Timeline for Skewness of the logarithm of a gamma random variable
Current License: CC BY-SA 3.0
33 events
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| Nov 16, 2017 at 12:28 | history | edited | Glen_b | CC BY-SA 3.0 |
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| Nov 16, 2017 at 11:55 | history | bounty ended | amoeba | ||
| Nov 14, 2017 at 9:08 | comment | added | Glen_b | Noice. I should have waited an hour to offer that bounty here; I cost myself a 200+-day. | |
| Nov 14, 2017 at 9:06 | comment | added | amoeba | Here is my follow-up question: stats.stackexchange.com/questions/313619. But this time I managed to answer it myself :) | |
| Nov 11, 2017 at 11:40 | comment | added | amoeba | Sure. I'm going to accept Francis' answer (as it gets to the skewness formula faster and does the full derivation) but I think both answers deserve a bounty. If you want to bounty Francis' one then go ahead (bounty can be started already). Thanks again for taking the time. It seems that I have one or two related questions arising that I might post soon. | |
| Nov 9, 2017 at 23:45 | comment | added | Glen_b | @amoeba On bounties; I have plans to bounty Francis' excellent answer here. | |
| Nov 9, 2017 at 23:28 | comment | added | Glen_b | Thanks; I think most of those are clear improvements. I had planned to introduce some headings next edit (not quite those but they'll do). The other edits are fine -- again, not quite what I'd have said but they're okay. I see a word missing now but I'm not editing for one word. | |
| Nov 9, 2017 at 22:20 | comment | added | amoeba | I took the liberty to edit for some formatting; I hope you will not mind but if you dislike some of my formatting choices feel free to change them back. Apart from purely formatting, I edited the paragraph towards the end about cumulants/moments confusion in Chan. Please take a look and make sure that I did not introduce further confusion! But it does make sense to me now. | |
| Nov 9, 2017 at 22:19 | history | edited | amoeba | CC BY-SA 3.0 |
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| Nov 9, 2017 at 15:13 | comment | added | Glen_b | I'm not talking about the function, I am talking about what to call the distribution. The point is to show different references using different names for the same object. | |
| Nov 9, 2017 at 15:12 | history | edited | Glen_b | CC BY-SA 3.0 |
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| Nov 9, 2017 at 15:11 | comment | added | J. M. is not a statistician |
Ah, then I suggest editing it to ExpGammaDistribution, to minimize confusion like mine.
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| Nov 9, 2017 at 15:10 | comment | added | Glen_b | "this" being the same this in the previous sentence -- the distribution. | |
| Nov 9, 2017 at 15:09 | comment | added | J. M. is not a statistician |
No, you said "It looks like Mathematica writes the moments of this distribution in terms of the polygamma function (Mathematica calls this ExpGamma)" and I was pointing out that the polygamma function is indeed called PolyGamma[] in Mathematica. The distribution being considered by the OP is not built-in, but can be expressed with TransformedDistribution[]. All probability distribution functions in Mathematica end in, well, Distribution.
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| Nov 9, 2017 at 15:07 | comment | added | Glen_b | @J.M.isnotastatistician You're saying that Mathematica calls the distribution of the log of a gamma random variable PolyGamma? That makes no sense. See here | |
| Nov 9, 2017 at 15:02 | comment | added | J. M. is not a statistician |
@Glen, Mathematica uses PolyGamma[], not ExpGamma[].
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| Nov 9, 2017 at 15:01 | comment | added | J. M. is not a statistician | I should probably add the note that the Hurwitz zeta function can be expressed in terms of the polygamma function, and vice versa: $$\psi^{(n)}(z)=(-1)^{n+1}\,\Gamma(n+1)\,\zeta(n+1,z)$$ So, the answer to the @amoeba's question of "will the tetragamma function appear?" is YES. | |
| Nov 9, 2017 at 14:09 | comment | added | amoeba | And I did not expect this question to kindle such a thorough discussion! Thanks a lot, again. I will certainly put a bounty here when it becomes possible (after 2 days). Regarding cumulants: am still confused by "the third cumulant is the skewness" -- Wikipedia tells me that 3rd cumulant is the third central moment, not the third standardized moment (skewness). Also, what is normally denoted by gammas? I don't know. | |
| Nov 9, 2017 at 13:56 | comment | added | Glen_b | Hopefully correct now. I did not mean to spend this many hours on this. | |
| Nov 9, 2017 at 13:56 | history | edited | Glen_b | CC BY-SA 3.0 |
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| Nov 9, 2017 at 13:51 | comment | added | Glen_b | Hmm, I mean to refer to the things normally denoted $\gamma_1$ and $\gamma_2$. I will fix. | |
| Nov 9, 2017 at 13:44 | history | edited | Glen_b | CC BY-SA 3.0 |
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| Nov 9, 2017 at 13:41 | history | edited | Glen_b | CC BY-SA 3.0 |
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| Nov 9, 2017 at 13:36 | history | edited | Glen_b | CC BY-SA 3.0 |
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| Nov 9, 2017 at 13:15 | comment | added | amoeba | Yes, the Encyclopedia does give the same formula for kurtosis. | |
| Nov 9, 2017 at 13:11 | history | edited | amoeba | CC BY-SA 3.0 |
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| Nov 9, 2017 at 13:00 | history | edited | Glen_b | CC BY-SA 3.0 |
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| Nov 9, 2017 at 12:55 | comment | added | Glen_b | Sorry only just now saw your comment (I've been editing for about an hour or so); that's correct, though if the Encyclopedia gives kurtosis the way Chan gives it in his thesis, it seems that it's wrong (as given above), but readily corrected. The neat formulas appear to be for cumulants rather than standardized central moments. | |
| Nov 9, 2017 at 12:51 | history | edited | Glen_b | CC BY-SA 3.0 |
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| Nov 9, 2017 at 11:36 | history | edited | Glen_b | CC BY-SA 3.0 |
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| Nov 9, 2017 at 11:34 | comment | added | amoeba | Cool. Thanks a lot! According to the encyclopedia entry that Stephan linked to above, the final answer for skewness is $\psi''(\alpha)/\psi'(\alpha)^{3/2}$ (which almost qualifies as "neat"!). So it seems that all the scary zetas will have to cancel out. | |
| Nov 9, 2017 at 11:29 | history | edited | Glen_b | CC BY-SA 3.0 |
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| Nov 9, 2017 at 11:07 | history | answered | Glen_b | CC BY-SA 3.0 |