# Talk:Heavy-tailed distribution

WikiProject Statistics (Rated Start-class, Mid-importance)

This article is within the scope of the WikiProject Statistics, a collaborative effort to improve the coverage of statistics on Wikipedia. If you would like to participate, please visit the project page or join the discussion.

Start  This article has been rated as Start-Class on the quality scale.
Mid  This article has been rated as Mid-importance on the importance scale.
WikiProject Mathematics (Rated Start-class, Low-importance)
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
 Start Class
 Low Importance
Field: Probability and statistics

## Comments from before 13 March 2007

This material was originally taken from the long-range dependency article, which conflated long-range dependent processes and the heavy-tailed distributions that can arise from them as if they were the same thing. -- The Anome 23:56, 23 November 2006 (UTC)

## Proposal: merge Heavy-tailed distribution with Power Law

I propose to redirect the Heavy-tailed distribution article name to the power law article (note that the editors of the power laws article are in the process of producing a dramatically better version that is currently public). The new power-laws article covers both power-law functions and power-law distributions (including distributions with power-law tails), and so information on Heavh-tailed distribution would naturally fit as as a subsection of that topic. In fact, it would be nice to have a section there on the relationship between power-tail tails and extreme value theory.Paresnah 20:14, 13 March 2007 (UTC)

If the Cauchy distribution is heavy tailed, then I don't see that there needs to be a Power Law merger (the power law article doesn't deal with two tails). (forgot to sign) --Henrygb 22:40, 18 March 2007 (UTC)
Why not just update the power law article to briefly discuss distributions with two tails? Paresnah 18:33, 20 March 2007 (UTC)
This article does not describe heavy-tailed distributions as the term is used by probabilists. Many heavy-tailed distributions have finite variance. Only a few follow power-law type laws (power law type because they can be of the form $l(x)x^{-\alpha}$ where $l(x)$ is a slowly varying function). Examples of non-power law heavy-tailed distributions are log-normal and Weibull, but there are many others. The article needs to differentiate between heavy tails, long tails and, most important, subexponential dstributions. Of the heavy-tailed distributions in actual practical use all the ones I know of are subexponential (which is a subclass of long-tailed distributions, which is a subclass of heavy-tailed distributions). The definition of a subexponential distribution implies its most important property: Let $X_1, X_2, \ldots$ be i.i.d. random variables with a subexponential distribution; then the probability that the sum of these exceeds some high level $x$ is asymptotically equal to the probability that the maximum of the $X_i$ exceeds $x$.PoochieR 17:47, 6 September 2007 (UTC)
I have some familiarity with heavy tailed distributions, and do not see that the topic can be covered naturally within the Power Law article, which seems to be about something different, which I don't fully care to understand. Here is a somewhat long explanation of where I come in from, which bears on why i think that heavy-tailed distributions should be discussed separately:
My experience with heavy tailed distributions is largely from having done an empirical study explaining and predicting 1-0 outcomes, which commonly are done using probit or logit maximum likelihood models. A commonly expressed difference between probit and logit models is that logit uses the logistical distribution that has heavier tails than the normal distribution assumed in the probit model. Both distributions are symmetric, two-tailed. In many empirical settings in economics and elsewhere, logit models perform better (achieve higher loglikelihood) because they better accomodate discordant outlier observations. Discordant outliers are points where an opposite-than-expected outcome happens at extreme values of independent values. At extreme values, it is very unlikely the opposite-than-expected outcome will occur. But in fact the predictive economic model is almost always misspecified, as the models are ad hoc and there are always omitted variables. In other words, there will be reasons not captured in a model, why an opposite outcome should occur. Probit, which uses normal distribution, allows extremely low probability for such outcomes. So, the estimation of parameters is highly affected by a single outlier observation. Logit allows higher likelihood, hence is less disrupted by an outlier, and estimation allowing for heavier and heavier tails might perform even better. In my application, I found that a Cauchy-based estimation performed significantly better than either probit or logit. Cauchy distribution is also a two-tailed, symmetric distribution.
The point of explaining this is to say that there are applications where what matters is how heavy-tailed the distribution is. This has little relation to whatever the power-law article is about. In my view it is sensible to have an article on the topic of heavy tailed distributions alone. It would naturally hold discussion of applications of heavy-tailed distributions, such as I describe, that would not fit in the power-law article. doncram 21:12, 11 September 2007 (UTC)
The problem with dropping the comments on one-sided tail heaviness is that the definitions I have given are only correct for right heavy tails. To do it in general just requires complete duplication using $\Pr(X \leq x)$; would this not double the length of the page to very little benefit? Most applications I have come across concern themselves with right tail heaviness: but I'm a probabilist rather than a statistician. PoochieR 13:58, 12 September 2007 (UTC)
Can't the generalization be explained in principle, or can't it otherwise be briefly noted? I guess I would hope it could be explained succinctly without getting bogged down. That's a challenge for the writing of the article, but not reason enough to limit the scope of the article when the full topic has merit for discussion. I'm confident you/we/wikipedians can do a good job of it eventually. I added a link to Stable distribution. See mention in that article to Mandelbrot's application of stable distributions to finance. There are many more applications; I will try to dig up what I can from materials of an American Statistical Association conference on Heavy-tailed distributions that I attended. doncram 20:49, 12 September 2007 (UTC)
Discussion on this topic has stalled as did efforts to revise the Power Law article. However, based on the consensus here, it is safe to say that this article will not be merged as it is too complex in nature. As such I am removing the merge tag. If you dispute this, please re-add the tag and reopen conversation. ~ Don4of4 [Talk] 04:30, 19 August 2011 (UTC)

