Heavy-tailed distribution
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In probability theory, heavy-tailed distributions are probability distributions whose tails are not exponentially bounded:[1] that is, they have heavier tails than the exponential distribution. In many applications it is the right tail of the distribution that is of interest, but a distribution may have a heavy left tail, or both tails may be heavy.
There are two important subclasses of heavy-tailed distributions, the long-tailed distributions and the subexponential distributions. In practice, all commonly used heavy-tailed distributions belong to the subexponential class.
There is still some discrepancy over the use of the term heavy-tailed. There are two other definitions in use. Some authors use the term to refer to those distributions which do not have all their power moments finite; and some others to those distributions that do not have a variance. The definition given in this article is the most general in use, and includes all distributions encompassed by the alternative definitions, as well as those distributions such as log-normal that possess all their power moments, yet which are generally acknowledged to be heavy-tailed.
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[edit] Definition of heavy-tailed distribution
The distribution of a random variable X with distribution function F is said to have a heavy right tail if[2]
This is also written in terms of the tail distribution function as
This is equivalent to the statement that the moment generating function of F, MF(t), is infinite for all t > 0[3].
The definitions of heavy-tailed for left-tailed or two tailed distributions are similar.
[edit] Definition of long-tailed distribution
The distribution of a random variable X with distribution function F is said to have a long right tail[4] if for all
or equivalently
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 bad, it is probably worse 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.
[edit] Subexponential distributions
Subexponentiality is defined in terms of convolutions of probability distributions. For two independent, identically distributed random variables X1,X2 with common distribution function F the convolution of F with itself, F * 2 is defined, using Lebesgue-Stieltjes integration, by:
The n-fold convolution F * n is defined in the same way. The tail distribution function is defined as .
A distribution F on the positive half-line is subexponential[5] if
This implies[6] that, for any ,
The probabilistic interpretation[7] of this is that, for a sum of n independent random variables with common distribution F,
This is often known as the principle of the single big jump[8].
A distribution F on the whole real line is subexponential if the distribution is[9]. Here is the indicator function of the positive half-line. Alternatively, a random variable X supported on the real line is subexponential if and only if X + = max(0,X) is subexponential.
All subexponential distributions are long-tailed, but examples can be constructed of long-tailed distributions that are not subexponential.
[edit] Common heavy-tailed distributions
All commonly used heavy-tailed distributions are subexponential.[10]
Those that are one-tailed include:
- the Pareto distribution;
- the Log-normal distribution;
- the Weibull distribution;
- the Burr distribution;
- the Log-gamma distribution.
Those that are two-tailed include:
- The Cauchy distribution, itself a special case of
- the t-distribution;
- all of the Stable Distribution family, excepting the special case of the normal distribution within that family. Stable distributions may be symmetric or not.
[edit] References
- ^ Asmussen, Applied Probability and Queues, 2003
- ^ Asmussen, Applied Probability and Queues, 2003
- ^ Rolski, Schmidli, Scmidt, Teugels, Stochastic Processes for Insurance and Finance, 1999
- ^ Asmussen, Applied Probability and Queues, 2003
- ^ Asmussen, Applied Probability and Queues, 2003
- ^ Embrechts, Kluppelberg, Mikosch, Modelling Extremal Events, 1997
- ^ Embrechts, Kluppelberg, Mikosch, Modelling Extremal Events, 1997
- ^ Foss, Konstantopolous, Zachary, "Discrete and continuous time modulated random walks with heavy-tailed increments", Journal of Theoretical Probability, 20 (2007), No.3, 581—612
- ^ Willekens, E. Subexponentiality on the real line. Technical Report, K.U. Leuven(1986)
- ^ Embrechts, Kluppelberg, Mikosch, Modelling Extremal Events, 1997