HIGH QUANTILE ESTIMATION AND THE PORT METHODOLOGY

Transcription

1 REVSTAT Statistical Journal Volume 7, Number 3, November 2009, HIGH QUANTILE ESTIMATION AND THE PORT METHODOLOGY Authors: Lígia Henriques-Rorigues Instituto Politécnico e Tomar an CEAUL, Portugal Ligia.Rorigues@aim.estt.ipt.pt M. Ivette Gomes Universiae e Lisboa, F.C.U.L. D.E.I.O. an C.E.A.U.L., Portugal ivette.gomes@fc.ul.pt Receive: August 2009 Revise: September 2009 Accepte: October 2009 Abstract: In many areas of application, a typical requirement is to estimate a high quantile χ 1 p of probability 1 p, a value, high enough, so that the chance of an exceeance of that value is equal to p, small. The semi-parametric estimation of high quantiles epens not only on the estimation of the tail inex γ, the primary parameter of extreme events, but also on an aequate estimation of a scale first orer parameter. The great majority of semi-parametric quantile estimators, in the literature, o not enjoy the aequate behaviour, in the sense that they o not suffer the appropriate linear shift in the presence of linear transformations of the ata. Recently, an for heavy tails γ > 0, a new class of quantile estimators was introuce with such a behaviour. They were name PORT-quantile estimators, with PORT staning for peas over ranom threshol. In this paper, also for heavy tails, we introuce a new class of PORT-quantile estimators with the above mentione behaviour, using the PORT methoology an incorporating Hill an moment PORT-classes of tail inex estimators in one of the most recent classes of quantile estimators suggeste in the literature. Uner convenient restrictions on the unerlying moel, these classes of estimators are consistent an asymptotically normal for aequate, the number of top orer statistics use in the semi-parametric estimation of χ 1 p. Key-Wors: statistics of extremes; heavy tails; high quantiles; semi-parametric estimation; PORT methoology; asymptotic behaviour. AMS Subject Classification: 62G32, 62E20.

2 246 Lígia Henriques-Rorigues an M. Ivette Gomes

3 High Quantile Estimation an the PORT Methoology INTRODUCTION A moel F is sai to have a heavy right-tail whenever the right tail function, F : 1 F, is a regularly varying function with a negative inex of regular variation α /γ, i.e., for every x > 0, lim t Ftx/Ft x /γ. Then we are in the omain of attraction for maxima of an extreme value EV istribution function.f., EV γ x exp + γx /γ, x > /γ, γ > 0, an we write F D M EV γ>0. The parameter γ is the tail inex, one of the primary parameters of rare events. In a context of heavy tails, an with the notation Ut : F 1 /t, t 1, F y : inf{x: Fx y} the generalize inverse function of the unerlying moel F, the first orer parameter or tail inex γ > 0 appears, for every x > 0, as the limiting value, as t, of the quotient lnutx lnut / lnx e Haan, Inee, with the usual notation RV α for the class of regularly varying functions with an inex of regularly variation α, i.e., positive measurable functions g such that gtx/gt x α, as t an for all x > 0, we can further say 1.1 F D M EV γ>0 iff U RV γ iff 1 F RV /γ Gneeno, Heavy-taile istributions have recently been accepte as realistic moels for various phenomena in economics, ecology, bibliometrics an biometry, among others. See, for instance, the recent boos on the topic by Marovich 2007 an Resnic For small values of p, we want to extrapolate beyon the sample, estimating a typical parameter in many areas of application, a high quantile χ 1 p, i.e., a value such that Fχ 1 p 1 p, or equivalently, 1.2 χ 1 p U1/p, p p n 0, K as n, K [0, 1]. We are going to base inference on the largest + 1 orer statistics, an as usual in semi-parametric estimation of parameters of extreme events, we shall assume that is an intermeiate sequence of integers in [1, n[, i.e., 1.3 n, /n 0, as n. In orer to erive the asymptotic non-egenerate behaviour of semi-parametric estimators of parameters of extreme events, we nee more than the firstorer conition, U RV γ, provie in 1.1. A convenient conition is the following secon-orer conition, which guarantees that 1.4 lim t lnutx lnut γ lnx At xρ 1 ρ,

