rodriguez-Avi J, Conde-Sanchez A, Saez-Castillo AJ, Olmo-Jimenez MJ (2007). “A generalization of the beta–binomial distribution.” Journal of the Royal Statistical Society Series C: Applied Statistics, 56(1), 51–61. Pearson JW (2009). Computation of hypergeometric functions. Ph.D. thesis, University of Oxford.

See Also

hypergeo_powerseries

Examples

#plotting the random variables and probability values
col <- rainbow(6)
a <- c(.1,.2,.3,1.5,2.1,3)
plot(0,0,main="GHGBB probability function graph",xlab="Binomial random variable",
ylab="Probability function values",xlim = c(0,7),ylim = c(0,0.9))
for (i in 1:6)
{
lines(0:7,dGHGBB(0:7,7,1+a[i],0.3,1+a[i])$pdf,col = col[i],lwd=2.85)
points(0:7,dGHGBB(0:7,7,1+a[i],0.3,1+a[i])$pdf,col = col[i],pch=16)
}

dGHGBB(0:7,7,1.3,0.3,1.3)$pdf      #extracting the pdf values
dGHGBB(0:7,7,1.3,0.3,1.3)$mean     #extracting the mean
dGHGBB(0:7,7,1.3,0.3,1.3)$var      #extracting the variance
dGHGBB(0:7,7,1.3,0.3,1.3)$over.dis.par  #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(0,7),ylim = c(0,1))
for (i in 1:4)
{
lines(0:7,pGHGBB(0:7,7,1+a[i],0.3,1+a[i]),col = col[i])
points(0:7,pGHGBB(0:7,7,1+a[i],0.3,1+a[i]),col = col[i])
}

pGHGBB(0:7,7,1.3,0.3,1.3)     #acquiring the cumulative probability values