## Factual error?

The examples seem to be wrong. The text states that a heavy-tailed distribution has no moments beyond the first one. However, the lognormal and Cauchy distribution are listed. These have moments. -- Zz (talk) 15:58, 15 January 2008 (UTC)

It is quite correct that the log-normal distribution has all its moments. But the text does NOT say that a heavy-tailed distribution has only its first moment (in fact the Cauchy doesn't even have a finite mean). Some authors do use the term in this way,but the most common usage, and the one that includes all the distributions implied by the other distributions is the one given at the start of the article: that the moment generating function is not defined for any positive value of the argument (assuming we are talking about a heavy right-tail). This is all clearly explained (I hope) in the introduction. PoochieR (talk) 09:50, 22 January 2008 (UTC)

## Proposal: merge heavy-tails with fat-tails

Although there are essentially three different usages of the term heavy-tailed in probability theory, this article deals with the most general definition. The term fat-tail doesn't have a similiar rigourousness of definition. The current fat-tail article deals only with regularly varying distributions with finite mean, ad as such is dealing with a narrow though important subclass of heavy-tailed distributions. Fat-tails could well be merged with power laws, but this article, dealing as it does with heavy-tailed Weibull, log-normal and other non-power law distributions is logically independent. PoochieR (talk) 07:00, 17 June 2008 (UTC)

## Fatter than normal

I've seen the term fat/heavy tails used to mean fatter than the normal distribution. I know I've seen it used that way in robust statistics. (See These lecture notes, which refer to the double exponential distribution as having heavy tails) and it's implicit in the discussion on the fat tail page itself, since the reason why heavy-tailed distributions are an issue is that using the normal distribution causes you to underestimate the risk of catastrophic losses. I edited the article to mention the usage, though perhaps the article should be revised more extensively. -- Walt Pohl (talk) 13:30, 8 March 2010 (UTC)

I have occasionally seen heavy-tailed used in this way, and it certainly is worth mentioning. It's just using heavy-tailed as a shorthand for heavier than the normal, but doesn't introduce any new mathematical ideas. I don't think there's anything else to say about it. Distributions that lie between the normal and the exponential in heaviness behave mathematically very similiarly to the normal. The reason heavy-tailed distributions as defined here are of interest is because there is a qualitative difference from 'normal' behaviour. — Preceding unsigned comment added by AndrewWR (talkcontribs) 09:32, 11 November 2010

## Does not define heavy

Article does not define what it means by a heavy distribution, or a heavy tail. Recursion results. ᛭ LokiClock (talk) 08:29, 11 June 2010 (UTC)

The article is about heavy-tailed distributuions. I'm not sure what a heavy distribution is, never having come across the term. The first line of the article says: heavy-tailed distributions are probability distributions whose tails are not exponentially bounded. There is also a precise mathematical definition of what this means for the case of right-tailed heaviness. The mathematical definition for left-tailed heaviness is almost identical and was omitted to avoid needless repetition.— Preceding unsigned comment added by AndrewWR (talkcontribs) 09:32, 11 November 2010

## Tsallis Distributions

Tsallis distributions are often used for their heavy/fat tails. They should be mentioned here as well. Purple Post-its (talk) 14:32, 27 June 2011 (UTC)

## Illustrations needed!

Illustrations would be so helpful here. Anybody? Vegard (talk) 19:25, 10 August 2012 (UTC)

Italic text== Incorrect definition? ==

I believe the exponential should have a leading negative sign, no?

No, here should not be a leading negative sign. The definition ensures (roughly speaking) that if the distribution is itself exponentail, or has a lighter tail then the limit is finite, so not a heavy distribution. But a distribution such as a Pareto where the tail distribution is a polynomial, then the limit will be infinite, and by the definition it will then be a heavy tailed distribution. AWRichards1 (talk) 21:26, 23 February 2013 (UTC)

I think we get unbounded behavior for any nonzero distribution give the current limiting definition proposed in the 'Definition of Heavy-tailed...'