Extreme Value Analysis : an Introduction

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Extreme Value Analysis : an Introduction

Myriam Charras-Garrido, Pascal Lezaud

To cite this version:

Myriam Charras-Garrido, Pascal Lezaud. Extreme Value Analysis : an Introduction. Journal de la Societe Française de Statistique, Societe Française de Statistique et Societe Mathematique de France, 2013, 154 (2), pp 66-97. �hal-00917995�

Journal de la Société Française de Statistique

Vol. 154No. 2 (2013)

Extreme Value Analysis: an Introduction

Titre:Introduction à l’analyse des valeurs extrêmes

Myriam Charras-Garrido¹ and Pascal Lezaud²

Abstract:We provide an overview of the probability and statistical tools underlying the extreme value theory, which aims to predict occurrence of rare events. Firstly, we explain that the asymptotic distribution of extreme values belongs, in some sense, to the family of the generalised extreme value distributions which depend on a real parameter, called the extreme value index. Secondly, we discuss statistical tail estimation methods based on estimators of the extreme value index.

Résumé :Nous donnons un aperçu des résultats probabilistes et statistiques utilisés dans la théorie des valeurs extrêmes, dont l’objectif est de prédire l’occurrence d’événements rares. Dans la première partie de l’article, nous expliquons que la distribution asymptotique des valeurs extrêmes appartient, dans un certain sens, à la famille des distributions des valeurs extrêmes généralisées qui dépendent d’un paramètre réel, appelé l’indice de valeur extrême. Dans la seconde partie, nous discutons des méthodes d’évaluation statistiques des queues basées sur l’estimation de l’indice des valeurs extrêmes

Keywords:extreme value theory, max stable distributions, extreme value index, distribution tail estimation

Mots-clés :théorie des valeurs extrêmes, lois max-stables, indice des valeurs extrêmes, estimation en queue de distribution

AMS 2000 subject classifications:60E07, 60G70, 62G32, 62E20

1. Introduction

The consideration of the major risks in our technological society has become vital because of the economic, environmental and human impacts of industrial disasters. One of the standard approaches to studying risks uses the extreme value theory; a branch of statistics dealing with the extreme deviations from the median of probability distributions. Of course, this approach is based on the language of probability theory and thus the first question to ask is whether a probability approach applies to the studied risk. For instance, can we use probabilities in order to study the disappearance of dinosaurs? More recently, the Fukushima disaster, only 25 years after that of Chernobyl, raises the question of the appropriateness of the probability methods used. Moreover, as explained inBouleau(1991), the extreme value theory aims to predict occurrence of rare events (e.g. earthquakes of large magnitude), outside the range of available data (e.g. earthquakes of magnitude less than 2). So, its use requires some precautions, and inBouleau(1991) the author concludes that

The approach attributing a precise numerical value for the probability of a rare phenomenon is suspect, unless the laws of nature governing the phenomenon are explicitly and exhaustively known [...] This does

1 INRA, UR346, F-63122 Saint-Genes-Champanelle, France.

E-mail:myriam.charras-garrido@clermont.inra.fr

2 ENAC, MAIAA, F-31055 Toulouse, France.

E-mail:lezaud@recherche.enac.fr

not mean that the use of probability or probability concepts should be rejected.

Nevertheless, the extreme value theory remains a well suited technique capable of predicting extreme events. Although the application of this theory in the real world always needs to be viewed with a critical eye, we suggest, in this article, an overview of the mathematical and statistical theories underlying it.

As already said before, the main objective of the extreme value theory is to know or predict the statistical probabilities of events that have never (or rarely) been observed. Firstly, the statistical analysis of extreme values has been developed in order to study flood levels. Nowadays, the domains of application include other meteorological events (such as precipitation or wind speed), industry (for example important malfunctions), finance (e.g.financial crises), insurance (for very large claims due to catastrophic events), environmental sciences (like concentration of ozone in the air), etc.

Formally, we consider the sampleX1, . . . ,Xnofnindependent and identically distributed (iid) random variables with common cumulative distribution function (cdf)F. We define the ordered sample byX_1,n≤X_2,n≤. . .≤X_n,n=M_n. We are interested in two related problems. The first one consists in estimating the tail of the survivor function ¯F =1−F: givenhn>Mn, we want to estimate p=F(h¯ n). This corresponds to the estimation of the risk to get out a zone, for example the probability to exceed the level of a dyke for a flood application. The second problem consists in estimating extreme quantiles: givenpn<1/n, we want to estimateh=F¯⁻¹(p_n). This corresponds to estimating the limit of a critical zone, as the level of a dyke for a flood application, to be exceeded with probabilityp_n. Note that since we are interested to extrapolate outside the range of available observations, we have to assume that the quantile probability depends onnand that limn→∞pn=0.

In both problems, the same difficulty arises: the cdfF is unknown and difficult to estimate beyond observed data. We want to get over the maximal observationM_n, that means to extrapolate outside the range of the available observations. Both parametric and non parametric usual estima- tion methods fail in this case. For the parametric method, models considered to give similar results in the sample range can diverge in the tail. This is illustrated in Figure1that presents the relative difference between quantiles from a Gaussian and a Student distribution. For the non parametric method, 1−Fb_n(x) =0 ifx>M_n, whereFbdenotes the empirical distribution function,i.e.it is estimated that outside the sample range nothing is likely to be observed. As we are interested in extreme values, an intuitive solution is to use only extreme values of the sample that may contain more information than the other observations on the tail behaviour. Formally, this solution leads to a semi-parametric approach that will be detailed later.

