Adaptive Reduced-Bias Tail Index and VaR Estimation via the Bootstrap Methodology

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1 This article was downloaded by: [b-on: Biblioteca do conhecimento online UL] On: 13 October 2011, At: 00:12 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Communications in Statistics - Theory and Methods Publication details, including instructions for authors and subscription information: Adaptive Reduced-Bias Tail Index and VaR Estimation via the Bootstrap Methodology M. Ivette Gomes a, Sandra Mendonça b & Dinis Pestana a a DEIO, CEAUL, and FCUL, University of Lisbon, Lisbon, Portugal b CEAUL, Lisbon, Portugal and Department of Mathematics and Engineering, University of Madeira, Funchal, Portugal Available online: 05 Jul 2011 To cite this article: M. Ivette Gomes, Sandra Mendonça & Dinis Pestana (2011): Adaptive Reduced-Bias Tail Index and VaR Estimation via the Bootstrap Methodology, Communications in Statistics - Theory and Methods, 40:16, To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Communications in Statistics Theory and Methods, 40: , 2011 Copyright Taylor & Francis Group, LLC ISSN: print/ x online DOI: / Adaptive Reduced-Bias Tail Index and VaR Estimation via the Bootstrap Methodology M. IVETTE GOMES 1, SANDRA MENDONÇA 2, AND DINIS PESTANA 1 1 DEIO, CEAUL, and FCUL, University of Lisbon, Lisbon, Portugal 2 CEAUL, Lisbon, Portugal and Department of Mathematics and Engineering, University of Madeira, Funchal, Portugal In this article, we deal with the estimation, under a semi-parametric framework, of a positive extreme value index, the primary parameter in Statistics of Extremes, and associated estimation of the Value at Risk (VaR) at a level p, the size of the loss occurred with a small probability p. We consider second-order minimum-variance reduced-bias (MVRB) estimators, and propose the use of bootstrap computerintensive methods for the adaptive choice of thresholds, both for and Var p. Applications in the fields of insurance and finance, as well as a small-scale simulation study of the bootstrap adaptive estimators behaviour, are also provided. Keywords Bootstrap methods; Semi-parametric estimation; Statistics of extremes; Value at risk. Mathematics Subject Classification Primary 62G32, 62E20; Secondary 65C Introduction and Preliminaries In this article, we will place ourselves under a semi-parametric framework, to deal with the estimation of a positive extreme value index, the primary parameter in Statistics of Extremes. We are also interested in the associated estimation of the Value at Risk at a level p, denoted VaR p, the size of the loss occurred with a fixed small probability p. Equivalently, we can say that we are interested in the associated estimation of a (high) quantile, 1 p = F 1 p, of a probability distribution function (df) F, where F y = inf x F x y denotes the generalized inverse function of a right heavy-tailed model F. Let us denote U t = F 1 1/t, t 1, a (reciprocal) quantile function such that VaR p = U 1/p. A model F is said to be heavy-tailed whenever the right tail function, F = 1 F, is a regularly varying function with a negative index of regular Received October 2009; Accepted May 2010 Address correspondence to M. Ivette Gomes, DEIO, CEAUL, and FCUL, University of Lisbon, Lisbon, Portugal; migomes@fc.ul.pt 2946

3 Adaptive Reduced-Bias Tail Index and VaR Estimation 2947 variation, denoted 1/, i.e., for every x>0, >0 such that F tx /F t t x 1/ F RV 1/ also equivalent to say that U RV. Then we write F M EV >0, meaning that we are in the domain of attraction for maxima of an extreme value (EV) distribution function (df), given by G x EV x = { exp 1 + x 1/ 1 + x > 0 if 0 exp exp x x if = 0 but with >0. The parameter is the tail index or extreme value index, one of the primary parameters of extreme events, strongly related with other parameters of extreme events like high quantiles and probabilities of exceedance of high thresholds. The extreme value index measures essentially the weight of the right tail F = 1 F, as can be seen in Fig. 1. If <0, the right tail is light, i.e., F has a finite right endpoint (x = sup x F x < 1 <+ ); if = 0, the right tail is of an exponential type. This happens for instance for the well-known normal df, with an associated probability density function x = e x2 /2 / 2, x ; if >0, the right tail is heavy of a negative polynomial type, i.e., F has an infinite right endpoint. Heavytailed models appear often in practice in fields like Telecommunications, Insurance, Finance, and Biostatistics. Power laws, such as the Pareto income distribution and the Zipf s law for city-size distribution, were observed a few decades ago in some important phenomena in Economics and Biology and have seriously attracted scientists in recent years. For heavy-tailed parents, given a sample X = X 1 X n and the associated sample of ascending order statistics (o.s. s), X 1 n X n n, the classical extreme Figure 1. Probability density function g x = dg x /dx, for = 1, = 0, and = 1, together with x.

4 2948 Gomes et al. value index estimator is the Hill estimator (Hill, 1975), here denoted H H k, and given by H k = 1 k k ln X n i+1 n ln X n k n (1.1) i=1 the average of the k log-excesses over a high random threshold X n k n. In order to achieve consistency of H k as an estimate of, X n k n needs to be an intermediate o.s., i.e., k needs to be such that k = k n and k/n 0 as n (1.2) With Q standing for quantile function, the classical Var p -estimator, based upon the Hill estimator H k, in (1.1), is given by Q p H k = X n k+1 n c H k k c k c k n p = k (1.3) np and has been introduced in Weissman (1978). But the Hill estimator in (1.1), as well as the associated VaR-estimator in (1.3), reveals usually a high asymptotic bias, i.e., k H k is asymptotically normal with variance 2 and a non null mean value, equal to / 1, whenever k A n/k 0, finite, with A the function in (1.4). This non null asymptotic bias, together with a rate of convergence of the order of 1/ k, leads to sample paths with a high variance for small k, a high bias for large k, and a very sharp mean squared error (MSE) pattern, as a function of k. Recently, several authors have been dealing with bias reduction in the field of extremes. In articles prior to 2005, among which we mention the pioneering articles by Peng (1998), Beirlant et al. (1999), Feuerverger and Hall, and Gomes et al. (2000), bias-reduction was achieved at expenses of an increasing asymptotic variance, always greater than 2, the asymptotic variance of Hill s estimators. For an overview of this subject, see (Reiss and Thomas, 2007, Ch. 6); see also Gomes et al. (2008a). Slightly more restrictively than the whole domain of attraction (for maxima), M EV >0, and in order to be able to theoretically guarantee adequate properties for the second-order parameters estimators, we shall consider parents such that U t = Ct ( 1 + A t / + o t ) A t = t (1.4) as t, where <0 and 0. The simplest class of second-order minimumvariance reduced-bias (MVRB) extreme value index estimators, with an asymptotic variance equal to 2, is the one in Caeiro et al. (2005), used for a semi-parameteric estimation of ln VaR p in Gomes and Pestana (2007b). This class, here denoted H H k, depends upon the estimation of the second-order parameters in (1.4). Its functional form is H k H ˆ ˆ k = H k ( 1 ˆ n/k ˆ / 1 ˆ ) (1.5) where ˆ ˆ is an adequate consistent estimator of. Algorithms for the estimation of, as well as heuristic methods for the choice of k in the estimation

