Alves, Isabel Fraga

Neves, Cláudia

Repositório Institucional da Universidade de Aveiro

Extreme Value Distributions ∗Isabel Fraga AlvesCEAUL, DEIO, Faculty of Sciences,University of Lisbon, PortugalCla´udia NevesUIMA, Department of MathematicsUniversity of Aveiro, Portugal1 IntroductionExtreme Value distributions arise as limiting distributions for maximums or minimums (extreme values) of asample of independent, identically distributed random variables, as the sample size increases. Extreme ValueTheory (EVT) is the theory of modelling and measuring events which occur with very small probability.This implies its usefulness in risk modelling as risky events per definition happen with low probability.Thus, these distributions are important in statistics. These models, along with the Generalized ExtremeValue distribution, are widely used in risk management, finance, insurance, economics, hydrology, materialsciences, telecommunications, and many other industries dealing with extreme events. The class of ExtremeValue Distributions (EVD’s) essentially involves three types of extreme value distributions, types I, II andIII, defined below.Definition 1 (Extreme Value Distributions for maxima).The following are the standard Extreme Value distribution functions:(i) Gumbel (type I): Λ(x) = exp{− exp(−x)}, x ∈ R;(ii) Fre´chet (type II): Φα(x) ={0, x ≤ 0;exp{−x−α}, x > 0, α > 0;(iii) Weibull (type III): Ψα(x) ={exp{−(−x)α}, x ≤ 0, α > 0;1, x > 0.The EVD families can be generalized with the incorporation of location (λ) and scale (δ) parameters,leading toΛ(x;λ, δ) = Λ((x− λ)/δ), Φα(x;λ, δ) = Φα((x− λ)/δ), Ψα(x;λ, δ) = Ψα((x− λ)/δ), λ ∈ R, δ > 0.Among these three families of distribution functions, the type I is the most commonly referred in discussionsof extreme values. Indeed, the Gumbel distribution {Λ(x;λ, δ) = Λ((x−λ)/δ);λ ∈ R, δ > 0}, is often coined“the” extreme value distribution.Proposition 1 (Moments and Mode of EVD).The mean, variance and mode of the EVD as in definition 1 are, respectively:(i) Gumbel – Λ: E[X] = γ = 0.5772 . . . = Euler′s constant; V ar[X] = pi2/6; Mode = 0;(ii) Fre´chet – Φα: E[X] = Γ(1 − 1/α), for α > 1; V ar[X] = Γ(1 − 2/α) − Γ2(1 − 1/α), for α > 2;Mode = (1 + 1/α)−1/α;∗This work has been partially supported by FCT/POCI 2010 project.1(iii) Weibull – Ψα: E[X] = −Γ(1 + 1/α); V ar[X] = Γ(1 + 2/α)− Γ2(1 + 1/α); Mode = −(1− 1/α)−1/α,for α > 1, and Mode = 0, for 0 < α ≤ 1;here Γ denotes the gamma function Γ(s) :=∫∞0 xs−1e−xdx, s > 0.Definition 2 (Extreme Value Distributions for minima).The standard converse EVD’s for minima are defined as: Λ∗(x) = 1 − Λ(−x), Φ∗α(x) = 1 − Φα(−x) andΨ∗α(x) = 1−Ψα(−x).The Gumbel distribution, named after one of the pioneer scientists in practical applicationsof the Extreme Value Theory (EVT), the German mathematician Emil Gumbel (1891-1966),has been extensively used in various fields including hydrology for modeling extreme events.Gumbel applied EVT on real world problems in engineering and in meteorological phenomenasuch as annual flood flows(Gumbel, 1958):“It seems that the rivers know the theory. It only remains to convince the engineers of the validity of this analysis.” Emil GumbelThe EVD of type II was named after Maurice Fre´chet (1878-1973), a French mathematicianwho devised one possible limiting distribution for a sequence of maxima, provided convenientscale normalization (Fre´chet, 1927). In applications to finance, the Fre´chet distribution hasbeen of great use apropos to the adequate modeling of market-returns which are often heavy-tailed. Maurice Fre´chetThe EVD of type III was named after Waloddi Weibull (1887-1979), a Swedish engineerand scientist well-known for his work on strength of materials and fatigue analysis (Weibull,1939). Even though the Weibull distribution was originally developed to address the problemsfor minima arising in material sciences, it is widely used in many other areas thanks to itsflexibility. If α = 1, the Weibull distribution function for minima, Ψ∗α, reduces to the Expo-nential model, whereas for α = 2 it mimics the Rayleigh distribution which is mainly usedin the telecommunications field. Furthermore, Ψ∗α resembles the Normal distribution whenα = 3.5.Waloddi WeibullOwing to the equalitymin(X1, . . . , Xn) = −max(−X1, . . . ,−Xn)it