Semiparametric Lower Bounds for Tail Index Estimation

Transcription

1 Semiparametric Lower Bouds for Tail Idex Estimatio Ja Beirlat Christel Bouquiaux Bas J.M. Werker Katholieke Uiversiteit Leuve Uiversité Libre de Bruxelles Tilburg Uiversity August 23, 2004 Abstract We cosider estimatio of the tail idex parameter from i.i.d. observatios i Pareto ad Weibull type models, usig a local ad asymptotic approach. The slowly varyig fuctio describig the o-tail behavior of the distributio is cosidered as a ifiite dimesioal uisace parameter. Without further regularity coditios, we derive a Local Asymptotic Normality LAN result for suitably chose parametric submodels of the full semiparametric model. From this result, we immediately obtai the optimal rate of covergece of tail idex parameter estimators for more specific models previously studied. O top of the optimal rate of covergece, our LAN result also gives the miimal limitig variace of estimators regular for our parametric model through the covolutio theorem. We show that the classical Hill estimator is regular for the submodels itroduced with limitig variace equal to the iduced covolutio theorem boud. We also discuss the Weibull model i this respect. Ruig title: Semiparametric tail idex estimatio Keywords: Extreme value theory, Local Asymptotic Normality, Pareto model, Weibull model AMS-classificatio: 62G32 Departmet Wiskude, Katholieke Uiversiteit Leuve, Celestijelaa 200 B, B-300, Leuve, Belgium. Ja.Beirlat@wis.kuleuve.ac.be Istitut de Statistique, Uiversité Libre de Bruxelles, Campus de la Plaie, CP20, B-050 Bruxelles, Belgium. Bouquiac@ulb.ac.be Fiace ad Ecoometrics group, CetER, Tilburg Uiversity, P.O.Box 9053, 5000 LE, Tilburg, The Netherlads. B.J.M.Werker@TilburgUiversity.l The authors thak Joh Eimahl ad two referees for their pertiet remarks.

2 Itroductio Cosider a i.i.d. sequece of radom variables X,..., X with commo distributio fuctio F. I this paper we assume that F is either of the Pareto type or of the Weibull type. More precisely, F is said to be of the Pareto type if F x = [xlx] /γ, x,. where γ > 0 is called the Pareto tail idex parameter ad l is some slowly varyig fuctio i the eighborhood of ifiity. Similarly, we say that F is of the Weibull type if log [ F x] = [xlx] /τ, x,.2 where τ > 0 is called the Weibull tail idex parameter ad, as before, l is some slowly varyig fuctio i the eighborhood of ifiity. I this paper, we will be iterested i the behavior of the distributios ear ifiity. We therefore, oly require. ad.2 for values of x. We aalyze the Pareto ad Weibull type models from a semiparametric poit of view i which we take the tail idex parameter γ for the Pareto case ad τ for the Weibull model as the parameter of iterest ad l as a fuctioal uisace parameter. A atural approach might be to use the taget space argumets for semiparametric models with i.i.d. observatios as set out i, e.g., Bickel, Klaasse, Ritov, ad Weller 993. However, these results are ot applicable i the model uder study due to the o-smoothess of the parameter of iterest as fuctioal of the uderlyig distributio. The taget space reasoigs are based o pathwise differetiability of the parameter of iterest with respect to the uderlyig distributio agaist the taget spaces. This differetiability, however, does ot hold for the extreme value idex. The preset paper offers the followig cotributios. First, we uify several ow results cocerig the optimal rate of covergece for tail idex estimators otably, the results of Hall ad Welsh 984 ad Drees 998 for the Pareto model. Without imposig further restrictios to. or.2, we costruct alteratives that are locally asymptotically ormal with respect to some fixed distributio which is ot ecessarily the strict Pareto ad that coverge at a arbitrary rate. Subsequetly, we show that the extra smoothess coditios imposed o the distributio i, e.g., Hall ad Welsh 984 or Drees 998, iduce immediately a boud o the rate of covergece ay uiformly cosistet estimator ca achieve. Give a rate of covergece we defie precisely what we mea by this i Sectio 5, oe may woder what is the miimal limitig variace of estimators attaiig this rate, i.e. a Cramer-Rao type boud. We itroduce suitably chose parametric submodels that are Locally Asymptotically Normal LAN. The covolutio theorem see, e.g, Le Cam ad Yag, 990 the gives lower bouds for the asymptotic precisio with which the tail idex parameter ca be estimated whe usig estimators that are regular with respect to these parametric submodels. For the Pareto model, we show that the widely-used Hill estimator has a limitig variace which equals the lower boud obtaied from the covolutio theorem. I these discussios we do ot cosider a possible adaptive choice of the rate of covergece, see, e.g., Hall ad Welsh 985. We also cosider Weibull type distributios. These distributios are much less studied tha the Pareto type distributios. However, the Weibull model offers some properties that 2

