In probability theory, heavytailed 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 three important subclasses of heavytailed distributions: the fattailed distributions, the longtailed distributions and the subexponential distributions. In practice, all commonly used heavytailed distributions belong to the subexponential class.
There is still some discrepancy over the use of the term heavytailed. 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 lognormal that possess all their power moments, yet which are generally considered to be heavytailed. (Occasionally, heavytailed is used for any distribution that has heavier tails than the normal distribution.)
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Transcription
Contents
Definitions
Definition of heavytailed distribution
The distribution of a random variable X with distribution function F is said to have a heavy (right) tail if the moment generating function of X, M_{X}(t), is infinite for all t > 0.^{[2]}
That means
 ^{[3]}
An implication of this is that
 ^{[4]}
This is also written in terms of the tail distribution function
as
Definition of longtailed distribution
The distribution of a random variable X with distribution function F is said to have a long right tail^{[1]} if for all t > 0,
or equivalently
This has the intuitive interpretation for a righttailed longtailed distributed quantity that if the longtailed quantity exceeds some high level, the probability approaches 1 that it will exceed any other higher level.
All longtailed distributions are heavytailed, but the converse is false, and it is possible to construct heavytailed distributions that are not longtailed.
Subexponential distributions
Subexponentiality is defined in terms of convolutions of probability distributions. For two independent, identically distributed random variables with common distribution function the convolution of with itself, is convolution square, using Lebesgue–Stieltjes integration, by:
The nfold convolution is defined in the same way. The tail distribution function is defined as .
A distribution on the positive halfline is subexponential^{[1]}^{[5]}^{[6]} if
This implies^{[7]} that, for any ,
The probabilistic interpretation^{[7]} of this is that, for a sum of independent random variables with common distribution ,
This is often known as the principle of the single big jump^{[8]} or catastrophe principle.^{[9]}
A distribution on the whole real line is subexponential if the distribution is.^{[10]} Here is the indicator function of the positive halfline. Alternatively, a random variable supported on the real line is subexponential if and only if is subexponential.
All subexponential distributions are longtailed, but examples can be constructed of longtailed distributions that are not subexponential.
Common heavytailed distributions
All commonly used heavytailed distributions are subexponential.^{[7]}
Those that are onetailed include:
 the Pareto distribution;
 the Lognormal distribution;
 the Lévy distribution;
 the Weibull distribution with shape parameter greater than 0 but less than 1;
 the Burr distribution;
 the loglogistic distribution;
 the loggamma distribution;
 the logCauchy distribution, sometimes described as having a "superheavy tail" because it exhibits logarithmic decay producing a heavier tail than the Pareto distribution.^{[11]}^{[12]}
Those that are twotailed include:
 The Cauchy distribution, itself a special case of both the stable distribution and the tdistribution;
 The family of stable distributions,^{[13]} excepting the special case of the normal distribution within that family. Some stable distributions are onesided (or supported by a halfline), see e.g. Lévy distribution. See also financial models with longtailed distributions and volatility clustering.
 The tdistribution.
 The skew lognormal cascade distribution.^{[14]}
Relationship to fattailed distributions
A fattailed distribution is a distribution for which the probability density function, for large x, goes to zero as a power . Since such a power is always bounded below by the probability density function of an exponential distribution, fattailed distributions are always heavytailed. Some distributions, however, have a tail which goes to zero slower than an exponential function (meaning they are heavytailed), but faster than a power (meaning they are not fattailed). An example is the lognormal distribution. Many other heavytailed distributions such as the loglogistic and Pareto distribution are, however, also fattailed.
Estimating the tailindex^{[when defined as?]}
There are parametric (see Embrechts et al.^{[7]}) and nonparametric (see, e.g., Novak^{[15]}) approaches to the problem of the tailindex estimation.
To estimate the tailindex using the parametric approach, some authors employ GEV distribution or Pareto distribution; they may apply the maximumlikelihood estimator (MLE).
Pickand's tailindex estimator
With a random sequence of independent and same density function , the Maximum Attraction Domain^{[16]} of the generalized extreme value density , where . If and , then the Pickands tailindex estimation is^{[7]}^{[16]}
where . This estimator converges in probability to .
Hill's tailindex estimator
Let be a sequence of independent and identically distributed random variables with distribution function , the maximum domain of attraction of the generalized extreme value distribution , where . The sample path is where is the sample size. If is an intermediate order sequence, i.e. , and , then the Hill tailindex estimator is^{[17]}
where is the th order statistic of . This estimator converges in probability to , and is asymptotically normal provided is restricted based on a higher order regular variation property^{[18]} .^{[19]} Consistency and asymptotic normality extend to a large class of dependent and heterogeneous sequences,^{[20]}^{[21]} irrespective of whether is observed, or a computed residual or filtered data from a large class of models and estimators, including misspecified models and models with errors that are dependent.^{[22]}^{[23]}^{[24]}
Ratio estimator of the tailindex
The ratio estimator (REestimator) of the tailindex was introduced by Goldie and Smith.^{[25]} It is constructed similarly to Hill's estimator but uses a nonrandom "tuning parameter".
A comparison of Hilltype and REtype estimators can be found in Novak.^{[15]}
Software
Estimation of heavytailed density
Nonparametric approaches to estimate heavy and superheavytailed probability density functions were given in Markovich.^{[27]} These are approaches based on variable bandwidth and longtailed 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 "piecingtogether approach" which provides a certain parametric model for the tail of the density and a nonparametric 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 datadriven methods of such selection are a crossvalidation and its modifications, methods based on the minimization of the mean squared error (MSE) and its asymptotic and their upper bounds.^{[28]} A discrepancy method which uses wellknown nonparametric statistics like KolmogorovSmirnov's, von Mises and AndersonDarling'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.^{[27]} Bootstrap is another tool to find smoothing parameters using approximations of unknown MSE by different schemes of resamples selection, see e.g.^{[29]}
See also
 Leptokurtic distribution
 Outlier
 Long tail
 Power law
 Seven states of randomness
 Fattailed distribution
References
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