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I have come upon different parameterizations of the Gamma Distribution, but not with regard to shape-scale or shape-rate. It is rather about the sign in the exponent.
Wolfram lists the pdf as being proportional to $$x^{a-1} \exp{-\frac{x}{b}}$$ https://reference.wolfram.com/language/ref/GammaDistribution.html

However, I saw some papers where the minus sign is missing such that, $$x^{a-1}\exp{\frac{x}{b}}$$ From my understanding, both parameters $a$ and $b$ have to be positive so this must make some kind of difference. Do I have some error in reasoning here?

Edit: Excerpt from a paper formula

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  • $\begingroup$ Could you provide a reference where you actually saw the formula without minus sign..? $\endgroup$ – Tim Sep 16 '16 at 14:46
  • $\begingroup$ Sure, for instance: Jumps in Equity Index Returns Before and During the Recent Financial Crisis: A Bayesian Analysis (2016, by Kou, Yu and Zhong) on p. 5 in the footnote $\endgroup$ – mscnvrsy Sep 16 '16 at 15:19
  • $\begingroup$ I added a screenshot for clarification. $\endgroup$ – mscnvrsy Sep 16 '16 at 17:19
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    $\begingroup$ This formulation is simply wrong. Have you tried plotting this function..? $\endgroup$ – Tim Sep 16 '16 at 18:25
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This is clearly a typo in the paper. You can verify that by seeing that the form of the pdf they present, does not integrate to something finite.

$$x^{\alpha - 1} e^{x/\beta} \to 0 \text{ as } x \to 0$$

$$x^{\alpha - 1} e^{x/\beta} \to \infty \text{ as } x \to \infty.$$

Thus, the pdf integrates to infinity. You can see that in wolfram alpha here.

Thus, the integral diverges, and the pdf presented is not a valid pdf.

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  • $\begingroup$ Hey, the paper is 2016, so there is still time to correct the typo! (Or, given the paper title, could this be a sly way to seed the next financial crisis by misleading the next gen of quants?) $\endgroup$ – GeoMatt22 Sep 16 '16 at 23:27
  • $\begingroup$ Okay, thank you very much! Astonishingly, I found another paper with the exact same typo but from 2008. $\endgroup$ – mscnvrsy Sep 17 '16 at 9:30

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