5 Adaptive Reduced-Bias Tail Index and VaR Estimation 2949 of through H k, are provided in Gomes and Pestana (2007a,b), and reformulated in Sec. 2 of this article. For a comparison of different reduced-bias estimators, see Gomes et al. (2007), among others. The estimator in (1.5) is indeed the simplest one and the most efficient for a large variety of underlying parents, being its behaviour illustrative of the behavior of other reduced-bias estimators of the extreme value index, either MVRB or not. We now rephrase Theorem 4.1 in Beirlant et al. (2008). Let ˆ k be any consistent estimator of the tail index, such that for k = k n intermediate and such that ka n/k, finite, as n, ) d k (ˆ k Normal bˆ 2ˆ (1.6) n Let us further assume that c k = k/np and ln c k / k 0, as n, and let us consider Q p ˆ k, with Q p H k provided in (1.3). Then, ( k Qp ˆ k /VaR p 1 ) d /ln c k Normal A bˆ n even if we work with reduced-bias tail index estimators like the one in (1.5), provided that we assume that (1.2) and (1.4) hold, ˆ ˆ is consistent for the estimation of and ˆ ln n/k = o p 1, asn. In (1.6), we have b H = 1/ 1 whereas b H = 0. The asymptotic variance in (1.6) is equal to 2 both for the Hill and the corrected-hill estimators, in (1.1) and (1.5), respectively. Consequently, the H-estimators outperform the H-estimators for all k. A similar remark applies to Q p H k, comparatively to Q p H k, with Q p H k given in (1.3). Reduced-bias estimators of high quantiles have also been studied in Matthys and Beirlant (2003), Matthys et al. (2004), Gomes and Figueiredo (2006), and Li et al. (2010), among others. In Sec. 2 of this article, we discuss the estimation of the second-order parameters and. In Sec. 3, we propose the use of bootstrap computer-intensive methods for the adaptive choice of k, not only for the use of H, as an estimator of the tail index, but also the use of Q p H as an estimator of VaR p. Applications in the fields of insurance and finance will be given in Sec. 4. A small-scale simulation study of the proposed bootstrap adaptive estimators is provided in Sec Estimation of Second-Order Parameters The reduced-bias tail index estimators require the estimation of the second-order parameters and in (1.4). Such an estimation will now be briefly discussed. 2ˆ 2.1. Estimation of the Shape Second-Order Parameter We shall consider a class of estimators, parameterized in a tuning real parameter, with the functional expression, ˆ k ˆ n k = min ( 0 3 Tn k 1 T n k 3 ) (2.1)

6 2950 Gomes et al. dependent on the statistics where T n k = ( M 1 n k ) ( M 2 n k /2) /2 ( 2 M n k /2 ) /2 ( 3 M n k /6 ) if 0 /3 ln ( Mn 1 1 ln( M 2 2 n k ) 1 ln( M 2 2 n k /2) k /2 ) 1 ln( M 3 ) if = 0 3 n k /6 M j n k = 1 k k ln X n i+1 n ln X n k n j i=1 j = Under mild restrictions on k, the statistics Tn k converge to 3 1 / 3, independently of the tuning parameter. Distributional properties of the estimators in (2.1) can be found in Fraga Alves et al. (2003). Consistency is achieved in the class of models in (1.4), for intermediate k-values, i.e., k-values such that (1.2) holds, and also such that ka n/k,as n. We have here decided for a heuristic choice of k, given by k 1 = [ n 1 ] = (2.2) where x denotes the integer part of x. With such a choice of k 1, and provided that k1 A n/k 1, we get ˆ = ˆ k 1 = o p 1/ ln n, a condition needed, in order not to have any increase in the asymptotic variance of the new bias-corrected Hill estimator in equation (1.5). Note that with the choice of k 1 in (2.2), we get k1 A n/k 1 if and only if >1/2 1/ 2 = 499 5, an almost irrelevant restriction, from a practical point of view. Other interesting alternative classes of -estimators have recently been introduced in Goegebeur et al. (2008, 2010) and Ciuperca and Mercadier (2010). Remark 2.1. Under adequate general conditions, and for an appropriate tuning parameter =, the -estimators in (2.1) show highly stable sample paths as functions of k, the number of top o.s. s used, for a range of large k-values see, for instance, the patterns of ˆ 0 k in Figs. 3 and 6. Remark 2.2. It is sensible to advise practitioners not to choose blindly the value of in (2.1): sample paths of ˆ k, as functions of k, for a few values of, should be drawn, in order to elect the value of which provides the highest stability for large k, by means of any stability criterion, like the one proposed in the Algorithm of Sec Estimation of the Scale Second-Order Parameter For the estimation of we shall consider ˆ ˆ k = ( k )ˆ n dˆ k D 0 k Dˆ k (2.3) dˆ k Dˆ k D 2ˆ k

7 Adaptive Reduced-Bias Tail Index and VaR Estimation 2951 dependent on the estimator ˆ = ˆ k 1 suggested in Sec. 2.1, with and where, for any 0, d k = 1 k k i=1 ( ) i and D k k = 1 k k i=1 ( ) i U k i with ( U i = i ln X ) n i+1 n 1 i k X n i n the scaled log-spacings. Details on the distributional behavior of the estimator in (2.3) can be found in Gomes and Martins (2002) and more recently in Gomes et al. (2008b) and Caeiro et al. (2009). Consistency is achieved for models in (1.4), k values such that (1.2) holds and ka n/k,asn, and estimators ˆ of such that ˆ = o p 1/ ln n. Alternative estimators of can be found in Caeiro and Gomes (2006) and Gomes et al. (2010) An Algorithm for the Second-Order Parameter Estimation Based on the above-mentioned comments, and on the algorithms proposed before, we now advance with the following simplified algorithm: Algorithm (Second-Order Estimates). 1. Given an observed sample x 1 x 2 x n, plot the observed values of ˆ k in (2.1), for the tuning parameters = 0 and = Consider ˆ k k, with the interval ( n n ), compute their median, denoted, and compute I = k ˆ k 2, = 0 1. Next choose the tuning parameter = 0ifI 0 I 1 ; otherwise, choose = Work with ˆ ˆ = ˆ k 1 and ˆ ˆ = ˆ ˆ k 1, ˆ k, k 1, and ˆ ˆ k given in (2.1), (2.2), and (2.3), respectively. Remark 2.3. If there are negative elements in the sample, the value of n, in the Algorithm, should be replaced by n +, the number of positive elements in the sample. Remark 2.4. This algorithm leads in almost all situations to the tuning parameter = 0 whenever 1 and = 1, otherwise. For details on this and similar algorithms; see Gomes and Pestana (2007a) Asymptotic Non-Degenerate Behaviour of the Estimators The asymptotic normality, as well as the asymptotic behaviour of bias, of the estimators ˆ n k and ˆ ˆ k in (2.1) and (2.3), respectively, is easier to state if we slightly restrict the class of models in (1.4). In this article, similarly to what has been done in Gomes et al. (2007), we consider a third-order framework where we merely make explicit a third-order term in (1.4), assuming that U t = Ct ( 1 + t / + t 2 + o t 2 ) (2.4)