3 are very useful i specific applicatios. We agai give a LAN result for suitably chose local alteratives for the slowly varyig uisace fuctio ad show that, uder some coditios, a estimator provided i Beirlat et al. 995 has a limitig variace which equals the lower boud iduced by the covolutio theorem i our parametric submodels. Related work o lower bouds for the speed of covergece ca be foud i the papers of Hall ad Welsh 984 ad Drees 998. Hall ad Welsh 984 establish the optimal rate of covergece for a specific semiparametric model. Drees 998 expads these results to a more geeral class of models ad to other maximal domais of attractio i.e., allowig γ IR. We uify the aforemetioed results for the positive γ case. Other papers usig the LAN paradigm i the case of extreme value idex estimatio are Falk 995, Wei 995, ad Maroh 997. I these papers, a LAN coditio is derived for the largest order statistics. We also fid that iferece ca be based o the largest values observed, sice oly these observatios appear i the cetral sequece of our parametric submodels. Both Wei 995 ad Maroh 997 assume that the upper-tail of the distributio essetially belogs to a parametric family. Drees 200 cosiders the estimatio problem from the related poit of view of covergece of experimets. While that paper is cocered with miimax bouds, we cosider covolutio theorem variace bouds. Compared to miimax results, results based o the covolutio theorem are stroger, but oly apply to estimators that are regular for the model uder cosideratio. Propositio 2. of Drees 200 ca be used to obtai covolutio theorem bouds i the viciity of the strict Pareto distributio. We cosider local alteratives to all distributio fuctios of the semiparametric model of iterest. The setup of the paper is as follows. I Sectio 2, we cosider the Pareto model ad obtai a LAN result for appropriately defied local alteratives. The LAN property yields lower bouds o the speed of covergece ad o the asymptotic dispersio of estimators that are regular with respect to the parametric models itroduced. This is detailed i Sectio 3. Applicatios of the geeral results to more specific Pareto type models are provided i Sectio 4. I Sectio 5, we show that the Hill estimator attais the variace lower boud iduced by the covolutio theorem applied to our parametric submodels. I Sectio 6 ad 7 we prove similar results for the Weibull model. Fially, the appedix gathers some techical proofs. 2 Local Asymptotic Normality of the Pareto Model Cosider a fixed cotiuous distributio fuctio F 0 of the Pareto type. with parameters γ 0 > 0 ad l 0, i.e., F 0 x = [xl 0 x] /γ 0, x. 2. As metioed i the itroductio, i this paper we take a semiparametric poit of view ad are iterested i the estimatio of the Pareto tail idex γ 0, while cosiderig the slowly varyig fuctio l 0 as uisace. I this sectio, we derive a Local Asymptotic Normality LAN result for appropriately defied local alteratives of the distributio fuctio F 0. This allows us ot oly to discuss optimal rates of covergece for semiparametric estimators, but also to discuss estimators i terms of their asymptotic variace. Formal results i this directio are discussed i geeral i Sectio 3 ad i Sectio 4 i particular. 3

4 The LAN coditio describes the asymptotic behavior of the likelihood ratio of local alteratives with respect to F 0. The rate of covergece is defied through a arbitrary positive sequece δ with δ 0 ad δ,. As log as o further assumptios like those discussed i Sectio 4 are made o the set of Pareto-type distributios, the sequece δ is arbitrary. The LAN coditio effectively gives the likelihood ratio for a model that cotais a parameter u IR that is used to localize the parameter of iterest γ 0. More precisely, for every u IR, we defie, for all 0 := mi{m IN : γ 0 + uδ > 0, m}, γ = γ 0 + uδ. 2.2 We also defie local alteratives for the uisace fuctio l 0 as follows { x γ /γ 0 l x = l 0 x γ/γ 0, x t l 0 xδ 2 γ γ 0, 2.3, x > t where := 0 mi{m IN : δ 2 >, m} ad t := U 0 δ 2, as, with U 0 t = F0 /t := if{s IR : F 0 s = /t}. Sice, F 0 is cotiuous, we have F 0 t = δ Remark 2. The alteratives costructed through 2.2 ad 2.3 are itroduced here i a ad hoc way. However, they are specific i the sese that the Hill estimator is regular with respect to these alteratives ad, at the same time, has a limitig variace which equals that of the lower boud iduced by the covolutio theorem for the alteratives. Details are discussed i Sectio 4. Remark 2.2 Drees 200 itroduces alteratives aroud the strict Pareto distributio i.e., fixig l 0 = of the form F t = t γ 0 hδ 2 exp uδ s ds, s where h is a fuctio satisfyig appropriate coditios. It remais a ope questio whether his results with the strict Pareto as ceter of localizatio ca be exteded to more geeral ceters of localizatio as i 2.3. The distributio fuctio correspodig to γ ad l is give by, for, { F0 x, x t F x = [xl x] /γ = [ F 0 x] γ 0/γ [ F 0 t ] γ 0/γ, x > t. 2.5 It is obvious that, for each fixed, F defies a cotiuous distributio fuctio such that F is regularly varyig at ifiity with idex /γ. Furthermore, ote that F is absolutely cotiuous w.r.t. F 0 ad desity df df 0 x = t, x t ] γ0 /γ. 2.6, x > t [ γ 0 F0 x γ F 0 t 4

