pGBeta1: Generalized Beta Type-1 Distribution

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

Generalized Beta Type-1 Distribution

Description

These functions provide the ability for generating probability density values, cumulative probability density values and moment about zero values for the Generalized Beta Type-1 Distribution bounded between [0,1].

Usage

pGBeta1(p,a,b,c)

Arguments

p	vector of probabilities.
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.

Details

The probability density function and cumulative density function of a unit bounded Generalized Beta Type-1 Distribution with random variable P are given by

g_{P}(p)= \frac{c}{B(a,b)} p^{ac-1} (1-p^c)^{b-1}

; 0 \le p \le 1

G_{P}(p)= \frac{p^{ac}}{aB(a,b)} 2F1(a,1-b;p^c;a+1)

0 \le p \le 1

a,b,c > 0

The mean and the variance are denoted by

E[P]= \frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})}

var[P]= \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

The moments about zero is denoted as

E[P^r]= \frac{B(a+b,\frac{r}{c})}{B(a,\frac{r}{c})}

r = 1,2,3,....

Defined as B(a,b) is Beta function. Defined as 2F1(a,b;c;d) is 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 pGBeta1 gives the cumulative density values in vector form.

References

\insertRef

manoj2013mcdonaldfitODBOD \insertRefjaniffer2014estimatingfitODBOD \insertRefroozegar2017mcdonaldfitODBOD

Examples

#plotting the random variables and probability values
col <- rainbow(5)
a <- c(.1,.2,.3,1.5,2.15)
plot(0,0,main="Probability density graph",xlab="Random variable",ylab="Probability density values",
xlim = c(0,1),ylim = c(0,10))
for (i in 1:5)
{
lines(seq(0,1,by=0.001),dGBeta1(seq(0,1,by=0.001),a[i],1,2*a[i])$pdf,col = col[i])
}

dGBeta1(seq(0,1,by=0.01),2,3,1)$pdf    #extracting the pdf values
dGBeta1(seq(0,1,by=0.01),2,3,1)$mean   #extracting the mean
dGBeta1(seq(0,1,by=0.01),2,3,1)$var    #extracting the variance

pGBeta1(0.04,2,3,4)        #acquiring the cdf values for a=2,b=3,c=4
mazGBeta1(1.4,3,2,2)              #acquiring the moment about zero values
mazGBeta1(2,3,2,2)-mazGBeta1(1,3,2,2)^2        #acquiring the variance for a=3,b=2,c=2

#only the integer value of moments is taken here because moments cannot be decimal
mazGBeta1(3.2,3,2,2)

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