3566:
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.
25:
1971:. 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
2549:
3565:
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,
3521:
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. Note that both
Pickand's and Hill's tail-index estimators commonly make use of logarithm of the order statistics.
3325:
430:
2268:
1380:
1257:
270:
629:
634:
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.
2678:
3096:
128:
that possess all their power moments, yet which are generally considered to be heavy-tailed. (Occasionally, heavy-tailed is used for any distribution that has heavier tails than the normal distribution.)
1053:
1612:
903:
2256:
1911:
3494:
is restricted based on a higher order regular variation property . Consistency and asymptotic normality extend to a large class of dependent and heterogeneous sequences, irrespective of whether
2196:
335:
538:
3010:
1140:
3530:
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".
124:. 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
2893:
2846:
2145:
1454:
3437:
3492:
3045:
2795:
2751:
2097:
2053:
1795:
1682:
1083:
1723:
3088:
691:
346:
3367:
2948:
2544:{\displaystyle \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),}
1286:
1969:
940:
761:
3519:
1294:
1171:
3457:
2698:
188:
549:
90:. 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.
3387:
2913:
2818:
2117:
1747:
1638:
1474:
1403:
1163:
731:
711:
638:
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.
1905:, 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.
3885:
4159:
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.
3998:
3951:
2557:
43:
4214:
Lee, Seyoon; Kim, Joseph H. T. (2019). "Exponentiated generalized Pareto distribution: Properties and applications towards extreme value theory".
4186:
Resnick, S. and
Starica, C. (1997). Asymptotic behavior of Hill’s estimator for autoregressive data. Comm. Statist. Stochastic Models 13, 703–721.
3320:{\displaystyle \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},}
1800:
All subexponential distributions are long-tailed, but examples can be constructed of long-tailed distributions that are not subexponential.
4406:
4401:
116:. There are two other definitions in use. Some authors use the term to refer to those distributions which do not have all their power
4013:
3966:
647:
4269:
948:
4045:
1482:
3580:
2798:
773:
4195:
Ling, S. and Peng, L. (2004). Hill’s estimator for the tail index of an ARMA model. J. Statist. Plann. Inference 123, 279–293.
4330:
4258:
Goldie C.M., Smith R.L. (1987) Slow variation with remainder: theory and applications. Quart. J. Math. Oxford, v. 38, 45–71.
4082:
3935:
3803:
3662:
2201:
4355:
4141:
Hill B.M. (1975) A simple general approach to inference about the tail of a distribution. Ann. Stat., v. 3, 1163–1174.
764:
3545:
2150:
4380:
61:
282:
448:
150:
464:
3585:
2953:
3753:"A Theorem on Sums of Independent Positive Random Variables and Its Applications to Branching Random Processes"
1088:
105:. In practice, all commonly used heavy-tailed distributions belong to the subexponential class, introduced by
4177:
Hill, J. (2010) On tail index estimation for dependent, heterogeneous data. Econometric Th., v. 26, 1398–1436.
4150:
Hall, P.(1982) On some estimates of an exponent of regular variation. J. R. Stat. Soc. Ser. B., v. 44, 37–42.
2851:
2823:
2122:
1919:
1413:
3396:
4411:
4204:
Hill, J. B. (2015). Tail index estimation for a filtered dependent time series. Stat. Sin. 25, 609–630.
3605:
157:
3462:
3015:
2756:
2753:
be a sequence of independent and identically distributed random variables with distribution function
2711:
2058:
2013:
1976:
1851:
1752:
1643:
1061:
3850:
1687:
425:{\displaystyle \lim _{x\to \infty }e^{tx}{\overline {F}}(x)=\infty \quad {\mbox{for all }}t>0.\,}
3575:
3553:
1941:
is a distribution for which the probability density function, for large x, goes to zero as a power
1405:
98:
87:
83:
39:
3682:
3619:
3610:
3050:
1972:
1938:
1879:
1872:
1823:
656:
94:
1991:
There are parametric and non-parametric approaches to the problem of the tail-index estimation.
1865:
3845:
3333:
2918:
1375:{\displaystyle {\overline {F^{*n}}}(x)\sim n{\overline {F}}(x)\quad {\mbox{as }}x\to \infty .}
1252:{\displaystyle {\overline {F^{*2}}}(x)\sim 2{\overline {F}}(x)\quad {\mbox{as }}x\to \infty .}
4168:
Hsing, T. (1991) On tail index estimation using dependent data. Ann. Stat., v. 19, 1547–1569.