5 The followig theorem gives the quadratic approximatio of the likelihood ratio of F with respect to F 0 for i.i.d. copies X,..., X of X with cdf F 0. It proves that the alteratives costructed are, without further regularity coditios, LAN ad idetifies the so-called cetral sequece below. Theorem 2. The log-likelihood ratio Λ = Λ X,..., X = log df X i df 0 X i of F with respect to F 0 for i.i.d. copies X,..., X of X with cdf F 0, satisfies where = δ γ 0 Λ = u u 2 2 γ0 2 + o IP, log F 0X i F 0 t Thus, the Fisher iformatio is give by /γ 2 0. The proof of this LAN result relies o a simple lemma. I{X i > t } Lemma 2. Give F 0 that is cotiuous, we have, for all k IN, t log F k 0x df 0 x = k k![ F 0 t ]. F 0 t L N 0, /γ Proof: Usig the trasformatio v = F 0 x/ F 0 t the itegral is reduced to a Gamma itegral. Proof of Theorem 2.: Sice δ 2 [ F 0 t ] =, a applicatio of Chebychev s iequality shows that, uder F 0, δ 2 I{X i > t } = + o IP, ad likewise, usig Lemma 2. with k = ad k = 2, [ ] δ 2 F0 X i log I{X i > t } = + o IP. F 0 t The quadratic approximatio 2.7 ow follows immediately, sice, uder F 0, we have Λ = = uδ γ 0 log γ [ ] 0 γ0 + γ γ = u 2 I{X i > t } + u2 2γ 2 0 u 2 γ0 2 + o IP log F 0X i F 0 t uδ log γ 0 5 I{X i > t } [ ] F0 X i I{X i > t } u2 F 0 t γ0 2 + o IP

6 It remais to show the covergece i distributio of the cetral sequece i 2.8. To this extet, defie, for i, ξ i = δ + log F 0X i I{X i > t } := δ + ai{x i > t }, F 0 t where a 0 whe X i > t. For fixed sufficietly large, the ξ i, i =,...,, are idepedet radom variables. Uder F 0, usig 2.4 ad Lemma 2., we get EI{X i > t } = F 0 t = /δ, E a I{X i > t } = EaI{X i > t } = F 0 t, E a 2 I{X i > t } = 2 F 0 t, E a 3 I{X i > t } = Ea 3 I{X i > t } = 6 F 0 t. Therefore, makig use of + a 3 + a 3, we fid Eξ i = 0, Var ξ i = Eξi 2 = δ [ F 0 t ] =, E ξ i 3 δ [ F 0 t ] = 6 δ. Sice, for, E ξ i 3 Var ξ i 3/2 6δ 0, the Liapuov Cetral Limit Theorem implies This completes the proof. = γ 0 ξ i L N 0, /γ 2 0. The cetral sequece obtaied i Theorem 2., is of the peak-over-threshold POT type. This meas that we oly look at observatios that exceed the determiistic threshold t. We will later be iterested i the behavior of Hill type estimators, where the threshold t is replaced by a appropriate empirical quatile of the observatios. The followig LAN result formalizes this. Let X i: deote the i-th order statistic of X,..., X. Moreover, defie the Hill estimator for a sequece, with ad <, as H k = log X i+:. 2.9 X : Theorem 2.2 Let be a sequece of itegers tedig to ifiity. Cosider the sequece δ = / ad the correspodig cetral sequece as defied i 2.8. The, still assumig δ i.e., / 0, we have, uder F 0, γ0 2 = γ 0 + log F 0X i+: F 0 X : 6 + o IP, 2.0

7 or, equivaletly, γ 2 0 H γ 0 = The proof beig more techical, it is left for the appedix. log l 0X i+: l 0 X : + o IP LAN, optimal rates of covergece, ad the covolutio theorem A LAN coditio as i Theorem 2. or 2.2 allows for the derivatio of bouds o the optimal rate of covergece of reasoable estimators for the tail-idex parameter γ. For various specific models see, e.g., Hall ad Welsh, 984, ad Drees, 998 such optimal rates of covergece are already ow ad Sectio 4 discusses i detail how these ow results ca easily be obtaied i the preset framework. But the LAN coditio allows for more precise lower bouds o the asymptotic behavior of estimators regular i the parametric model tha the rate of covergece aloe. Through the so-called covolutio theorem, oe obtais lower bouds for the asymptotic distributio of these estimators whose rate of covergece is optimal. I particular, this gives a lower boud for the variace of the asymptotic distributio. All geeral cosequeces of the LAN coditio discussed i this sectio are well ow, but repeated for the reader s coveiece. A proof of all results ca be foud i, e.g., Le Cam ad Yag 990 or Bickel et al Optimal rates of covergece follow from the fact that sequeces of probability measures that are LAN, are automatically cotiguous. Lemma 3. If the product measures based o i.i.d. copies of F ad F 0 are LAN as i Theorem 2. ad 2.2, the they are cotiguous. We use cotiguity i this paper i the sese of Theorem 3...b of Le Cam ad Yag 990, i.e. for ay sequece of radom variables r = r X,..., X, we have r = O IP, uder F, if ad oly if r = O IP, uder F 0. Let P deote a arbitrary class of distributios of the Pareto type 2.. More specific examples for the Pareto case will be cosidered i Sectio 4. Fix a distributio F 0 P ad a sequece δ such that δ. The sequece δ provides a upper boud o the rate of covergece of a estimator, provided that the local alteratives F costructed from γ i 2.2 ad l i 2.3 belog to the model P ad provided that we require the estimator to be uiformly cosistet over P. Theorem 3. Suppose that the local alteratives F costructed i 2.2 ad 2.3 are such that F P. Let ˆγ be a estimator of γ for which lim M lim sup sup IP F {α ˆγ γ > M} = 0, 3. F P the α = Oδ