8 2952 Gomes et al. as t, with C > 0, 0, <0. Note that to assume (2.4) is equivalent to say that the more general third-order condition lim t ln U tx ln U t ln x A t B t x 1 = x (2.5) holds with = < 0 and that we may choose, in (2.5), A t = t = t B t = t = A t 0 with and scale second- and third-order parameters, respectively. Under the third-order framework in (2.4), if (1.2) holds and, with A given in (1.4), ka 2 n/k A, finite, as n, ka n/k (ˆ n k ) is asymptotically normal with an asymptotic bias proportional to A. Moreover, if we additionally guarantee that ˆ = o p 1/ ln n, ka n/k (ˆ ˆ k ) is also asymptotically normal with an asymptotic bias also proportional to A,asn (further details on the subject can be seen in Caeiro et al. (2009). Remark 2.5. Despite of the fact that we have not mentioned before the possible dependency on t of the scale second- and third-order parameters and in (2.4), we can indeed have = t and = t, with and both slowly varying functions, i.e., positive measurable functions, say g, such that lim t g tx /g t = 1 for all x>0. Remark 2.6. Several common heavy-tailed models belong to the class in (2.4). Among them we mention: 1. the Fréchet model, with df F x = exp x 1/, x 0, >0, for which = = 1, = 0 5 and = 5/6; 2. the Extreme Value (EV) model, with df F x = exp ( 1 + x 1/ ), x> 1/, > 0, provided that = 1or 2. Then = = 1. For = 1/2 we have = = 0 5, and we are also in (2.4). The parameter is equal to 1 for = 1/2, 3/2 for = 1 and 1/2 for 2. The parameter is 1/4 for = 1/2, 1/12 for = 1, 11/12 for = 2 and 3 5 /24 for >2; 3. the Generalized Pareto (GP) model, with df F x = x 1/, x 0, >0, for which = = and = = 1; 4. the Burr model, with df F x = x / 1/, x 0, >0, = <0 and, as for the GP model, = = 1; 5. the Student s t -model with degrees of freedom, with a probability density function f t t = + 1 /2 ( 1 + t 2 / ) +1 /2 / /2 t > 0 for which = 1/ and = = 2/. For an explicit expression of and as a function of ; see Caeiro and Gomes (2008). Remark 2.7. The EV model in Remark 2.6, item 2, does not belong to the class in (2.4) if ( 0 2 ) \ { 1 1}, but it belongs to the class in (2.5). If 0 < <1, we get 2

9 Adaptive Reduced-Bias Tail Index and VaR Estimation 2953 =, = 1, = 1 and = /2. If 1 < <2, we get = 1, = 1/2, = 1, and = The Bootstrap Methodology and Adaptive Reduced-Bias Tail Index and VaR Estimation Let E i denote a sequence of independent, identically distributed standard exponential random variables, and define Z k = 1 k k E i and Z k = k ( Z k 1 ) (3.1) i= Asymptotic Behaviour of the MVRB Estimator We now refer the following result, a particular case of Theorem 2.1 in Caeiro et al. (2009): for models in (2.4), with A and Z k given in (1.4) and (3.1), respectively, and under mild restrictions on k, we can guarantee that H ˆ ˆ k = d + Z ( ( k A n/k + b H A 2 n/k + O p )) 1 + o p 1 (3.2) k k where, with = /, b H = 1 ( ) (3.3) 1 2 Consequently, even if k A n/k, with ka 2 n/k A, finite, the type of levels k where the mean squared error of H k is minimum, ( d k H ˆ ˆ k ) Normal ( A b H 2 = 2) n H Moreover, and regarding the VaR p -estimation through Q p H, with Q p H and c k given in (1.3), we have k ln c k ( ) Qp H k d 1 Normal ( VaR A b H 2 = 2) (3.4) n H p If ka 2 n/k, H ˆ ˆ k is O p A 2 n/k The Bootstrap Methodology in the Estimation of Optimal Sample Fractions The bootstrap methodology can enable us to estimate the optimal sample fraction (OSF) k0 H n /n = arg min k MSE H k /n, in a way similar to the one used for the classical tail index estimators, now through the use of the auxiliary statistic, T H n k = H k/2 H k k = 2 n 1 (3.5) which converges in probability to zero, for intermediate k, i.e., whenever (1.2) holds.

10 2954 Gomes et al. Under a semi-parametric framework, with denoting the mean value operator, a possible substitute for the MSE of H k is, cf. Eq. (3.2), AMSE H k = ( ) 2 Z k + b H A 2 n/k = 2 k k + b2 H A4 n/k depending on n and k, and with AMSE standing for asymptotic mean squared error. We get k 0 H n = arg min AMSE ( H k ) k = ( 4 b 2 H 2 4 n 4 ) 1/ 1 4 = k H 0 n 1 + o 1 (3.6) The results in Gomes et al. (2000) and Gomes and Oliveira (2001) enable us to get, now under the third-order framework in (2.4), the following asymptotic distributional representation for Tn H k in (3.5): T H n k = d P k + b H A 2 n/k + O p A n/k / k k with P k asymptotically standard normal, and b H given in (3.3). The AMSE of Tn H k is thus minimal at a level k 0 T n such that ka 2 n/k A 0, i.e., a level of the type of the one in (3.6), with b H replaced by b H 2 2 1, and we consequently have k 0 H n = k 0 T n ( ) o 1 (3.7) Remark 3.1. Note that a similar procedure works for the MVRB estimators in Gomes et al. (2007, 2008b), as well as for any reduced-bias estimator in the literature, provided that we assume the validity of condition (2.4). How does the bootstrap methodology then work? Given the sample X n = X 1 X n from an unknown model F, and the functional Tn H k = k X n,1 k<n, define I A as the indicator function of the event A, and consider for any n 1 = O n 1, with 0 < <1, the bootstrap sample X n 1 = X1 X n 1, from the model Fn x = 1 n n i=1 I X i x, the empirical df associated with the original sample X n. Next, associate to that bootstrap sample the corresponding bootstrap auxiliary statistic, related with Tn H, in (3.5), denoted T n 1 k = k X n 1,1 k<n 1. Then, with the obvious notation k 0 T n 1 k n 0 Tn H 1 = arg min k AMSE ( Tn 1 k ),weget k 0 T n 1 ( k 0 T n = n1 ) o n p 1 as n Consequently, for another sample size n 2, and for every >1, we have [ k 0 T n 1 ] ( ) n 1 n { = k 0 T n k0 2 n n T n } op 1 2

