A NOTE ON SECOND ORDER CONDITIONS IN EXTREME VALUE THEORY: LINKING GENERAL AND HEAVY TAIL CONDITIONS

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1 REVSTAT Statistical Journal Volume 5, Number 3, November 2007, A NOTE ON SECOND ORDER CONDITIONS IN EXTREME VALUE THEORY: LINKING GENERAL AND HEAVY TAIL CONDITIONS Authors: M. Isabel Fraga Alves CEAUL, DEIO, Faculty of Science, University of Lisbon, Portugal isabel.alves@fc.ul.pt M. Ivette Gomes CEAUL, DEIO, Faculty of Science, University of Lisbon, Portugal ivette.gomes@fc.ul.pt Laurens de Haan Department of Economics, Erasmus University Rotterdam, The Netherlands ldhaan@few.eur.nl Cláudia Neves UIMA, Department of Mathematics, University of Aveiro, Portugal claudia.neves@ua.pt Received: July 2007 Revised: October 2007 Accepted: October 2007 Abstract: Second order conditions ruling the rate of convergence in any first order condition involving regular variation and assuring a unified extreme value limiting distribution function for the sequence of maximum values, linearly normalized, have appeared in several contexts whenever researchers are working either with a general tail, i.e., R, or with heavy tails, with an extreme value index > 0. In this paper we shall clarify the link between the second order parameters, say ρ and ρ that have appeared in the two above mentioned set-ups, i.e., for a general tail and for heavy tails, respectively. We illustrate the theory with some examples and, for heavy tails, we provide a link with a third order framework. Key-Words: extreme value index; regular variation; semi-parametric estimation. AMS Subject Classification: Primary 62G32, 62E20, 26A12.

2 286 Fraga Alves, Gomes, de Haan and Neves

3 Second Order Conditions in EVT INTRODUCTION Let X 1, X 2,..., X n be an independent, identically distributed i.i.d. sample from an unknown distribution function d.f. F. It is well-known from Gnedenko s seminal work Gnedenko, 1943 that if there exist normalizing constants a n > 0, b n R and a non-degenerate d.f. G such that, for all x, { lim P a } n n maxx1,..., X n b n x = Gx, G is, up to scale and location, an Extreme Value d.f., dependent on a shape parameter R, and given by exp + x /, 1+ x > 0 if G x := exp exp x, x R if = 0. We then say that F is in the domain of attraction for maxima of the d.f. G in 1.1 and write F D M G. 2. FIRST AND SECOND ORDER CONDITIONS 2.1. A general tail R The following extended regular variation property de Haan, 1984, denoted ERV, is a well-known necessary and sufficient condition for F D M G : 2.1 lim Utx Ut at = x if 0 lnx if = 0, for every x > 0 and some positive measurable function a. For the case > 0 we see easily from 2.9 that we can choose at = Ut. Apart from the first order condition in 2.1, we shall consider the most common second order condition, specifying the rate of convergence in 2.1. We shall assume the existence of a function At, possibly not changing in sign and tending to zero as t, such that 2.2 lim Utx Ut at At x = H,ρ x := 1 ρ x +ρ +ρ x for all x > 0, where ρ 0 is also a second order parameter controlling the speed of convergence of maximum values, linearly normalized, towards the limit law in