1265:
265:{\displaystyle \int _{-\infty }^{\infty }e^{tx}\,dF(x)=\infty \quad {\mbox{for all }}t>0.}
1944:
915:
736:
624:{\displaystyle {\overline {F}}(x+t)\sim {\overline {F}}(x)\quad {\mbox{as }}x\to \infty .\,}
3497:
1902:
1837:
117:
4020:
3973:
3442:
2683:
1906:
1830:
8:
1999:
1980:
1895:
1816:
4297:
4289:
4241:
4223:
4124:
3945:
3863:
3835:
3615:
3372:
2898:
2803:
2102:
1883:
1858:
1732:
1726:
1623:
1459:
1388:
1148:
716:
696:
75:
4049:
3820:
1617:
This is often known as the principle of the single big jump or catastrophe principle.
4376:
4351:
4326:
4078:
3992:
3931:
3799:
3752:
3658:
1844:
86:
whose tails are not exponentially bounded: that is, they have heavier tails than the
4245:
4293:
4281:
4233:
4114:
3867:
3855:
3791:
3698:
3650:
1995:
1408:
651:
4237:
3821:"Discrete and Continuous Time Modulated Random Walks with Heavy-Tailed Increments"
3549:
3390:
441:
143:
3790:. Stochastic Modelling and Applied Probability. Vol. 33. Berlin: Springer.
3649:. Stochastic Modelling and Applied Probability. Vol. 51. pp. 266–301.
1994:
To estimate the tail-index using the parametric approach, some authors employ
1898:, itself a special case of both the stable distribution and the t-distribution;
4285:
3859:
3795:
4416:
4395:
4323:
Nonparametric
Analysis of Univariate Heavy-Tailed data: Research and Practice
4119:
4102:
3703:
3654:
2673:{\displaystyle X_{(n-k(n)+1,n)}=\max \left(X_{n-k(n)+1},\ldots ,X_{n}\right)}
106:
3881:
3542:
3964:
2099:, the Maximum Attraction Domain of the generalized extreme value density
3886:"Catastrophes, Conspiracies, and Subexponential Distributions (Part III)"
1912:
financial models with long-tailed distributions and volatility clustering
1882:, sometimes described as having a "super-heavy tail" because it exhibits
120:
finite; and some others to those distributions that do not have a finite
3533:
A comparison of Hill-type and RE-type estimators can be found in Novak.
93:
There are three important subclasses of heavy-tailed distributions: the
4128:
125:
3967:"Statistical inference for heavy and super-heavy tailed distributions"
3840:
3785:
3694:
3600:
3595:
3925:
4228:
121:
3590:
1808:
All commonly used heavy-tailed distributions are subexponential.
3739:
An
Introduction to Heavy-Tailed and Subexponential Distributions
276:
This is also written in terms of the tail distribution function
1385:
The probabilistic interpretation of this is that, for a sum of
3965:
Alves, M.I.F., de Haan, L. & Neves, C. (March 10, 2006).
3645:
Asmussen, S. R. (2003). "Steady-State
Properties of GI/G/1".
1729:
of the positive half-line. Alternatively, a random variable
1640:
on the whole real line is subexponential if the distribution
3911:
Willekens, E. (1986). "Subexponentiality on the real line".
2055:
a random sequence of independent and same density function
1749:
supported on the real line is subexponential if and only if
1048:{\displaystyle F^{*n}(x)=\int _{0}^{x}F(x-y)\,dF^{*n-1}(y).}
1932:
3818:
1607:{\displaystyle \Pr\sim \Pr\quad {\text{as }}x\to \infty .}
137:
4270:"Estimating the Heavy Tail Index from Scaling Properties"
2002:; they may apply the maximum-likelihood estimator (MLE).
898:{\displaystyle \Pr=F^{*2}(x)=\int _{0}^{x}F(x-y)\,dF(y),}
435:
112:
There is still some discrepancy over the use of the term
2251:{\displaystyle \lim _{n\to \infty }{\frac {k(n)}{n}}=0}
4103:"Statistical Inference Using Extreme Order Statistics"
1886:
producing a heavier tail than the Pareto distribution.
1354:
1231:
602:
406:
247:
4345:
3560:
3500:
3465:
3445:
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3375:
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2901:
2854:
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2759:
2714:
2686:
2560:
2271:
2204:
2153:
2125:
2105:
2061:
2016:
1975:. Many other heavy-tailed distributions such as the
1947:
1755:
1735:
1690:
1646:
1626:
1485:
1462:
1416:
1391:
1297:
1268:
1174:
1151:
1091:
1064:
951:
918:
776:
739:
719:
699:
659:
552:
467:
349:
285:
191:
4014:"Stable Distributions: Models for Heavy Tailed Data"
3741:, Springer Science & Business Media, 21 May 2013
3525:
1840:
with shape parameter greater than 0 but less than 1;
1803:
763:
and called the convolution square, is defined using
3819:Foss, S.; Konstantopoulos, T.; Zachary, S. (2007).
3788:
Modelling extremal events for insurance and finance
34:
may be too technical for most readers to understand
4075:Extreme value methods with applications to finance
3786:Embrechts P.; Klueppelberg C.; Mikosch T. (1997).