Before starting with the description of the estimation procedures, we need to introduce the probability background which is based on the elegant theory of max-stable distribution functions, the counterpart of the (alpha) stable distributions, seeFeller(1971). The stable distributions are concerned with the limit behaviour of the partial sumSn=X1+X2+···+Xn, asn→∞, whereas the theory of sample extremes is related to the limit behaviour ofMn. The main result is the Fisher- Tippett-Gnedenko Theorem2.3which claims thatM_n, after proper normalisation, converges in distribution to one of three possible distributions, the Gumbel distribution, the Fréchet distribution, or the Reversed Weibull distribution. In fact, it is possible to combine these three distributions together in a single family of continuous cdfs, known as the generalized extreme value (GEV)

FIGURE1. Relative distance between quantiles of order p computed withN(0,1)and Student(4) models.

distributions. A GEV is characterized by a real parameterγ, the extreme value index, as a stable distribution is it by a characteristic exponentα ∈]0,2]. Let us mention the similarity with the Gaussian Law, a stable distribution withα=2, and the Central Limit Theorem. Next we have to find some conditions to determine for a given cdfFthe limiting distribution ofM_n. The best tools suited to address that are the tail quantile function (cf. (3) for the definition) and the slowly varying functions. Finally, these results will be widened to some stationary time series.

The paper is articulated in two main Sections. In Section2, we will set up the context in order to state the Fisher-Tippett-Gnedenko Theorem in Subsection2.1. In this paper, we will follow closely the approach presented inBeirlant et al.(2004b), which transfers the convergence in distribution to the convergence of expectations for the class of real, bounded and continuous functions. Other recent texts includeEmbrechts et al.(2003) andReiss and Thomas(1997). In Subsection2.2, some equivalent conditions in terms ofF will be given, since it is not easy to compute the tail quantile function. Finally, in Subsection2.3the condition about the independence between theX_i will be relaxed in order to adapt the previous result for stationary time series satisfying a weak dependence condition. The main result of this part is Theorem2.12.

Section3addresses the statistical point of view. Subsection3.1gives asymptotic properties of extreme order statistics and related quantities and explains how they are used for this extrapolation to the distribution tail. Subsection3.2presents tail and quantile estimations using these extrapola- tions. In Subsection3.3, different optimal control procedures on the quality of the estimates are explored, including graphical procedures, tests and confidence intervals.

2. The Probability theory of Extreme Values

Let us consider the sampleX1, . . . ,X_nofniid random variables with common cdfF. We define the ordered sample byX_1,n≤X_2,n≤. . .≤X_n,n=M_n, and we are interested in the asymptotic distribution of the maximaMnasn→∞. The distribution ofMnis easy to write down, since

P(M_n≤x) =P(X₁≤x, . . . ,X_n≤x) =Fⁿ(x).

Intuitively extremes, which correspond to events with very small probability, happen near the upper end of the support ofF, hence the asymptotic behaviour ofMnmust be related to the right tail of the distribution near the right endpoint. We denote byω(F) =inf{x∈R:F(x)≥1}, the right endpoint of F and by ¯F(x) =1−F(x) =P(X >x) the survivor function of F. We obtain that for all x <ω(F), P(Mn≤x) =Fⁿ(x) →0 as n→∞, whereas for all x≥ω(F) P(M_n≤x) =Fⁿ(x) =1.

ThusM_nconverges in probability toω(F)asn→∞, and since the sequenceM_nis increasing, Mnconverge almost surely toω(F). Of course, this information is not very useful, so we want to investigate the fluctuations ofM_nin the similar way the Central Limit Theorem (CLT) is derived for the sumS_n=∑_iX_i. More precisely, we look after conditions onF which ensure that there exists a sequence of numbers{bn,n≥1}and a sequence of positive numbers{an,n≥1}such that for all real valuesx

P

M_n−b_n an ≤x

=Fⁿ(a_nx+b_n)→G(x) (1) asn→∞, whereGis a non-degenerate distribution (i.e. without Dirac mass). If (1) holds,Fis said to belong to thedomain of attractionofGand we will writeF∈D(G). The problem is twofold: (i) find all possible (non-degenerate) distributionsGthat can appear as a limit in (1), (ii) characterize the distributionsF for which there exists sequences(an)and(bn)such that (1) holds.

Introducing the thresholdun=un(x):=anx+bngives the more understanding interpretation of our problem, since

P(M_n≤u_n) =Fⁿ(u_n) =

1−nF(u¯ n) n

n

.

Hence, we need rather conditions on the tail ¯Fto ensure thatP(Mn≤un)converges to a non-trivial limit. The first result you obtain is the following:

Proposition 2.1. For a givenτ∈[0,∞]and a sequence(u_n)of real numbers the two assertions (i) nF(u¯ _n)→τ, and (ii)P(M_n≤un)→e^−τ are equivalent.

Clearly, Poisson’s limit Theorem is the key behind this Proposition. Indeed, we assume for simplicity that 0<τ<∞and we letK_n(u_n) =∑ⁿ_i=1I_{Xi>un}; it is the number of excesses over the thresholdu_nin the sampleX₁, . . . ,X_n. This quantity has a binomial distribution with parametersn andp=F(u¯ n);

P(K_n(u_n) =k) = n

k

p^k(1−p)^n−k.

The Poisson’s limit Theorem yields thatKn(un)converges in law to a Poisson distribution with parameterτ if and only ifEK_n(u_n)→τ; this is nothing but Proposition2.1.

Now, let us assume thatX1>unand consider the discrete timeT(un)such thatX_1+T(un)>un

andX_i≤u_nfor all 1<i≤T(u_n), i.e.T(u_n) =min{i≥1 :X_i+1>u_n}. In order to hope for a limit distribution, we will have to normalizeT(un)by the factorn(soT(un)/n∈(0,1]); then

P n⁻¹T(un)>k/n

=P(X2≤un,···,X_k+1≤un|X1>un) =F(un)^n(k/n). Letx>0, then fork=⌊nx⌋

P(n⁻¹T(u_n)>x) =P(n⁻¹T(u_n)>k/n) = (1−F¯(u_n))^n(k/n),

hence ifnF(u¯ _n)→τ asn→∞, we haveP(n⁻¹T(u_n)>x)→e^−τx, that means the excess times are asymptotically distributed according to an exponential law with parameterτ. The precise approach of this result requires the introduction of the point process of exceedances(N_n)defined by:

Nn(B) =

n

∑

i=1

δ_i/n(B)I_{Xi>un}=♯{i/n∈B:Xi>un},

whereBis a Borel set on(0,1]andδ_i/n(B) =1 ifi/n∈Band 0 else. Then we have the following result (seeResnick(1987)):

Proposition 2.2. Let(un)n∈N be threshold values tending toω(F) as n→∞. Then, we have limn→∞nF(u¯ n) =τ∈(0,∞), if and only if(Nn)converges in distribution to a Poisson process N with parameterτ as n→∞.

