Armelle Guillou

Jean Diebolt

Pierre Ribereau

Diebolt, Jean

Guillou, Armelle

Ribereau, Pierre

Numérisation de Documents Anciens Mathématiques

1/dpernic,imate thee establish-delà d’unlissons leursdent des esti-C. R. Acad. Sci. Paris, Ser. I 341 (2005) 307–312http://france.elsevier.com/direct/CRASSStatisticsAsymptotic normality of the extreme quantile estimator baseon the POT methodJean Diebolta, Armelle Guilloub, Pierre Ribereauba CNRS, université de Marne-la-Vallée, équipe d’analyse et de mathématiques appliquées, 5, boulevard Descartes, bâtiment CoChamps-sur-Marne, 77454 Marne-la-Vallée cedex 2, Franceb Université Paris VI, laboratoire de statistique théorique et appliquée, boîte 158, 175, rue du Chevaleret, 75013 Paris, FranceReceived 6 February 2005; accepted after revision 29 June 2005Available online 30 August 2005Presented by Paul DeheuvelsAbstractThe POT (Peaks-Over-Threshold) approach consists in using the generalized Pareto distribution (GPD) to approxdistribution of excesses over thresholds. In this Note, we propose extreme quantile estimators based on this method. Wtheir asymptotic normality under suitable general assumptions.To cite this article: J. Diebolt et al., C. R. Acad. Sci. Paris, Ser. I341 (2005). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméNormalité asymptotique de l’estimateur d’un quantile extrême basé sur la méthode POT. La méthode POT (pics audelà d’un seuil) consiste à utiliser une distribution de Pareto généralisée (GPD) pour approximer la loi des excès au-seuil. Dans cette Note, nous proposons des estimateurs de quantiles extrêmes basés sur cette méthode. Nous étabnormalités asymptotiques sous des hypothèses générales.Pour citer cet article : J. Diebolt et al., C. R. Acad. Sci. Paris, Ser. I341 (2005). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.Version française abrégéeLe principe de la méthode POT est d’estimer la distribution des excès au-delà d’un seuilu par une loi GPDdépendant de deux paramètres(γ, σ ) après estimation de ces derniers à partir de la loi des excès au-delàu.En utilisant cette approche, nous pouvons proposer des estimateurs de quantiles extrêmes, qui dépendeE-mail addresses:diebolt@math.univ-mlv.fr (J. Diebolt), guillou@ccr.jussieu.fr (A. Guillou).1631-073X/$ – see front matter 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crma.2005.06.032308 J. Diebolt et al. / C. R. Acad. Sci. Paris, Ser. I 341 (2005) 307–312généralesé sur unre résul-empleomematethe data.ucial pa-ko [6],excessesmateurs de(γ, σ ). Nous établissons, sous des hypothèses convenables, en particulier une condition trèsdu second ordre (Condition 2.1 ci-dessous), la normalité asymptotique de ces estimateurs quantiles baseuil aléatoire. Notre démarche s’appuie sur un formalisme développé par Drees [4]. Nous illustrons nottat dans le cas où le couple(γ, σ ) est estimé par la méthode du maximum de vraisemblance. Un autre exd’application, celui des moments pondérés généralisés, est donné dans Diebolt et al. [3].1. IntroductionLet X1, . . . ,Xn be a sample ofn independent and identically distributed (i.i.d.) random variables from scontinuous distribution functionF . Now the question is how to obtain with such a limited sample a good estifor a quantileF−1(1− p) = inf{y: F(y)  1− p},wherep is small such that the quantile to be estimated is situated on the