4 248 Lígia Henriques-Rorigues an M. Ivette Gomes which we assume to hol for every x > 0, being ρ 0 the shape or, more properly, the generalize shape secon orer parameter. The limit function in 1.4 is necessarily of this given form an A RV ρ Gelu an e Haan, Sometimes, only for the sae of simplicity, we shall assume to be woring in a sub-class of Hall-Welsh class of moels Hall an Welsh, 1985, where there exist γ > 0, ρ < 0, C > 0 an β 0, such that, as t, 1.5 Ut C t γ 1 + At 1 + o1, with At γβt ρ. ρ Typical heavy-taile moels, such as the Fréchet, the Generalize Pareto an the Stuent-t ν belong to such a class. Then, the secon-orer conition in equation 1.4 hols, with At γβt ρ, β 0, ρ < 0. The parameters β an ρ are the so-calle generalize scale an shape secon-orer parameters, respectively. If conition 1.5 hols, Ut Ct γ, as t, an from 1.2, we have χ 1 p U1/p Cp γ, as p 0. An obvious estimator of χ 1 p is thus Ĉp ˆγ, with Ĉ an ˆγ any consistent estimators of C an γ, respectively. Given a sample X 1, X 2,..., X n, let us enote X 1:n X 2:n X n:n the set of associate ascening orer statistics. Denoting Y a stanar Pareto moel, i.e., a moel such that F Y y 1 1/y, y > 1, the universal uniform transformation an the fact that Y n :n p n/ for intermeiate, enables us to write X n :n p Cn/ γ, as n, where the notation X n p Yn means that X n /Y n converges in probability to one, as n. Consequently, an obvious estimator of C, propose in Hall 1982, is an Ĉ C,n,ˆγ : X n :n /nˆγ Q,pn,ˆγ Ĉ p ˆγ n X n :n / ˆγ is the obvious quantile-estimator at the level p Weissman, The semiparametric estimation of high quantiles epens thus strongly on the estimation of the tail inex γ, the primary parameter of extreme events. In the classical approach, we often consier for ˆγ either the Hill estimator Hill, 1975 or the moment estimator Deers et al., 1989, both base on the +1 top orer statistics, enote X : X n :n,, X n:n. The Hill estimator is the average of the log-excesses, 1.6 H,n H n X γ,n,h : 1 lnx n i+1:n lnx n :n, i1

5 High Quantile Estimation an the PORT Methoology 249 an the moment estimator has the functional expression, 1.7 M,n M n X γ,n,m : M 1,n { 1 M 1 2/M 2 2,n with M α,n efine by,n}, 1.8 M α,n Mα n X : 1 lnx n i+1:n lnx n :n α, α 1, 2. i1 Uner the secon orer framewor in 1.4 an for any intermeiate sequence, i.e. whenever 1.3 hols, we have for the Hill estimator H, in 1.6, an for the moment estimator M, in 1.7, generally enote by T, the valiity of the following asymptotic istributional representation, 1.9 γ,n,t γ + σ T P,T + b T An/ 1 + o p 1, where P,T is asymptotically stanar normal an 1.10 σ 2 : γ2, b H H : 1 1 ρ, σ2 : 1 + γ1 ρ + ρ γ2 an b M M : γ1 ρ 2. Most of the semi-parametric quantile estimators in the literature, lie the ones in Gomes an Figueireo 2006, Gomes an Pestana 2007, Beirlant et al. 2008, Caeiro an Gomes 2008, as well as in other papers on semi-parametric quantile estimation prior to 2005 see also, e Haan an Ferreira, 2006, o not enjoy the aequate behaviour in the presence of linear transformations of the ata, a behaviour relate with the fact that for any quantile χ 1 p we have 1.11 χ 1 p s + δx s + δ χ 1 p X for any moel X, real s an positive δ. Recently, an for γ > 0, Araújo Santos et al provie quantile estimators with the linear property in 1.11, base upon a sample of excesses over a ranom threshol X nq:n, enote 1.12 X q : X n:n X nq:n,, X nq+1:n X nq:n, nq : [nq] + 1, where [x] enotes, as usual, the integer part of x, with: 0 < q < 1, for istributions with a left enpoint, x F : inf{x: Fx > 0}, finite or infinite the ranom threshol X nq:n is an empirical quantile; q 0, for istributions with a finite left enpoint x F the ranom threshol is the minimum, X 1:n. Such estimators were name PORT-quantile estimators, with PORT staning for peas over ranom threshol, an are base on the PORT-Hill an PORTmoment estimators, generically enote Tq T,n q : T n X q for T H

