sgr (version 1.3.1)

dgBeta: Generalized Beta Distribution.

Description

The generalized beta distribution extends the classical beta distribution beyond the [0,1] range (Whitby, 1971).

Usage

dgBeta(x, a = min(x), b = max(x), gam = 1, del = 1)

Arguments

x

Vector of quantilies.

a

Minimum of range of r.v. \(X\).

b

Maximum of range of r.v. \(X\).

gam

Gamma parameter.

del

Delta parameter.

Value

Gives the density.

Details

The Generalized Beta Distribution is defined as follows:

$$G(x;a,b,\gamma,\delta) = \frac{1}{B(\gamma,\delta)(b-a)^{\gamma+\delta-1}} (x-a)^{\gamma-1}(b-x)^{\delta-1}$$

where \(B(\gamma,\delta)\) is the beta function. The parameters \(a \in R\) and \(b \in R\) (with \(a < b\)) are the left and right end points, respectively. The parameters \(\gamma > 0\) and \(\delta > 0\) are the governing shape parameters for \(a\) and \(b\) respectively. For all the values of the r.v. \(X\) that fall outside the interval \([a, b]\), \(G\) simply takes value 0. The generalized beta distribution reduces to the beta distribution when \(a = 0\) and \(b = 1\).

References

Whitby, O. (1971). Estimation of parameters in the generalized beta distribution (Technical Report NO. 29). Department of Statistics: Standford University.

See Also

dgBetaD

Examples

# NOT RUN {
curve(dgBeta(x))
curve(dgBeta(x,gam=3,del=3))
curve(dgBeta(x,gam=1.5,del=2.5))

x <- 1:7
GA <- c(1,3,1.5,8); DE <- c(1,3,4,2.5)
par(mfrow=c(2,2))
for (j in 1:4) {
  plot(x,dgBeta(x,gam=GA[j],del=DE[j]),type="h",
       panel.first=points(x,dgBeta(x,gam=GA[j],del=DE[j]),pch=19),
       main=paste("gamma=",GA[j]," delta=",DE[j],sep=""),ylim=c(0,.6),
       ylab="dgBeta(x)")  
}
# }