4 288 Fraga Alves, Gomes, de Haan and Neves 1.1, for a general R. We then say that the function U is of second order extended regular variation, and use the notation U 2ERV,ρ. In 2.2, the cases = 0 and ρ = 0 are obtained by continuity arguments. More specifically, we can write H,ρ x = 1 x ρ ρ ρ 1 lnx x lnx x if = 0, ρ 0 ln 2 x if = ρ = 0. 2 We remark that A RV ρ. For a large variety of models we have ρ < 0 thus making sensible to simplify 2.2. We now state: if 0, ρ = 0 Proposition 2.1 Gomes and Neves, Let us assume that there exist a and A such that 2.2 holds, with ρ < 0. Then, there exist a 0 and A 0 such that 2.3 lim with Utx Ut a 0 t A 0 t x = x+ρ +ρ 2.4 A 0 t = At/ρ, a 0 t = at 1 A 0 t. From Theorem A in Draisma de Haan, Peng and Pereira 1999, with slight additions in Ferreira, de Haan and Peng 2003 and in de Haan and Ferreira 2006, we state the following: Theorem 2.1. Suppose the right endpoint x F := U > 0 and there exist a and A such that 2.2 holds, with ρ 0, ρ. Define at 2.5 At := Ut +, + := max0,. Then for +ρ < l := lim Ut at and the following holds At 0 and exists and is finite At At c, with 0 if < ρ c = if 0 ρ < or 0 < < ρ and l = 0 +ρ ± if +ρ = 0 or 0 < < ρ and l 0 or ρ < 0.

5 Second Order Conditions in EVT Heavy tails > 0 The most typical first order condition for heavy tails, i.e., for the case > 0 in 1.1, comes also from Gnedenko For any real τ, let us denote by RV τ the class of regularly varying functions with an index of regular variation τ, i.e., positive measurable functions g such that lim gtx/gt = x τ for all x > 0. Then, for > 0, 2.8 F D M G F = 1 F RV /. Equivalently, and with U standing for a quantile type function associated to F and defined by Ut := 1/1 F { t = inf x: Fx 1 1 t}, de Haan 1970 established that 2.9 F D M G U RV. To measure the rate of convergence in 2.9, it is then sensible to consider one of the following conditions: 2.10 lim Utx Ut x Ãt = x x ρ ρ lnutx lnut lnx lim Ãt = x ρ ρ, for all x > 0, where ρ 0 is a second order parameter controlling the speed of convergence of maximum values, linearly normalized, towards the limit law in 1.1 pertaining to > 0. Under these circumstances, we say that the function U is of regular variation of second order, and use the notation U 2RV, ρ. We remark that à RV ρ. 3. THE LINK BETWEEN THE SECOND ORDER CONDITION FOR A HEAVY AND FOR A GENERAL TAIL The following results hold with any measurable eventually positive function U. Lemma 3.1. If 2.1 holds for some R, then the auxiliary function at in 2.1 is of regular variation at infinity with index, i.e., a RV and at lim Ut = + := max0,.

6 290 Fraga Alves, Gomes, de Haan and Neves Moreover, if > 0, both functions a and U belong to RV ; if < 0, then x F = U <, lim at/ x F Ut = and x F U RV. Furthermore, with := min,0, and provided that U > 0, 3.1 lim lnutx lnut at/ut = x, for every x > 0. Proof: The first part of the lemma comes from Theorems 1.9 and 1.10 in Geluk and de Haan The limit in 3.1 follows easily when we distinguish between the cases > 0 and 0. For the derivation of asymptotic properties of semi-parametric estimators of, a topic out of the scope of this paper, it is important to know, for all x > 0, not only the rate of convergence of lnutx lnut, but also of Utx/Ut and of Ut/Utx, as t. We shall now see in more detail for the different relevant subspaces of the semi-plane, ρ R R 0, the limiting behaviour, as t, of Utx/Ut and Ut/Utx. The limit behavior of lnutx lnut has been analyzed e.g. in Appendix B of de Haan and Ferreira write Lemma 3.2. Assume that 2.2 holds, i.e., U 2ERV,ρ. Then, we may 3.2 Utx Ut = x + + At [ x + At ax, t;, ρ 1+ o1 ], where ax, t;, ρ = ln 2 x 2 1 x lnx x 1 x +ρ x ρ +ρ x +ρ x ρ At +ρ 1 x lnx x At if = ρ = 0 if < ρ = 0 if 0, ρ < 0 if > 0, ρ < 0 if ρ = 0 <. Proof: Directly from 2.2, we get { Utx Ut 1 = at x + At x +ρ x 1+ } o1. Ut ρ +ρ

