In probability theory, heavy-tailed distributions are probability distributions whose tails are not exponentially bounded: 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.
Contents
- Definition of heavy tailed distribution
- Definition of long tailed distribution
- Subexponential distributions
- Common heavy tailed distributions
- Relationship to fat tailed distributions
- Estimating the tail index
- Pickands tail index estimator
- Hills tail index estimator
- Ratio estimator of the tail index
- Software
- Estimation of heavy tailed density
- References
There are three important subclasses of heavy-tailed distributions: the fat-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 finite 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. (Occasionally, heavy-tailed is used for any distribution that has heavier tails than the normal distribution.)
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
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.
The definitions of heavy-tailed for left-tailed or two tailed distributions are similar.
Definition of long-tailed distribution
The distribution of a random variable X with distribution function F is said to have a long right tail if for all t > 0,
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 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.
Subexponential distributions
Subexponentiality is defined in terms of convolutions of probability distributions. For two independent, identically distributed random variables
The n-fold convolution
A distribution
This implies that, for any
The probabilistic interpretation of this is that, for a sum of
This is often known as the principle of the single big jump or catastrophe principle.
A distribution
All subexponential distributions are long-tailed, but examples can be constructed of long-tailed distributions that are not subexponential.
Common heavy-tailed distributions
All commonly used heavy-tailed distributions are subexponential.
Those that are one-tailed include:
Those that are two-tailed include:
Relationship to fat-tailed distributions
A fat-tailed distribution is a distribution for which the probability density function, for large x, goes to zero as a power
Estimating the tail-index
There are parametric (see Embrechts et al.) and non-parametric (see, e.g., Novak) approaches to the problem of the tail-index estimation.
To estimate the tail-index using the parametric approach, some authors employ GEV distribution or Pareto distribution; they may apply the maximum-likelihood estimator (MLE).
Pickand's tail-index estimator
With
where
Hill's tail-index estimator
Let
where
Ratio estimator of the tail-index
The ratio estimator (RE-estimator) of the tail-index was introduced by Goldie and Smith. It is constructed similarly to Hill's estimator but uses a non-random "tuning parameter".
A comparison of Hill-type and RE-type estimators can be found in Novak.
Software
Estimation of heavy-tailed density
Nonparametric approaches to estimate heavy- and superheavy-tailed probability density functions were given in Markovich. These are approaches based on variable bandwidth and long-tailed kernel estimators; on the preliminary data transform to a new random variable at finite or infinite intervals which is more convenient for the estimation and then inverse transform of the obtained density estimate; and "piecing-together approach" which provides a certain parametric model for the tail of the density and a non-parametric model to approximate the mode of the density. Nonparametric estimators require an appropriate selection of tuning (smoothing) parameters like a bandwidth of kernel estimators and the bin width of the histogram. The well known data-driven methods of such selection are a cross-validation and its modifications, methods based on the minimization of the mean squared error (MSE) and its asymptotic and their upper bounds. A discrepancy method which uses well-known nonparametric statistics like Kolmogorov-Smirnov's, von Mises and Anderson-Darling's ones as a metric in the space of distribution functions (dfs) and quantiles of the later statistics as a known uncertainty or a discrepancy value can be found in. Bootstrap is another tool to find smoothing parameters using approximations of unknown MSE by different schemes of re-samples selection, see e.g.