11 Adaptive Reduced-Bias Tail Index and VaR Estimation 2955 It is then enough to choose n 2 = n ( n 1 ) n in the expression above in order to have independence of. The simplest choice is n 2 = n 2 1 /n, i.e., = 2. We then have [ k 0 T n 1 ] 2 k 0 T n = k 0 T n 1 + o p 1 as n (3.8) 2 On the basis of (3.8) we are thus able to estimate k 0 T, and next, Eq. (3.7) enables us to estimate k 0 H n provided that we know how to estimate. We shall use here the estimator ˆ proposed in Step 3 of the Algorithm in Sec Then, with ˆk 0 T denoting the sample counterpart of k 0 T, the usual notation x for the integer part of x, and taking into account (3.6), we have the k 0 -estimate, ˆk H 0 and the -estimate ( [ ( ) 2 ˆk H 0 n n 1 2 2ˆ 1 = min n 1 ˆk 0 T n2 1 /n ˆ (ˆk 0 T n 1 ) 2 ] ) + 1 (3.9) H H n n 1 = H ˆ ˆ ˆk H 0 n n 1 (3.10) Note also that, on the basis of (3.4), and with c k given in (1.3), we get k 0 Qp k 0 Qp H = arg min AMSE ( Q p H k ) k = arg min k ln c k 2 AMSE ( H k ) (3.11) A few practical questions may be raised under the set-up developed: How does the asymptotic method work for moderate sample sizes? What is the type of the sample path of the new estimator for different values of n 1? Is the method strongly dependent on the choice of n 1? What is the sensitivity of the method with respect to the choice of the -estimate? Although aware of the theoretical need to have n 1 = o n, what happens if we choose n 1 = n? Answers to these questions are not a long way from the ones given for classical estimation (see Hall, 1990; Draisma et al., 1999; Danielsson et al., 2001; Gomes and Oliveira, 2001), and some of them will be given in Secs. 4 and 5 of this article An Algorithm for the Adaptive MVRB Estimation The Algorithm goes on: 4. Compute H ˆ ˆ k, k = 1 2 n 1, with H ˆ ˆ k given in (1.5). 5. Next, consider sub-sample sizes n 1 = o n and n 2 = n 2 1 /n For l from 1 until B = 250, generate independently B bootstrap samples x1 x n 2 and x1 x n 2 xn 2 +1 x n 1, of sizes n 2 and n 1, respectively, from the empirical df Fn x = 1 n n i=1 I X i x associated with the observed sample x 1 x n. 7. Denoting Tn k the bootstrap counterpart of T n H k, in (3.5), obtain tn 1 l k t n 2 l k, 1 l B, the observed values of the statistic T n i k i = 1 2.

12 2956 Gomes et al. For k = 1 2 n i 1, compute as well as MSE 1 n i k = 1 B B l=1 ( t n i l k ) 2 i = 1 2 (3.12) MSE 2 n i k = ln 2 c k MSE 1 n i k i = Obtain, for i = 1 2, ˆk 0 T n i = arg min 1 k n i 1 MSE 1 n i k ˆk 0 Q p n i = arg min 1 k n i 1 MSE 2 n i k 9. Compute the threshold estimate ˆk 0 H, in (3.9). 10. Obtain H H ) n n 1 = H ˆ ˆ (ˆk H 0, already provided in (3.10). 11. Compute also ( [ ( ) 2 ˆk Q p H 0 ˆk Q 1 2 2ˆ p H 0 n n 1 = min n 1 ˆk 0 Q p n 2 1 /n Finally, compute Q p Q p H = Q p H Remarks. (ˆk Q p H 0 ). 1 4ˆ (ˆk 0 Q p n 1 ) 2 ] ) + 1 (3.13) (i) If there are negative elements in the sample, and apart from the computation of c k = k/ np, used in the quantile estimator Q p H, with Q p H given in (1.3), the value of n should again be replaced by n + = n i=1 I X i >0 (the number of positive values in the sample), with n possibly denoting also n 1 or n 2. (ii) The use of the sample of size n 2, x1 x n 2, and of the extended sample of size n 1, x1 x n 2 xn 2 +1 x n 1, lead us to increase the precision of the result with a smaller B, the number of bootstrap samples generated in Step 6 of the Algorithm. This is quite similar to the use of the simulation technique of Common Random Numbers in comparison problems, when we want to decrease the variance of a final answer to z = y 1 y 2, inducing a positive dependence between y 1 and y 2. (iii) The Monte Carlo procedure in the Algorithm of this section can be replicated if we want to associate easily a standard error to the estimated parameters, i.e., we can repeat steps from 6. until 12. r times, obtain (ˆk 0 j H H j ˆk Q p H 0 j Q p j),1 j r, and take as overall estimates the averages ( ˆk H 0 j = 1 r r ˆk H 0 j H j = 1 r j=1 r j=1 H j ˆk Q p H 0 j = 1 r r j=1 ˆk Q p H 0 j Q p j = 1 r r Q p j j=1 ) together with associated standard errors. The value of B can also be adequately chosen. (iv) Although aware of the fact that k must be intermediate, we have used the entire region of search of arg min k MSE k, the region 1 k n 1, with n possibly

13 Adaptive Reduced-Bias Tail Index and VaR Estimation 2957 denoting n 1 or n 2 and MSE k standing for any mean squared error structure dependent on k. Indeed, whenever working with small values of n, k = 1or k = n 1 are also possible candidates to the value of an intermediate sequence for such n. Here, we have again the old controversy between theoreticians and practioners: k n = c ln n is intermediate for every constant c, and if we take for instance c = 1/5, k n = 1 for every n Also, Hall s formula of the asymptotically optimal level, given by k H 0 n = ( 1 2 n 2 / ( 2 2)) 1/ 1 2 valid for Hill s estimator and for models in (1.4), may lead, for a fixed n, and for several choices of the parameters and, to values of k0 H n either equal to1orton 1according as is close to 0 or quite small, respectively. (v) Although aware of the need to have n 1 = o n, it seems that, once again, like in the classical semi-parametric estimation, we get good results up till n. A possible choice seems then to be for instance n 1 = [ n 0 95] or n 1 = [ n 0 955], the one used later on in Secs. 4.1 and Applications in the Fields of Finance and Insurance 4.1. An Application to Financial Data We will now consider an illustration of the performance of the adaptive MVRB estimates under study, through the analysis of one of the sets of financial data considered in Gomes and Pestana (2007b), the log-returns of the daily closing values of the Microsoft Corp. (MSFT) stock, collected from January 4, 1999 through November 17, This corresponds to a sample of size n = 1762, with n + = 882 positive values. Figure 2 shows a box-plot (left) and a histogram (right) of the available data. Particularly, the box-plot clearly provides evidence on the heaviness of both left and right tails. Figure 2. Box-plot (left) and histogram (right) of the MSFT log-returns.