3513:
3486:
3451:
3431:
3381:
3361:
3319:
3082:
3039:
3004:
2942:
2907:
2887:
2840:
2812:
2789:
2745:
2692:
2672:
2543:
2250:
2190:
2139:
2111:
2091:
2047:
1963:
1789:
1741:
1717:
1676:
1632:
1606:
1468:
1448:
1397:
1374:
1280:
1251:
1157:
1134:
1077:
1047:
934:
897:
755:
725:
705:
685:
623:
532:
424:
329:
264:
4216:Communications in Statistics - Theory and Methods
2005:
4393:
4316:
4314:
4274:Methodology and Computing in Applied Probability
2607:
2206:
2191:{\displaystyle \lim _{n\to \infty }k(n)=\infty }
2155:
1769:
1539:
1533:
1486:
777:
484:
469:
351:
308:
3928:Laws of Small Numbers: Extremes and Rare Events
1165:on the positive half-line is subexponential if
650:. For two independent, identically distributed
641:
4320:
3726:Stochastic Processes for Insurance and Finance
2703:
330:{\displaystyle {\overline {F}}(x)\equiv \Pr\,}
4311:
4267:
4011:
3926:Falk, M., Hüsler, J. & Reiss, R. (2010).
3439:. This estimator converges in probability to
2680:. This estimator converges in probability to
454:is said to have a long right tail if for all
4100:
4068:
4066:
4043:
3997:: CS1 maint: multiple names: authors list (
3950:: CS1 maint: multiple names: authors list (
2996:
2972:
2937:
2922:
1986:
1983:distribution are, however, also fat-tailed.
533:{\displaystyle \lim _{x\to \infty }\Pr=1,\,}
156:is said to have a heavy (right) tail if the
4339:
4096:
4094:
3781:
3779:
3777:
3775:
3773:
3683:"The Class of Subexponential Distributions"
4072:
3750:
3640:
3638:
3636:
3005:{\displaystyle k(n)\in \{1,\ldots ,n-1\},}
2797:, the maximum domain of attraction of the
4227:
4118:
4063:
3910:
3849:
3839:
3718:
3702:
3556:tool for estimating the heavy-tail index.
2834:
2133:
1925:The skew log-normal cascade distribution.
1135:{\displaystyle {\overline {F}}(x)=1-F(x)}
1010:
876:
648:convolutions of probability distributions
646:Subexponentiality is defined in terms of
620:
529:
421:
326:
223:
62:Learn how and when to remove this message
46:, without removing the technical details.
4364:
4091:
3770:
3676:
3674:
3644:
3459:, and is asymptotically normal provided
3090:, then the Hill tail-index estimator is
2950:is an intermediate order sequence, i.e.
1933:Relationship to fat-tailed distributions
4370:
3880:
3680:
3633:
138:Definition of heavy-tailed distribution
4394:
4268:Crovella, M. E.; Taqqu, M. S. (1999).
4213:
3581:Generalized extreme value distribution
2799:generalized extreme value distribution
436:Definition of long-tailed distribution
4373:The Bootstrap and Edgeworth Expansion
4046:"Skew Lognormal Cascade Distribution"
3671:
2888:{\displaystyle {X_{t}:1\leq t\leq n}}
44:make it understandable to non-experts
2841:{\displaystyle \xi \in \mathbb {R} }
2140:{\displaystyle \xi \in \mathbb {R} }
942:is defined inductively by the rule:
693:with a common distribution function
18:
3737:S. Foss, D. Korshunov, S. Zachary,
3724:Rolski, Schmidli, Scmidt, Teugels,
1890:Those that are two-tailed include:
1811:Those that are one-tailed include:
1449:{\displaystyle X_{1},\ldots ,X_{n}}
13:
4407:Types of probability distributions
4402:Tails of probability distributions
3828:Journal of Theoretical Probability
3561:Estimation of heavy-tailed density
3481:
3432:{\displaystyle X_{1},\dots ,X_{n}}
3034:
2216:
2185:
2165:
1706:
1665:
1598:
1366:
1243:
614:
479:
401:
361:
242:
205:
200:
14:
4428:
3526:Ratio estimator of the tail-index
1804:Common heavy-tailed distributions
4101:Pickands III, James (Jan 1975).
23:
4261:
4252:
4207:
4198:
4189:
4180:
4171:
4162:
4153:
4144:
4135:
4037:
4005:
3958:
3919:
3586:Generalized Pareto distribution
3487:{\displaystyle k(n)\to \infty }
3040:{\displaystyle k(n)\to \infty }
2790:{\displaystyle F\in D(H(\xi ))}
2746:{\displaystyle (X_{t},t\geq 1)}
2092:{\displaystyle F\in D(H(\xi ))}
2048:{\displaystyle (X_{n},n\geq 1)}
1790:{\displaystyle X^{+}=\max(0,X)}
1677:{\displaystyle FI([0,\infty ))}
1586:
1352:
1229:
1078:{\displaystyle {\overline {F}}}
1058:The tail distribution function
600:
404:
245:
4350:. New York: Chapman and Hall.
