2
$\begingroup$

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

Improve this question
$\endgroup$
4
  • $\begingroup$ Could you provide a reference where you actually saw the formula without minus sign..? $\endgroup$
    – Tim
    Commented Sep 16, 2016 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
    Commented Sep 16, 2016 at 15:19
  • $\begingroup$ I added a screenshot for clarification. $\endgroup$
    – mscnvrsy
    Commented Sep 16, 2016 at 17:19
  • 2
    $\begingroup$ This formulation is simply wrong. Have you tried plotting this function..? $\endgroup$
    – Tim
    Commented Sep 16, 2016 at 18:25

1 Answer 1

Reset to default
5
$\begingroup$

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.

Improve this answer
$\endgroup$
2
  • $\begingroup$ 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
    Commented Sep 16, 2016 at 23:27
  • $\begingroup$ Okay, thank you very much! Astonishingly, I found another paper with the exact same typo but from 2008. $\endgroup$
    – mscnvrsy
    Commented Sep 17, 2016 at 9:30

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.