2.1. The possible limits

Hereafter, we work under the assumption that the underlying cdfF is continuous and strictly increasing. What are the possible non-degenerate limit laws for the maximaM_n? Firstly, the limit law of a sequence of random variables is uniquely determined up to changes of location and scale (seeResnick(1987)), that means if there exists sequences(an)and(bn)such that

P

Xn−bn

a_n ≤x

→G(x), then the relation

P

X_n−β_n α_n ≤x

→H(x), holds for the sequences(βn)and(αn)if and only if

nlim→∞an/αn=σ∈[0,∞), lim

n→∞(bn−βn)/αn=µ ∈R.

In that case,H(x) =G((x−µ)/σ)and we say thatH andGare of thesame type. Thus, a cdfF cannot be in the domain of attraction of more than one type of cdf.

Furthermore, the question turns out to be closely related to the following property, identified byFisher and Tippett (1928). Assume that the properly normalized and centred maxima Mn

converges in distribution toGand letn=mr, withm,n,r∈N. Hence, asn→∞, we have Fⁿ(a_mx+b_m) = [F^m(a_mx+b_m)]^r→G^r(x).

From the previous discussion, it follows that there existar>0 andbrsuch thatG^r(x) =G(arx+ b_r); we say that the cdfGismax-stable.

To emphasize the role played by the tail function, we define an equivalence relation between cdfs in this way. Two cdfsF andHare calledtail-equivalentif they have the same right end-point, i.e. ifω(F) =ω(H) =x₀, and

limx↑x0

1−F(x) 1−H(x) =A,

for some constantA. Using the previous discussion, it can be shown (seeResnick(1987)) that F∈D(G)if and only ifH∈D(G); moreover, we can take the same norming constants.

The main result of this Section is the Theorem of Fisher, Tippet and Gnedenko which charac- terizes the max-stable distribution functions.

Theorem 2.3(Fisher-Tippett-Gnedenko Theorem). Let(Xn)be a sequence of iid random vari- ables. If there exist norming constants a_n>0, b_n∈Rand some non degenerate cdf G such that a⁻_n¹(Mn−bn)converges in distribution to G, then G belongs to the type of one of the following three cdfs:

Gumbel: G0(x) =exp(−e^−x), x∈R,

Fréchet: G_1,α(x) =exp(−x^−α), x≥0,α>0, Reversed Weibull: G_2,α(x) =exp(−(−x)^−α), x≤0,α<0.

Figure2shows the convergence of(Mn−bn)/anto its extreme value limit in case of a uniform distributionU[0,1].

FIGURE2. Plot ofP((Xn,n−bn)/an≤x) = (1+ (x−1)/n)ⁿfor n=5(dashed line) and n=10(dotted line) and its limit exp(−(1−x))(solid line) as n→∞for a U[0,1]distribution F.

The three types of cdfs given in Theorem (2.3) can be thought of as members of a single family of cdfs. For that, let us introduce the new parameterγ=1/α and the cdf

Gγ(x) =exp(−(1+γx)^−1/γ)), 1+γx>0. (2) The limiting caseγ→0 corresponds to the Gumbel distribution. The cdfG_γ(x)is known as the generalized extreme valueor as theextreme valuecdf in the von Mises form, and the parameter γis called theextreme value index. Figure3gives examples of Gumbel, Fréchet and Reversed Weibull distributions.

FIGURE3. Examples of Gumbel (γ=0in solid line), Fréchet (forγ=1in dashed line) and Reversed Weibull (for γ=−1in dotted line) cdfs.

Now, we will present the sketch of the Theorem’s proof, following the approach ofBeirlant et al.(2004b) which transfers the convergence in distribution to the convergence of expectations for the class of real, bounded and continuous functions (see Helly-Bray Theorem inBillingsley (1995)).

Let us introduce thetail quantile function

U(t):=inf{x : F(x)≥1−1/t}, (3) which is non-decreasing over the interval[1,∞). Then, for any real, bounded and continuous functions f,

E

f a⁻_n¹(M_n−b_n)

=n Z ∞

−∞

f

x−b_n an

Fⁿ⁻¹(x)dF(x),

= Z _n

0

f

U(n/v)−bn

an

1−v

n n−1

dv.

Now observe that(1−v/n)ⁿ⁻¹→e^−v, as n→∞, while the interval of integration extends to [0,∞). To obtain a limit for the left-hand term, we can makea⁻_n¹(U(n/v)−b_n)convergent for all positivev. Considering the casev=1 suggests thatbn=U(n)is an appropriate choice. Thereby, the natural condition to be imposed is that for some positive functionaand anyu>0

xlim→∞

U(xu)−U(x)

a(x) =h(u)exists, (C)

with the limit functionhnot identically equal to zero. We have the following Proposition (Propo- sition 2.2 in Section 2.1 inBeirlant et al.(2004b))

Proposition 2.4. The possible limits in(C)are given by (h_γ(u) =c^uγ_γ⁻¹ γ6=0

h0(u) =clogu , where c≥0andγis real.

The casec=0 has to be excluded since it leads to a degenerate limit, and the casec>0 can be reduced to the casec=1 by incorporatingcin the functiona. Hence, we replace the condition (C)by

xlim→∞

U(xu)−U(x)

a(x) =h_γ(u)exists, (C_γ). (4)

The above result entails that under(C_γ), we find that withbn=U(n)andan=a(n) E

f a⁻_n¹(Mn−bn)

→ Z ∞

0

f h_γ(1/v)

e^−vdv:=

Z ∞

−∞f(u)dG_γ(u), asn→∞, withG_γ given by (2).

