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?
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1Which three parameter gamma? I've seen more than one. Do you mean the one that's simply a shifted two parameter gamma? – Glen_b Oct 18 '13 at 19:16
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The meaning of "location parameter" and "scale parameter" is independent of the distribution. Also, what do you mean by "structure of its density"? – jbowman Oct 19 '13 at 00:00
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It refers to the probability density function – Vani Oct 19 '13 at 04:42
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@jbowman Physicists like to say density function, as actually adding the term "probability" to make 'probability density function' adds no extra meaning. – Carl Dec 18 '16 at 09:04
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@Glen_b More likely means the generalized gamma distribution. – Carl Dec 18 '16 at 09:10
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2@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. – Glen_b Dec 19 '16 at 00:28
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@Glen_b My son is a physicist and is quite upset if I add the adjective. I think that only definite integrals of the density function can be thought of as having probabilities, not the function itself. For example, we could call a CDF a probability function, but not a PDF, and yet we do the opposite. – Carl Dec 19 '16 at 00:46
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3I 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. – Glen_b Dec 19 '16 at 00:55
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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 $\text{mean}=\text{location}+\text{shape}\cdot \text{scale}$.