suffices to consider henceforth only the EVD’s for maxima featuring in Definition 1. In probability theoryand statistics, the Generalized Extreme Value (GEV) distribution is a family of continuous probability dis-tributions developed under the extreme value theory in order to combine the Gumbel, Fre´chet and Weibullfamilies. The GEV distribution arises from the extreme value theorem (Fisher-Tippett, 1928 and Gne-denko, 1943) as the limiting distribution of properly normalized maxima of a sequence of independent andidentically distributed (i.i.d.) random variables. Because of this, the GEV distribution is fairly used as anapproximation to model the maxima of long (finite) sequences of random variables. In some fields of ap-plication the GEV distribution is in fact known as the Fisher-Tippett distribution, named after Sir RonaldAylmer Fisher (1890-1962) and Leonard Henry Caleb Tippett (1902-1985) who recognized the only threepossible limiting functions outlined above in Definition 1.2 Extreme Value Theory and Max-StabilityRichard von Mises (1883-1953) studied the EVT in 1936, giving in particular the von Mises sufficientconditions on the hazard rate (assuming the density exists) in order to give a situation in which EVTbehaviour occurs, leading to one of the above three types of limit law that is, giving an extremal domainof attraction D(G) for the extreme-value distribution G. Later on, and motivated by a storm surge in theNorth Sea (31 January-1 February 1953) which caused extensive flooding and many deaths, the NetherlandsGovernment gave top priority to understanding the causes of such tragedies with a view to risk mitigation.Since it is the maximum sea level which is the danger, EVT became a Netherlands scientific priority. Arelevant work in the field is the doctoral thesis of Laurens de Haan in 1970.2The fundamental extreme value theorem (Fisher-Tippett 1928; Gnedenko, 1943) ascertains the Gener-alized Extreme Value distribution in the von Mises-Jenkinson parametrization (von Mises, 1936; Jenkinson,1955) as an unified version of all possible non-degenerate weak limits of partial maxima of sequences com-prising i.i.d. random variables X1, X2, . . .. That is:Theorem 1 (Fisher-Tippett 1928; Gnedenko, 1943).If there exist normalizing constants an > 0 and bn ∈ R such thatlimn→∞P{a−1n (max(X1, . . . , Xn)− bn) ≤ x}= G(x),for some non-degenerate distribution function G, then it is possible to redefine the normalizing constants insuch a way thatG(x) = Gξ(x) := exp(−(1 + ξx)−1/ξ),for all x such that 1 + ξx > 0, with extreme value index ξ ∈ R. Taking ξ → 0, then Gξ(x) reduces to Λ(x)for all x ∈ R (cf. Definition 1). Thus the distribution function F belongs to the domain of attraction of Gξ,which is denoted by F ∈ D(Gξ).Remark 1. Note that, as n→∞, the max(X1, . . . , Xn) detached of any normalization converges in distri-bution to a degenerate law assigning probability one to the right endpoint of F , xF := sup{x : F (x) < 1}.For ξ < 0, ξ = 0 and ξ > 0, the Gξ distributionfunction reduces to Weibull, Gumbel and Fre´chet distribu-tions, respectively. More precisely,Λ(x) ≡ G0(x),Φα(x) ≡ G1/α(α(x− 1)),andΨα(x) ≡ G−1/α(α(1 + x)).For exhaustive details on EVD see Chapter 22 of Johnson,Balakrishnan, and Kotz (1995).Proposition 2 (Moments and Mode of GEV).The mean, variance and mode of the GEV as in Theorem 1 are, respectively:E[X] = −1ξ [1−ξ(1−ξ)], for ξ < 1; V ar[X] = 1ξ2 [Γ(1−2ξ)−Γ2(1−ξ)], for ξ < 1/2; Mode = 1ξ [(1+ξ)−ξ−1],for ξ 6= 0.Proposition 3 (Skewness of GEV).The skewness coefficient of GEV distribution, defined as skewGξ := E[{X −E[X]}3]/{V ar[X]}3/2, is equalto zero at ξ0 ' −2.8. Moreover, skewGξ > 0, for ξ > ξ0, and skewGξ < 0, for ξ < ξ0. Furthermore, for theGumbel distribution, skewG0 ' 1.14.The Fre´chet domain of attraction contains distributions with polynomially decay tails. All distribu-tion functions belonging to Weibull domain of attraction are light-tailed with finite right endpoint. Theintermediate case ξ = 0 is of particular interest in many applied sciences where extremes are relevant, notonly because of the simplicity of inference within the Gumbel domain G0 but also for the great variety ofdistributions possessing an exponential tail whether having finite right endpoint or