border of or beyond the range ofEstimating such high quantiles is directly linked to the accurate modelling of the tail of the distributionF(x) := P(X > x)for large thresholdsx.From extreme value theory, the behaviour of such extreme quantiles is known to be governed by one crrameter (γ ) of the underlying distribution, called the extreme value index (EVI). Indeed, as shown by Gnedenfor someγ ∈ R, there exists sequences of constants(αn) ∈ R and(σn) ∈ R+ such thatP(Xn,n − αnσn x)d−→ Hγ (x) :={exp(−(1+ γ x)−1/γ ) for γ = 0,exp(−exp(−x)) for γ = 0. (1)In such a case, we say thatF is in the maximum domain of attraction ofHγ , denoted byF ∈ MDA(Hγ ).Since this approach is only based on a set of maxima, another method consists in considering theY1, . . . , YNn over a high thresholdu. HereNn is the number of excesses andYj = Xij − u > 0 where theij ’sare thei, 1 i  n, such thatXi > u. The limiting distribution of these excesses over the thresholdu is then thegeneralized Pareto distribution (GPD) (see Pickands, [7]). This approach is called the GPD approach.More precisely, denote byτF ∈ (0,∞] the right endpoint ofF and define the excess distribution function byFu(x) := P(X − u  x | X > u) for 0< u < τF and 0< x < τF − u.ThenF ∈ MDA(Hγ ) iff ∃σ(u) > 0: limu→τFsup0<x<τF −u∣∣Fu(x) − Gγ,σ(u)(x)∣∣ = 0,whereGγ,σ (x) := 1− Gγ,σ (x) =1−(1+ γ xσ)−1/γfor γ = 0 and 1+ γ xσ> 0,1− exp(− xσ)for γ = 0,denotes the distribution function of the GPD (γ,σ ).Therefore, for large thresholdsu one expects that the excess distributionFu is well approximated by a GPDwith shape parameterγ equal to the EVI ofF . Schematically, the POT method then works as follows:• Given a sampleX1, . . . ,Xn, select a high thresholdu. Let Nu be the number of observationsXi1, . . . ,XiNuexceedingu and denote the excessesYj = Xij − u  0.J. Diebolt et al. / C. R. Acad. Sci. Paris, Ser. I 341 (2005) 307–312 309s.POTisults.lg• Fit a GPDGγ,σ to the excessesY1, . . . , YNu to obtain estimateŝγ andσ̂ for the shape and scale parameter• As F(x) = F(u) · Fu(x − u) for x > u, one can estimate the tail ofF bŷF(x) = Nun(1+ γ̂ x − uσ̂)−1/γ̂for x > u. (2)• Estimates for quantilesxp = F−1(1− p) > u can be obtained by inverting (2):x̂p = u + σ̂ (Nu/np)γ̂ − 1γ̂.In practice we fixu at the(k+1) largest observationXn−k,n. A GPD is then fitted to thek excesses(Xn−k+1,n −Xn−k,n), . . . , (Xn,n −Xn−k,n). We denote the resulting parameter estimates byγ̂ POTk andσ̂POTk . The correspondingPOT quantile estimator is thenx̂POTp,k := Xn−k,n + σ̂POTk(k/(np))γ̂POTk − 1γ̂ POTkfor p <kn. (3)The remainder of this Note is the following. In Section 2, we establish the asymptotic normality of thisestimator under general assumptions. The example where(γ̂ POTk , σ̂POTk ) are the maximum likelihood estimatorsgiven in Section 3.2. Asymptotic normality for the POT quantile estimatorIn order to state our main result, we need some formalism, which is essentially based on Drees’ [4] resIf the estimatorγ̂ is based on the(kn + 1) largest observations, then it can be rewritten asγ̂ = Tn(Qn) whereQn,kn(t) := Qn(t) := F−1n(1− knnt)= Xn−[knt],n, t ∈ [0,1],is the empirical tail quantile function,Tn a smooth functional defined on a suitable space andFn the classicaempirical distribution function of the original sampleX1, . . . ,Xn.In order to ensure consistency ofTn(Qn), (kn)n∈N will be an