If we writea(x) =x^γℓ(x), then the limiting conditiona(xu)/a(x)→u^γleads toℓ(xu)/ℓ(x)→1.

This kind of condition refers to the notion of regular variation.

Definition 2.5. A positive measurable functionℓon(0,∞)which satisfies

xlim→∞

ℓ(xu)

ℓ(x) =1, u>0,

is calledslowly varyingat∞(we writeℓ∈R₀).

A positive measurable functionhon(0,∞)isregularly varyingat∞of indexγ∈R(we write h∈R_γ) if

xlim→∞

h(xu)

h(x) =x^γ, u>0.

The slowly varying functions play a fundamental role in probability theory, good references are the books ofFeller(1971),Bingham et al.(1989) andKorevaar(2004). In particular, we have the following result due toKaramata(1933):ℓ∈R₀if and only if it can be represented in the form

ℓ(x) =c(x)exp

_{Z x}

1

ε(u) u du

,

wherec(x)→c∈(0,∞)andε(x)→0 asx→∞. Typical examples areℓ(x) = (logx)^βfor arbitrary β andℓ(x) =exp{(logx)^β}, whereβ <1. Furthermore, ifh∈R_γ withγ>0, thenh(x)→∞, while forγ<0,h(x)→0, asx↑∞.

Because of their intrinsic importance, we distinguish between the three cases whereγ>0, γ<0 and the intermediate case whereγ=0. We have the following result (see Theorem 2.3 in Section 2.6 inBeirlant et al.(2004b))

Theorem 2.6. Let(C_γ)hold

(i) Fréchet case:γ>0. Hereω(F) =∞, the ratio a(x)/U(x)→γ as x→∞and U is of the same regular variation as the auxiliary function a: moreover,(C_γ)is equivalent with the existence of a slowly varying functionℓUfor which U(x) =x^γℓU(x).

(ii) Gumbel case:γ=0. The ratios a(x)/U(x)→0and a(x)/{ω(F)−U(x)} →0whenω(F) is finite.

(iii) Reversed Weibull case:γ<0. Hereω(F)is finite, the ratio a(x)/{ω(F)−U(x)} → −γand {ω(F)−U(x)}is of the same regular variation as the auxiliary function a: moreover,(C_γ) is equivalent with the existence of a slowly varying functionℓUfor whichω(F)−U(x) = x^γℓU(x).

2.2. Equivalent conditions in terms ofF

Until now, only necessary and sufficient conditions onU have been given in such a way that F∈D(G_γ). Nevertheless, it is not always easy to compute the tail quantile function of a cdfF.

So, it could be preferable to express the relation between(C_γ)to the underlying distributionF.

The link between the tail ofFand its tail quantile functionUdepends on the concept of thede Bruyn conjugate(see Proposition 2.5 in Section 2.9.3 inBeirlant et al.(2004b)).

Proposition 2.7. Ifℓ∈R₀, then there existsℓ^∗∈R₀, the de Bruyn conjugate ofℓ, such that

ℓ(x)ℓ^∗(xℓ(x))→1, x↑∞.

Moreover,ℓ^∗is asymptotically unique in the sense that if alsoℓbis slowly varying andℓ(x)ℓ(xℓ(x))b → 1, thenℓ^∗∼ℓ. Furthermore,b (ℓ^∗)^∗∼ℓ.

This yields the full equivalence between the statements

1−F(x) =x^−1/γℓF(x), and U(x) =x^γℓU(x),

where the two slowly varying functionsℓ_F andℓ_Uare linked together via the de Bruyn conjugation.

So, according to Theorem2.6(i) and (iii) we get that

Theorem 2.8. Referring to the notation of Theorem2.6, we have:

(i) Fréchet case:γ>0. F∈D(G_γ)if and only if there exists a slowly varying functionℓF for whichF(x) =¯ x^−1/γℓ_F(x). Moreover, the two slowly varying functionsℓ_Uandℓ_F are linked together via the de Bruyn conjugation.

(ii) Reversed Weibull case: γ<0. F ∈D(G_γ) if and only if there exists a slowly varying functionℓFfor whichF¯ ω(F)−x⁻¹

∼x^1/γℓF(x),x↑∞. Moreover, the two slowly varying functionsℓ_U andℓ_F are linked together via the de Bruyn conjugation.

When the cdfF has a density f, it is possible to derive sufficient conditions in terms of the hazard functionr(x) = f(x)/(1−F(x)). These conditions, which are due tovon Mises(1975), are known as the von Mises conditions. In particular, the calculations involved on checking the attraction condition toG0are often tedious, in this respect, the von Mises criterion can be particularly useful.

Proposition 2.9(von Mises’ Theorem). Sufficient conditions on the density of a distribution for it belongs toD(G_γ)are the following:

(i) Fréchet case:γ>0. Ifω(F) =∞andlim_x↑∞xr(x) =1/γ, then F∈D(G_γ),

(ii) Gumbel case:γ=0. r(x)is ultimately positive in the neighbourhood ofω(F), is differen- tiable there and satisfieslim_x↑ω(F)^dr(x)_dx =0, then F ∈D(G₀)

(iii) Reversed Weibull case:γ <0.ω(F)<∞andlim_x↑ω(F)(ω(F)−x)r(x) =1/γ, then F∈ D(G_γ).

Some examples of distributions which belong to the Fréchet, the Reversed Weibull and the Gumbel domain are given in respectively Table1, Table2and Table3. For more details about the norming constantsanandbn, seeEmbrechts et al.(2003). We also recall that the choice of these constants is not unique, for example we can chooseα_ninstead ofa_nif lim_n→∞a_n/α_n=1 (see the beginning of the Section2.1).