8 Proof: The cosistecy coditio 3. implies i particular that α ˆγ γ = O IP, uder F. By the cotiguity followig from Lemma 3., this implies that α ˆγ γ = O IP uder F 0. Sice we obviously also have from 3. that α ˆγ γ 0 = O IP uder F 0, we obtai immediately α γ γ 0 = O. Usig 2.2, this completes the proof. If the model P is take as all distributio fuctios of the form 2., the, as we have see i Sectio 2, the alteratives F belog to P whatever the sequece δ. Thus, give a possible sequece α, oe ca always fid a sequece δ, covergig to zero very slowly, such that 3.2 does ot hold, i.e., such that lim sup α δ =. This implies that there caot exist a uiformly cosistet estimator of γ i the full Pareto model, o matter how weak the cosistecy requiremet i 3., i.e., o matter how slowly α coverges to ifiity. Eve if P is take as a subset of the full semiparametric model cosistig of all distributio fuctios of the form 2., uiformly cosistet estimatio is ot possible if the iterior of P with respect to the variatioal distace is ot empty. This follows alog the same lies as the proof of Theorem 3. upo otig that the variatioal distace betwee F ad F 0 is bouded by 2[ F 0 t ] ad, hece, coverges to zero. The same result ca easily be obtaied by direct methods, but it is also a immediate cosequece of our geeral LAN result. Cocludig, if meaigful optimal rates of covergece are to be foud, oe must restrict the model by imposig extra regularity o the slowly varyig fuctio l i 2.. This will be cosidered for previously studied models i Sectio 4. Aother importat cosequece of the LAN property is the so-called covolutio theorem see, e.g., Le Cam ad Yag 990, page 85. This theorem gives a lower boud for the asymptotic variace of regular estimators, give a fixed rate of covergece α = δ. Theorem 3.2 Suppose that the product measures based o i.i.d. copies of F ad F 0 are LAN as i Theorem 2. ad 2.2. Suppose, moreover, that ˆγ is a regular estimator for γ i the sese, for, δ ˆγ γ 0 δ ˆγ γ δ ˆγ γ 0 γ 2 0 L U, uder F 0, ad 3.3 L U, uder F, where U deotes a arbitrary radom variable. The, we have, uder F 0, γ0 2 L V Z, 3.4 where V N0, γ 2 0 ad Z are idepedetly distributed. Uder F, the same covergeec of the sequece of vectors i 3.4 holds, but with V Nu, γ 2 0. The covolutio theorem states that, give regularity of the estimator as defied above, the most cocetrated limitig distributio possible for estimatig γ, is a N0, γ0 2 distributio. All regular estimators have a limitig distributio that is the covolutio of this N0, γ0 2 ad some other distributio. If this other distributio is ot degeerated, the limitig distributio is more spread out tha the N0, γ0 2 distributio, i the sese that it gives rise to a larger asymptotic variace. I Sectio 5, we show that the Hill estimator with = δ /2 is, uder some coditios, regular for the alteratives itroduced ad has a limitig variace equal to γ0 2. Sectio 7 shows the aalogous result for a estimator itroduced i Beirlat et al. 995 for the Weibull model. 8

9 4 More specific Pareto type models We illustrate the geeral theory of the previous sectios by reviewig two examples from the literature. I these examples, more specific assumptios are made o the slowly varyig fuctio l. We will cosider i this sectio the models itroduced i Hall ad Welsh 984 ad Drees 998. Example 4. Hall ad Welsh 984 cosider the model described by all desities of the form fx = C x/γ [ + rx], γ > 0, C > γ The model P cosidered i Hall ad Welsh 984 is defied startig from fixed γ 0 > 0, ρ > 0, C 0 > 0, ad ε > 0, as the set of distributio fuctios, satisfyig 4., for which γ γ 0 ε, C C 0 ε, ad sup x ρ/γ rx, 4.2 x is bouded over P. For this model, estimators which are uiformly cosistet i the sese of 3. ca be costructed, provided that α coverges ot too quickly to ifiity i.e., if δ coverges ot too slowly to zero. To be precise, cosider the alteratives F costructed aroud the strict Pareto distributio, i.e., F 0 x = x /γ 0, x, 4.3 for some γ 0 > 0. I the otatio of 4., we have C 0 = ad r 0 x = 0. Oe easily verifies that the alteratives F as costructed i 2.2 ad 2.3 are such that 4. holds with Sice r x = { γ γ0 x/t /γ /γ 0, x t x > t sup x ρ/γ r x = O t ρ/γ x γ γ 0 we fid that sup x x ρ/γ r x remais bouded as if ad oly if t ρ/γ δ = O, i.e., if ad oly if δ = O ρ 2ρ+,. From Theorem 3. we ow obtai that α ˆγ γ = O IP uiformly over the Hall ad Welsh 984 model implies α = O ρ 2ρ+,. 4.5 I this example, we assumed that l 0 x = C 0 =, but it ca easily be exteded to cover the case l 0 x = C 0. Example 4.2 Drees 998 imposes that the slowly varyig fuctio l is ormalized, i.e., for some η : [, IR, x lx = C exp ηz/zdz

