Long-Tailed Distribution
A Long-Tailed Distribution is a heavy-tailed distribution that ...
- Context:
- It can (often) be used to model a Long-Tailed Dataset.
- …
- Counter-Example(s):
- See: Probability Distribution, Rank-Size Distribution, Rank-Frequency Distribution, Power Law.
References
2016
- (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/Long_tail Retrieved:2016-9-11.
- In statistics and business, a long tail of some distributions of numbers is the portion of the distribution having a large number of occurrences far from the "head" or central part of the distribution. The distribution could involve popularities, random numbers of occurrences of events with various probabilities, etc. The term is often used loosely, with no definition or arbitrary definition, but precise definitions are possible. In statistics, the term long-tailed distribution has a narrow technical meaning, and is a subtype of heavy-tailed distribution; see that article for details. [1] Intuitively, a distribution is (right) long-tailed if, for any fixed amount, when a quantity exceeds a high level, it almost certainly exceeds it by at least that amount: big quantities are probably even bigger.Template:Efn Note that statistically, there is no sense of the "long tail" of a distribution, but only the property of a distribution being long-tailed.
In business, the term long tail is applied to rank-size distributions or rank-frequency distributions (primarily of popularity), which often form power laws and are thus long-tailed distributions in the statistical sense. This is used to describe the retailing strategy of selling a large number of unique items with relatively small quantities sold of each (the "long tail") — usually in addition to selling fewer popular items in large quantities (the "head"). Sometimes an intermediate category is also included, variously called the body, belly, torso, or middle. The specific cutoff of what part of a distribution is the "long tail" is often arbitrary, but in some cases may be specified objectively; see segmentation of rank-size distributions.
The long tail concept has found some ground for application, research, and experimentation. It is a term used in online business, mass media, micro-finance (Grameen Bank, for example), user-driven innovation (Eric von Hippel), and social network mechanisms (e.g. crowdsourcing, crowdcasting, peer-to-peer), economic models, and marketing (viral marketing).
- In statistics and business, a long tail of some distributions of numbers is the portion of the distribution having a large number of occurrences far from the "head" or central part of the distribution. The distribution could involve popularities, random numbers of occurrences of events with various probabilities, etc. The term is often used loosely, with no definition or arbitrary definition, but precise definitions are possible. In statistics, the term long-tailed distribution has a narrow technical meaning, and is a subtype of heavy-tailed distribution; see that article for details. [1] Intuitively, a distribution is (right) long-tailed if, for any fixed amount, when a quantity exceeds a high level, it almost certainly exceeds it by at least that amount: big quantities are probably even bigger.Template:Efn Note that statistically, there is no sense of the "long tail" of a distribution, but only the property of a distribution being long-tailed.
- ↑ Levine, David M.; Stephan, David; Krehbiel, Timothy C.; Berenson, Mark L. Statistics for Managers using Microsoft Excel. 3rd edition. Prentice Hall, 2002, p. 124.
- (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/heavy-tailed_distribution#Definition_of_long-tailed_distribution Retrieved:2016-9-11.
- The distribution of a random variable X with distribution function F is said to have a long right tail if for all t > 0, : [math]\displaystyle{ \lim_{x \to \infty} \Pr[X\gt x+t|X\gt x] =1, \, }[/math] or equivalently : [math]\displaystyle{ \overline{F}(x+t) \sim \overline{F}(x) \quad \mbox{as } x \to \infty. \, }[/math] This has the intuitive interpretation for a right-tailed long-tailed distributed quantity that if the long-tailed quantity exceeds some high level, the probability approaches 1 that it will exceed any other higher level: if you know the situation is good, it is probably better than you think.
All long-tailed distributions are heavy-tailed, but the converse is false, and it is possible to construct heavy-tailed distributions that are not long-tailed.
- The distribution of a random variable X with distribution function F is said to have a long right tail if for all t > 0, : [math]\displaystyle{ \lim_{x \to \infty} \Pr[X\gt x+t|X\gt x] =1, \, }[/math] or equivalently : [math]\displaystyle{ \overline{F}(x+t) \sim \overline{F}(x) \quad \mbox{as } x \to \infty. \, }[/math] This has the intuitive interpretation for a right-tailed long-tailed distributed quantity that if the long-tailed quantity exceeds some high level, the probability approaches 1 that it will exceed any other higher level: if you know the situation is good, it is probably better than you think.