6 250 Lígia Henriques-Rorigues an M. Ivette Gomes or M, < n n q, an with H n X an M n X provie in 1.6 an 1.7, respectively. They are given by 1.13 Q,pn,Tq : Tq X n :n X nq:n + X nq:n, where Tq can more generally be any consistent estimator of the tail inex γ, mae location/scale invariant by using any of the transforme samples X q, in The PORT-Hill an the PORT-moment estimators have been stuie by simulation in Gomes et al. 2008a. The class of estimators suggeste here is also a function of the sample of the excesses X q in We use the PORT methoology an incorporate Hill an moment PORT-classes of tail inex estimators in one of the classes of quantile estimators suggeste in Caeiro an Gomes 2008, slightly moifie in orer to satisfy the linear property in More specifically, we shall consier quantile estimators of the type, 1.14 Q,pn,Tq : X n [/2]:n X n :n 2 Tq 1 Tq + X nq:n. Uner convenient restrictions on the unerlying moel, these classes of estimators are consistent an asymptotically normal for aequate, the number of top orer statistics use in the semi-parametric estimation of χ 1 p. In Section 2 of this paper, we shall present a few introuctory technical etails an asymptotic preliminary results associate with the PORT methoology. The asymptotic behaviour of the PORT-classes of tail inex estimators uner stuy, together with the asymptotic comparison of the PORT-Hill an the PORT-moment estimators at optimal levels, will be erive in Section 3. In Section 4, we erive the asymptotic behaviour of the new classes of PORT-quantile estimators. Finally, in Section 5, we raw some overall conclusions. 2. TECHNICAL DETAILS RELATED WITH THE PORT METHODOLOGY 2.1. The secon orer framewor for heavy-taile moels uner a real shift If we introuce a eterministic shift, i.e. a new location, s 0, in the unerlying moel X, with quantile function U X t, the transforme ranom variable r.v. Y X+s has an associate quantile function given by U s t U Y t U X t + s

7 High Quantile Estimation an the PORT Methoology 251 an conition 1.4 can be rewritten as 2.1 lim t lnu s tx lnu s t γ lnx A s t for all x > 0, with A s RV ρs. xρs 1 ρ s, Let F be a moel with quantile function Ut U X t, given in 1.5. Then U Y t C t γ 1 + At ρ + s C t γ + ot ρ, as t. Therefore both U s t U Y t an Ut U X t are asymptotically equivalent to C t γ, but { ρ if ρ > γ ρ s γ if ρ γ. The function A s t in 2.1 can be chosen as γs Ut, if ρ < γ A s t : At γs Ut, if ρ γ At, if ρ > γ Asymptotic preliminary results in the PORT methoology In this subsection we begin with the presentation of the asymptotic results for the statistics M α,q,n M n α X q, < n n q, base on the sample of excesses X q, 0 q < 1, in 1.12 an with M n α X provie in 1.8. In the following, χ q enotes the q-quantile of F: Fχ q q by convention χ 0 x F, whenever finite so that, 2.2 X nq:n p n χ q for 0 q < 1. We present, without proof, the following Lemma: Lemma 2.1 Araújo Santos et al., If the secon orer conition 1.4 hols, if n is an intermeiate sequence, i.e. 1.3 hols, then, for any real q, 0 q < 1, with Fχ q q χ 0 x F, whenever finite, an for α 1, 2, M α,q,n 1 i1 ln X n i+1:n χ q X n :n χ q α 1 o p. Un/