7 Second Order Conditions in EVT 291 With the notation in 2.5, i.e., at/ut = + + At, we may write Utx x Ut 1 = x + + At + At x +ρ x + + At 1+ o1 ρ +ρ and 3.2 follows for any ρ < 0. If ρ = 0 and 0, then, also directly from 2.2, and by continuity arguments, { Utx Ut 1 = at x + At x lnx x 1+ } o1, Ut and things work as before, with At/ρ replaced by At/ and x+ρ +ρ replaced by x lnx. The case = ρ = 0 comes again directly from 2.2 and by continuity arguments. Theorem 3.1. Let U ERV,ρ as introduced in 2.2. Let c be the limit in 2.7. i If > 0, 3.3 lim Ut Utx x Ãt = K,ρ x := x xρ ρ x x for all x > 0, where, with At given in 2.5, At if c = 3.4 Ãt := +ρ +ρ At if c = ±. if c = +ρ if c = ±, Necessarily, Ã RV ρ, with ρ if c = 3.5 ρ = +ρ if c = ±. ii If 0, we need further to assume that ρ. Then, x lnx if < ρ = lim Ut a t 1 Ut Utx A t x = K,ρx = x +ρ +ρ if < ρ < 0 x 2 if ρ < < 0 2 ln 2 x if ρ < = 0,

8 292 Fraga Alves, Gomes, de Haan and Neves where 3.7 A t = At At ρ if < ρ = 0 if < ρ < 0 2 At if ρ < < 0 At if ρ < = 0 and 3.8 a t = { at if ρ < = 0 at 1 A t otherwise. Necessarily, A RV ρ, with 3.9 ρ = { ρ if < ρ 0 if ρ < 0. Proof: We shall consider the cases i and ii separately. Case i: If > 0, ρ < 0 and 2.2 holds, i.e., U 2ERV,ρ, we have from 3.2, Utx x Ut x = At + At x +ρ x + o At. ρ +ρ If c = ±, then At = o At, Utx Ut x = x x At + o At and Utx Ut x At x x. If c = / +ρ, we get At = At +ρ 1+ o1. Since in this region ρ, we may further write Utx Ut x = x x At x ρ = At +ρ x ρ + At +ρ + o At. x ρ ρ x + o At Consequently, 3.10 lim Utx Ut x Ãt = x xρ ρ x x if c = +ρ if c = ± = x 2 K,ρ x,

9 Second Order Conditions in EVT 293 with K,ρ x and Ãt defined in 3.3 and 3.4, respectively. Finally, 3.10, together with the fact that Utx Ut x = x Utx Ut Ut 1+ Ut Utx x = x 2 Utx x o1, leads us to the limit in 3.3, with Ãt and ρ given in 3.4 and 3.5, respectively. If > 0 and ρ = 0, we get, again from 3.2, Utx Ut x = At x x = x At + At x lnx x + At lnx x + o At + o At. But if > 0 and ρ = 0, then c = / +ρ = 1, At = At+o At, and Utx Ut x = At x lnx + o At. Consequently, 3.3 holds, with Ãt = At At/ +ρ and ρ = ρ = 0. Case ii: If < ρ = 0, we get, again from 3.2, Ut 1 Ut at Utx and with a t = at = x = x 1 At, + At 1 At Ut a 1 Ut = x + At t Utx x lnx x + At + o At x lnx + o At x lnx + o At. Consequently, 3.6, 3.7, 3.8 and 3.9 follow in this region of the, ρ-plane. If < ρ < 0, at/ut At = oat, and again from 3.2, Ut 1 Ut at Utx = x = x + At x +ρ x ρ +ρ 1 At + At x +ρ ρ ρ +ρ + o At + o At