14 2958 Gomes et al. Figure 3. Estimates of the shape second-order parameter and of the scale second-order parameter for the MSFT log-returns. In Fig. 3, we present the sample path of ˆ k in (2.1), as a function of k, for = 0 and = 1, together with the sample paths of the -estimators in (2.3), also for = 0 and = 1. Note that the -estimates associated with = 0 and = 1 lead us to choose, on the basis of any stability criterion for large k, the estimate associated with = 0. The algorithm here presented led us to the -estimate ˆ 0 = 0 72, obtained at the level k 1 = [ n ] = 876. The associated -estimate is ˆ 0 = The methodology is quite resistant to different choices of k 1, leading practically to no changes in the sample paths of the MVRB estimators. In Fig. 4, we present, at the left, the estimates of the tail index, provided by the H and the H estimators, in (1.1) and (1.5), respectively. At the right, we present the corresponding quantile estimates associated with p = 1/ 2n in the quantile estimators Q H Q p H and Q H Q p H, with Q p H k provided in (1.3). Figure 4. Estimates of the tail index (left) and of the quantile 1 p VaR p, associated to p = 1/ 2n (right) for the MSFT log-returns.

15 Adaptive Reduced-Bias Tail Index and VaR Estimation 2959 Regarding the tail index estimation, note that whereas the Hill estimator is unbiased for the estimation of the tail index when the underlying model is a strict Pareto model, it exhibits a relevant bias when we have only Pareto-like tails, as happens here, and may be seen from Fig. 4 (left). A similar comment applies to the VaR-estimation, as can be seen from Fig. 4 (right). The MVRB estimators, which are asymptotically unbiased, have a smaller bias, exhibit more stable sample paths as functions of k, and enable us to take a decision upon the estimate of and VaR, even with the help of any heuristic stability criterion, like the largest run method suggested in Gomes and Figueiredo (2006). For the Hill estimator, as we know how to estimate the second-order parameters and, we have simple techniques to estimate the OSF. Indeed, we get ˆk 0 H = [( ( 1 ˆ 0 2 n + 2ˆ 0 / 2 ˆ 0 ˆ 0)) 2 1/ 1 2ˆ 0 ] + 1 = 73, and an associated -estimate equal to Up until now, we did not have the possibility of adaptively estimate the OSF associated to the MVRB estimates or, more generally, associated to any of the reduced-bias estimators available in the literature. The algorithm presented in this article helps us to provide such an adaptive choice. For a sub-sample size n 1 = n = 650, and B = 250 bootstrap generations, we have got ˆk 0 H = 164 and the MVRB tail index estimate H = 0 331, the value pictured in Fig. 4 (left). For the estimation of VaR 1/ 2n, the estimated threshold was 136, and Q 1/ 2n = , the value pictured in Fig. 4 (right) An Application to Insurance Data We shall next consider an illustration of the performance of the adaptive MVRB estimates under study, through the analysis of automobile claim amounts exceeding 1,200,000 Euro over the period , gathered from several European insurance companies co-operating with the same re-insurer (Secura Belgian Re). This data set was already studied in Beirlant et al. (2004), Vandewalle and Beirlant (2006), Beirlant et al. (2008), and Gomes et al. (2009), as an example to excessof-loss reinsurance rating and heavy-tailed distributions in car insurance. First, we have performed a preliminary graphical analysis of the data, x i,1 i n, n = 371. Figure 5 is equivalent to Fig. 2, but for the SECURA Belgian Re data. It is clear from the graph in Fig. 5, that data have been censored to the left and that the right-tail of the underlying model is quite heavy. Figures 6 and 7 are equivalent to Figs. 3 and 4, respectively, but for the Secura Belgian Re data. Figure 5. Box-plot (left) and Histogram (right) of the Secura data.

16 2960 Gomes et al. Figure 6. Estimates of the shape second-order parameter and of the scale second-order parameter for the Secura Belgian Re data. Once again, the sample paths of the -estimates associated with = 0 and = 1 lead us to choose, on the basis of any stability criterion for large k, the estimate associated with = 0. The algorithm here presented led us to the -estimate ˆ 0 = 0 74, obtained at the level k 1 = n = 368. The associated -estimate is ˆ 0 = The comments made in Sec. 4.1 on the classical versus reduced-bias tail index and VaR estimation apply here as well. For the Hill estimator, we get the estimate ˆk 0 H = 55, and an associated -estimate equal to The application of the algorithm presented in this article, with a sub-sample size n 1 = n = 284, and B = 250 bootstrap generations, led us to ˆk 0 H = 109 and to the adaptive MVRB tail index estimate H = 0 237, the value pictured in Fig. 7 (left). For the estimation of VaR 1/ 2n, the estimated threshold was 107, and Q 1/ 2n = , the value pictured in Fig. 7 (right). Figure 7. Estimates of the tail index (left) and of the quantile 1 p VaR p, associated to p = 1/ 2n (right) for the Secura Belgian Re data.