TABLE1.A list of distributions in the Fréchet domain

Distribution 1−F(x) Extreme value

index

Pareto ∼Kx^−α,K,α>0 _α¹

F(m,n)

R∞ x

Γ(^m+n₂ )

Γ(^m₂)Γ(ⁿ₂)ω^m2−1 1+^m_nω₋^m+n₂ dω x>0;m,n>0

2 n

Fréchet 1−exp(−x^−α)

x>0;α>0

1 α

Tn

R∞ x

2Γ(ⁿ⁺¹₂ )

√nπΓ(ⁿ₂)

1+^ω_n²₋ⁿ⁺¹₂ dω x>0;m,n>0

1 n

TABLE2.A list of distributions in the Reversed Weibull domain Distribution 1−F ω(F)−¹x

Extreme value

index Uniform

1 x

x>1 −1

Beta(p,q)

R1 1−¹x

Γ(p+q)

Γ(p)Γ(q)u^p−1(1−u)^q−1du

x>1; p,q>0 −¹_q Reversed Weibull 1−exp(−x^−α)

x>0;α>0 −_α¹

Finally, we give an alternative condition for(C_γ)(Proposition 2.1 in Section 2.6 inBeirlant et al.(2004b)). It constitutes the basis for numerous statistical techniques to be discussed in Section3.

TABLE3.A list of distributions in the Gumbel domain

Distribution 1−F(x)

Weibull exp(−λx^τ), x>0;λ,τ>0 Exponential exp(−λx), x>0;λ>0

Gamma _Γ(m)^{λm R}_x^∞u^m−1exp(−λu)du, x>0;α,m>0

Logistic 1/(1+exp(x)), x∈R

Normal ^R_x^∞√¹

2πσ²exp

−^(x_2σ^−µ)2²

, x∈R;σ>0,µ∈R

Log-normal ^R_x^∞√ ¹ 2πσ²uexp

−_2σ¹2(logu−µ)²

du, x>0; µ∈R,σ>0

Proposition 2.10. The distribution F belongs toD(G_γ)if and only if for some auxiliary function b and1+γv>0

1−F(y+b(y)v)

1−F(y) →(1+γv)^−1/γ, (C^∗

γ) as y→ω(F). Then

b(y+vb(y))

b(y) →1+γv.

Condition(C∗

γ)has an interesting probabilistic interpretation. Indeed,(C∗

γ)reformulates as lim

x↑ω(F)

P

X−v

b(v) >xX >v

= (1+γv)^−1/γ.

Hence, the condition(C_γ^∗) gives a distributional approximation for the scaled excesses over the high thresholdv, and the appropriate scaling factor isb(v). This motivates the following definitions.

LetXbe a random variable with cdfFand right endpointω(F). For a fixedu<ω(F), Fu(x) =P(X−u≤x|X>u), x≥0 (5) is theexcesscdf of the random variableXover the thresholdu. The function

e(u) =E(X−u|X>u)

is called themean excessfunction ofX. The functioneuniquely determinesF. Indeed, whenever Fis continuous, we have

1−F(x) =e(0) e(x)exp

− Z x

0

1 e(u)du

, x>0.

Define the cdfHγ by

H_γ(x) =

(1−(1+γx)^−1/γ, ifγ6=0, 1−e^−x, ifγ=0,

where x≥0 ifγ ≥0 and 0≤x≤ −1/γ if γ<0. H_γ is called a standard generalised Pareto distribution (GPD). In order to take into account a scale factorσ, we will denote

H_γ,σ(x) =

(1−(1+γ(x/σ))^−1/γ, ifγ6=0,

1−e^−x/σ, ifγ=0, (6)

which is defined forx∈IR⁺ ifγ ≥0 andx∈[0,−σ/γ[ifγ<0. Then, condition(C∗

γ)above suggests a GPD as appropriate approximation of the excess cdfF_ufor largeu. This result is often formulated as follows inPickands(1975): for some functionσ to be estimated from the data

F¯u(x)≈H_γ,σ(u)(x).

2.3. Extremes of Stationary Time Series

Beforehand, we restricted ourselves to iid random variables. However, in reality extremal events often tend to occur in clusters caused by local dependence. This requires a modification of standard methods for analysing extremes. We say that the sequence of random variable(X_i)isstrictly stationaryif for any integerh≥0 andn≥1, the distribution of the random vector(X_h+1, . . . ,Xh+n) does not depend on h. We seek the limiting distribution of (Mn−bn)/an for some choice of normalizing constantsa_n>0 andb_n. However, the limit distribution needs not to be the same as for the maximumMenof the associated independent sequence(Xei)_1≤i≤nwith the same marginal distribution as(Xi). For instance, starting with an iid sequence(Yi,1≤i≤n+1)of random variables with common cdfH, we define a new sequence of random variables(X_i,1≤i≤n)by Xi=max(Y_i,Y_i+1). We see that the dependence causes large values to occur in pairs. Indeed, the random variablesXiare distributed according to the cdfF=H²; so ifF satisfies the equivalent conditions in Proposition2.1, we conclude thatnH(u_n)→τ/2. Consequently, the maximum M_n=X_n,nsatisfies

nlim→∞P(Mn≤un) =e^−τ/2.

To hope for the existence of a limiting distribution of(Mn−bn)/an, the long-range dependence at extreme levels needs to be suitably restricted. To measure the long-range dependence,Leadbetter (1974) introduced a weak dependence condition known as theD(u_n)condition. Before setting out this condition, let us introduce some notations as inBeirlant et al.(2004b). For a setJof positive integers, letM(J) =max_i∈JX_i(withM(/0) =−∞)). IfI={i₁, . . . ,i_p},J={j₁, . . . ,j_q}, we write thatI≺Jif and only if

1≤i1<···<i_p< j1···< j_q≤n, and the distanced(I,J)betweenIandJis given byd(I,J) = j₁−i_p.

Condition 2.11(D(un)). For any two disjoint subsetsI, J of{1, . . . ,n} such thatI ≺J and d(I,J)≥l_nwe have

P({M(I)≤u_n} ∩ {M(J)≤u_n})−P(M(I)≤u_n)P(M(J)≤u_n)≤α_n,ln andα_n,ln →0 asn→∞for some positive integer sequencel_nsuch thatl_n=o(n).

TheD(u_n) condition says that any two events of the form{M(I)≤un} and {M(J)≤un} become asymptotically independent asnincreases when the index setsI andJare separated by a relatively short distancel_n=o(n). This condition is much weaker than the standard forms of mixing condition (such as strong mixing).

