Heavy-tailed distribution

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 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.)

Definitions

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^[1]

\lim _{x\to \infty }e^{\lambda x}\Pr[X>x]=\infty \quad {\mbox{for all }}\lambda >0.\,

This is also written in terms of the tail distribution function

\overline{F}(x) \equiv \Pr[X>x] \,

\lim_{x \to \infty} e^{\lambda x}\overline{F}(x) = \infty \quad \mbox{for all } \lambda>0.\,

This is equivalent to the statement that the moment generating function of F, M_F(t), is infinite for all t > 0.^[2]

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^[1] if for all t > 0,

\lim _{x\to \infty }\Pr[X>x+t\mid X>x]=1,\,

or equivalently

\overline{F}(x+t) \sim \overline{F}(x) \quad \mbox{as } x \to \infty. \,

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 $X_1,X_2$ with common distribution function $F$ the convolution of $F$ with itself, $F^{*2}$ is defined, using Lebesgue–Stieltjes integration, by:

\Pr[X_1+X_2 \leq x] = F^{*2}(x) = \int_{- \infty}^\infty F(x-y)\,dF(y).

The n-fold convolution $F^{*n}$ is defined in the same way. The tail distribution function ${\overline {F}}$ is defined as $\overline{F}(x) = 1-F(x)$ .

A distribution $F$ on the positive half-line is subexponential ^[1]^[3]^[4] if

\overline{F^{*2}}(x) \sim 2\overline{F}(x) \quad \mbox{as } x \to \infty.

This implies^[5] that, for any $n\geq 1$ ,

\overline{F^{*n}}(x) \sim n\overline{F}(x) \quad \mbox{as } x \to \infty.

The probabilistic interpretation^[5] of this is that, for a sum of $n$ independent random variables $X_{1},\ldots ,X_{n}$ with common distribution $F$ ,

\Pr[X_1+ \cdots +X_n>x] \sim \Pr[\max(X_1, \ldots,X_n)>x] \quad \text{as } x \to \infty.

This is often known as the principle of the single big jump^[6] or catastrophe principle.^[7]

A distribution $F$ on the whole real line is subexponential if the distribution $F I([0,\infty))$ is.^[8] Here $I([0,\infty))$ 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.

Common heavy-tailed distributions

All commonly used heavy-tailed distributions are subexponential.^[5]

Those that are one-tailed include:

the Pareto distribution;
the Log-normal distribution;
the Lévy distribution;
the Weibull distribution with shape parameter greater than 0;
the Burr distribution;
the log-gamma distribution;
the log-Cauchy distribution, sometimes described as having a "super-heavy tail" because it exhibits logarithmic decay producing a heavier tail than the Pareto distribution.^[9]^[10]

Those that are two-tailed include:

The Cauchy distribution, itself a special case of both the stable distribution and the t-distribution;
The family of stable distributions,^[11] excepting the special case of the normal distribution within that family. Some stable distributions are one-sided (or supported by a half-line), see e.g. Lévy distribution. See also financial models with long-tailed distributions and volatility clustering.
The t-distribution.
The skew lognormal cascade distribution.^[12]

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 $x^{-a}$ . Since such a power is always bounded below by the probability density function of an exponential distribution, fat-tailed distributions are always heavy-tailed. Some distributions however have a tail which goes to zero slower than an exponential function (meaning they are heavy-tailed), but faster than a power (meaning they are not fat-tailed). An example is the log-normal distribution. Many other heavy-tailed distributions such as the log-logistic and Pareto distribution are however also fat-tailed.

Estimating the tail-index

There are parametric (see Embrechts et al.^[5]) and non-parametric (see, e.g., Novak^[13]) 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 $(X_n , n \geq 1)$ a random sequence of independent and same density function $F \in D(H(\xi))$ , the Maximum Attraction Domain^[14] of the generalized extreme value density $H$ , where $\xi \in \mathbb{R}$ . If $\lim_{n\to\infty} k(n) = \infty$ and $\lim_{n\to\infty} \frac{k(n)}{n}= 0$ , then the Pickands tail-index estimation is^[5]^[14]

\xi _{(k(n),n)}^{\text{Pickands}}={\frac {1}{\ln 2}}\ln \left({\frac {X_{(n-k(n)+1,n)}-X_{(n-2k(n)+1,n)}}{X_{(n-2k(n)+1,n)}-X_{(n-4k(n)+1,n)}}}\right)

where $X_{(n-k(n)+1,n)}=\max \left(X_{n-k(n)+1},\ldots ,X_{n}\right)$ . This estimator converge in probability to $\xi$ .

Hill's tail-index estimator

Let $(X_{t},t\geq 1)$ be a sequence of independent and identically distributed random variables with distribution function $F \in D(H(\xi))$ , the maximum domain of attraction of the generalized extreme value distribution $H$ , where $\xi \in \mathbb{R}$ . The sample path is ${X_{t}:1\leq t\leq n}$ where $n$ is the sample size. If $\{k(n)\}$ is an intermediate order sequence, i.e. $k(n)\in \{1,\ldots ,n-1\},$ , $k(n)\to \infty$ and $k(n)/n\to 0$ , then the Hill tail-index estimator is^[15]

\xi _{(k(n),n)}^{\text{Hill}}=\left({\frac {1}{k(n)}}\sum _{i=n-k(n)+1}^{n}\ln(X_{(i,n)})-\ln(X_{(n-k(n)+1,n)})\right)^{-1},

where $X_{(n-k(n)+1,n)}=\min \left(X_{n-k(n)+1},\ldots ,X_{n}\right)$ . This estimator converges in probability to $\xi$ , and is asymptotically normal provided $k(n)\to \infty$ is restricted based on a higher order regular variation property^[16] .^[17] Consistency and asymptotic normality extend to a large class of dependent and heterogeneous sequences,^[18]^[19] irrespective of whether $X_{t}$ is observed, or a computed residual or filtered data from a large class of models and estimators, including mis-specified models and models with errors that are dependent.^[20]^[21]^[22]