4346:Wand M.P., Jones M.C. (1995).
3904:
3874:
3812:
3744:
3731:
3647:Applied Probability and Queues
3478:
3475:
3469:
3354:
3342:
3297:
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3251:
3239:
3234:
3222:
3214:
3192:
3186:
3160:
3154:
3126:
3117:
3111:
3105:
3074:
3063:
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3028:
3022:
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2934:
2928:
2784:
2781:
2775:
2769:
2740:
2715:
2635:
2629:
2599:
2584:
2578:
2566:
2526:
2511:
2505:
2490:
2477:
2462:
2456:
2441:
2429:
2414:
2408:
2393:
2380:
2365:
2359:
2347:
2298:
2289:
2283:
2277:
2233:
2227:
2213:
2179:
2173:
2162:
2086:
2083:
2077:
2071:
2042:
2017:
2006:Pickand's tail-index estimator
1784:
1772:
1718:{\displaystyle I([0,\infty ))}
1712:
1709:
1697:
1694:
1671:
1668:
1656:
1653:
1595:
1583:
1574:
1542:
1536:
1527:
1489:
1363:
1349:
1343:
1324:
1318:
1240:
1226:
1220:
1201:
1195:
1129:
1123:
1108:
1102:
1039:
1033:
1007:
995:
971:
965:
889:
883:
873:
861:
837:
831:
812:
780:
765:Lebesgue–Stieltjes integration
611:
597:
591:
575:
563:
517:
487:
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389:
358:
323:
311:
302:
296:
236:
230:
132:
1:
4238:10.1080/03610926.2018.1441418
3627:
1338:
1313:
1215:
1190:
1097:
1070:
642:Subexponential distributions
586:
558:
384:
291:
103:subexponential distributions
7:
3569:
3536:
3083:{\displaystyle k(n)/n\to 0}
2704:Hill's tail-index estimator
1262:This implies that, for any
686:{\displaystyle X_{1},X_{2}}
10:
4433:
3751:Chistyakov, V. P. (1964).
3681:Teugels, Jozef L. (1975).
3606:Seven states of randomness
158:moment generating function
80:heavy-tailed distributions
3860:10.1007/s10959-007-0081-2
3796:10.1007/978-3-642-33483-2
3362:{\displaystyle X_{(i,n)}}
2262:tail-index estimation is
1987:Estimating the tail-index
1852:log-logistic distribution
1456:with common distribution
99:long-tailed distributions
84:probability distributions
4107:The Annals of Statistics
3930:. Springer. p. 80.
3655:10.1007/0-387-21525-5_10
3576:Leptokurtic distribution
2943:{\displaystyle \{k(n)\}}
2915:is the sample size. If
95:fat-tailed distributions
88:exponential distribution
16:Probability distribution
4321:Markovich N.M. (2007).
4286:10.1023/A:1010012224103
3620:Holy grail distribution
3611:Fat-tailed distribution
1973:log-normal distribution
1939:fat-tailed distribution
1880:log-Cauchy distribution
1873:q-Gaussian distribution
1824:Log-normal distribution
1281:{\displaystyle n\geq 1}
175:), is infinite for all
4120:10.1214/aos/1176343003
4012:John P. Nolan (2009).
3890:Rigor + Relevance blog
3704:10.1214/aop/1176996225
3515:
3488:
3453:
3433:
3383:
3363:
3321:
3207:
3084:
3041:
3006:
2944:
2909:
2889:
2842:
2814:
2791:
2747:
2694:
2674:
2545:
2252:
2192:
2141:
2113:
2093:
2049:
1965:
1964:{\displaystyle x^{-a}}
1791:
1743:
1719:
1678:
1634:
1608:
1470:
1450:
1399:
1376:
1282:
1253:
1159:
1136:
1079:
1049:
936:
935:{\displaystyle F^{*n}}
899:
757:
756:{\displaystyle F^{*2}}
727:
707:
687:
625:
534:
440:The distribution of a
426:
331:
266:
142:The distribution of a
4044:Stephen Lihn (2009).
3695:University of Louvain
3687:Annals of Probability
3516:
3514:{\displaystyle X_{t}}
3489:
3454:
3434:
3384:
3364:
3322:
3166:
3085:
3042:
3007:
2945:
2910:
2890:
2848:. The sample path is
2843:
2815:
2792:
2748:
2695:
2675:
2546:
2253:
2193:
2142:
2114:
2094:
2050:
1966:
1792:
1744:
1720:
1679:
1635:
1609:
1471:
1451:
1400:
1377:
1283:
1254:
1160:
1137:
1080:
1050:
937:
900:
758:
733:with itself, written
728:
713:, the convolution of
708:
688:
626:
535:
449:distribution function
427:
332:
267:
151:distribution function
4325:. Chitester: Wiley.