10 The model P cosidered i Drees 998 is ow defied as all distributios satisfyig 2. ad 4.6 such that sup ηz /hz 4.7 z is bouded over P for some give cotiuous, positive, ad decreasig fuctio h. As i the Hall ad Welsh 984 model, this model does allow for uiformly cosistet estimators i the sese of 3.. Fix F 0, C 0 > 0, γ 0 > 0, ad η 0 accordig to 4.6. The alteratives F costructed i 2.2 ad 2.3 ow also satisfy 4.6 with C = C 0 ad η z = [ γ0 γ + Sice h is decreasig, we fid sup z η z hz γ ] 0 I{z t } η 0 z + γ γ γ 0 max{, γ 0 η 0 z } sup γ z hz + γ γ 0 I{z t }. 4.8 ht The first term o the right-had side is bouded as. I order that the secod term is bouded as, we eed δ /ht = O,. 4.9 I the special case that hz = z ρ/γ 0 ad η 0 z = 0, the coditio 4.9 traslates to the requiremet that δ /[δ] 2 ρ is bouded, i.e., ρ δ = O 2ρ+ 4.0 The preset example is i fact a variatio of the Drees 998 model. Drees 998 imposes the coditios 4.6 ad 4.7 o the slowly varyig part of the fuctio U as defied i Sectio 2. It is possible to cosider exactly Drees 998 model i our framework i the eighborhood of the strict Pareto distributio. More precisely, cosider U 0 t = F0 /t = t γ 0, t. The fuctio U defied by U t = F /t, t, with F defied i 2.5, for give sequece δ, γ = γ 0 + uδ, ad t = δ 2 γ 0, is the easily see to be give by with t U t = t γ exp η z dz, z η z = { γ0 γ, z δ 2 0, z > δ 2 I this case, the coditio that sup z η z /hz remais bouded as implies that δ hδ 2 = O,. For hz = z ρ we fid the same coditio 4.0. Note that Drees 998 cosiders the o-pareto case, i.e., where the tail-idex γ may be zero or egative. This is a o-trivial extesio that is ot covered by our preset results.. 0

11 5 The Hill estimator Sectio 2 provides a LAN result for suitably chose parametric families of the semiparametric Pareto type model. I the previous sectio, we have see how this result immediately yields the optimal rates of covergece i more specific Pareto type models, like those of Hall ad Welsh 984 ad a model ispired by Drees 998. Furthermore, the LAN result, via the covolutio theorem, gives a lower boud o the asymptotic variace of estimators which are regular for the alteratives itroduced. I this sectio, we show that, give a fixed rate of covergece, ad apart from a well-ow asymptotic bias, the Hill estimator, uder a regularity coditio, attais this lower boud. Thus, throughout this sectio, we fix a sequece of oegative itegers with ad / 0 as ad the correspodig sequece δ = /. Let P deote a arbitrary class of distributios of the Pareto type 2.. Cosider a Pareto type distributio F P. We may decompose the iverse of / F as follows: t := if{s : F s = /t} = t γ Lt, t >, 5. F with L slowly varyig at ifiity. I order to study the asymptotic behavior of the Hill estimator, we have to impose like Smith 982 a secod order coditio which specifies the rate of covergece of Ltx/Lt to. More precisely, let c be some costat ad g : 0, 0, a ρ-varyig fuctio with ρ 0. Cosider the followig asymptotic coditio SR2 x > : Ltx x Lt = + cgt v ρ dv + ogt, as t. 5.2 The SR2-coditio is widely accepted as a appropriate coditio to specify the slowly varyig part of the model. i a semi-parametric way. Uder the SR2-coditio, we have the followig result. Theorem 5. Suppose that F is of the Pareto type. ad satisfies the SR2-coditio with g/ A, 5.3 for some A IR. The, uder the local alteratives defied by γ = γ 0 + uδ ad 2.3, with δ = /, we have ad H L γ 0 NcA/ ρ + u, γ0, 2 H L γ NcA/ ρ, γ0. 2 The limitig behavior of the Hill estimator for u = 0, i.e., uder F 0 i Theorem 5. is wellow see, e.g., Hall, 982, Haeusler ad Teugels, 985, or the more recet papers Csörgő ad Viharos, 998, de Haa ad Resick, 998, ad de Haa ad Peg, 998. However, we describe its behavior uder our local alteratives as well. We provide a proof i the appedix that is effectively based o Theorem 2.2. Note that Theorem 5. is ot at odds

12 with Theorem 2.2 of Drees 998 which cosiders the estimator H with k /. Such a estimator is ot regular at the rate δ = / that we cosider. Observe that, if the SR2-coditio is satisfied, the it is also satisfied by the local alteratives costructed i Sectio 2. More precisely, if the iverse of / F 0 evaluated i t > ca be writte as t γ L 0 t where L 0 satisfies the SR2-coditio, say L 0 tx/l 0 t = + c 0 g 0 t x u ρ 0 du + og 0 t, the the same is true for the alteratives F, i.e. the correspodig slowly varyig fuctio L ca be costructed such that with L tx/l t = + c g t c = c 0 γ γ 0, ρ = ρ 0 γ γ 0, x u ρ du + og t, g t = g 0 t γ /γ 0 δ 2 γ /γ 0. Note that g δ 2 = g 0 δ. 2 The above ca be prove by otig that the iverse of / F is give by see 2.5 { U0 t for t δ 2 U t =, U 0 t γ/γ 0 F 0 t γ/γ0 for t > δ, 2 where, as before, U 0 t = F 0 /t. Thus, for t > δ 2, L t = δ 2 γ 0 γ L 0 t γ/γ 0 δ 2 γ/γ 0 Note, however, that coditio 5.3 is ot ecessarily satisfied by the alteratives F. 6 Local Asymptotic Normality of the Weibull Model The Pareto model, while popular i practice, is ot always the best choice i some applicatios, see, e.g., Keller ad Klüppelberg 99 or Klüppelberg ad Villaseñor 993. See furthermore Chapter 4 i the Beirlat, Teugels, Vyckier 996 moograph. Fix τ 0 > 0 ad a slowly varyig fuctio l 0 ad cosider the distributio F 0 give by log[ F 0 x] = [xl 0 x] /τ 0, x. 6. As for the Pareto type model, we cosider local alteratives based o a arbitrary positive sequece δ with δ 0 ad δ = olog as. For every u IR, we defie the local alteratives F through.2 with τ = τ 0 + uδ, 6.2. l x = { x τ /τ 0 [l 0 x] τ /τ 0, x t l 0 x[logδ 2 ] τ τ 0, x > t, 6.3 2