not. In fact, separatingstatistical inference procedures according to the most suitable domain of attraction for the sampled distribu-tion has become an usual practice. In this respect we refer to Neves and Fraga Alves (2008) and referencestherein.Definition 3 (Univariate Max-Stable Distributions).A random variable X with distribution function F is max-stable if there are normalizing sequences {an >0} and {bn ∈ R} such that the independent copies X1, X2, · · · , Xn satisfy the equality in distributionmax(X1, · · · , Xn) d=anX + bn. Equivalently, F is a max-stable distribution function if [F (x)]n = F ((x −bn)/an), all n ∈ N.3The class GEV, up to location and scale parameters, {Gξ(x;λ, δ) = Gξ((x− λ)/δ), λ ∈ R, δ > 0}, repre-sents the only possible max-stable distributions.Additional information can be found in Kotz and Nadarajah (2000), a monograph which describes in anorganized manner the central ideas and results of probabilistic extreme-value theory and related extreme-value distributions – both univariate and multivariate – and their applications, and it is aimed mainlyat a novice in the field. De Haan and Ferreira (2006) constitutes an excellent introduction to EVT at thegraduate level, however requiring some mathematical maturity in regular variation, point processes, empiricaldistribution functions, and Brownian motion. Reference Books in Extreme Value Theory and in the fieldof real world applications of EVD’s and Extremal Domains of Attraction are: Embrechts, Klu¨ppelbergand Mikosch (2001), Beirlant, Goegebeur, Segers and Teugels (2004), David and Nagaraja (2003), Gumbel(1958), Castillo, Hadi, Balakrishnan, and Sarabia (2005) and Reiss and Thomas (2007).ReferencesBeirlant, J., Goegebeur, Y., Segers, J., Teugels, J.: Statistics of Extremes: Theory and Applications. Wiley, England(2004).Castillo, E., Hadi A.S. Balakrishnan, N. and Sarabia, J.M. Extreme Value and Related Models with Applications inEngineering and Science, John Wiley & Sons, Hoboken, New Jersey, (2005).David, H.A., Nagaraja, H.N.: Order Statistics, 3rd edition. Wiley, Hoboken, New Jersey (2003).Embrechts, P., Klu¨ppelberg, C., Mikosch, T.: Modelling Extremal Events for Insurance and Finance. Springer, Berlin,Heidelberg (3rd Ed. 2001).Fisher, R.A. and Tippett, L.H.C. (1928). Limiting forms of the frequency distribution of the largest and smallestmember of a sample, Proc. Camb. Phil. Soc., 24, 180–190.Fre´chet, M . (1927): Sur la loi de probabilit de l’e´cart maximum, Ann. Soc. Polon. Math. (Cracovie), 6, 93–116.Gnedenko, B. V. (1943). Sur la distribution limite du terme maximum d’une se´rie ale´atoire. Ann. Math., 44, 423–453.Gumbel, E.J. Statistics of Extremes, Columbia University Press, New York, (1958). (375 pages).de Haan, L.(1970). On Regular Variation and its Application to the Weak Convergence of Sample extremes, Math.Centre Tracts vol. 32, Mathematisch Centrum, Amsterdam.De Haan, L. e Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer Series in Operations Researchand Financial Engineering. Boston.Jenkinson, A.F. (1955). The frequency distribution of the annual maximum (or minimum) values of meteorologicalelements, Quart. J. Roy. Meteo. Soc., 81, 158–171.Johnson, N.L., Balakrishnan, N. and Kotz, S. Continuous Univariate Distributions - Volume 2, Second edition , JohnWiley & Sons, New York, 1995 (719 pages).Kotz, S., Nadarajah, S. (2000). ”Extreme Value Distributions: Theory and Applications.” London: Imperial CollegePress.von Mises, R. (1936). La distribution de la plus grande de n valeurs. Reprinted in Selected Papers Volumen II,American Mathematical Society, Providence, R.I., 1954, pp. 271–294.Neves, C. and M. I. Fraga Alves, M. I. (2008). Testing extreme value conditions - an overview and recent approaches.REVSTAT - Statistical Journal, Vol 6,1, 83–100. Special issue “Statistics of Extremes and Related Fields” editedby Jan Beirlant, Isabel Fraga Alves, Ross Leadbetter.Reiss, R.-D., Thomas, M. Statistical Analysis of Extreme Values, with Application to Insurance, Finance, Hydrologyand Other Fields, 2nd edition; 3rd edition, Birkhuser Verlag (2001; 2007).4

Extreme Value Distributions

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Extreme Value Distributions

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