pKumBin: Kumaraswamy Binomial Distribution

Description Usage Arguments Details Value References Examples

View source: R/Kumaraswamy.R

Description

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

Usage

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pKumBin(x,n,a,b,it=25000)

Arguments

x

vector of binomial random variables

n

single value for no of binomial trial

a

single value for shape parameter alpha representing a

b

single value for shape parameter beta representing b

it

number of iterations to converge as a proper probability function replacing infinity

Details

Mixing kumaraswamy distribution with binomial distribution will create the Kumaraswamy 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_{KumBin}(x)= ab{n \choose x} ∑_{j=0}^{it} (-1)^j{b-1 \choose j}B(x+a+aj,n-x+1)

a,b > 0

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

n = 1,2,3,...

it > 0

The mean, variance and over dispersion are denoted as

E_{KumBin}[x]= nbB(1+\frac{1}{a},b)

Var_{KumBin}[x]= (n^2)b(B(1+\frac{2}{a},b)-bB(1+\frac{1}{a},b)^2)+ nb(B(1+\frac{1}{a},b)-B(1+\frac{2}{a},b))

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

Defined as B(a,b) is the beta function

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

Value

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

References

Li, X. H., Huang, Y. Y., & Zhao, X. Y. (2011). The Kumaraswamy Binomial Distribution. Chinese Journal of Applied Probability and Statistics, 27(5), 511-521.

Examples

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## Not run: 
#plotting the random variables and probability values
col<-rainbow(5)
a<-c(1,2,5,10,.85)
plot(0,0,main="Kumaraswamy 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,dKumBin(0:10,10,a[i],a[i])$pdf,col = col[i],lwd=2.85)
points(0:10,dKumBin(0:10,10,a[i],a[i])$pdf,col = col[i],pch=16)
}

## End(Not run)
dKumBin(0:10,10,4,2)$pdf  #extracting the pdf values
dKumBin(0:10,10,4,2)$mean #extracting the mean
dKumBin(0:10,10,4,2)$var  #extracting the variance
dKumBin(0:10,10,4,2)$over.dis.para #extracting the over dispersion value

#plotting the random variables and cumulative probability values
col<-rainbow(5)
a<-c(1,2,5,10,.85)
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 1:5)
{
lines(0:10,pKumBin(0:10,10,a[i],a[i]),col = col[i])
points(0:10,pKumBin(0:10,10,a[i],a[i]),col = col[i])
}

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

Amalan-ConStat/R-fitODBOD documentation built on Oct. 1, 2018, 7:13 p.m.