3498:
3463:
3452:{\displaystyle \xi }
3443:
3397:
3373:
3334:
3097:
3051:
3016:
2954:
2919:
2899:
2852:
2824:
2804:
2757:
2712:
2693:{\displaystyle \xi }
2684:
2558:
2269:
2202:
2151:
2123:
2103:
2059:
2014:
1945:
1903:stable distributions
1866:Fréchet distribution
1838:Weibull distribution
1753:
1733:
1688:
1644:
1624:
1483:
1460:
1414:
1389:
1295:
1266:
1172:
1149:
1089:
1062:
949:
916:
774:
737:
717:
697:
657:
550:
465:
347:
283:
189:
4073:Novak S.Y. (2011).
3884:(January 9, 2014).
3135:
2307:
2000:Pareto distribution
1896:Cauchy distribution
1817:Pareto distribution
1797:is subexponential.
991:
857:
458: > 0,
209:
179: > 0.
3616:Taleb distribution
3548:2020-11-25 at the
3511:
3484:
3449:
3429:
3379:
3359:
3317:
3100:
3080:
3037:
3002:
2940:
2905:
2885:
2838:
2810:
2787:
2743:
2690:
2670:
2541:
2272:
2248:
2220:
2188:
2169:
2137:
2109:
2089:
2045:
1961:
1859:gamma distribution
1787:
1739:
1727:indicator function
1715:
1674:
1630:
1604:
1466:
1446:
1395:
1372:
1358:
1278:
1249:
1235:
1155:
1132:
1075:
1045:
977:
932:
912:-fold convolution
895:
843:
753:
723:
703:
683:
621:
606:
530:
483:
422:
410:
365:
327:
262:
251:
192:
76:probability theory
4412:Actuarial science
4332:978-0-470-72359-3
4084:978-1-43983-574-6
3937:978-3-0348-0008-2
3805:978-3-642-08242-9
3664:978-0-387-00211-8
3382:{\displaystyle i}
3164:
3133:
2908:{\displaystyle n}
2813:{\displaystyle H}
2532:
2327:
2305:
2240:
2205:
2154:
2112:{\displaystyle H}
1907:Lévy distribution
1884:logarithmic decay
1845:Burr distribution
1831:Lévy distribution
1742:{\displaystyle X}
1633:{\displaystyle F}
1590:
1469:{\displaystyle F}
1398:{\displaystyle n}
1357:
1341:
1316:
1234:
1218:
1193:
1158:{\displaystyle F}
1100:
1073:
726:{\displaystyle F}
706:{\displaystyle F}
605:
589:
561:
468:
409:
387:
350:
294:
250:
72:
71:
64:
4424:
4387:
4386:
4371:Hall P. (1992).
4368:
4362:
4361:
4348:Kernel smoothing
4343:
4337:
4336:
4318:
4309:
4308:
4306:
4305:
4296:. Archived from
4265:
4259:
4256:
4250:
4249:
4231:
4222:(8): 2014–2038.
4211:
4205:
4202:
4196:
4193:
4187:
4184:
4178:
4175:
4169:
4166:
4160:
4157:
4151:
4148:
4142:
4139:
4133:
4132:
4122:
4098:
4089:
4088:
4070:
4061:
4060:
4058:
4057:
4048:. Archived from
4041:
4035:
4034:
4032:
4031:
4025:
4019:. Archived from
4018:
4009:
4003:
4002:
3996:
3988:
3986:
3984:
3979:on June 23, 2007
3978:
3972:. Archived from
3971:
3962:
3956:
3955:
3949:
3941:
3923:
3917:
3916:
3913:Technical Report
3908:
3902:
3901:
3899:
3897:
3878:
3872:
3871:
3853:
3843:
3825:
3816:
3810:
3809:
3783:
3768:
3767:
3765:
3763:
3748:
3742:
3735:
3729:
3722:
3716:
3715:
3713:
3711:
3706:
3678:
3669:
3668:
3642:
3520:
3518:
3517:
3512:
3510:
3509:
3493:
3491:
3490:
3485:
3458:
3456:
3455:
3450:
3438:
3436:
3435:
3430:
3428:
3427:
3409:
3408:
3388:
3386:
3385:
3380:
3368:
3366:
3365:
3360:
3358:
3357:
3326:
3324:
3323:
3318:
3313:
3312:
3304:
3300:
3296:
3295:
3238:
3237:
3206:
3201:
3165:
3163:
3146:
3134:
3131:
3129:
3089:
3087:
3086:
3081:
3070:
3046:
3044:
3043:
3038:
3011:
3009:
3008:
3003:
2949:
2947:
2946:
2941:
2914:
2912:
2911:
2906:
2894:
2892:
2891:
2886:
2884:
2865:
2864:
2847:
2845:
2844:
2839:
2837:
2819:
2817:
2816:
2811:
2796:
2794:
2793:
2788:
2752:
2750:
2749:
2744:
2727:
2726:
2699:
2697:
2696:
2691:
2679:
2677:
2676:
2671:
2669:
2665:
2664:
2663:
2645:
2644:
2603:
2602:
2550:
2548:
2547:
2542:
2537:
2533:
2531:
2530:
2529:
2481:
2480:
2434:
2433:
2432:
2384:
2383:
2340:
2328:
2326:
2312:
2306:
2303:
2301:
2257:
2255:
2254:
2249:
2241:
2236:
2222:
2219:
2197:
2195:
2194:
2189:
2168:
2146:
2144:
2143:
2138:
2136:
2118:
2116:
2115:
2110:
2098:
2096:
2095:
2090:
2054:
2052:
2051:
2046:
2029:
2028:
1996:GEV distribution
1970:
1968:
1967:
1962:
1960:
1959:
1796:
1794:
1793:
1788:
1765:
1764:
1748:
1746:
1745:
1740:
1724:
1722:
1721:
1716:
1683:
1681:
1680:
1675:
1639:
1637:
1636:
1631:
1613:
1611:
1610:
1605:
1591:
1588:
1573:
1572:
1554:
1553:
1520:
1519:
1501:
1500:
1475:
1473:
1472:
1467:
1455:
1453:
1452:
1447:
1445:
1444:
1426:
1425:
1409:random variables
1404:
1402:
1401:
1396:
1381:
1379:
1378:
1373:
1359:
1355:
1342:
1334:
1317:
1312:
1311:
1299:
1287:
1285:
1284:
1279:
1258:
1256:
1255:
1250:
1236:
1232:
1219:
1211:
1194:
1189:
1188:
1176:
1164:
1162:
1161:
1156:
1141:
1139:
1138:
1133:
1101:
1093:
1084:
1082:
1081:
1076:
1074:
1066:
1054:
1052:
1051:
1046:
1032:
1031:
990:
985:
964:
963:
941:
939:
938:
933:
931:
930:
904:
902:
901:
896:
856:
851:
830:
829:
805:
804:
792:
791:
762:
760:
759:
754:
752:
751:
732:
730:
729:
724:
712:
710:
709:
704:
692:
690:
689:
684:
682:
681:
669:
668:
652:random variables
630:
628:
627:
622:
607:
603:
590:
582:
562:
554:
543:or equivalently
539:
537:
536:
531:
482:
431:
429:
428:
423:
411:
407:
388:
380:
378:
377:
364:
336:
334:
333:
328:
295:
287:
271:
269:
268:
263:
252:
248:
222:
221:
208:
203:
67:
60:
56:
53:
47:
27:
26:
19:
4432:
4431:
4427:
4426:
4425:
4423:
4422:
4421:
4392:
4391:
4390:
4383:
4369:
4365:
4358:
4344:
4340:
4333:
4319:
4312:
4303:
4301:
4266:
4262:
4257:
4253:
4212:
4208:
4203:
4199:
4194:
4190:
4185:
4181:
4176:
4172:
4167:
4163:
4158:
4154:
4149:
4145:
4140:
4136:
4099:
4092:
4085:
4077:. London: CRC.
