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According to Wikipedia the most extreme case of a fat tail follows a power law:

The most extreme case of a fat tail is given by a distribution whose tail decays like a power law.

That is, if the complementary cumulative distribution of a random variable X can be expressed as

$$Pr[X>x] \sim x^{-\alpha} \quad \text{as} \quad x \to \infty, \quad \alpha >0$$

For these cases we have that for some sample size of size at least $n$ there are order statistics that have a finite expectation value.

However, in relation to a question about infinite/finite expectation values of order statistics, I got to think of a special case of distributions for which there is no size $n$ such that the order statistic will have a finite expectation value. This occurs when the quantile function has an essential singularity.

An example is $$Q(p) = e^{1/(1-p)} - e$$ for which the distribution function is

$$F(x) = \begin{cases} 0 \quad &\text{if} &\quad x<0 \\ 1 - \frac{1}{\log(x+e)} \quad &\text{if} &\quad x\geq 0 \\ \end{cases}$$ or $$f(x) = \begin{cases} 0 \quad &\text{if} &\quad x<0 \\ \frac{1}{(x+e)\log(x+e)^2} \quad &\text{if} &\quad x\geq 0 \\ \end{cases}$$

another case is discussed here: https://stats.stackexchange.com/a/417418/164061 the distribution functions that approach a power law can be bounded above by a linear function on a log-log plot, functions that are not like that will have in some sense 'more fat' tails than a distribution function that approaches a power law.

So it seems that we can think of distributions that have even more extreme tails than $Pr[X>x] \sim x^{-\alpha}$

Are there descriptions of fat tailed distributions that have this property? For instance do they have a particular name? (I suggest ultra-fat tailed distribution, if none exists yet)

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Sextus Empiricus
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  • Apparently there is a concept of super-heavy-tailed and the log-Cauchy distribution is an example, but I do not know an original reference or where the term originates. – Sextus Empiricus May 29 '20 at 10:30
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    Falk et al. (2011) devote a section to Super-Heavy Tail analysis (section 2.7) with a subsection called "Super-Heavy Tails in the Literature" and more. – COOLSerdash May 29 '20 at 10:48
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    In this case, Wiki is misleading. Tails heavier than an exponential power law can be called super-exponential tails. In other words, they grow faster than an exponential tail. One way to identify them is in a log-log graph which linearizes exponential growth. In this case, super-exponential growth curves sharply away (either upward or downward) from the linearized exponential. The current Covid-19 pandemic is probably one example of such super-exponential growth, one description of which is here...https://en.wikipedia.org/wiki/Dragon_king_theory –  May 29 '20 at 12:49
  • @user332577 that is an interesting link. But I am not talking about super-exponential tails, I am talking about super-power-law tails, which are even more fat. – Sextus Empiricus May 29 '20 at 13:28
  • You need to clarify the distinction. Why wouldn't they have the same mechanism? –  May 29 '20 at 14:43
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    @user332577, a power law decay dominates an exponential decay (for sufficiently large values the area of the tail is larger). Distributions with exponential tails have always finite moments. Distributions with power law tails do not always, but for some sufficiently large sample the order statistics will have finite moments. The distributions with 'super-heavy' tails are even more fat than the power law distributions, such that for no order statistic the moments are finite. – Sextus Empiricus May 29 '20 at 15:18
  • Ok, but note that these are theoretical assumptions which are very difficult to discriminate or distinguish empirically. See *So you think you have a power law, do you?" by Cosma Shalizi...http://www.stat.cmu.edu/~cshalizi/2010-10-18-Meetup.pdf –  May 29 '20 at 16:19
  • @user332577 I will read that presentation, but I do not follow your line of thoughts/comments. What is your point about "wiki is misleading", about "supra-extraponential" while this is about 'supra-power-law', and about "the current Covid-19 pandemic is probably one example of such super-exponential growth"? – Sextus Empiricus May 29 '20 at 17:42
  • To add to the discussion, as noted, Cauchy distributions are extreme valued. In addition, there are Levy distributions which can be heavier than an exponential power law...https://en.wikipedia.org/wiki/L%C3%A9vy_distribution Also, tweedie distributions...https://en.wikipedia.org/wiki/Tweedie_distribution#Related_distributions The problem with all of these is that the crisp theoretical assumptions do not result in crisp empirical distinctions, e.g., tail index estimators are many and much criticized..http://www.math.ucsd.edu/~politis/PAPER/jspi-tail.pdf –  May 29 '20 at 17:43
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    I am unfamiliar with an exponential power law distribution. I know about distributions with exponential tails and distributions with power law tails. But what are distributions with exponential power law distributions? Is that what you meant by super-exponential in your first comment? – Sextus Empiricus May 29 '20 at 17:46
  • I do not get why you mention Cauchy and Levy distributions. They have tails that are approximately power laws. And for an order statistic like the median of a sample with size $\geq 3/5$ their expectation value will already be finite. I am speaking about distributions that are heavier than that. – Sextus Empiricus May 29 '20 at 17:49
  • Also confusing is "One way to identify them is in a log-log graph which linearizes exponential growth" log-log linearizes power-law and not exponential law. So that is already where I loose your point and do not understand what you speak about. – Sextus Empiricus May 29 '20 at 17:59
  • I do accept that super-exponential distribution can also be a super-power law distribution. So, what is your point? Maybe you are making too many assumptions about my thoughts? Can you start again explaining your three points? https://stats.stackexchange.com/questions/469236/how-do-we-call-a-more-extreme-case-of-fat-tails-than-a-power-law?noredirect=1#comment866846_469236 – Sextus Empiricus May 29 '20 at 18:56
  • "That is where Levy and Tweedie distributions fit, super-power law distributions are just a subset of that broader class." I do not see your point. Levy and Tweedie distributions do not contain distributions that have tails that are more fat than a power law. The distributions that I describe are not a subset of the Levy and Tweedie distributions. – Sextus Empiricus May 29 '20 at 20:13

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