8 252 Lígia Henriques-Rorigues an M. Ivette Gomes Remar 2.1. Note that if q 0, 1, X nq:n χ q O p 1/ n an we can assure that M α,q,n 1 i1 lnxn i+1:n χ q lnx n :n χ q α / O p /n Un/ op 1, for α 1, 2. Proposition 2.1. If the secon orer conition 1.4 hols an n is an intermeiate sequence, i.e. 1.3 hols, the statistics M α,q,n Mα n X q, with < n n q, M n α X given in 1.8 an X q given in 1.12, satisfy for α 1, 2, 2.3 M 1,q,n M 1,n + γ χ q 1 + op 1, 1 + γ Un/ 2.4 M 2,q,n M 2,n + 2 γ2 2 + γ χ q 1 + op 1 + γ 2 1. Un/ M 1,q,n Proof: The first moment of the log-excesses can be rewritten as 1 Xn i+1:n X nq:n ln X n :n X nq:n i1 M 1,n + 1 ln 1 i1 Xnq:n X n i+1:n 1 Xnq:n X n :n. Since ln1 + x x, as x 0, 1 ln 1 i1 Xnq:n X n i+1:n 1 Xnq:n X n :n p 1 i1 Xnq:n X n :n Xnq:n X n :n 1 i1 X n q:n X n i+1:n 1 X n :n. X n i+1:n If n is intermeiate, i.e. 1.3 hols, an {Y i } i1,..., is a sequence of inepenent an ientically istribute i.i.. stanar Pareto r.v. s, then Y n :n p n/ an M 1,q,n M 1,n + M 1,n + χ q Un/ χ q Un/ 1 1 i1 i1 1 X n :n X n i+1:n 1 Y γ i 1 + op 1. Given that EY γ 1/1 + γ an by the wea law of large numbers we get 2.3.

9 High Quantile Estimation an the PORT Methoology 253 For α 2 an using similar evelopments, we have M 2,q,n 1 i1 M 2,n + M 2,n + ln X n i+1:n X n :n + χ q Un/ 2 χ q Un/ 2 χ q Un/ 1 i1 1 Y γ i 1 + op 1 2 ln X n i+1:n 1 Y γ i 1 + op 1 X n :n M 1,n 1 i1 γ lnyi Y γ i 1 + op 1. Since ElnY Y γ 1/1 + γ 2 an by the wea law of large numbers we get 2.4. Remar 2.2. It has been prove in Gomes an Martins 2001 that, uner the secon orer framewor, in 1.4, an for levels such that 1.3 hols, we get for M α,n Mα n X, in 1.8, an asymptotic istributional representation of the type M 1,n M 2,n γ + γz1 + An/ 1 + op 1, 1 ρ 2γ γ 2 Z γ 1 1 ρ 2 ρ1 ρ 2 An/ 1 + o p 1, where, with {E i } i 1 a sequence of i.i.. stanar exponential r.v. s, Z α 1 Γ2α + 1 Γ 2 α + 1 Ei α Γα + 1, α 1, 2, is asymptotically stanar normal. Moreover, the covariance structure of Z α is given by Cov Z α, Zβ i1 Γα + β +1 Γα +1 Γβ +1 Γ2α +1 Γ 2 α +1 Γ2β +1 Γ 2 β ASYMPTOTIC BEHAVIOUR OF THE PORT-CLASSES OF TAIL INDEX ESTIMATORS In this section we present, uner the valiity of the secon orer conition in 1.4, the asymptotic istributional representations of the PORT-Hill estimators, H,n q : H n X q, an the PORT-moment estimators, M,n q : M n X q,

10 254 Lígia Henriques-Rorigues an M. Ivette Gomes with functional expressions given by an H,n q ˆγ,n,Hq 1 Xn i+1:n X nq:n ln X n :n X nq:n i1 M,n q ˆγ,n,Mq M 1,q,n respectively, < n n q, M α,q,n 1.8 an 1.12, respectively. Mα n { 1 M 1,q,n } 2/M 2,q,n, X q, M α X an X q provie in The following theorem has been prove in Araújo Santos et al n Theorem 3.1 Araújo Santos et al., If the secon orer conition 1.4 hols, n is an intermeiate sequence of positive integers, i.e. 1.3 hols, an for any real q, 0 q < 1, we have for T,n q, with T enoting either H or M, an asymptotic istributional representation of the type 3.1 Tq T,n q γ + σ T P,T + b T An/ + c T χ q 1 + op 1, Un/ where P,T, given in 1.9, is asymptotically stanar normal, σ 2 T provie in 1.10, an b T are 3.2 c H : γ 1 + γ an c M : γ γ 2. For simplicity of notation, let us now istinguish the following regions: R 1 : γ + ρ < 0 χ q 0. R 2 : γ + ρ > 0 γ + ρ 0 χ q 0. R 3 : γ + ρ 0 χ q 0. Corollary 3.1 Araújo Santos et al., Uner the conitions of Theorem 3.1, the following results hol: In R 1, T,n q γ + σ T P,T / + c T χ q 1+op 1 / Un/. Consequently, if / Un/ λ 1 finite, then T,n q γ n Normal λ 1 c T χ q, σ 2 T. In R 2, T,n q γ + σ T P,T / + b T An/ 1 + o p 1. Consequently, if An/ λ 2 finite, then T,n q γ n Normal λ 2 b T, σ 2 T.