10 294 Fraga Alves, Gomes, de Haan and Neves and with a t = at 1 At ρ, Ut a 1 Ut = x + At t Utx ρ and the results in the proposition hold. x +ρ +ρ If ρ < 0, At = o At, and also from 3.2, we get o At, Ut x Utx = 1 At + A 2 x 2 t 1+ o1. Consequently, for < 0, since Ut 1 Ut at Utx and with a t = at = x = x 1+ 2 At x 2= 2 x 2 2 x 2 At x 2 x 1+ o At 2 At x 2 1+ o1, 2, Ut a 1 Ut = x 2 At t Utx and the results in the proposition follow. x 2 1+ o1, If ρ < = 0, then from 3.11, we get Ut 1 Ut = lnx At ln 2 x 1+ o1, at Utx and the result in the proposition follows as well. 2 Corollary 3.1. Under the conditions and notations of Proposition 2.1, and for > 0, 3.12 lim lnutx lnut lnx Ãt for every x > 0, and with à provided in 3.4. = K,ρ x := x ρ ρ x if c = +ρ if c = ±, Proof: The proof follows immediately from relation 3.3.

11 Second Order Conditions in EVT 295 Remark 3.1. Note that the second order condition in 3.12 is the usual second order condition for heavy tails, i.e., the second order condition provided in Remark 3.2. Note next that the region {, ρ: 0 < < ρ and l 0 } in the, ρ-plane, jointly with the line ρ =, are transformed in the line ρ = in the, ρ-plane. There we have c = ±. Outside that line we have c = / + ρ = / +ρ. Remark 3.3. For > 0, the rate of convergence in 3.1, i.e., the rate of convergence of lnutx lnut / at/ut lnx towards zero, is measured by Ãt in 3.4 only if ρ 0. If ρ = 0, the rate of convergence in 3.1 can be of a smaller order than Ãt as may be seen in Example 4.1. For 0, Lemma 3.2 gives rise to 3.1 in a similar way as it yields Corollary EXAMPLES AND SOME ADDITIONAL COMMENTS Example 4.1. A model with ρ = ρ = 0 and > 0. Consider the model Ut = t 1+ lnt. Then x Utx Ut = t lnt +1 + x lnx, x > 0, lnt +1 i.e., ρ = 0 in 2.2, since Utx Ut t ln t+1 x 1/ lnt +1 = x lnx. Notice that H,0 x = x lnx x /, meaning that 2.2 is equivalent to Utx Ut at 1 At/ x At/ as stated in 2.3. Consequently we should choose At = at 1 1 lnt +1 RV 0, x lnx, 1 lnt +1 = t lnt +1. Theorem 2.1 yields c = 1 while Theorem 3.1 determines ρ = 0 and Ãt = At. Indeed, we have lnutx lnut lnx = lnx lnt +1 + ln 2 x lnt o ln 2, t

12 296 Fraga Alves, Gomes, de Haan and Neves as t, thus making suitable to take At = lnt +1 in the left hand side of lnutx lnut lnx lim At = lnx, x > 0 and 3.12 holds in fact with Ãt = At. Furthermore, after a few manipulations of 4.1, we get ln Utx ln Ut lnx 1 1+ ln t ln 2 t ln2 x. Therefore, the rate of convergence in 3.1 is of the order of 1/ ln 2 t = o At, as mentioned in Remark 3.3. Example 4.2. For the Fréchet model, Fx = exp x /, x 0 > 0, we get successively, Ut = ln 1 1 t = t t t 2 + o t 2 = t t 24 t 2 + o t 2. Hence, Utx Ut = x t x + o t 2 t if 1 t x 1 12 t 2 x + o t 2 if = 1. If we make correspondence with condition 2.3, we see that ρ = Likewise, 2.4 can be set as 1 if 1 a 0 t = t 2 t and A 0 t = 1 12 t 2 if = 1. According to Proposition 2.1, if we choose At = ρa 0 t = 2 t if t 2 if = 1 { if 1 2 if = 1.

