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Suppose $Y=\max\{X_1, X_2,\dots,X_N\}$ where all $X_i$ are independent and follows gamma distribution. I know that extreme value theory deals with maximum of random variables. Can anybody tell me, hopefully with reference, $Y$ will follow which extreme value distribution (Gumbel, Weibull or Frechet) ?

upol94
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  • [This](http://math.stackexchange.com/questions/847595/problem-with-the-expectation-of-a-maximum-of-independent-gamma-distributed-rando) may be helpful. – Christoph Hanck Apr 21 '15 at 05:45
  • Why are we limited to 3 choices? – wolfies Apr 21 '15 at 05:45
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    @wolfies That's what the theory has found. – Alecos Papadopoulos Apr 21 '15 at 11:26
  • It seems you are looking for the **limit** law (as $N \to \infty$). Perhaps better to mention that in the question. – P.Windridge Apr 21 '15 at 11:39
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    @AlecosPapadopoulos Quite the opposite, the question mentions nothing about asymptotic behaviour ... and none of the 3 posited distributions are a valid answer to the question posed. – wolfies Apr 21 '15 at 13:42
  • @wolfies Indeed, it doesn't, but it created (to me) that impression, exactly because he referred to the three possible asymptotic distributions. So the OP should clarify the point. – Alecos Papadopoulos Apr 21 '15 at 14:06
  • @upol94 As wolfies points out, the correct answer to the question you have posed is "none of the above". If you're seeking asymptotic results you need to make that clear in your question. – Glen_b Jun 20 '17 at 00:58

2 Answers2

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The gamma distribution is in the Gumbel domain of attraction. You can refer to the book by L. de Haan and A Ferreira, Extreme Values Theory, an Introduction. See therein theorem 1.1.8 and exercise 1.7 for its application to the gamma distribution. Another very useful book even provides explicit values for the two sequences required in the normalisation: P. Embrechts, C. Klüppelberg and T. Mikosch Modelling Extremal Events for Inusurance and Finance; This is in section 3.4, p. 156 in my edition.

Yves
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Taken from
David, H. A., & Nagaraja, H. N. (2003). Order statistics 3d ed., ch 10. p 296

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The "two references that give the most complete and rigorous discussion of the problem" are

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Alecos Papadopoulos
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    This answer reads like your justification for saying Frechet is that it has non-negative support. But the extreme value limit laws are **after** normalisation. E.g. for the maximum of Exponential random variables (which are non-negative), the limit is Gumbel. – P.Windridge Apr 21 '15 at 11:44
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    More precisely, if $X_1, \ldots$ are i.i.d Exp($\lambda$), then you can show (bare hands) that $$\mathbb{P}( \lambda \cdot \max_{i = 1,\ldots,n} X_i \le \ln(n) + x) \to e^{-e^{-x}}$$, and Theorem 10.5.1 applies with $a_n =1/\lambda$, $b_n = \ln(n)/\lambda$. (Just to clarify-- I have no idea which class the Gamma limits belong to, but it is not as simple as looking at the supports). – P.Windridge Apr 21 '15 at 11:53
  • @P.Windridge Thanks, that's a standard source of confusion (for me that is). I deleted the reference to Frechet while I am digging this. – Alecos Papadopoulos Apr 21 '15 at 11:57
  • Cool, I'll be interested in the answer. I suspect it is Gumbel, just because the Gamma and Exponential distributions are so closely related... but that's only speculation :) – P.Windridge Apr 21 '15 at 12:09
  • @P.Windridge, Thanks, I am also confused on the same point. Is gamma also follow Gumbel distribution ? – upol94 Apr 21 '15 at 12:18
  • @upol94 the exponential distribution is a special case of the gamma. So, for some parameters, the limit class is certainly Gumbel... but off the top of my head I don't know whether this holds generally. – P.Windridge Apr 21 '15 at 12:27
  • The question does not ask about the asymptotic distribution of $Y$ ... it seeks the distribution of $Y$. – wolfies Apr 21 '15 at 13:43
  • That's a pretty extensive quote ... and not readable by anyone using a screen reader (as a few of our sight-impaired users do). Could you consider whether you can take a few parts - whatever you need most - and write them as actual quotes rather than screenshots? \[You might be able to make reference to one of these wikipedia articles [(1)](https://en.wikipedia.org/wiki/Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem), [(2)](https://en.wikipedia.org/wiki/Extreme_value_theory), [(3)](https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution) for part of it, perhaps.] – Glen_b Jun 20 '17 at 01:17