11 High Quantile Estimation an the PORT Methoology 255 In R 3, T,n q γ + σ T P,T / + b T An/ + c T χ q / Un/ 1 + op 1. Consequently, if / Un/ λ 1 an An/ λ 2, with λ 1 an λ 2 both finite, then T,n q γ n Normal λ 1 c T χ q + λ 2 b T, σ 2 T Asymptotic comparison at optimal levels We now procee to an asymptotic comparison of the estimators at their optimal levels in the lines of e Haan an Peng 1998, Gomes an Martins 2001, Gomes et al. 2005, 2007b an Gomes an Neves Suppose that γ,n, q, now enote γ q, is a general semi-parametric PORT-tail inex estimator, with istributional representation, 3.3 γ q γ + σ P, + χ q 1 b An/ + c + op 1, Un/ which hols for any intermeiate, an where P, is an asymptotically stanar normal r.v. Given the results presente in Corollary 3.1, the Asymptotic Mean Square Error AMSE of γ q is AMSE γ q : σ 2 + χ q 2 c2 U 2 n/, in R 1 σ 2 where Var γ q : σ 2 / an + b2 A 2 n/, in R 2 σ 2 + χ 2 q b + c A 2 n/, in R 3, Cγβ / : 1 Un/, in R1 χ q c Un/ Bias γ q : b An/ : 2 An/, in R 2 b + c χ q Cγβ An/ : 3 An/, in R 3. Let 0, q : arg min AMSE γ q be the so-calle optimal level for the estimation of γ through γ q, i.e., the level associate with a minimum asymptotic mean square error, an let us enote γ n0, q : γ q 0, q, the estimator compute at its optimal level. The use of regular variation theory enable Deers an e Haan 1989 to prove that, whenever i 0, i 1, 2, 3 in this stuy, there exists a function ϕn ϕn; ρ, γ, epenent only on the unerlying moel, an not on the estimator, but epenent here on i 1, 2, 3,

12 256 Lígia Henriques-Rorigues an M. Ivette Gomes such that lim n ϕnamse γ n0, q : LMSE γn0, q exists, with LMSE staning for limiting mean-square error. Moreover, 1 + 2γ 2γ σ 2 1+2γ γ 2 γ, in R LMSE γ n0, q 2ρ 1 2ρ 2ρ 1 2ρ σ 2 2ρ 1 2ρ σ 2 2ρ 1 2ρ ρ, in R ρ, in R 3. It is then sensible to consier the following: Definition 3.1. Given γ n0,t1 q γ T1 q 0,T1 q an γn0,t2 q γ T2 q 0,T2 q, two biase PORT-estimators γ an γ T1 q T 2 for which istributional representations of the type 3.3 hol with constants q σ T1, T1 an σt2, T2, T1, T2 0, respectively, both compute at their optimal levels, the Asymptotic Root Efficiency AREFF of γ relatively to γ is T1 q T 2 q AREFF T1 q T 2 q AREFFˆγ T1 q ˆγ T 2 q : with LMSE given in 3.4. LMSE γ n0,t2 q LMSE γ, n0,t1 q Remar 3.1. Note that this measure was evise so that the higher the AREFF measure is, the better the first estimator is. Remar 3.2. The optimal levels 0,Tq for the estimation of γ through γ Tq, with T enoting either H or M are enote by 0,Hq an 0,Mq an are given in Table 1. Table 1: Optimal levels for the estimation of γ through PORT-Hill an PORT-moment estimators. Region 0,Hq 0,Mq C1+ γn γ 2/1+2γ C 1+ γ 2 1+ γ 2 n γ 2/1+2γ R 1 χ q 2γ γ 2 χ q 2γ 1 ρn ρ 2/1 2ρ 1+ γ2 1 ρ 2 n ρ 2/1 2ρ R 2 R 3 β 2ρ C1 ρn ρ βc + χ q 2ρ γ1 ρ + ρ β 2ρ 2/1 2ρ C 1+ ρ 2 1 ρ 2 n ρ ρ 2 βc + χ q 2ρ 2/1 2ρ