13 Second Order Conditions in EVT 297 and at = t / 1 A 0 t = we get the limiting result in t +1 2 t t 3 12 t 2 1 if 1 if = 1, We will derive that 3.12 holds for Ãt = /2t, with ρ = ρ = 2 for = 1, and ρ = = ρ for 1. As seen before regarding the limit in 2.2, we have whenever 1, Ut = t t 24 t 2 + o t 2 ; Then, At = at Ut = at = 2 t +1 /2t+ +ρ ; At = ρ +ρ/2t. = 2 t t + + ρ 2 t 24 t t 2 + o t 2 2 t 2t + 2 t 2t + + ρ 2 2 t t 2 2t + + ρ + o t 2 ρ = 2 t + + ρ ρ t + o t 0, and At At = 2 t +ρ2t+ +ρ o t 12 t +ρ. Let us think on Ut at = Ut At 2ρ t +ρ = t t 2t+ +ρ 24 t 2 + o t 2 0 =: l, if 0< <1. Hence, we conclude that c = / +ρ, for 1. If we consider the case = 1, at Ut = 12 t2 12 t t t t 2 + o t 2 = t t 2 + o t 2.

14 298 Fraga Alves, Gomes, de Haan and Neves Consequently, and as was expected from Theorem 2.1, At = at Ut 1 = t 6 t + o t and At At = 3 t t + o t Ut at = 1 2 0,, i.e., c = t 12 t t + o t 1 2 = l. Since this limit l is different from zero and = 1 < ρ = 2, we indeed expected to have c = ±, as actually happens. Now, from Theorem 3.1, ρ = = and we may choose Ãt = At = t 6 t + o t, or more simply Ãt = 1/2t. Indeed, and as mentioned before for = 1, 3.12 holds true with Ãt = /2t and ρ = ρ = 2. Example 4.3. Consider the extreme value model with d.f. G x = exp + x /, 1 + x > 0, R. For this model, ln 1 1 t 1 Ut = = t 1 t t 24 t 2 + o t 2 Then = Utx Ut = 1 t 1 t + t 2 + o t if < 0 lnt 1 2 t + o t if = 0 t 1 t 2 t + o t if 0 < < 1 t t 1 12 t 2 + o t 2 if = t + o t if > 1. t x t x 2 t t 2 x x o t if 1 + o t 2 if = 1,

15 Second Order Conditions in EVT 299 i.e., we may choose, in 2.3, a 0 t = t and A 0 t = Since we get, from 2.4, 2 t 1 A 0 t = at = if t 2 if = 1 a 0t 1 A 0 t = 2 t t, with ρ = if 1 12 t 2 12 t 2 if = 1, 2 t +1 2 t t 3 12 t 2 1 if 1 if = 1 { if 1 2 if = 1. and Then at Ut = 1 lnt if 1 2 t At = ρ A 0 t = 1 6 t 2 if = 1. 1 t t t + o t + t 1+ o1 if < 0 if = t + o t if 0 < < t + o t if = t + o t if > 1, and consequently, At = at Ut + = t 1 + o1 if < o1 if = 0 lnt t + o t if 0 < < t + o t if = 1 2 t + o t if > 1.

16 300 Fraga Alves, Gomes, de Haan and Neves Then At At = 2 t o1 if < 0 2 t 1 + o1 if = 0 lnt 2 1 t1 1 + o1 if 0 < < 1 9 t 1 + o1 if = o1 if > 1 0 if < if < 1 = if > 1 + ρ =: c. Note that for = ρ = we get a finite limit At/At and different from / +ρ = 1/2. Let us now compute for 0 < < ρ, Ut at = t t + o t if 0 < < 1 t 3 2 t + o t if = if 0 < < 1 if = 1 =: l, in agreement with Theorem 2.1. Note however that l 0 for all 0 < < ρ and c = ± for all region 0 < < ρ. On another side, for heavy tails, i.e., for > 0, { lnutx lnut lnx x ρ if 0 < 1, ρ = Ãt ρ ρ = if > 1, t if 0 < < 1 3 Ãt = if = 1 2 t = At 1 + o1, if > 1 2 t now in agreement with the results in Corollary 3.1.