13 where t is give by log F 0 t = logδ 2. Elemetary calculatios show that F is absolutely cotiuous with respect to F 0, where F ad F 0 coicide for x t ad for x > t we have log df x = log τ 0 τ0 + log log[ F 0x] df 0 τ τ log[ F 0 t ] { [ ] log[ F0 x] τ0 /τ } log F 0 x. 6.4 log[ F 0 t ] To state the LAN result for the Weibull model, we defie the log-likelihood ratio of the i.i.d. variables X,..., X of F with respect to F 0 : Λ = Λ X,..., X = [ ] df log X i. df 0 Theorem 6. The log-likelihood ratio Λ satisfies, uder F 0, Λ = u u 2 2 τ0 2 + o IP, 6.5 where = δ τ 0 L N 0, /τ 2 0. log F 0X i F 0 t I {X i > t } 6.6 The proof of this LAN result is similar to that for the Pareto case. Observe that, from Lemma 2., we obtai t by dividig by log F 0 t k. log[ F0 x] k log[ F 0 t ] F 0 t df 0 x = k! log[ F 0 t ] k, 6.7 Proof of Theorem 6. From 6.7, with k = ad k = 2, we get This implies δ Moreover, combiig the iequality log[ F0 X i ] log[ F 0 t ] I {X i > t } = o IP. τ 0 log log[ F 0X i ] τ log[ F 0 t ] I {X i > t } = o IP. t >, a < 2 : t a + at aa t 2 3

14 with 6.7 for k = 2 ad k = 3 gives { log[ F0 X i ] τ0 /τ } log[ F 0 X i ] I {X i > t } log[ F 0 t ] = log[ F 0 X i ] τ 0 log[ F0 X i ] τ log[ F 0 t ] I {X i > t } + o IP. This last expressio ca be writte as the sum of ad τ 0 log[ F0 X i ] 2 log[ F 0 t ] τ log[ F 0 t ] I {X i > t }, τ 0 log[ F0 X i ] log[ F 0 t ] τ log[ F 0 t ] I {X i > t }, of which the first part vaishes asymptotically i view of 6.7. The above results imply that we may write Note Λ = τ 0 log[ F0 X i ] log[ F 0 t ] τ log[ F 0 t ] I {X i > t } + log τ 0 I {X i > t } + o IP. τ ad, i virtue of 6.7, δ 2 log[ F 0 t ] δ 2 I{X i > t } = + o IP log[ F0 X i ] log[ F 0 t ] I {X i > t } = + o IP, which proves the quadratic expasio for the log-likelihood ratio. Let log[ F0 X i ] ξ i = δ log[ F 0 t ] log[ F 0 t ] I {X i > t }. The limitig distributio of the cetral sequece follows from the Liapuov Cetral Limit Theorem, usig 6.7 to obtai Eξ i = 0, Var ξ i =, E ξ i 3 6 δ /. 4

15 The LAN result of Theorem 6. is based o a cetral sequece of the POT-type, i.e. the cetral sequece cosists oly of those observatios that exceed a give determiistic threshold t. As i the Pareto case, we ca also for the Weibull model provide a cetral sequece based o order statistics. Theorem 6.2 Let be a sequece of itegers tedig to ifiity with = olog. Cosider the sequece δ = / ad the correspodig cetral sequece. The, we may write τ0 2 = τ 0 log F 0X i+: F 0 X : + o IP, ad τ 2 0 ˆτ τ 0 = log/ X i+: X : l0 X i+: l 0 X : + o IP, where the estimator ˆτ is defied by ˆτ = log/ The proof is agai left for the appedix. X i+: X :. 7 Estimatio i the Weibull model Beirlat et al. 995 provide the limitig distributio of the estimator ˆτ = log/ X i+: X :. Our results agai allow us to study the behavior of this estimator uder the local alteratives costructed. We itroduce the followig otatio. Let K 0 deote the geeralized iverse of log F 0. The, we may write K 0 t = t τ 0 L 0 t with L 0 slowly varyig. Theorem 7. Suppose L 0 defied above satisfies SR2. Let be a sequece of itegers tedig to ifiity with = olog ad glog/ A. Now, uder the local alteratives defied by τ = τ 0 + uδ ad 6.3, with δ = /, we fid ad ˆτ L τ 0 N ca + u, /τ0 2, ˆτ L τ N ca, /τ0 2. Theorem 6. ad Theorem 7. impose /δ = olog. This implies that the δ are relatively large ad the alteratives F are far from F 0. We cojecture, but were uable to prove formally, that, e.g., geometric rates of covergece ca ot be obtaied i the Weibull model. This cojecture is based o two cosideratios. First, a small chage i 5