Ratio estimator of the tail-index

The ratio estimator (RE-estimator) of the tail-index was introduced by Goldie and Smith.^[23] 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.^[13]

Software

aest, C tool for estimating the heavy-tail index.^[24]

Estimation of heavy-tailed density

Nonparametric approaches to estimate heavy- and superheavy-tailed probability density functions were given in Markovich.^[25] 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.^[26] 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.^[25] Bootstrap is another tool to find smoothing parameters using approximations of unknown MSE by different schemes of re-samples selection, see e.g.^[27]

References

1 2 3 4 Asmussen, S. R. (2003). "Steady-State Properties of GI/G/1". Applied Probability and Queues. Stochastic Modelling and Applied Probability. 51. pp. 266–301. ISBN 978-0-387-00211-8. doi:10.1007/0-387-21525-5_10.
↑ Rolski, Schmidli, Scmidt, Teugels, Stochastic Processes for Insurance and Finance, 1999
↑ V. P. Chistyakov, A Theorem on Sums of Independent Positive Random Variables and Its Applications to Branching Random Processes, Theory of Probability and Its Applications 1964 https://www.researchgate.net/publication/242637603_A_Theorem_on_Sums_of_Independent_Positive_Random_Variables_and_Its_Applications_to_Branching_Random_Processes
↑ J.L. Teugels, The Class of Subexponential Distributions, Annals of Probability 1975 http://projecteuclid.org/download/pdf_1/euclid.aop/1176996225
1 2 3 4 5 Embrechts P.; Klueppelberg C.; Mikosch T. (1997). Modelling extremal events for insurance and finance. Stochastic Modelling and Applied Probability. 33. Berlin: Springer. doi:10.1007/978-3-642-33483-2.
↑ Foss, S.; Konstantopoulos, T.; Zachary, S. (2007). "Discrete and Continuous Time Modulated Random Walks with Heavy-Tailed Increments" (PDF). Journal of Theoretical Probability. 20 (3): 581. arXiv:math/0509605 . doi:10.1007/s10959-007-0081-2.
↑ Wierman, Adam (January 9, 2014). "Catastrophes, Conspiracies, and Subexponential Distributions (Part III)". Rigor + Relevance blog. RSRG, Caltech. Retrieved January 9, 2014.
↑ Willekens, E. (1986). "Subexponentiality on the real line". Technical Report. K.U. Leuven.
↑ Falk, M., Hüsler, J. & Reiss, R. (2010). Laws of Small Numbers: Extremes and Rare Events. Springer. p. 80. ISBN 978-3-0348-0008-2.
↑ Alves, M.I.F., de Haan, L. & Neves, C. (March 10, 2006). "Statistical inference for heavy and super-heavy tailed distributions" (PDF).
↑ John P. Nolan (2009). "Stable Distributions: Models for Heavy Tailed Data" (PDF). Retrieved 2009-02-21.
↑ Stephen Lihn (2009). "Skew Lognormal Cascade Distribution".
1 2 Novak S.Y. (2011). Extreme value methods with applications to finance. London: CRC. ISBN 978-1-43983-574-6.
1 2 Pickands III, James (Jan 1975). "Statistical Inference Using Extreme Order Statistics". The Annals of Statistics. 3 (1): 119–131. JSTOR 2958083. doi:10.1214/aos/1176343003.
↑ Hill B.M. (1975) A simple general approach to inference about the tail of a distribution. Ann. Stat., v. 3, 1163–1174.
↑ Hall, P.(1982) On some estimates of an exponent of regular variation. J. R. Stat. Soc. Ser. B., v. 44, 37–42.
↑ Haeusler, E. and J. L. Teugels (1985) On asymptotic normality of Hill's estimator for the exponent of regular variation. Ann. Stat., v. 13, 743–756.
↑ Hsing, T. (1991) On tail index estimation using dependent data. Ann. Stat., v. 19, 1547–1569.
↑ Hill, J. (2010) On tail index estimation for dependent, heterogeneous data. Econometric Th., v. 26, 1398–1436.
↑ Resnick, S. and Starica, C. (1997). Asymptotic behavior of Hill’s estimator for autoregressive data. Comm. Statist. Stochastic Models 13, 703–721.
↑ Ling, S. and Peng, L. (2004). Hill’s estimator for the tail index of an ARMA model. J. Statist. Plann. Inference 123, 279–293.
↑ Hill, J. B. (2015). Tail index estimation for a filtered dependent time series. Stat. Sin. 25, 609–630.
↑ Goldie C.M., Smith R.L. (1987) Slow variation with remainder: theory and applications. Quart. J. Math. Oxford, v. 38, 45–71.
↑ Crovella, M. E.; Taqqu, M. S. (1999). "Estimating the Heavy Tail Index from Scaling Properties". Methodology and Computing in Applied Probability. 1: 55. doi:10.1023/A:1010012224103.
1 2 Markovich N.M. (2007). Nonparametric Analysis of Univariate Heavy-Tailed data: Research and Practice. Chitester: Wiley. ISBN 978-0-470-72359-3.
↑ Wand M.P., Jones M.C. (1995). Kernel smoothing. New York: Chapman and Hall,. ISBN 978-0412552700.
↑ Hall P. (1992). The Bootstrap and Edgeworth Expansion. Springer,. ISBN 9780387945088.

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