intermediate sequence, i.e.kn → ∞ andkn/n → 0.However, to obtain a non degenerate limiting distribution, one has to impose stronger restrictions forkn that dependon the second-order behaviour of the underlying distribution functionF . The following expansion of the underlyinquantile function, extensively studied by de Haan and Stadtmüller [2], is the most general one.Condition 2.1. There exist measurable, locally bounded functionsa, Φ : (0,1) → (0,∞) andΨ : (0,∞) → R suchthatF−1(1− tx) − F−1(1− t)a(t)= x−γ − 1γ+ Φ(t)Ψ (x) + R(t, x)for all t ∈ (0,1) andx > 0. (By convention,(x−γ − 1)/γ := − logx if γ = 0.) Here(i) Ψ ≡ 0 andR(t, x) = o(1) as t ↘ 0, for all x > 0, or(ii) x 	→ γΨ (x)/(x−γ − 1) is not constant,Φ(t) = o(1) andR(t, x) = o(Φ(t)) as t ↘ 0, for all x > 0.Remark 1. Condition 2.1(i) is equivalent to our basic assumptionF ∈ MDA(Hγ ) given in (1). In Condition 2.1(ii),Φ is regularly varying at zero with order−ρ for ρ  0, that islimΦ(tx) = x−ρt→0 Φ(t)310 J. Diebolt et al. / C. R. Acad. Sci. Paris, Ser. I 341 (2005) 307–312etheefor all x > 0. Moreover, according to de Haan and Stadtmüller ([2], Remark 3(ii)), one hasΨ (x) = cΨ ×G−1γ+ρ,1(x) if ρ < 0,−x−γ logx/γ if γ = −ρ = 0,log2(x) if γ = −ρ = 0,(4)for some constantcΨ = 0 if the normalizing functiona and the functionΦ are suitably chosen. This will bassumed in all the sequel and imply a weighted uniform convergence result for the remainder termR(t, x) in aneighbourhood of 0 (see Lemma 2.1 in Drees, [4]).Now, in order to give our main result, we need some classical assumptions.Suppose thatF is three times differentiable. We denotef (t) := et (1−γ )U ′(et ),whereU is the tail quantile function defined asU(t) = F−1(1− 1t).Let V andM be two functions defined asV (t) = F−1(e−t ) and M(t) = V′′(ln t)V ′(ln t)− γ.We assume the first and second-order following conditionslimt→+∞M(t) = 0, (5)M has a constant sign at infinityand there existsρ  0 such thatM is regularly varying at infinity with orderρ. (6)We also suppose thatk = kn → ∞, kn→ 0, knp↗ ∞ such thatlog(k/(np))√k→ 0, (7)andlimn→∞(logknp)supvlog(n/k)∣∣∣∣f ′′(v)f ′(v) − β∣∣∣∣ = 0, whereβ  0. (8)Remark 2. The quotientf ′′/f ′ is defined to be zero whenf ′ is zero. For many usual distributions, such asUniform, Weibull, Fréchet or standard Normal distributions, the condition (8) is satisfied withβ = 0. However, inthe case of the Cauchy distribution,β = −2.We are now able to state our main result.Theorem 2.2. Suppose thatF is three times differentiable and that assumptions(5)–(8) and those ensuring thasymptotic normality of√kγ̂ POTk − γσ̂POTka(k/n)− 1Xn−k,n − F−1(1− k/n) =G(1)kG(2)kG(3)k −→d N(B,Γ ) (9)a(k/n)J. Diebolt et al. / C. R. Acad. Sci. Paris, Ser. I 341 (2005) 307–312 311ponentsfiedsmp-are satisfied. Then, under the additional assumption√kΦ(kn)−→ λ ∈ [0,∞), (10)we have• under Condition2.1(ii), if γ < 0:√k(x̂POTp,k − xp)σ̂POTk∫ k/np1 sγ̂ POTk −1 logs ds= G(1)k − γG(2)k + γ 2G(3)k + cΨγ 2γ + ρ λ1{ρ<0} + oP(1), (11)• under Condition2.1(ii), if γ  0, by choosingΦ(t) = M(1/t)c̃Ψ·1ρif ρ < 0,1 if γ = −ρ = 0,12 if γ = −ρ = 0,(12)√k(x̂POTp,k − xp)σ̂POTk∫ k/np1 sγ̂ POTk −1 logs ds= G(1)k − c̃Ψ λ1{γ0,β=0} ·{ρ if ρ < 0,1 if γ = −ρ = 02 if γ = −ρ = 0,+ oP(1), (13)wherec̃Ψ = ±cΨ in order to ensure thatΦ takes its values in(0,∞).In these two cases, the limiting distribution is a normal distribution whose