Now, we partition the integers{1, . . . ,n}intokndisjoint blocksIj={(j−1)rn+1, . . . ,jrn}of sizer_n=o(n)withk_n= [n/r_n]and, in casek_nr_n<n, a remainder block,I_kn+1={k_nr_n+1, . . . ,n}.

A crucial point is that the events{Xi>un}are sufficiently rare for the probability of an exceedance occurring near the ends of the blocksI_jto be negligible. Therefore, if we drop out the remainder block and the terminal sub-blocksI^′_j={jrn−ln+1, . . . ,jrn}of sizeln, we can consider only the sub-blockI^∗_j ={(j−1)rn+1, . . . ,jrn−ln}which are approximatively independent. Thus we get

P(M_n≤u_n) =P

kn

\

j=1

{M(I^∗_j)≤u_n}

!

+o(1).

Finally, using conditionD(u_n)withk_nα_n,ln →0, we obtain

P

kn

\

j=1

{M(I^∗_j)≤u_n}

!

−P^kn({M(I₁^∗)≤u_n})

≤k_nαn,ln→0,

asn→∞. Now, we observe that if thresholdsunincrease at a rate such that lim supnF(un)>∞, then

P^kn(M(I₁^∗)≤un)−P^kn(Mrn ≤un)≤kn|P(M(I₁^∗)≤un)−P(Mrn ≤un)|

=knP(M(I₁^∗)≤un<M(I₁^′))

≤knlnP(X₁>un)→0.

So, under theD(u_n), we obtain the appropriate condition

P(M_n≤u_n)−P^kn(M_rn ≤u_n)→0 (7) from which the following fundamental results were derived, seeLeadbetter(1974,1983) Theorem 2.12. Let(X_n)be a stationary sequence for which there exist sequences of constants a_n>0and b_nand a non-degenerate distribution function G such that

P

M_n−b_n an ≤x

→G(x), n→∞.

If D(u_n)holds with u_n=a_nx+b_nfor each x such that G(x)>0, then G is an extreme value distribution function.

Theorem 2.13. If there exist sequences of constants an>0and bnand a non-degenerate distri- bution functionG such thate

P Men−bn

a_n ≤x

!

→G(x),e n→∞,

if D(anx+bn)holds for each x such thatG(x)e >0and ifP[(Mn−bn)/an≤x]converges for some x, then we have

P

Mn−bn

an ≤x

→G(x):=Ge^θ(x), n→∞, for some constantθ∈[0,1].

Theorem2.12shows that the possible limiting distributions for maxima of stationary sequences satisfying the D(u_n) condition are the same as those for maxima of independent sequences.

Nevertheless Theorem2.12does not mean that the relationsMn∈D(G)andMen∈D(G)e hold withGe=G. In fact,Gis often of the formGe^θ for someθ∈[0,1](see for instance the introductory example). This is precisely what Theorem2.13claims.

The constantθ is calledextremal indexand always belongs to the interval[0,1]. For instance, if we consider the max-autoregressive process of order one defined by the recursion

Xi=max{αXi−1,(1−α)Z_i}

where 0≤α <1 and where theZ_i are independent Fréchet random variables. Then it can be proved that (cf.Beirlant et al.(2004b) Section 10.2.1)

P(Mn≤x) =P(X1≤x)[P(Z1≤x/(1−α)]→exp[−(1−α)/x]:=G(x)

Whereas,G(x) =e exp(−1/x), soθ=1−α. This example shows that any number in(0,1]can be an extremal index. The caseθ =0 is pathological, it entails that sample maximaMnof the process are of smaller order than sample maximaMe_n. We refer toLeadbetter et al.(1983) and Denzel and O’Brien(1975) for some examples. Moreover,θ>1 is impossible; this follows from the following argument (seeEmbrechts et al.(2003) Section 8.1.1):

P(Mn≤un) =1−P [n i=1

{Xi>un}

!

≥1−nF(un).

The left-hand side converges toe^{−θ τ} whereas the right-hand side has limit 1−τ, hencee^{−θ τ}≥ 1−τ for allτ>0 which is possible only ifθ≤1. A case in which there is no extremal index is given inO’Brien(1974). In this article, eachXnis uniform over[0,1],X1,X3, . . .being independent andX_2na certain function ofX_2n−1for eachn. Finally, a case whereD(u_n)does not hold but the extremal index exists is given by the following example ofDavis(1982). LetY₁,Y₂, . . ., be iid, and define the sequence

(X₁,X₂,X₃, . . .) = (Y₁,Y₂,Y₂,Y₃,Y₃, . . .)or(Y₁,Y₁,Y₂,Y₂, . . .)

each with probability 1/2. It follows fromDavis(1982) that the sequence (Xn) has extremal index 1/2. HoweverD(u_n)does not hold: for example, ifX₁=X₂thenX_n=X_n+1ifnis odd and Xn6=Xn+1ifnis even. For more details, we refer toLeadbetter(1983).

To sum up, unlessθis equal to one, the limiting distributions for the independent and stationary sequences are not the same. Moreover, ifθ>0 thenGis an extreme value distribution, but with different parameters thanG. Thus ife

G(x) =exp −

1+γx−µ σ

₋1/γ! ,

then we have

G(x) =e exp −

1+γx−µe e σ

₋1/γ! ,

withµ=µe−σe(1−θ^γ)/γandσ=σ θe ^γ(ifγ=0,σ =σeandµ=µe+σlogθ).

Under some regularity assumptions the limiting expected number of exceedances overunin a block containing at least one such exceedance is equal to 1/θ (ifθ >0). In fact, using the notations previously introduced, we obtain (seeBeirlant et al.(2004b) Section 10.2.3)

1 θ =lim

n→∞

rnF(u¯ n) P(Mrn >un) =E

"_r

n

∑

i=1

1(Xi>un)Mrn>un

# .

We can have an insight into this result with the following approach: let us assume thatu_nis a threshold sequence such thatnF(un)→τ andP(Mn≤un)→exp(−θ τ), then from (7) (with k_n=⌊n/r_n⌋) we get

n

r_nP(M_rn >u_n)→θ τ, and conclude that

θ=lim

n→∞

P(Mrn >un) r_nF(u_n) .