17 Adaptive Reduced-Bias Tail Index and VaR Estimation 2961 Remark 4.1. Note that the estimates obtained are not a long way from the ones obtained before in Beirlant et al. (2008). Note also that bootstrap confidence intervals are easily associated with the estimates presented Resistance of the Methodology to Changes in the Sub-Sample Size n 1 Figures 8 and 9 are associated with the data in Secs. 4.1 and 4.2, respectively. In these figures, we picture at the left, and as a function of the sub-sample size n 1, ranging, in Fig. 8, from n 1 = [ n ] = 628 until n 1 = [ n ] = 881, the estimates of the OSF for the adaptive bootstrap estimation of through the MVRB estimator H in (1.5). The associated -estimates are pictured at the right. For the MSFT log-returns, the bootstrap tail index estimates are quite stable as a function of the sub-sample size n 1 (see Fig. 8, at the right), varying from a minimum value equal to until 0.348, with a median equal to 0.329, not a long way from the -estimate obtained in Sec. 4.1, equal to and associated with the arbitrarily chosen sub sample size n 1 = [ n = 650. The bootstrap estimates of the OSF for the estimation of through H, in (1.5), vary from 7% until 12%. For the Secura Belgian Re data, and for sub-sample sizes n 1, ranging from n 1 = [ n 0 95] = 275 until n 1 = [ n ] = 370, the estimates of the OSF are extremely volatile, ranging from 24.8% until 99.7%. This obviously gives rise to estimates of ranging from until 0.291, with a median equal to 0.240, close by chance to the -estimate obtained in Sec. 4.2, at the arbitrarily chosen sub-sample size n 1 = [ n 0 955] = 284. The above-mentioned volatility of the k0 H -estimates is due to the behavior of the estimates MSE1 n i k k = 1 2 n i 1 i= 1 2, in (3.12), with two minima, for a few values of either n 1 or n 2 (11 pairs n 1 n 2 out of 96), being the global minima achieved at the largest k-value. In Fig. 10, left, we can see that only 11 pairs (ˆk 0 T n 1 ˆk 0 T n 2 ), the ones associated with peaks, are neatly wrong estimates of ( k 0 T n 1 k 0 T n 2 ). If we restrict the search to the first minimum (something easy to perform for a single data set, but not quite feasible in a simulation), we get the graph in Fig. 10, right. The associated estimates of k0 H /n vary then from 24.8% until 35%, with a median equal to 0.29 and the adaptive bootstrap estimates H n n 1 range from Figure 8. Estimates of the OSF s ˆk 0 H /n (left) and the bootstrap adaptive extreme value index estimates H (right), as functions of the sub-sample size n 1, for the MSFT log-returns.

18 2962 Gomes et al. Figure 9. Estimates of the OSF s ˆk 0 H /n (left) and the bootstrap adaptive extreme value index estimates H (right), as functions of the sub-sample size n 1, for the Secura Belgian Re data. Figure 10. Estimates (ˆk 0 T n 1 ˆk 0 T n 2 ) for the Secura Belgian Re data, associated to the global minima of MSE1 n i i = 1 2, in (3.12) (left), and to the first minimum of both MSE s (right). Figure 11. Estimates of the OSF s ˆk 0 H /n (left) and the bootstrap adaptive extreme value index estimates H (right), as functions of the sub-sample size n 1, for the Secura Belgian Re data, and a search of the first minima in the bootstrap estimates MSE1 n i, i = 1 2, in (3.12).

19 Adaptive Reduced-Bias Tail Index and VaR Estimation 2963 Table 1 Adaptive bootstrap estimates of k H 0,, kq 1/ 2n H 0, and VaR 1/ 2n Data ˆk H 0 M H ˆk Q p H 0 M Q 1/2n MSFT [628, 881] SECURA [275, 370] ,869,911 Table 2 Adaptive bootstrap estimates and 95% confidence intervals for and VaR 1/ 2n Data ˆ 95% CI s for VaR 1/ 2n 95% CI s for VaR 1/ 2n MSFT (0.308, 0.346) (16.75, 20.79) SECURA (0.225, 0.291) 9,158,849 (8,381,519, 11,696,720) Table 3 OSF s for the proposed bootstrap adaptive estimation through the MVRB estimator H in (1.5), with = 0, and the MVRB estimator Q p=1/ 2n H, with Q p H given in in (1.3), for underlying Burr parents, with = 0 5 H H Q 1/ 2n H Q 1/ 2n H n = (0.052) (0.125) (0.048) (0.144) (0.030, 0.215) (0.100, 0.600) (0.015, 0.185) (0.025, 0.540) (0.066) (0.112) (0.063) (0.149) (0.030, 0.255) (0.220, 0.630) (0.015, 0.240) (0.045, 0.620) (0.073) (0.118) (0.079) (0.137) (0.050, 0.335) (0.270, 0.735) (0.020, 0.300) (0.185, 0.680) n = (0.038) (0.109) (0.037) (0.119) (0.028, 0.172) (0.086, 0.486) (0.006, 0.126) (0.022, 0.436) (0.043) (0.112) (0.050) (0.127) (0.030, 0.194) (0.160, 0.548) (0.006, 0.194) (0.086, 0.542) (0.048) (0.108) (0.051) (0.124) (0.038, 0.222) (0.216, 0.612) (0.012, 0.220) (0.176, 0.612) n = (0.032) (0.096) (0.034) (0.101) (0.022, 0.139) (0.078, 0.415) (0.003, 0.138) (0.036, 0.415) (0.037) (0.100) (0.039) (0.109) (0.029, 0.167) (0.146, 0.514) (0.015, 0.149) (0.099, 0.513) (0.044) (0.097) (0.048) (0.109) (0.045, 0.204) (0.180, 0.583) (0.024, 0.200) (0.170, 0.578)

20 2964 Gomes et al until 0.243, with a median equal to 0.240, as can be seen in Fig. 11, where we have chosen the same scales as in Fig. 9, for easier comparison. The procedure needs thus to be used with care and we think it is not sensible to choose an ad hoc value of n 1, as we did before in Secs. 4.1 and 4.2. We suggest here the replacement of Step 5. in the Algorithm by 5. Consider values of n 1 ranging, for instance, in the above-mentioned closed interval, = [ n n ], and put n 2 = n 2 1 /n + 1. After running Steps 6 8 for the different pairs n 1 n 2 in Step 5, replace Steps 9 12 by 9. Compute the median, denoted ˆk H 0 M, of the estimates ˆk 0 H n n 1, n 1, with ˆk 0 H given in (3.9). 10. Consider as overall -estimate, H = H (ˆk 0 M) H, with H k provided in (1.5). 11. Compute the median, denoted ˆk Q p H 0 M,of ˆk Q p H 0 n n 1, n 1, with ˆk Q p H 0 given in (3.13). 12. Finally, compute Q p = Q p H (ˆk Q ) p H 0 M. Table 4 Bootstrap adaptive estimates of through the MVRB estimator H in (1.5), with = 0, and of VaR 1/ 2n, through the MVRB estimator Q p=1/ 2n H, with Q p H given in in (1.3), for underlying Burr parents, with = 0 5 H H Q 1/ 2n H Q 1/ 2n H VaR 1/ 2n n = (0.162) (0.101) (16.50) (6.38) (0.367, 0.953) (0.334, 0.776) (7.28, 65.44) (6.76, 32.26) (0.131) (0.062) (12.73) (5.04) (0.308, 0.846) (0.411, 0.645) (8.06, 52.44) (7.81, 31.43) (0.113) (0.055) (11.24) (3.95) (0.297, 0.765) (0.359, 0.599) (9.37, 54.60) (7.79, 25.02) n = (0.124) (0.077) (19.99) (8.72) (0.424, 0.829) (0.418, 0.718) (9.66, 76.22) (9.93, 43.46) (0.097) (0.050) (16.05) (5.50) (0.365, 0.710) (0.386, 0.587) (10.45, 74.83) (12.40, 36.10) (0.081) (0.038) (12.49) (4.04) (0.350, 0.638) (0.398, 0.551) (12.37, 61.44) (14.11, 30.55) n = (0.091) (0.066) (26.46) (12.10) (0.441, 0.762) (0.424, 0.706) (17.69, 115.5) (19.61, 70.55) (0.071) (0.039) (17.08) (6.41) (0.398, 0.664) (0.419, 0.585) (23.31, 87.94) (23.45, ) (0.058) (0.030) (15.65) (4.25) (0.388, 0.622) (0.412, 0.519) (22.04, 80.28) (21.59, 36.98)