variance depends on the comof Γ .Proof of Theorem 2.2. We rewrite the differencêxPOTp,k − xp as follows:x̂POTp,k − xp =({(k/(np))γ̂POTk − 1γ̂ POTk− (k/(np))γ − 1γ}σ̂POTk)+((k/(np))γ − 1γ{σ̂POTk − a(kn)})+(a(kn)G(3)k√k)+(U(nk)− U(1p)+ a(kn)(k/(np))γ − 1γ).The three first terms in brackets can be treated as in de Haan and Rootzén [1]. For the last term, ifγ < 0 we have touse Lemma 2.1 in Drees [4], while ifγ  0, we can prove, by a Taylor expansion, that Condition 2.1(ii) is satisfor a(t) := 1tU ′(1t)[1+ cM(1t)] with c a suitable constant andΦ(·) satisfying (12). Then, following again the lineof proof of de Haan and Rootzén [1], Theorem 2.2 follows. See Diebolt et al. [3] for more details.3. ExampleOur key point is Corollary 2.1 in Drees [4]. According to this result, if we defineDn :=F−1(1− knn)− a(knn){1{γ =0}γ+ cΨγ + ρ Φ(knn)· 1{γ =−ρ>0}}if γ  −12,Xn,n if γ < −12,then, under Condition 2.1(ii) witha, Φ and Ψ satisfying Lemma 2.1 in Drees [4] and the additional assution (10), we have√kn(Qn − Dna(kn/n)− G−1γ,1 −1γ1{γ =0})−→ Vγ + λ(Ψ + cΨγ + ρ 1{γ =−ρ>0})(14)weakly in a suitable space, whereVγ := t−(γ+1)W(t), t ∈ [0,1], with W a standard Brownian motion.312 J. Diebolt et al. / C. R. Acad. Sci. Paris, Ser. I 341 (2005) 307–312like-esem 2.2, foromparepproach,(2004)Now, we use Theorem 2.1 in Drees et al. [5] which gives the asymptotic normality of the maximumlihood estimators under Condition 2.1(ii) forγ > −1/2 and the additional assumption (10). Combining thresults with (14), we deduce that the convergence (9) holds. Hence, under the assumptions of our Theoreγ > −1/2, (11) and (13) are satisfied and the limiting distribution is aN(λC,σ 2) distribution, whereσ 2 = (1+ γ )21{γ0} + (1+ 4γ + 5γ 2 + 2γ 3 + 4γ 4)1{γ<0},andC =[cΨρ(γ + 1)(1− ρ)(1+ γ − ρ) − c̃Ψ ρ1{β=0}]1{γ0,ρ<0} + [cΨ − c̃Ψ 1{β=0}]1{γ>0,ρ=0}+ [2cΨ − 2c̃Ψ 1{β=0}]1{γ=0,ρ=0} +[cΨ ρ2 1+ 3γ + 2γ 2(1− ρ)(γ + ρ)(1+ γ − ρ)]1{γ<0,ρ<0}.Another example of application has been proposed in Diebolt et al. [3]. It concerns the case where(γ̂ POTk , σ̂POTk ) arethe generalized probability-weighted moments estimators. Also a simulation study is provided in order to cthese two POT quantile estimators to classical estimators: the Weissman and the moment ones.References[1] L. de Haan, H. Rootzén, On the estimation of high quantiles, J. Statist. Plann. Inference 35 (1993) 1–13.[2] L. de Haan, U. Stadtmüller, Generalized regular variation of second order, J. Austral. Math. Soc. Ser. A 61 (1996) 381–395.[3] J. Diebolt, A. Guillou, P. Ribereau, Asymptotic normality of extreme quantile estimators based on the peaks-over-threshold asubmitted for publication, 2005.[4] H. Drees, On smooth statistical tail functionals, Scand. J. Statist. 25 (1) (1998) 187–210.[5] H. Drees, A. Ferreira, L. de Haan, On maximum likelihood estimation of the extreme value index, Ann. Appl. Probab. 14 (3)1179–1201.[6] B.V. Gnedenko, Sur la distribution limite du terme maximum d’une série aléatoire, Ann. Math. 44 (1943) 423–453.[7] J. Pickands III, Statistical inference using extreme order statistics, Ann. Statist. 3 (1975) 119–131.

Asymptotic normality of the extreme quantile estimator based on the POT method

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Asymptotic normality of the extreme quantile estimator based on the POT method

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