mazGHGBeta {fitODBOD} | R Documentation |
These functions provide the ability for generating probability density values, cumulative probability density values and moment about zero values for the Gaussian Hypergeometric Generalized Beta distribution bounded between [0,1].
mazGHGBeta(r,n,a,b,c)
r |
vector of moments. |
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 lambda representing as c. |
The probability density function and cumulative density function of a unit bounded Gaussian Hypergeometric Generalized Beta Distribution with random variable P are given by
g_{P}(p)= \frac{1}{B(a,b)}\frac{2F1(-n,a;-b-n+1;1)}{2F1(-n,a;-b-n+1;c)} p^{a-1}(1-p)^{b-1} \frac{c^{b+n}}{(c+(1-c)p)^{a+b+n}}
;
0 \le p \le 1
G_{P}(p)= \int^p_0 \frac{1}{B(a,b)}\frac{2F1(-n,a;-b-n+1;1)}{2F1(-n,a;-b-n+1;c)} t^{a-1}(1-t)^{b-1}\frac{c^{b+n}}{(c+(1-c)t)^{a+b+n}} \,dt
;
0 \le p \le 1
a,b,c > 0
n = 1,2,3,...
The mean and the variance are denoted by
E[P]= \int^1_0 \frac{p}{B(a,b)}\frac{2F1(-n,a;-b-n+1;1)}{2F1(-n,a;-b-n+1;c)} p^{a-1}(1-p)^{b-1}\frac{c^{b+n}}{(c+(1-c)p)^{a+b+n}} \,dp
var[P]= \int^1_0 \frac{p^2}{B(a,b)}\frac{2F1(-n,a;-b-n+1;1)}{2F1(-n,a;-b-n+1;c)} p^{a-1}(1-p)^{b-1}\frac{c^{b+n}}{(c+(1-c)p)^{a+b+n}} \,dp - (E[p])^2
The moments about zero is denoted as
E[P^r]= \int^1_0 \frac{p^r}{B(a,b)}\frac{2F1(-n,a;-b-n+1;1)}{2F1(-n,a;-b-n+1;c)} p^{a-1}(1-p)^{b-1}\frac{c^{b+n}}{(c+(1-c)p)^{a+b+n}} \,dp
r = 1,2,3,...
Defined as B(a,b)
as the beta function.
Defined as 2F1(a,b;c;d)
as the Gaussian Hypergeometric function.
NOTE : If input parameters are not in given domain conditions necessary error messages will be provided to go further.
The output of mazGHGBeta
give the moments about zero in vector form.
Rodriguez-Avi, J., Conde-Sanchez, A., Saez-Castillo, A. J., & Olmo-Jimenez, M. J. (2007). A generalization of the beta-binomial distribution. Journal of the Royal Statistical Society. Series C (Applied Statistics), 56(1), 51-61.
Pearson, J., 2009. Computation of Hypergeometric Functions. Transformation, (September), p.1–123.
#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),dGHGBeta(seq(0,1,by=0.001),7,1+a[i],0.3,1+a[i])$pdf,col = col[i])
}
dGHGBeta(seq(0,1,by=0.01),7,1.6312,0.3913,0.6659)$pdf #extracting the pdf values
dGHGBeta(seq(0,1,by=0.01),7,1.6312,0.3913,0.6659)$mean #extracting the mean
dGHGBeta(seq(0,1,by=0.01),7,1.6312,0.3913,0.6659)$var #extracting the variance
#plotting the random variables and cumulative probability values
col <- rainbow(6)
a <- c(.1,.2,.3,1.5,2.1,3)
plot(0,0,main="Cumulative density graph",xlab="Random variable",ylab="Cumulative density values",
xlim = c(0,1),ylim = c(0,1))
for (i in 1:6)
{
lines(seq(0.01,1,by=0.001),pGHGBeta(seq(0.01,1,by=0.001),7,1+a[i],0.3,1+a[i]),col=col[i])
}
pGHGBeta(seq(0,1,by=0.01),7,1.6312,0.3913,0.6659) #acquiring the cumulative probability values
mazGHGBeta(1.4,7,1.6312,0.3913,0.6659) #acquiring the moment about zero values
#acquiring the variance for a=1.6312,b=0.3913,c=0.6659
mazGHGBeta(2,7,1.6312,0.3913,0.6659)-mazGHGBeta(1,7,1.6312,0.3913,0.6659)^2
#only the integer value of moments is taken here because moments cannot be decimal
mazGHGBeta(1.9,15,5,6,1)