Another interpretation of extremal event, due toO’Brien(1987), is that under some assumptions θ represents the limiting probability that an exceedance is followed by a run of observations below the threshold

θ= lim

n→∞P(max{X2,X3, . . . ,Xrn} ≤un|X1>un).

So, both interpretations identifyθ=1 with exceedances occurring singly in the limit, unlike θ<1 which implies that exceedances tend to occur in clusters.

The caseθ=1 can be checked by using the following sufficient conditionD^′(u_n)introduced byLeadbetter(1974), when allied withD(u_n),

Condition 2.14(D^′(u_n)).

klim→∞lim sup

n→∞

n

⌊n/k⌋

∑

j=2

P(X₁>u_n,X_j>u_n) =0.

Notice thatD^′(u_n)implies E

"

∑

1≤i<j≤⌊n/k⌋

1(X_i>u_n,X_j>u_n)

#

≤ ⌊n/k⌋

⌊n/k⌋

∑

j=2

E[1(X₁>u_n,X_j>u_n)→0], so that, in the mean, joint exceedances ofunby pairs(Xi,Xj)become unlikely for largen.

Verifying the conditions D(un) and D^′(un) is, in general, tedious, except in the case of a Gaussian stationary sequence. Indeed, letr(n) =cov(X0,Xn) be the auto-covariance function, then the so called Berman’s conditionr(n)logn→∞allied with lim sup_n→∞nΦ(u_n)<∞, where Φis the normal distribution, are sufficient to imply the both conditionsD(u_n)andD^′(u_n)(see Leadbetter et al.(1983)). Let recall that the normal distributionΦis in the Gumbel maximum domain of attraction.

3. The Statistical point of view of Extreme Values Theory

As mentioned in the Introduction, the cdfF is unknown and difficult to estimate beyond observed data, so we need to extrapolate outside the range of the available observations. In this Section, using the properties developed in Section2, we will introduce and discuss different procedures capable of carrying out this extrapolation.

3.1. Extrapolation to the distribution tail

Firstly, we can use the properties of the maximumMngiven in Section2for this extrapolation, as presented in Subsection3.1.1. We can also base our extrapolation to the distribution tail on the excesses or peaks over a threshold as presented in Subsection3.1.2. Both extrapolation procedures are derived from asymptotic procedures that correspond to a first order approximation of the distribution tail. Second order conditions as presented in Subsection3.1.3may help to improve this approximation.

3.1.1. Using maxima

Theorem2.3gives the asymptotic distribution of the maximumMn. Then we use the approximation of the distribution ofM_nby the generalized extreme value (GEV) cdf (2) to write

F(x) =P(M_n≤x)^1/n∼G^1/n_γ

x−b_n an

, x→ω(F).

This gives a semi-parametric approximation of the tail of the cdf F. This approximation is illustrated in Figure4for a uniform distribution and different values ofn, using the theoretical values ofan,bnandγ. Let recall that the uniform distribution is in the Reversed Weibull maximum domain of attraction, and that in this caseγ=−1 (cf. Table2).

We can equivalently approximate an extreme quantile by F¯⁻¹(pn) =U(1/pn)∼bn+an

γ (−ln(1−pn)ⁿ)^−γ−1

, pn→0 whenn→∞. In these two approximations, appear three quantitiesa_n,b_n andγ whose theoretical values are only known when the cdfF is known. In practice, these quantities are unknown.a_ncorresponds to a shape parameter,bnto a scale parameter, andγis the extreme value index. These parameters would be estimated in Subsection3.2to produce semi-parametric estimations of the distribution tail. In this case, this estimation would be performed using a block maxima sample.

3.1.2. Using Peaks Over a Threshold: The POT method

Modelling block maxima is a wasteful approach to extreme value analysis if other data on extremes are available. A natural alternative is to regard observations that exceed some high thresholdu, smaller than the right endpointω(F)ofF as extreme events.

Excesses occur conditioned on the event that an observation is larger than a thresholdu. They are denoted by(Y₁, . . .)and represented in Figure5. The excess cdfF_udefined in (5) expresses

FIGURE4. ComparingF(x)¯ (solid line) and1−G^1/n₋₁((x−bn)/an)with an=n⁻¹and bn=1for n=50(dashed line) and n=100(dotted line) for a uniform distribution F.

also as

Fu(y) =P(X≤u+y|X≥u) =1−F(u+y)

F(u) , y>0.

Pickands’ Theorem (Pickands (1975)) implies thatF_u can be approximated by a generalized Pareto distribution (GPD) function given by (6). Parameterγ is the extreme value index, and σ =an+γ(u−bn). In Section 3.1.1, approximating the distribution of the maximum by an EVD leads to semi-parametric estimations of the tail of the cdf F and an extreme quantile.

Equivalently, approximating the distribution of the excesses over a thresholdumay lead to the following semi-parametric approximations. For the tail of the cdfF, we have the semi-parametric approximationF(x)∼1−F(u)H_γ,σ(x−u),x→ω(F). And for an extreme quantile, we obtain the semi-parametric approximation

F¯⁻¹(pn)∼u+σ γ

"

p F(u)

₋γ

−1

#

, pn→0 whenn→∞.

Again, we have three unknown parameters γ, σ and u to be estimated (see Subsection 3.2).

Note that in practice, u<Mn corresponds to a quantile inside the sample range that can be easily estimated by an observation (a quantile of the empirical distribution function). In practice, we choose ub=Xn−k+1,n, where k is the number of excesses. However, this does not avoid the estimation of a parameter sincek has to be accurately chosen. This choice is detailed in Subsection3.3.1.

FIGURE5. Excesses(Y₁, . . .)over a threshold u.