16 the parameter τ i 6. leads to a much larger chage i the distributio F, tha a similar chage i γ i 2.. As a cosequece, iferece about τ i the Weibull model is much more difficult tha iferece about γ i the Pareto model. Formally, for geometric rates δ = α with α > 0, we expect the log-likelihood ratio i 6.5 to coverge to zero. The secod cosideratio regards the estimator discussed i Theorem 7. above. I case = /δ 2 is chose too large, the bias A teds to ifiity. This suggest that there is o covergece i distributio of ˆτ τ 0. A Some proofs This appedix cotais three proofs that were omitted from the mai text i order to improve readability. Proof of Theorem 2.2: Let U :... U : be the order statistics of i.i.d. uiformly over the iterval [0, ] distributed r.v. s U,..., U. Usig the quatile trasformatio, we obtai γ0 2 + γ 0 + log F 0X i+: F 0 X : d = γ 0 + log U i I {U i > /} / + γ 0 + log U i+: U : We decompose the latter expressio ito T + T 2 + T T = γ 0 log [ Ui /. + γ 0 [ ] U i+: log, k / T 2 = γ 0 T 3 = γ 0 log / Sice = o, we have, by Chebyshev s iequality, Sice IP{U i = U j ; i j} = 0, we have T 3, with ] I{U i > /} I{U i > /}, [ ] U :. U : = / + O P /. d = γ 0 + γ 0 [ Ui log / [ Ui log / 6 ] I{U i > /} ] I{U i > U :}. A.

17 Now, for ay d 0,, put T d = γ 0 log U i / { I / d / U i / + d } /. Note that we have, usig A., { T lim lim sup IP > T } d d Hece, i order to prove lim d it is sufficiet to show, for each d 0,, { lim sup IP U : / > d } / = 0. T = o IP, T d = o IP. A.2 But, A.2 follows easily from the Markov iequality, sice E T d 2dγ 0 max log [ + d/ ], log [ d/ ] 0. It remais to cosider T 2 + T 3. We start by rewritig T 3. Applyig a Taylor series expasio, we fid, for θ betwee / ad U : ad usig A., T 3 = γ 0 U : / + γ 0 2 θ 2 = γ 0 U : / + o IP. To complete the proof, we defie the uiform empirical process ad the uiform quatile process α s = G s s, for 0 s, β s = s U s, for 0 s, U : / 2 A.3 where ad G s = #{k : k, U k s}, U s = Usig A.3, the sum of T 2 ad T γ 0 α { Uk: if k / < s k/, U : if s = 0. 3 ca ow be writte as k β + o IP From Corollary 2.3 i Csörgő et al. 986, with λ =, oe fids α k β k = o IP. This completes the proof of Theorem

18 I order to prove Theorem 5., we eed two techical lemma s. Lemma A. Let Y,..., Y be idepedet radom variables with commo distributio fuctio Gy = /y, y. Let Y :,..., Y : deote the order statistics of Y,..., Y. Let be a sequece of itegers with ad,. The, as ad for all β <, k β Y i+: IP β, ad Y : log Y i+: Y : IP. Proof: The first result is Lemma 2.4 of Dekkers et al The secod result follows easily from the law of large umbers upo otig that log [Y i+: /Y :] is distributed as the order statistics of a stadard expoetial sample of size. Hece, the result follows from the cosistecy of the Hill estimator for the strict Pareto case. The secod lemma we eed ca be foud i Smith 982. Lemma A.2 Suppose L satisfies the SR2-coditio with ρ 0. If ρ < 0, the for all ε > 0 there exists a t ε such that we have Ltx x log Lt cgt u ρ du εgt, A.4 wheever t t ε ad x >. If ρ = 0, the the same result holds with the right-had side replaced by εgtx ε. We ow may prove Theorem 5.. Proof of Theorem 5.: We first cosider the behavior of the Hill estimator uder the ull hypothesis F 0. I the literature, may proofs exist of the asymptotic behavior of the Hill estimator uder the ull. We preset the proof for completeess oly. I virtue of Theorem 2.2 ad usig the quatile trasformatio, we eed to prove that log lf U i+: lf U : A.5 teds to ca/ρ i probability, where U :,..., U : deote the order statistics of a uiform sample of size. Now t = F F t [ ] /γ = F tlf t [ /γ = / t γ L/ tlf t], 8

19 implies lf t = /L/ t. Sice / U i: d = Y i:, we have A.5 = d log LY i+: LY :. Moreover, Y : = / + o IP ad, sice g is regularly varyig, this implies Thus, usig Coditio 5.3, gy :/g/ = + o IP. Provided that gy : = A + o IP. gy : log LY i+: LY : teds to c/ρ i probability, the desired result follows. I case ρ < 0, sice ɛ is arbitrary i A.4, it follows that A.6 A.7 A.7 = Applyig Lemma A. for ρ < 0, we ideed fid Y i+: /Y : c u ρ du + o IP. A.7 = c/ρ + o IP. The case ρ = 0 follows similarly usig agai Lemma A. ad otig that the extra factor x ε i the right-had side of A.4 does t affect the coclusio. The behavior of the Hill estimator uder the local alteratives 2.2 ad 2.3 ow follows immediately from Le Cam s third lemma see, e.g., Bickel et al. 993, p Before provig Theorem 6.2, we first establish the followig lemma. Lemma A.3 Let 0 < with ad = olog/. Let ω :,..., ω : be order statistics of a sample of size from the stadard expoetial distributio. The, for m IN, m ω i+: = o IP /log/ m ω : Proof: Note that ω i+: ω :, i =,..., are distributed as the order statistics of a stadard expoetial sample of size. Hece, from the law of large umbers we get k ω i+: ω : m IP m!. 9