17 Second Order Conditions in EVT 301 Example 4.4. The most common heavy-tailed models with ρ = and 0 < < ρ then necessarily with l 0, are such that Ut = C t 1 + A t + B t 2 + o t 2, A, B 0, C > 0. For these models, and Utx Ut = C t x Utx Ut C t B t 2 x x B t 2 + o t 2, x i.e., ρ + =, or equivalently, ρ = 2. Then, 2.2 holds, provided that we choose at = C t, At = 2B t 2 1+ B t 2 and at Ut = 1 A t 2 B A 2 t 2 + o t 2. Consequently, with At, l and c provided in 2.5, 2.6 and 2.7, respectively, At = A t 1+ Ot, i.e., c = ± and Ut at, At At = A 1+ Ot 2 B t ±, = C t A t + 2B t 2 + o t 2 AC 0, i.e., l = AC 0, as mentioned at the very beginning of this example. Indeed, we could also have written Ut = l + C t 1 + B t 2 + o t 2, as t. 5. THE SECOND ORDER CONDITION FOR A GENERAL TAIL, HEAVY TAIL AND A THIRD ORDER FRAMEWORK Note that for heavy-tailed models, the second order condition 2.2 implies a third order behaviour of the function lnut, whenever we are in the region 0 < ρ, and l 0, a region where At = o At. Also, since A RV, A RV ρ and A 2 RV 2, then A dominates A 2 if ρ > 2, but A 2 dominates A if ρ < 2. From the Proof of Theorem 3.1, Case i, the third order behaviour of lnut may be written as lim ln Utx ln Ut ln x At Bt x = H ρ, ηx,

18 302 Fraga Alves, Gomes, de Haan and Neves where H is defined in 2.2, At if 0 < < ρ 2 Bt := At if ρ At 2 < < ρ and the second and third order parameters ρ and η, respectively, are given by if 0 < < ρ ρ =, η = 2 +ρ if ρ 2 < < ρ. Note that in the region ρ/2 < < ρ we get ρ η. Remark 5.1. For the case = ρ/2, excluded from this note, everything depends on the relative behaviour of A and A 2, both regularly varying functions with the same index of regular variation ρ. Note also that the situation η = ρ is the one that most often happens in practice, for standard heavy-tailed models like Fréchet, Burr, the Generalized Pareto and Student s t d.f. s. For these d.f. s, 2.2 holds with ρ = 2. However, if we induce a shift l 0 in the above mentioned models, this relation between and ρ no longer exists and we may cover all region 0 < < ρ. Finally, we mention that for the extreme value model with d.f. G, we get { if 0 1/2 ρ =, ρ = and η = if 1/2 < < 1. For more details on the way the third order framework may be used in Statistics of Extremes, see, for heavy tails, Gomes, de Haan and Peng 2002 and Fraga Alves, Gomes and de Haan 2003a, papers dealing with the estimation of the second order parameter ρ, and Gomes, Caeiro and Figueiredo 2004, a paper dealing with reduced bias extreme value index estimation. For details on the general third order framework, see Fraga Alves, de Haan and Lin 2003b, Appendix; As a final remark, we would like to emphasise the importance of all these conditions in Statistics of Extremes. The first order conditions in 2.1, 2.8 and 2.9, together with additional light conditions on k, the number of top order statistics used in the estimation of a first order parameter, enable us to guarantee consistency of semi-parametric estimators of such a parameter. The primary first order parameter is the extreme value index, but we can refer other relevant parameters of extreme events, like high quantiles, return periods or probabilities of exceedances of high levels, among others. To obtain a Central Limit Theorem for these estimators, or consistency of any estimator of a second order parameter, like the shape second order parameters ρ or ρ, discussed in this paper, it is convenient to assume a second order condition, like the ones in 2.2 and 2.10.