3.1.3. Second order conditions

The first order condition (C_γ), or equivalently (C_γ^∗), relies to the convergence in distribution of the maximum M_n. We are now interested in the convergence rate for the distribution of the maximumM_n to the extreme value distribution. It corresponds to derive a remainder (see for examplede Haan and Ferreira(2006) Section 2.3 orBeirlant et al.(2004a) Section 3.3) of the limit expressed by the first order condition (C_γ).

The functionU (or the corresponding probability distribution) is said to satisfy thesecond order conditionif for some positive functionaand some positive or negative functionAwith lim_t→∞A(t) =0

tlim→∞

U(tx)−U(t) a(t) −^xγ_γ⁻¹

A(t) =Ψ(x), x>0, (8)

whereΨis some function that is not a multiple of the function(x^γ−1)/γ. FunctionsaandAare sometimes referred to as respectively first order and second order auxiliary functions. However, note that forAidentically one, we obtain the first order condition (C_γ) withΨidentically zero. The second order condition has been used to prove the asymptotic normality of different estimators and to define some of the estimators detailed in the following Section.

The following result (seede Haan and Ferreira(2006) Section 2.3) gives more insights on the functionsa,AandΨ.

Theorem 3.1. Suppose that the second order condition (8) holds. Then there exists constants c₁,

c2∈IR and some parameterρ≤0such that Ψ(x) =c1

Z x 1

s^γ−1 Z s

1

u^ρ−1duds+c2

Z x 1

s^γ+ρ−1. (9)

Moreover, for x>0,

tlim→∞ a(tx)

a(t) −x^γ

A(t) =c₁x^γx^γ−1

γ (10)

and

tlim→∞

A(tx)

A(t) =x^ρ. (11)

Equation (11) means that functionAis regularly varying with indexρ, while equation (10) gives a link between functionsaandA. Forρ6=0, the limiting functionΨcan be expressed as

Ψ(X) = c₁ ρ

x^γ+ρ−1

γ+ρ −x^γ−1 γ

+c₂x^γ+ρ−1 γ+ρ .

Ifρ=0 andγ6=0,Ψcan be written as Ψ(X) = c1

γ

x^γlog(x)−x^γ−1 γ

+c2

x^γ−1 γ . Finally, forρ=0 andγ=0,Ψcan be written as

Ψ(X) =c1

2 (log(x))²+c2log(x).

There are several equivalent expressions for these quantities that can be founde.g.inde Haan and Ferreira(2006) Section 2.3 orBeirlant et al.(2004a) Section 3.3.

3.2. Estimation

We present the estimation procedure both for the block maxima and peak over threshold methods.

Thus, in order to be general, we express the estimates from the original sample (X1, . . . ,Xn). We detail different estimates including maximum likelihood, moment, Pickands, Hill, regression and Bayesian estimates. In all cases, we focus on estimating the extreme value indexγ. Other parameters can be deduced and are not detailed.

3.2.1. Maximum likelihood estimates

Maximum likelihood is usually one of the most natural estimates, largely used owing to its good properties and simple computation. However, in the case of extreme estimates, the support of the EVD (or the GPD) depends on the unknown parameter values. Then, as detailed bySmith(1985), the usual regularity conditions underlying the asymptotic properties of maximum likelihood estimators are not satisfied. In caseγ>−1/2, the usual properties of consistency, asymptotic efficiency and asymptotic normality hold. But there is no analytic expression for the maximum

likelihood estimates. Then, maximization of the log-likelihood may be performed by standard numerical optimization algorithms, seee.g.Prescott and Walden(1980,1983),Hosking(2013) or Macleod(1989). An iterative formula is also available and presented inCastillo et al.(2004).

Moreover, remark that standard convergence properties are valuable for estimating using a sample issued from an EVD (or a GPD). Nevertheless, Fisher-Tippet-Gnedenko Theorem2.3 (or Pickands’ Theorem inPickands(1975)) only guarantees that the maximumMn(or the peaks over threshold) is approximately EVD (or GPD). Their accuracy in the context of extremes is more difficult to assess. However, asymptotic normality has been first proved forγ >−1/2, seee.g.de Haan and Ferreira(2006) Section 3.4. More recently,Zhou(2009,2010) proves the asymptotic normality forγ>−1 and the non-consistency forγ<−1. Confidence intervals follow immediately from this approximate normality of the estimator. But these properties are limited to the rangeγ>−1 concerning the quantity to be estimated. In practice, the potential range of value of the parameter is unknown and thus the accuracy of the estimation cannot be assessed. Then, alternative estimates have been proposed.

3.2.2. Moment and probability weighted moment estimates

The probability weighted moments of a random variableX with cdfF, introduced byGreenwood et al.(1979), are the quantitiesM_p,r,s=E(X^pF^r(X)(1−F(X))^s), for realp,rands. The standard moments are obtained forr=s=0. Moments and probability weighted moments do not exist for γ≥1. Forγ<1, we obtain for the EVD, setting p=1 ands=0,

M1,r,0= 1 r+1

b−a

γ[1−(r+1)^γΓ(1−γ)]

,

and for the GPD, settingp=1 andr=0,

M1,0,s= σ

(s+1)(s+1−γ).

By estimating these moments from a sample of block maxima or excesses over a threshold, we obtain estimates of the parametersa,b,σ,γ. Note that for block maxima and EVD, there is no analytic expression for the estimate ofγthat has to be computed numerically. Conversely, for peaks over threshold and GPD, we have the following analytic expression given inHosking and Wallis(1987)

b

γ_{PW M}(k) =2− Mb_1,0,0

Mb_1,0,0−2Mb_1,0,1 withMb_1,0,s=1 k

k

∑

i=1

1− i

k+1 s

Y_i,n.

Its conceptual simplicity, its easy implementation and its good performance for small samples make this approach still very popular. However, this does not apply for strong heavy tails and in this case again, the range limitationγ<1 concerns the quantity to be estimated. Moreover, the asymptotic normality is only valid forγ∈]−1,1/2[, seeHosking and Wallis(1987) orde Haan and Ferreira(2006) Section 3.6.

To overcome these drawbacks, generalized probability weighted moment estimates have been proposed byDiebolt et al.(2007) for the parameters of the GDP distribution that exist forγ<2