Solved – Three Parameter Gamma Distribution

gamma distribution

What is the motivation for the three parameter gamma distribution and the resulting structure of its density?

What is the meaning of the location, scale and shape parameters here?

Best Answer

The three parameter gamma distribution is needed only when you need to shift the distribution itself.
In the two-parameter gamma distribution, you could read the shape parameter as a proxy of the most probable value of the distribution, and the scale parameter of how "long" is its tail.

The mean of the distribution is given exactly by the product of the shape and the scale parameters.

If you need to shift the distribution, that's where you use the three-parameters gamma distribution. In this instance, the mean of the distribution is simply $\text{mean}=\text{location}+\text{shape}\cdot \text{scale}$.

Related Solutions

Solved – What are the mean and variance for the Gamma distribution

If the shape parameter is $k>0$ and the scale is $\theta>0$, one parameterization has density function

$$p(x) = x^{k-1} \frac{ e^{-x/\theta} }{\theta^{k} \Gamma(k)}$$

where the argument, $x$, is non-negative. A random variable with this density has mean $k \theta$ and variance $k \theta^{2}$ (this parameterization is the one used on the wikipedia page about the gamma distribution).

An alternative parameterization uses $\vartheta = 1/\theta$ as the rate parameter (inverse scale parameter) and has density

$$p(x) = x^{k-1} \frac{ \vartheta^{k} e^{-x \vartheta} }{\Gamma(k)}$$

Under this choice, the mean is $k/\vartheta$ and the variance is $k/\vartheta^{2}$.

R – How to Perform Maximum Likelihood Estimation of Inverse Gamma Distribution in R or RPy

Since you know the density, you can just use fitdistr.

# Sample data
library(LaplacesDemon) 
x <- rinvgamma(1000, 1,2)

library(MASS)
f <- function(x, rho, a, s)
  1/(a*gamma(rho)) * (a / (x-s))^(rho+1) * exp( - a/(x-s) )
fitdistr( x, f, list(rho=1, a=1, s=0) )

Best Answer

Related Solutions

Solved – What are the mean and variance for the Gamma distribution

R – How to Perform Maximum Likelihood Estimation of Inverse Gamma Distribution in R or RPy

Related Question