20 Now, log/ m = log/ ad the desired result follows from log/ ω : ω i+: ω : m m ω : = log/ + o IP. ω i+: ω : m, Proof of Theorem 6.2: The proof will follow the same lies as that of Theorem 2.2. Usig the quatile trasformatio, we obtai τ0 2 τ 0 d = τ 0 log F 0X i+: F 0 X k: log[ U i ] + log[ /] I {U i > /} τ 0 log[ U i+: ] + log[ U k: ]. We decompose this expressio ito three terms T = τ 0 log[ U i ] + log[ /]I{U i > /} τ 0 log[ U i+: ] + log[ /], T 2 = τ 0 I{U i > /}, T 3 = τ 0 log[ U :] log[ /]. The terms T 2 ad T 3 are equal to the terms appearig i the proof of Theorem 2.2. The term T is somewhat differet, but ca be hadled aalogously. More precisely, for ay d 0,, we defie T d = τ 0 log[/ U i ] I { / d / U i / + d } / Sice for each d 0,, T d = o IP, 20

21 we get Furthermore, τ 2 0 ˆτ τ 0 T = o IP. log/ X i+: X : l0 X i+: l 0 X : = τ 0 log[ F 0 X i+: ] + log[ F 0 X k: ] ˆτ k τ 0 log/ X i+: X : which is distributed as k τ 0 ω i+: ω k: log/ K 0ω i+: K 0 ω : l0 X i+: l 0 X : + o IP log/ l0 K 0 ω i+: l 0 K 0 ω : K0 ω i+: K 0 ω : τ 0 = τ 0 K0 ω ω i+: ω k: log/ i+: l 0 K 0 ω i+: K 0 ω :l 0 K 0 ω. : Sice l 0 K 0 t = /L 0 t, this expressio ca be simplified ito τ 0 ω i+: ω k: log/ ω i+: ω : τ0. A.8 Applyig a Taylor expasio of t τ, for t > ad aroud of order max τ, ad usig Lemma A.3, we get A.8 = τ 0 ω i+: ω k: τ 0 log/ = τ 0 ω : log/ = o IP. ω i+: ω : ω i+: ω : Proof of Theorem 7.: Agai, we start by cosiderig the asymptotic behavior of ˆτ uder the ull. Uder F 0, we eed to establish that log[/ ] X i+: X : 2 l0 X i+: l 0 X :

22 coverges to ca, i probability. As before, we use the quatile trasformatio. Let ω :,..., ω : be the order statistics of a sample of size from the stadard expoetial distributio. Now, log[/ ] τ ω i+: ω : L 0ω i+: L 0 ω : coverges to ca, i probability, i view of the results i the proof of Theorem 3.2i of Beirlat et al A applicatio of Le Cam s third lemma the completes the proof. Refereces Beirlat, J., M. Broiatowski, J.L. Teugels, ad P. Vyckier 995, The mea residual life fuctio at great age: Applicatios to tail estimatio, Joural of Statistical Plaig ad Iferece, 45, Beirlat, J., J.L. Teugels, ad P. Vyckier 996, Practical Aalysis of Extreme Values, Leuve Uiversity Press, Leuve. Bickel, P.J., C.A.J. Klaasse, Y. Ritov, ad J.A. Weller 993, Efficiet ad Adaptive Statistical Iferece for Semiparametric Models, Joh Hopkis Uiversity Press, Baltimore. Csörgő, M., S. Csörgő, L. Horvath, ad D.M. Maso 986, Weighted empirical ad quatile processes, Aals of Probability, 4, Csörgő, S. ad L. Viharos 998, Estimatig the tail idex, Asymptotic Methods i Probability ad Statistics, , Elsevier. De Haa, L. ad L. Peg 998, Compariso of tail idex estimators, Statistica Neerladica, 52, De Haa, L. ad S. Resick 998, O asymptotic ormality of the Hill estimator, Stochastic Models, 4, Dekkers, A.L.M., J.H.J. Eimahl, ad L. De Haa 989, A momet estimator for the idex of a extreme-value distributio, Aals of Statistics, 7, Drees, H. 998, Optimal rates of covergece for estimates of the extreme value idex, Aals of Statistics, 26, Drees, H. 200, Miimax Risk Bouds i Extreme Value Theory, Aals of Statistics, 29, Falk, M.995, O testig the extreme value idex via the POT-method, Aals of Statistics, 23, Haeusler, E. ad J.L. Teugels 985 O asymptotic ormality of Hill s estimator for the expoet of regular variatio, Aals of Statistics, 3, Hall, P. 982, O some simple estimates of a expoet of regular variatio, Joural of the Royal Statistical Society Series B, 44,

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