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?

  • 1
    $\begingroup$ Which three parameter gamma? I've seen more than one. Do you mean the one that's simply a shifted two parameter gamma? $\endgroup$
    – Glen_b
    Commented Oct 18, 2013 at 19:16
  • $\begingroup$ The meaning of "location parameter" and "scale parameter" is independent of the distribution. Also, what do you mean by "structure of its density"? $\endgroup$
    – jbowman
    Commented Oct 19, 2013 at 0:00
  • $\begingroup$ It refers to the probability density function $\endgroup$
    – Vani
    Commented Oct 19, 2013 at 4:42
  • 2
    $\begingroup$ @Carl That must be very confusing for physicists when they want to talk about a function that represents actual mass-density. I can understand a statistician dropping the adjective, but for a physicist it would seem to be necessary to keep it, since they might regularly need to refer to more than one kind of density. $\endgroup$
    – Glen_b
    Commented Dec 19, 2016 at 0:28
  • 3
    $\begingroup$ I wouldn't call a pdf a probability function. It's a density function, but since more than one thing can have density, we need to be able to express what it's a density of (and that's probability). The only think I'd normally call a probability function is the function that represents P(X=x) for a discrete random variable. $\endgroup$
    – Glen_b
    Commented Dec 19, 2016 at 0:55

1 Answer 1

  1. The three parameter gamma distribution is needed only when you need to shift the distribution itself.

  2. 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 $\text{mean}=\text{location}+\text{shape}\cdot \text{scale}$.


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