dgamma3: The gamma Distribution (3 Parameter)

In qualityTools: Statistical Methods for Quality Science

Description

Density function, distribution function and quantile function for the gamma distribution.

Usage

dgamma3(x, shape, scale, threshold)
pgamma3(q, shape, scale, threshold)
qgamma3(p, shape, scale, threshold, ...)

Arguments

x, q	vector of quantiles
p	vector of probabilities
shape	shape parameter by default 1
scale	scale parameter by default 1
threshold	threshold parameter by default 0
...	Arguments that can be passed into uniroot.

Details

The gamma distribution with ‘scale’ parameter alpha, ‘shape’ parameter c and ‘threshold’ parameter zeta has density given by

f(x) = (c/alpha) (((x-zeta)/alpha)^(c-1)) exp(-((x-zeta)/alpha)^c)

The cumulative distribution function is given by

F(x) = 1 - exp(-((x-zeta)/alpha)^c)

Value

dgamma3 gives the density, pgamma3 gives the distribution function and qgamma3 gives the quantile function.

Note

qgamma3 calls uniroot for each value of the argument ‘p’. The solution is consequently not exact; the ... can be used to obtain a more accurate solution if necessary.

Author(s)

Thomas Roth [e-mail]
Etienne Stockhausen [e-mail]

References

Johnson, L., Kotz, S., Balakrishnan, N. (1995) Continuous Univariate Distributions-Volume 1, 2nd ed. New York: John Wiley & Sons.

See Also

uniroot

Examples

##Simple Example
dgamma3(x=1,scale=1,shape=5,threshold=0)
temp=pgamma3(q=1,scale=1,shape=5,threshold=0)
temp
qgamma3(p=temp,scale=1,shape=5,threshold=0)
#
##Visualized Example
##prepare screen
#dev.new()
#split.screen(matrix(c(0,0.5,0,1, 0.5,1,0,1),byrow=TRUE,ncol=4))
##generate values
#x=seq(0,3,length=1000)
##plot different density functions
#screen(1)
#plot(x,y=dgamma3(x,threshold=0,shape=0.5,scale=1),col="green",
#     xlim=c(0,2.5),ylim=c(0,2.5),type="l",lwd=2,xlab="x",
#     ylab="f(x)",main="Density Function of gamma-Distribution")
#lines(x,y=dgamma3(x,threshold=0,shape=1,scale=1),lwd=2,col="red")
#lines(x,y=dgamma3(x,threshold=0,shape=1.5,scale=2),lwd=2,col="blue")
#lines(x,y=dgamma3(x,threshold=0,shape=5,scale=1),lwd=2,col="orange")
##add legend
#legend("topright",legend=c(expression(paste(alpha, " = 1 ")*
#       paste(c, " = 0.5 ")*paste(zeta," = 0")),
#       expression(paste(alpha, " = 1 ")*paste(c, " = 1 ")*
#       paste(zeta," = 0")),expression(paste(alpha, " = 2 ")*
#       paste(c, " = 1.5 ")*paste(zeta," = 0")),
#       expression(paste(alpha, " = 1 ")*paste(c, " = 5 ")*
#       paste(zeta," = 0"))),col=c("green","red","blue","orange"),
#       text.col="black",lwd=2,bty="0",inset=0.04)
#abline(v=0,lty=2,col="grey")
#abline(h=0,lty=2,col="grey")
##plot different distribution functions
#screen(2)
#plot(x,y=pgamma3(x,threshold=0,shape=0.5,scale=1),col="green",
#     xlim=c(0,2.5),ylim=c(0,1),type="l",lwd=2,xlab="x",ylab="F(x)",
#     main="Cumulative Distribution Function of gamma-Distribution")
#lines(x,y=pgamma3(x,threshold=0,shape=1,scale=1),lwd=2,col="red")
#lines(x,y=pgamma3(x,threshold=0,shape=1.5,scale=2),lwd=2,col="blue")
#lines(x,y=pgamma3(x,threshold=0,shape=5,scale=1),lwd=2,col="orange")
##add legend
#legend("bottomright",legend=c(expression(paste(alpha, " = 1 ")*
#       paste(c, " = 0.5 ")*paste(zeta," = 0")),
#       expression(paste(alpha, " = 1 ")*paste(c, " = 1 ")*
#       paste(zeta," = 0")),expression(paste(alpha, " = 2 ")*
#       paste(c, " = 1.5 ")*paste(zeta," = 0")),
#       expression(paste(alpha, " = 1 ")*paste(c, " = 5 ")*
#       paste(zeta," = 0"))),col=c("green","red","blue","orange"),
#       text.col="black",lwd=2,bty="0",inset=0.04)
#abline(v=0,lty=2,col="grey")
#abline(h=0,lty=2,col="grey")
#close.screen(all=TRUE)

Example output

Loading required package: Rsolnp
Loading required package: MASS

Attaching package: ‘qualityTools’

The following object is masked from ‘package:stats’:

    sigma

[1] 0.01532831
[1] 0.003659847
[1] 0.9999981

qualityTools documentation built on May 2, 2019, 10:21 a.m.