After making the assumption that monetary losses could be well represented by a gamma distribution (Boland, 2007), mostly negatively skewed, and being interested in loss ratios (ie. lost value / total value), I performed a number of simulations sampling from a gamma distribution. I constrained the shape and scale parameters in such a way that the minimum and maximum values that could be sampled lied between 0 and 1.

Later on, I realized I could have just sampled directly from a beta distribution given that it is already defined between 0 and 1 - thus representing directly the fact I was looking at loss ratios only.

But this then raised the question: given that both gamma and beta pdfs can take a range of similar "shapes" (based on the values of their parameters), is it equivalent to use a gamma distribution (by somewhat constraining the parameters) and a beta distribution?

I have been trying the understand the relationship between the 2 distributions, but apart from the fact that if X is gamma distributed with parameters (a,r) and Y is gamma distributed with parameters (b,r) then X/(X+Y) is beta distributed with parameters (a,b), I did not find anything I could relate to the case I am interested in.

Ref. Boland, Statistical and Probabilistic Methods in Actuarial Science (2007), CRC Press.


closed as unclear what you're asking by whuber Jan 25 '15 at 21:02

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    $\begingroup$ The nature of the constraint you applied is not clear. It's one thing to scale data so that observed data in a particular application fall within [0,1], but that does not amount to converting a gamma into a beta. $\endgroup$ – Nick Cox Jan 25 '15 at 15:06
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    $\begingroup$ I don't understand the "mostly negatively skewed" comment; all gamma distributions are right skew. You should clarify what you mean about 'constrained the scale and shape', since all gamma distributions have some probability to the right of 1 and so it should also be clear that no gamma distribution is the same as a beta distribution. Your definition of loss ratio seems to be wrong - the definition at investopedia matches many other sources. With the usual definition, loss ratios can exceed one. You should make your question clearer. $\endgroup$ – Glen_b -Reinstate Monica Jan 25 '15 at 15:37

Gamma distributions have support on an (infinite) half-interval of the real line. The parameters in a (two-parameter) Gamma distribution do not constrain the support and so it is not possible to strictly contain all the density on the unit-interval by manipulating these parameters and maintaining a non-degenerate distribution.

However you think you have done so seems, regrettably, erroneous. The version of the Beta that you mention (from a transformation involving two independent Gamma variables) is essentially the canonical definition of a Beta. It is the format of this transform (ratio of component in a sum to the value of the sum) that constrains the support, not the parameters.


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