These functions provide the ability for generating probability function values and cumulative probability function values for the Kumaraswamy Binomial Distribution.
dKumBin(x,n,a,b,it=25000)
| 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 |
The output of dKumBin gives a list format consisting
pdf probability function values in vector form.
mean mean of the Kumaraswamy Binomial Distribution.
var variance of the Kumaraswamy Binomial Distribution.
over.dis.para over dispersion value of the Kumaraswamy Distribution.
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.
$$P_{KumBin}(x)= ab{n \choose x} \sum_{j=0}^{it} (-1)^j{b-1 \choose j}B(x+a+aj,n-x+1) $$ $$a,b > 0$$ $$x = 0,1,2,...n$$ $$n = 1,2,3,...$$ $$it > 0$$
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.
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.
# 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) } 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 # }