4071:
4064:
4055:
4053:
4042:
4038:
4029:
4027:
4023:
4016:
4010:
4006:
3990:
3989:
3982:
3980:
3976:
3969:
3963:
3959:
3943:
3942:
3938:
3924:
3920:
3909:
3905:
3895:
3893:
3892:. RSRG, Caltech
3879:
3875:
3851:10.1.1.210.1699
3823:
3817:
3813:
3806:
3784:
3771:
3761:
3759:
3749:
3745:
3736:
3732:
3723:
3719:
3709:
3707:
3679:
3672:
3665:
3643:
3634:
3630:
3572:
3563:
3550:Wayback Machine
3539:
3528:
3505:
3501:
3499:
3496:
3495:
3464:
3461:
3460:
3444:
3441:
3440:
3423:
3419:
3404:
3400:
3398:
3395:
3394:
3391:order statistic
3374:
3371:
3370:
3341:
3337:
3335:
3332:
3331:
3305:
3258:
3254:
3221:
3217:
3202:
3170:
3150:
3145:
3144:
3140:
3139:
3130:
3104:
3098:
3095:
3094:
3066:
3052:
3049:
3048:
3017:
3014:
3013:
2955:
2952:
2951:
2920:
2917:
2916:
2900:
2897:
2896:
2860:
2856:
2855:
2853:
2850:
2849:
2833:
2825:
2822:
2821:
2805:
2802:
2801:
2758:
2755:
2754:
2722:
2718:
2713:
2710:
2709:
2706:
2685:
2682:
2681:
2659:
2655:
2619:
2615:
2614:
2610:
2565:
2561:
2559:
2556:
2555:
2489:
2485:
2440:
2436:
2435:
2392:
2388:
2346:
2342:
2341:
2339:
2335:
2316:
2311:
2302:
2276:
2270:
2267:
2266:
2223:
2221:
2209:
2203:
2200:
2199:
2158:
2152:
2149:
2148:
2132:
2124:
2121:
2120:
2104:
2101:
2100:
2060:
2057:
2056:
2024:
2020:
2015:
2012:
2011:
2008:
1989:
1952:
1948:
1946:
1943:
1942:
1935:
1929:
1901:The family of
1806:
1760:
1756:
1754:
1751:
1750:
1734:
1731:
1730:
1689:
1686:
1685:
1645:
1642:
1641:
1625:
1622:
1621:
1620:A distribution
1587:
1568:
1564:
1549:
1545:
1515:
1511:
1496:
1492:
1484:
1481:
1480:
1461:
1458:
1457:
1440:
1436:
1421:
1417:
1415:
1412:
1411:
1390:
1387:
1386:
1353:
1333:
1304:
1300:
1298:
1296:
1293:
1292:
1267:
1264:
1263:
1230:
1210:
1181:
1177:
1175:
1173:
1170:
1169:
1150:
1147:
1146:
1145:A distribution
1092:
1090:
1087:
1086:
1065:
1063:
1060:
1059:
1018:
1014:
986:
981:
956:
952:
950:
947:
946:
923:
919:
917:
914:
913:
852:
847:
822:
818:
800:
796:
787:
783:
775:
772:
771:
744:
740:
738:
735:
734:
718:
715:
714:
698:
695:
694:
677:
673:
664:
660:
658:
655:
654:
644:
601:
581:
553:
551:
548:
547:
472:
466:
463:
462:
442:random variable
438:
405:
379:
370:
366:
354:
348:
345:
344:
286:
284:
281:
280:
275:
246:
214:
210:
204:
196:
190:
187:
186:
169:
144:random variable
140:
135:
68:
57:
51:
48:
40:help improve it
37:
28:
24:
17:
12:
11:
5:
4430:
4420:
4419:
4414:
4409:
4404:
4389:
4388:
4381:
4363:
4357:978-0412552700
4356:
4338:
4331:
4310:
4260:
4251:
4206:
4197:
4188:
4179:
4170:
4161:
4152:
4143:
4134:
4113:(1): 119–131.
4090:
4083:
4062:
4036:
4004:
3957:
3936:
3918:
3915:. K.U. Leuven.
3903:
3873:
3811:
3804:
3769:
3743:
3730:
3717:
3670:
3663:
3631:
3629:
3626:
3625:
3624:
3623:
3622:
3608:
3603:
3598:
3593:
3588:
3583:
3578:
3571:
3568:
3562:
3559:
3558:
3557:
3538:
3535:
3527:
3524:
3508:
3504:
3483:
3480:
3477:
3474:
3471:
3468:
3448:
3426:
3422:
3418:
3415:
3412:
3407:
3403:
3378:
3356:
3353:
3350:
3347:
3344:
3340:
3328:
3327:
3316:
3311:
3308:
3303:
3299:
3294:
3291:
3288:
3285:
3282:
3279:
3276:
3273:
3270:
3267:
3264:
3261:
3257:
3253:
3250:
3247:
3244:
3241:
3236:
3233:
3230:
3227:
3224:
3220:
3216:
3213:
3210:
3205:
3200:
3197:
3194:
3191:
3188:
3185:
3182:
3179:
3176:
3173:
3169:
3162:
3159:
3156:
3153:
3149:
3143:
3138:
3128:
3125:
3122:
3119:
3116:
3113:
3110:
3107:
3103:
3079:
3076:
3073:
3069:
3065:
3062:
3059:
3056:
3036:
3033:
3030:
3027:
3024:
3021:
3001:
2998:
2995:
2992:
2989:
2986:
2983:
2980:
2977:
2974:
2971:
2968:
2965:
2962:
2959:
2939:
2936:
2933:
2930:
2927:
2924:
2904:
2883:
2880:
2877:
2874:
2871:
2868:
2863:
2859:
2836:
2832:
2829:
2809:
2786:
2783:
2780:
2777:
2774:
2771:
2768:
2765:
2762:
2742:
2739:
2736:
2733:
2730:
2725:
2721:
2717:
2705:
2702:
2689:
2668:
2662:
2658:
2654:
2651:
2648:
2643:
2640:
2637:
2634:
2631:
2628:
2625:
2622:
2618:
2613:
2609:
2606:
2601:
2598:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2574:
2571:
2568:
2564:
2552:
2551:
2540:
2536:
2528:
2525:
2522:
2519:
2516:
2513:
2510:
2507:
2504:
2501:
2498:
2495:
2492:
2488:
2484:
2479:
2476:
2473:
2470:
2467:
2464:
2461:
2458:
2455:
2452:
2449:
2446:
2443:
2439:
2431:
2428:
2425:
2422:
2419:
2416:
2413:
2410:
2407:
2404:
2401:
2398:
2395:
2391:
2387:
2382:
2379:
2376:
2373:
2370:
2367:
2364:
2361:
2358:
2355:
2352:
2349:
2345:
2338:
2334:
2331:
2325:
2322:
2319:
2315:
2310:
2300:
2297:
2294:
2291:
2288:
2285:
2282:
2279:
2275:
2247:
2244:
2239:
2235:
2232:
2229:
2226:
2218:
2215:
2212:
2208:
2187:
2184:
2181:
2178:
2175:
2172:
2167:
2164:
2161:
2157:
2135:
2131:
2128:
2108:
2088:
2085:
2082:
2079:
2076:
2073:
2070:
2067:
2064:
2044:
2041:
2038:
2035:
2032:
2027:
2023:
2019:
2007:
2004:
1988:
1985:
1958:
1955:
1951:
1934:
1931:
1927:
1926:
1923:
1920:t-distribution
1916:
1899:
1888:
1887:
1876:
1869:
1862:
1855:
1848:
1841:
1834:
1827:
1820:
1805:
1802:
1786:
1783:
1780:
1777:
1774:
1771:
1768:
1763:
1759:
1738:
1714:
1711:
1708:
1705:
1702:
1699:
1696:
1693:
1673:
1670:
1667:
1664:
1661:
1658:
1655:
1652:
1649:
1629:
1615:
1614:
1603:
1600:
1597:
1594:
1585:
1582:
1579:
1576:
1571:
1567:
1563:
1560:
1557:
1552:
1548:
1544:
1541:
1538:
1535:
1532:
1529:
1526:
1523:
1518:
1514:
1510:
1507:
1504:
1499:
1495:
1491:
1488:
1465:
1443:
1439:
1435:
1432:
1429:
1424:
1420:
1394:
1383:
1382:
1371:
1368:
1365:
1362:
1351:
1348:
1345:
1340:
1337:
1332:
1329:
1326:
1323:
1320:
1315:
1310:
1307:
1303:
1277:
1274:
1271:
1260:
1259:
1248:
1245:
1242:
1239:
1228:
1225:
1222:
1217:
1214:
1209:
1206:
1203:
1200:
1197:
1192:
1187:
1184:
1180:
1154:
1131:
1128:
1125:
1122:
1119:
1116:
1113:
1110:
1107:
1104:
1099:
1096:
1085:is defined as
1072:
1069:
1056:
1055:
1044:
1041:
1038:
1035:
1030:
1027:
1024:
1021:
1017:
1013:
1009:
1006:
1003:
1000:
997:
994:
989:
984:
980:
976:
973:
970:
967:
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4052:on 2014-04-07
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4026:on 2011-07-17
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3882:Wierman, Adam
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104:
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32:This article
30:
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4375:. Springer.
4372:
4366:
4347:
4341:
4322:
4302:. Retrieved
4298:the original
4277:
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4050:the original
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4021:the original
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3981:. Retrieved
3974:the original
3960:
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3894:. Retrieved
3889:
3876:
3841:math/0509605
3831:
3827:
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3787:
3760:. Retrieved
3757:ResearchGate
3756:
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3720:
3708:. Retrieved
3690:
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114:heavy-tailed
113:
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3983:November 1,
2258:, then the
1909:. See also
1406:independent
182:That means
133:Definitions
4396:Categories
4304:2015-09-03
4229:1708.01686
4056:2009-06-12
4030:2009-02-21
3896:January 9,
3834:(3): 581.
3628:References
126:log-normal
101:, and the
4280:: 55–79.
3946:cite book
3846:CiteSeerX
3601:Power law
3596:Long tail
3482:∞
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4246:88514574
3993:cite web
3762:April 7,
3710:April 7,
3570:See also
3546:Archived
3537:Software
2820:, where
2304:Pickands
2260:Pickands
2119:, where
1857:the log-
1589:as
1356:as
1233:as
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122:variance
52:May 2020
4294:8917289
4129:2958083
3868:3047753
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118:moments
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2895:where
2554:where
1981:Pareto
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4290:S2CID
4242:S2CID
4224:arXiv
4125:JSTOR
4024:(PDF)
4017:(PDF)
3977:(PDF)
3970:(PDF)
3864:S2CID
3836:arXiv
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3047:and
2198:and
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4079:ISBN
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3898:2014
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59:(
54:)
50:(
36:.
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