21 Adaptive Reduced-Bias Tail Index and VaR Estimation 2965 In Table 1, we present, jointly with the set in Step 5, the above-mentioned estimates, in Steps 9 12, for the two sets of data under analysis in Secs. 4.1 and 4.2. The running of the above mentioned algorithm 100 times provided the average estimates and the 95% bootstrap confidence intervals for and Var 1/ 2n presented in Table 2. The 95% confidence intervals are based on the quantiles with probability and of the 100 replicates. 5. A Small-Scale Simulation Study We have also run the above mentioned algorithm r = 100 times for Burr models, F x = x / 1/ x 0 > 0 < 0 with = 0 5 and = and 1, for sample sizes n = 200, 500, and 1,000. An equivalent algorithm, with the adequate modifications, was run for the classical estimator H, in (1.1), and associated VaR 1/ 2n estimator, Q 1/ 2n H, with Q p H given in (1.3). The overall estimates of the OSF s ( k0 H/n kh 0 /n kq p H 0 /n k Q p H 0 /n ) ( ) and of VaR1/ 2n are the averages of the corresponding r partial estimates and are provided in Tables 3 and 4, respectively. Associated standard errors are provided Table 5 OSF s for the proposed bootstrap adaptive estimation through the MVRB estimator H in (1.5), with = 0, and the MVRB estimator Q p=1/ 2n H, with Q p H given in in (1.3), for underlying EV parents H H Q 1/ 2n H Q 1/ 2n H n = (0.035) (0.092) (0.030) (0.101) (0.015, 0.140) (0.065, 0.395) (0.015, 0.120) (0.020, 0.385) (0.044) (0.098) (0.038) (0.118) (0.020, 0.205) (0.100, 0.440) (0.015, 0.145) (0.025, 0.425) (0.054) (0.100) (0.051) (0.122) (0.030, 0.255) (0.120, 0.495) (0.015, 0.220) (0.045, 0.475) n = (0.031) (0.071) (0.032) (0.084) (0.010, 0.130) (0.066, 0.326) (0.006, 0.112) (0.010, 0.308) (0.034) (0.070) (0.040) (0.084) (0.016, 0.160) (0.112, 0.366) (0.006, 0.138) (0.010, 0.364) (0.039) (0.068) (0.046) (0.079) (0.024, 0.176) (0.142, 0.398) (0.006, 0.170) (0.114, 0.396) n = (0.026) (0.065) (0.027) (0.074) (0.015, 0.123) (0.056, 0.299) (0.003, 0.096) (0.026, 0.299) (0.031) (0.067) (0.034) (0.077) (0.019, 0.152) (0.112, 0.352) (0.005, 0.134) (0.054, 0.352) (0.034) (0.062) (0.038) (0.071) (0.031, 0.164) (0.168, 0.390) (0.014, 0.164) (0.118, 0.383)

22 2966 Gomes et al. Table 6 Bootstrap adaptive estimates of through the MVRB estimator H in (1.5), with = 0, and of VaR 1/ 2n, through the MVRB estimator Q p=1/ 2n H, with Q p H given in in (1.3), for underlying EV parents H H Q 1/ 2n H Q 1/ 2n H VaR 1/ 2n n = (0.179) (0.120) (31.37) (13.66) (0.200, 0.903) (0.316, 0.801) (13.25, ) (11.61, 64.23) (0.217) (0.144) (133.5) (54.62) (0.427, 1.189) (0.435, 0.991) (29.48, 670.2) (25.67, 218.7) (0.265) (0.160) (613.7) (156.4) (0.522, 1.533) (0.571, 1.230) (64.80, ) (46.65, 663.2) n = (0.155) (0.092) (48.00) (17.49) (0.175, 0.922) (0.368, 0.710) (19.60, 169.7) (20.75, 91.01) (0.170) (0.084) (256.5) (57.60) (0.427, 1.153) (0.565, 0.897) (45.96, ) (47.32, 262.2) (0.200) (0.093) (1415.8) (227.8) (0.565, 1.410) (0.774, 1.116) (115.5, ) (150.4, 992.2) n = (0.110) (0.068) (67.33) (20.50) (0.392, 0.853) (0.385, 0.679) (33.98, 339.9) (37.09, 114.0) (0.127) (0.068) (417.2) (80.30) (0.578, 1.071) (0.603, 0.861) (98.11, ) (127.8, 460.3) (0.141) (0.066) (2370.6) (371.2) (0.743, 1.276) (0.809, 1.071) (312.7, ) (377.5, ) between parenthesis close to those estimates, both presented at the first row of each entry. In the second row of each entry, we present the 95% confidence intervals based on the quantiles with probability and of the 100 partial estimates. Tables 5 and 6 are equivalent to Tables 3 and 4, respectively, but for the Extreme Value df, EV x = exp 1 + x 1/ 1 + x > 0 with = , and Note that the df EV 0 75 does not belong to the class of models in (2.4). This was done in order to understand the robustness of the method to models with Some Overall Remarks Note first that the results for the Burr model hold true for any value, provided that we scale the estimates adequately. Regarding the OSF s, and as expected, the optimal number of o.s. s involved in the estimation of any of the parameters when we use the MVRB estimator H is always bigger than the corresponding OSF associated with the classical