13 High Quantile Estimation an the PORT Methoology 257 Proposition 3.1. The AREFF-inicator of γ Mq relatively to γ Hq is: γ 2 γ 1 1+2γ 1+ γ 1+2γ, in R1 γ 1+ γ 2 AREFF Mq Hq γ 2 1+ γ 2 ρ 2 1+ ρ 2 ρ 1 1 2ρ γ1 ρ 1 2ρ, γ1 ρ + ρ ρ 1 2ρ 1 ρ ρ in R ρ, in R3. This AREFF-measure is presente in Figure 1, where we can see that the gain in efficiency for the PORT-moment estimator happens for a large region of values of γ, ρ <AREFF Mq Hq <AREFF Mq Hq 1.5 AREFF Mq Hq >1.5 Figure 1: Asymptotic efficiency of γ Mq relatively to γ Hq in the γ,ρ-plane whenever χ q ASYMPTOTIC BEHAVIOUR OF THE PORT-QUANTILE ESTIMATORS We first present the following result, prove in Ferreira et al. 2003, on the asymptotic behaviour of intermeiate orer statistics:

14 258 Lígia Henriques-Rorigues an M. Ivette Gomes Proposition 4.1 Ferreira et al., Uner the secon orer framewor in 1.4 an for intermeiate sequences of positive integers, i.e. if 1.3 hols, X n :n Un/ 1 + γb + o p An/, where B is asymptotically stanar normal, an CovB i, B j 1 j/n ij, i < j. j We shall now consier an stuy the new PORT-quantile estimator, in We can state the following result: Theorem 4.1. Let us assume that the secon orer conition in 1.4 hols, with At γβ t ρ, that is an intermeiate sequence of integers, i.e. 1.3 hols, an that ln / 0, as n, with p n given in 1.2. Then, for any real q, 0 q < 1, an with T enoting either H or M, 4.1 σ T P,T + An/ ln ln b T An/ + c T Q,pn,Tq 1 χ q Un/ + b T 2 γ ln2 2 γ 2γ+ρ ρ2 γ 1 + op 1 χq 2 Un/ ln γ ln 2 1+ op 2 γ 1, with b T, σ T an c T given in 1.10 an 3.2, respectively, an where P,T is asymptotically stanar normal. Proof: The PORT-quantile estimator in 1.14 can be written as { Xn [/2]:n 1 Tq Q,pn,Tq : X n :n 1 X n :n 2 Tq + X } n q:n. 1 X n :n Therefore, Q,pn,Tq Xn [/2]:n 1 Tq X n :n X n :n 2 Tq + X n q:n χ 1 p n. X n :n X n :n As 2.2 hols, we can say that X nq:n/x n :n o p 1, an using the secon orer conition in 1.4, we can guarantee that X n [/2]:n U 2 n X n :n U n 2 γ 1 + 2ρ An/ 1 + o1. ρ

15 High Quantile Estimation an the PORT Methoology 259 Since U1/p n, the result in Proposition 4.1 enables us to write X n :n U n U U n n X n :n γ 1 An/ ρ γ 1 γb An/ ρ an the use of the elta metho leas us to 1 + o1 1 γb + o p An/ 1 + op 1, Tq 2 Tq Therefore γ 2 γ 1 + Tq γ ln 2γ ln2 1 + op 2 γ 1. Q,pn,Tq γ X n :n { Tq γ ln 2γ ln2 1 + op 2 γ 1 + γb + An/2γ+ρ 1 + op ρ2 γ 1 } γ Un/ 1 + γb + An/ 1 + op 1 ρ { Tq γ ln 2γ ln2 1 + op 2 γ 1 + γb + An/2γ+ρ 1 + op ρ2 γ 1 }, using also the result presente in Proposition 4.1. Since then γb / + An/ / ρ 1 + o p 1 o p 1/, γ Un/ Q,pn,Tq + γb + An/2γ+ρ ρ2 γ { Tq γ ln 2γ ln2 1 + op 2 γ op 1 }.

16 260 Lígia Henriques-Rorigues an M. Ivette Gomes Notice that U1/p n / γ Un/, an then ln Q,pn,Tq Tq γ + γb ln + ln ln Tq γ 2 γ ln2 2 γ An/2 γ+ρ ρ2 γ. Using the istributional representation of Tq in 3.1 an since ln /, 4.1 follows. hol: Corollary 4.1. Uner the conitions of Theorem 4.1, the following results In R 1, i.e. for values of γ + ρ < 0 an χ q 0, ln Q,pn,Tq σ T P,T + c T χq 2 Un/ ln γ ln2 2 γ χ q 1 + op 1 Un/ 1 + op 1. If / Un/ λ 1 finite, then ln Q,pn,Tq Normal λ 1 c n T χ q, σ 2 T. In R 2, i.e. for values of γ + ρ > 0 or γ + ρ 0 an χ q 0, ln Q,pn,Tq σ T P,T + b T An/ 1 + o p 1 An/ ln 2 γ ln2 b T 2 γ 2γ+ρ 1 + op ρ2 γ 1, If An/ λ 2 finite, then ln Q,pn,Tq Normal λ 2 b n T, σ 2 T.

17 High Quantile Estimation an the PORT Methoology 261 In R 3, i.e. for values of γ + ρ 0 an χ q 0, Q,pn,Tq ln σ T P,T + χ q 1 b T An/ + c T + op 1 Un/ An/ 2 ln γ ln2 b T 2 γ 2γ+ρ χq 2 ρ2 γ + Un/ln γ ln2 1+op 2 γ 1, If / Un/ λ 1 an An/ λ 2, with λ 1 an λ 2 both finite, then ln Q,pn,Tq Normal λ 1 c n T χ q + λ 2 b T, σ 2 T. Remar 4.1. Notice that, uner a secon orer framewor, the mean value an the variance of the r.v. γ,n,tq γ, provie in Corollary 3.1, are equal to the ones of Q,pn,Tq/ 1 / ln /. Since ln / goes to infinity very slowly, we can state a pre-asymptotic istributional representation, for moerate an n: Corollary 4.2. Uner the conitions of Theorem 4.1 an for moerate values of an n, the following pre-asymptotic results hol: In R 1, if / Un/ λ 1, finite, an with µ 1 : λ 1 c T χ q 1 2γ ln2 2 γ Q,pn,Tq ln / Normal µ 1, σ 2 T 1 c T ln /, 1+ γ 2 σ 2 ln2 /np T n. In R 2, if An/ λ 2, finite, an with µ 2 : λ 2 b T ln / γ ln2 2 γ 2γ+ρ ρ2 γ b T Q,pn,Tq ln / Normal µ 2, σ 2 T 1+, γ 2 σ 2 T ln2 /.

18 262 Lígia Henriques-Rorigues an M. Ivette Gomes In R 3, if / Un/ λ 1 an An/ λ 2, with λ 1 an λ 2 both finite, then, with µ 1 + µ 2 λ 1 c T χ q 1 2γ ln2 1 2 γ c T ln / + λ 2 b T 1+ 1 ln / 2 γ ln2 2 γ 2γ+ρ ρ2 γ, b T Q,pn,Tq ln / Normal µ 1 +µ 2, σ 2 γ 2 1+ T σ 2 ln2 /np T n. 5. CONCLUSIONS The new PORT-quantile estimator, efine in 1.14, is asymptotically equivalent to the PORT-quantile estimator in 1.13, stuie in Araújo Santos et al Consequently, an for finite sample sizes, we o not expect a much better behaviour of this new estimator comparatively to the one in However, the use, in 1.14, of a PORT-version of a minimum-variance reuce-bias extreme value inex estimator, lie the ones in Caeiro et al. 2005, Gomes et al. 2007a an Gomes et al. 2008b, leas to quantile estimators which overpass the estimator in Caeiro an Gomes 2008 an enjoy the aequate behaviour in the presence of linear transformations of the ata. Also, the use of subsampling proceures, similar to the ones in Hall an Scotto 2008, can improve this estimation proceure. These are however topics out of the scope of this paper. ACKNOWLEDGMENTS Research partially supporte by FCT/POCI 2010 an PTDC/FEDER.

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