19 Second Order Conditions in EVT 303 For the derivation of an asymptotic non-degenerate behaviour of estimators of second order parameters, we further need to assume a third order condition, ruling the rate of convergence in 2.2 or in Such a type of condition is also quite useful for the study of any second-order reduced-bias estimators, particularly if we want to have information on the bias of such estimators. For details on this type of extreme value index estimators and the importance of third order conditions see, for instance, the most recent papers on the subject Caeiro, Gomes and Pestana, 2005; Gomes, de Haan and Henriques Rodrigues, 2007b; Gomes, Martins and Neves, 2007c. In these papers, the adequate external estimation of second order parameters leads to reduced-bias estimators with the same asymptotic variance as the biased classical estimator for heavy tails, the Hill estimator Hill, For overviews on second-order reduced-bias estimation see Reiss and Thomas Chapter 6 and Gomes, Canto e Castro, Fraga Alves and Pestana 2007a. ACKNOWLEDGMENTS Research supported by FCT/POCI 2010 and POCI ERAS Project MAT/ 58876/2004. We also acknowledge the valuable suggestions from the referees. REFERENCES [1] Caeiro, F.; Gomes, M.I. and Pestana, D Direct reduction of bias of the classical Hill estimator, Revstat, 32, [2] Draisma, G.; Haan, L. de; Peng, L. and Pereira, T.T A bootstrapbased method to achieve optimality in estimating the extreme value index, Extremes, 24, [3] Ferreira, A.; Haan, L. de and Peng, L On optimizing the estimation of high quantiles of a probability distribution, Statistics, 375, [4] Fraga Alves, M.I.; Gomes, M.I. and Haan, L. de 2003a. A new class of semi-parametric estimators of the second order parameter, Portugaliae Mathematica, 602, [5] Fraga Alves, M.I.; Haan, L. de and Lin, T. 2003b. Estimation of the parameter controlling the speed of convergence in extreme value theory, Math. Methods of Statist., 12, [6] Fraga Alves, M.I.; Haan, L. de and Lin, T Third order extended regular variation, Publications de l Institut Mathématique, 8094,

20 304 Fraga Alves, Gomes, de Haan and Neves [7] Geluk, J. and Haan, L. de Regular Variation, Extensions and Tauberian Theorems, CWI Tract 40, Center for Mathematics and Computer Science, Amsterdam, The Netherlands. [8] Gnedenko, B.V Sur la distribution limite du terme maximum d une série aléatoire, Ann. Math., 44, [9] Gomes, M.I.; Caeiro, F. and Figueiredo, F Bias reduction of a tail index estimator trough an external estimation of the second order parameter, Statistics, 386, [10] Gomes, M.I.; Canto e Castro, L.; Fraga Alves, M.I. and Pestana, D. 2007a. Statistics of extremes for IID data and breakthroughs in the estimation of the extreme value index: Laurens de Haan leading contributions, Extremes, in press. [11] Gomes, M.I.; Haan, L. de and Henriques Rodrigues, L. 2007b. Tail index estimation for heavy-tailed models: accommodation of bias in the weighted log-excesses, J. Royal Statistical Society, B, in press. [12] Gomes, M.I.; Haan, L. de and Peng, L Semi-parametric estimation of the second order parameter asymptotic and finite sample behaviour, Extremes, 54, [13] Gomes, M.I.; Martins, M.J. and Neves, M. 2007c. Improving second order reduced bias extreme value index estimation, Revstat, 52, [14] Gomes, M.I. and Neves, C Asymptotic comparison of the mixed moment and classical extreme value index estimators, Statistics and Probability Letters, in press. [15] Haan, L. de On Regular Variation and its Application to Weak Convergence of Sample Extremes, Mathematisch Centrum Amsterdam. [16] Haan, L. de Slow variation and characterization of domains of attraction. In Statistical Extremes and Applications Tiago de Oliveira, Ed., D. Reidel, Dordrecht, [17] Hill, B.M A simple general approach to inference about the tail of a distribution, Ann. Statist., 35, [18] Haan, L. de and Ferreira, A Extreme Value Theory: an Introduction, Springer Series in Operations Research and Financial Engineering. [19] Reiss, R.-D. and Thomas, M Statistical Analysis of Extreme Values, with Application to Insurance, Finance, Hydrology and Other Fields, 3 rd edition, Birkhäuser Verlag.