23 Adaptive Reduced-Bias Tail Index and VaR Estimation 2967 H-estimator. Also, the OSF associated with VaR-estimation is always smaler than the corresponding one for the extreme value index estimation. Also, as expected, the confidence intervals based on H are always of a smaller size than the ones based on H, both for the extreme value index and for the high quantile. However, the estimates of the quantiles associated with the MVRB estimator are dangerously under estimating the true value of the quantile when becomes small, say close to 1 (see the entries = 1(n = 500 and n = 1 000) for the Burr model and the entries = 1(n = 500 and n = 1 000) for the EV model), providing bootstrap confidence intervals with an upper limit below the true value of the parameter. For small values of the estimates of based on the MVRB estimator overestimate the true (as happens also, and usually even more drastically, with the estimates associated with H). Possibly, the choice = 1 in (1.5) would be more sensible in these cases. The associated bootstrap confidence intervals would then cover the true value of VaR 1/ 2n. Despite of the fact that for the EV model, with = 0 75 (we have then = 0 75 and = 0 25), we could not find any difference in the pattern of the adaptive estimates. This shows some resistance of the methodology to changes in the model in (2.4). The algorithm performs in a similar way for any other reduced-bias estimator, either MVRB or not. Acknowledgment Research partially supported by FCT/POCI and PPDCT/FEDER. References Beirlant, J., Dierckx, G., Goegebeur, Y., Matthys, G. (1999). Tail index estimation and an exponential regression model. Extremes 2: Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J. (2004). Statistics of Extremes. Theory and Applications. New York: Wiley. Beirlant, J., Figueiredo, F., Gomes, M. I., Vandewalle, B. (2008). Improved reduced-bias tail index and quantile estimators. J. Statist. Plann. Infer. 138(6): Caeiro, F., Gomes, M. I. (2006). A new class of estimators of a scale second order parameter. Extremes 9: Caeiro, F., Gomes, M. I. (2008). Minumum-variance reduced-bias tail index and high quantile estimation. Revstat 6(1):1 20. Caeiro, F., Gomes, M. I., Pestana, D. D. (2005). Direct reduction of bias of the classical Hill estimator. Revstat 3(2): Caeiro, F., Gomes, M. I., Henriques-Rodrigues, L. (2009). Reduced-bias tail index estimators under a third order framework. Commun. Statist. Theor. Meth. 38(7): Ciuperca, G., Mercadier, C. (2010). Semi-parametric estimation for heavy tailed distributions. Extremes 13(1): Danielsson, J., de Haan, L., Peng, L., de Vries, C. G. (2001). Using a bootstrap method to choose the sample fraction in the tail index estimation. J. Multivariate Anal. 76: Draisma, G., de Haan, L., Peng, L., Pereira, T. (1999). A bootstrap-based method to achieve optimality in estimating the extreme value index. Extremes 2(4): Feuerverger, A., Hall, P. (1999). Estimating a tail exponent by modelling departure from a Pareto distribution. Ann. Statist. 27:

24 2968 Gomes et al. Fraga Alves, M. I., Gomes M. I., de Haan, L. (2003). A new class of semi-parametric estimators of the second order parameter. Portugaliae Mathematica 60(2): Goegebeur, Y., Beirlant, J., de Wet, T. (2008). Linking Pareto-tail kernel goodness-of-fit statistics with tail index at optimal threshold and second order estimation. Revstat. 6(1): Goegebeur, Y., Beirlant, J., de Wet, T. (2010). Kernel estimators for the second order parameter in extreme value statistics. J. Statist. Plann. Infer. 140(9): Gomes, M. I., Figueiredo, F. (2006). Bias reduction in risk modelling: Semi-parametric quantile estimation. Test 15(2): Gomes, M. I., Martins, M. J. (2002). Asymptotically unbiased estimators of the tail index based on external estimation of the second order parameter. Extremes 5(1):5 31. Gomes, M. I., Oliveira, O. (2001). The bootstrap methodology in Statistics of Extremes: Choice of the optimal sample fraction. Extremes 4(4): Gomes, M. I., Pestana, D. D. (2007a). A simple second order reduced-bias tail index estimator. J. Statist. Computat. Simul. 77(6): Gomes, M. I., Pestana, D. D. (2007b). A sturdy reduced-bias extreme quantile (VaR) estimator. J. Amer. Statist. Assoc. 102(477): Gomes, M. I., Martins, M. J., Neves, M. M. (2000). Alternatives to a semi-parametric estimator of parameters of rare events the Jackknife methodology. Extremes 3(3): Gomes, M. I., Martins, M. J., Neves, M. M. (2007). Improving second order reduced bias extreme value index estimation. Revstat 5(2): Gomes, M. I., Canto e Castro, L., Fraga Alves, M. I., Pestana, D. (2008a). Statistics of extremes for iid data and breakthroughs in the estimation of the extreme value index: Laurens de Haan leading contributions. Extremes 11(1):3 34. Gomes, M. I., de Haan, L., Henriques-Rodrigues, L. (2008b). Tail Index estimation for heavy-tailed models: Accommodation of bias in weighted log-excesses. J. Roy. Statist. Soc. B70(1): Gomes, M. I., Mendonca, S., Pestana, D. D. (2009). Adaptive reduced-bias tail index and value-at-risk estimation. In: Sakalauskas, L., Skiadas, C., Zavadskas, E. K., eds. Applied Stochastic Models and Data Analysis IMI and VGTU Editions. Vilnius, Lithuania: Vilnius Gediminas Technical University, pp Gomes, M. I., Henriques-Rodrigues, L., Pereira, H., Pestana, D. (2010). Tail index and second order parameters semi-parametric estimation based on the log-excesses. J. Statist. Computat. Simul. 80(6): Hall, P. (1990). Using bootstrap to estimate mean squared error and selecting parameter in nonparametric problems. J. Multivariate Anal. 32: Hill, B. (1975). A simple general approach to inference about the tail of a distribution. Ann. Statist. 3: Li, D., Peng, L., Yang, J. (2010). Bias reduction for high quantiles. J. Statist. Plann. Infer. 140: Matthys, G., Beirlant, J. (2003). Estimating the extreme value index and high quantiles with exponential regression models. Statistica Sinica 13: Matthys, G., Delafosse, M., Guillou, A., Beirlant, J. (2004). Estimating catastrophic quantile levels for heavy-tailed distributions. Insur. Math. Econ. 34: Peng, L. (1998). Asymptotically unbiased estimator for the extreme-value index. Statist. Probab. Lett. 38(2): Reiss, R.-D., Thomas, M. (2007). Statistical Analysis of Extreme Values, with Application to Insurance, Finance, Hydrology and Other Fields. 3rd ed. Basel, Switzerland: Birkhäuser Verlag. Vandewalle, B., Beirlant, J. (2006). On univariate extreme value statistics and the estimation of reinsurance premiums. Insur. Math. Econ. 38(3): Weissman, I. (1978). Estimation of parameters and large quantilesbased on the k largest observations. J. Amer. Statist. Assoc. 73: