Bias Reduction in the Estimation of a Shape Second-order Parameter of a Heavy Right Tail Model

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1 Bias Reduction in the Estimation of a Shape Second-order Parameter of a Heavy Right Tail Model Frederico Caeiro Universidade Nova de Lisboa, FCT and CMA M. Ivette Gomes Universidade de Lisboa, DEIO, CEAUL and FCUL October 16, 2012 Abstract: In extreme value theory, the shape second-order parameter is an important parameter related to the speed of convergence of maximum values, linearly normalised, towards its limit law. The adequate estimation of this parameter is vital for improving the estimation of the extreme value index, the primary parameter of interest in statistics of extremes. In this article we consider a recent class of semi-parametric estimators of the shape second-order parameter for heavy right tailed models. This class of estimators, based on largest order statistics, depends on a real tuning parameter, which maes them highly flexible and possibly unbiased for several models. In this article, we are interested in the adaptive choice of such tuning parameter and the number of top order statistics used in the estimation procedure. We propose an heuristic algorithm, for the adaptive choice of the tuning parameter and for the adaptive estimation of the second-order shape parameter. An application to simulated data is also provided. Keywords and phrases. Bias reduction, heavy tails, second-order parameter, semi-parametric estimation. 1 Introduction In statistics of extremes we are usually concerned in maing inference about characteristics related to the tails of the distribution F underlying the sample data, such as a high quantile, a small probability of exceedance or the return period of a high level. When we wor with right tails, which is the most common situation, many tail characteristics depend on the extreme value index 1

2 (EVI), denoted γ, which is the shape parameter in the extreme value (EV) distribution function (d.f.), exp( (1+γx) 1/γ ), 1+γx > 0 if γ 0 EV γ (x) := exp( exp( x)), x R if γ = 0. (1.1) The EV d.f. is the limiting d.f. of the maximum suitably linearly normalised of independent, identically distributed (i.i.d.), or possibly wealy dependent random variables (r.v. s), whenever such a non-degenerate limit exists. We then say that F is in the domain of attraction for maximum values of the EV γ d.f. and use the notation F D M (EV γ ). We shall deal with heavy-tails, i.e. a positive EVI. Then (de Haan 1984), the right-tail reciprocal quantile function defined as U(t) := F (1 1/t) = inf{x : F(x) 1 1/t}, t > 1, is of regular variation with an index of regular variation equal to γ, i.e. F D M (EV γ ), γ > 0 U(tx) lim t U(t) = xγ, x > 0. (1.2) The adequate EVI-estimation is vital for the estimation of other important quantities such as extreme high quantiles, upper tail probabilities or the right end-point when the EVI is non-positive. Several classical EVI-estimators are based on the largest order statistics. The choice of the number of top-order statistics to use is not easy, since those classical estimators have a strong asymptotic bias for moderate up to large values of and a big variance for small. To improve the estimation of the EVI or other related parameters, we usually need to deal with the estimation of a secondorder parameter,. This parameter appears in a second-order condition which rules the rate of convergence of the normalised sequence of maximum values towards the limiting law EV γ, in Eq. (1.1), i.e. lnu(tx) lnu(t) γlnx lim t A(t) = x 1, x > 0, (1.3) where A must then be of regular variation with index (Gelu and de Haan 1987). To get the non-degenerate asymptotic behaviour of the -estimators, we need further to impose a third-order condition, ruling the rate of convergence in (1.3), through the limit relation, lim t lnu(tx) lnu(t) γlnx A(t) B(t) x 1 = x+ 1 +, x > 0, (1.4) where B(t) must then be of regular variation with index 0. 2

3 We shall assume that the third-order condition holds with < 0 and that we can choose A(t) = γβt, B(t) = β t, with β 0 and β 0 scale second and third-order parameters, respectively. Remar 1.1. Notice that the third-order condition, given in(1.4), holds for most of the heavy-tailed models used in applications: For the Fréchet model, with d.f. F(x) = exp( x 1/γ ), x > 0, γ > 0, we have = = 1, β = 1/2 and β = 5/6. For the Burr model, with d.f. F(x) = 1 (1+x /γ ) 1/, x > 0, γ > 0, < 0, we have = and β = β = 1; For the Student s-t ν, where ν is real positive parameter, not necessary integer, with d.f. F(x) = F(x ν) = Γ((ν+1)/2 Γ(ν/2) πν x (1+ z2 ν ) (ν+1)/2 dz, x R, ν > 0, (1.5) we have γ = 1/ν, = = 2/ν, β = (ν +1)c 2 ν/(ν +2), β = (ν 2 +4ν +2)c 2 ν/ ( (ν +2)(ν +4) ), with c ν = (νb(ν/2,1/2)) 1/ν (c 1 = π leading to the usually called Cauchy d.f.), where B is the complete Beta function; For the Half-t ν model, i.e., the absolute value of a Student s-t distribution with d.f. 2(1 F(x ν)), x > 0 and F defined in Eq. (1.5), we have γ = 1/ν, = = 2/ν, β = (ν +1)d 2 ν/(ν +2) and β = (ν 2 +4ν +2)d 2 ν/ ( (ν +2)(ν +4) ), with d ν = ( ν 2 B(ν/2,1/2))1/ν. Remar 1.2. Although = for all these classical models in Remar 1.1, we can have < after a change in location. If X is our original parent and the second-order framewor (1.3) holds for U X (t), with a second-order shape parameter, then if Y = X+a, U Y (t) = U X (t)+a is also under the second-order framewor (Araújo Santos et al. 2006), but with a second-order shape parameter given by γ, γ a :=, γ >. The estimation of the second-order parameter in (1.3) appeared for the first time in Hall and Welsh (1985). We have again to deal with the bias-variance tradeoff problem, when chosing the level for the -estimators. Next we mention the class of -estimators in Gomes et al. (2002) and 3

4 Fraga Alves et al. (2003), based on the the α-moment of the log-excesses used in Deers et al. (1989), M (α) n, := 1 (lnx n i+1:n lnx n :n ) α, α > 0, i=1 where X 1:n X 2:n... X n:n denotes the ascending order statistics associated to a random sample of size n, (X 1,X 2,...,X n ). We shall consider here the particular members of the class of estimators in Fraga Alves et al. (2003). Such a class of estimators has been first parameterized by a tuning parameter τ 0, that can be straightforwardly considered as a real number (Caeiro and Gomes 2006), and is defined as where ˆ FAGH n () = ˆ FAGH(τ) n T (τ) n, := 3(T () := min 0, ( ) (1) τ ( ) M n, M (2) τ/2 n, /2 (τ) n, 1) (T (τ) n, 3) ( M (2) n, /2 ) τ/2 ( M (3) n, /6 ) τ/3, τ R,, (1.6) with the notation a bτ = blna if τ = 0. The parameter τ does not affect the asymptotic variance and allow us to change the dominant component of asymptotic bias (Caeiro and Gomes 2008). If the parameter τ is properly chosen we can even remove the dominant component of asymptotic bias in a large set of heavy tailed models. Consequently we can have a stable sample path of the estimator, as a function of the number of top order statistics to be considered, for large. Such a behaviour maes the choice of an optimal, in the sense of minimal mean squared error (MSE), less important. Recently, Ciuperca and Mercadier (2010) extended the estimators in Gomes et al. (2002) and Fraga Alves et al. (2003), and Goegebeur et al. (2010) introduced a new class of ernel estimators based on the scaled log-spacings U i, defined by U i := i{lnx n i+1:n lnx n i:n }. (1.7) Despite of these recent classes of -estimators, the class in Fraga Alves et al. (2003), with the adequate choice of τ, is still very efficient for the most common values of (usually 1 0.5). More recently, Caeiro and Gomes (2012) considered consistent estimators of γ > 0 defined by adequate linear combinations of the scaled log-spacings in Eq. (1.7), N (α) n, := α ( ) i α 1 U i, α 1, i=1 4

5 and proposed a new class of -estimators with functional expression, ˆ CG n () = ˆ CG(τ) n () := min 0, 1+ 1, τ R, (1.8) where R (τ) n, = ( ( N (1) n, N (3/2) n, ) τ ( ) τ ( again with the notation a τ = lna whenever τ = 0. ) N (3/2) τ n, N (2) n, Remar 1.3. Notice that we could also have wored with ( ) τ ( R (τ,α 1,α 2,α 3 ) n, = ( N (α 1) n, N (α 2) n, ) τ ( 1 R (τ) n, ) τ, τ R, (1.9) ) N (α τ 2) n, N (α 3) n, ) τ, τ R, α i α j, 1 i < j 3 and min 1 i 3 (α i ) 1. But then we had to deal with the choice of the values of additional tuning parameters. If we try to minimise the asymptotic variance, the optimal (α 1,α 2,α 3 ) will obviously depend on the parameter. As an illustrative example we can say that the choice (α 1,α 2,α 3 )=(1,1.5,2) is optimal for = Further details on this topic can be found in Gomes et al. (2008) and Beirlant et al. (2012). In Section 2 we derive the asymptotic behaviour of the -estimators in (1.6) and (1.8), for intermediate values of, i.e., sequences of integers = n such that = n, n = o(n), as n, (1.10) proving their consistency and normality whenever we go further in the tail, assuming that is such that lim n A(n/) =. We also study the optimal choice of the tuning parameter τ. In Section 3, we illustrate the behaviour of these -estimators through a Monte Carlo simulation. Section 4 of this article is dedicated to the heuristic choice of the tuning parameter τ and the adaptive estimation of the shape second-order parameter. Finally, in Section 5, we draw some concluding remars. 2 Asymptotic Properties 2.1 Non Degenerate Asymptotic Behaviour Theorem 2.1 (Caeiro and Gomes 2012). Let ˆ n() denote any of the -estimators defined in (1.6) or (1.8). If the second-order condition (1.3) holds, with < 0, is intermediate and such 5

6 that A(n/), as n, then ˆ n() converges in probability to. Under the third-order framewor in (1.4), ˆ n() ( d γσ W ) = + +b 1A(n/)+b 2B(n/) (1+o p (1)), (2.1) A(n/) with W an asymptotically standard normal r.v., σ FAGH σ CG = (1 ) , (2.2) = (1 )(2 )(3 2) , (2.3) and b FAGH 1 = [ τ(1 2) 2 (3 )(3 2)+6 ( 4(2 )(1 ) 2 1 )] 12γ(1 ) 2 (1 2) 2, b CG 1 = (τ 1), b CG 2 = 2γ b FAGH 2 = (+ )(1 ) 3 (1 ) 3, (1 )(2 )(3 2) (+ ) (1 )(2 )(3 2 2 ). Corollary 2.1. Moreover, if A(n/), A 2 (n/) λ A (λ A = 0 if < ) and A(n/)B(n/) λb (both finite), as n, then (ˆ n() ) = O p (1/( A(n/))) and A(n/)(ˆ n () ) d N(λ A b 1 +λ B b 2,(γσ ) 2 ), as n. In Figure 1 we plot the asymptotic standard deviation components σ FAGH in Eqs. (2.2) and (2.3), respectively. We have σ CG < σ FAGH and σ CG, given if < We also have argmin (σ CG ) = and argmin (σ FAGH ) = 1/3. Moreover, σ.5106 CG < σfagh 1/3. Corollary 2.2. Under the assumptions of Corollary 2.1, the optimal level 1, the level that minimises the asymptotic mean square error of the estimator ˆ n(), is of the order of n 2(+ )/(1 2(+ )), i.e., 1 = O(n 2(+ )/(1 2(+ )) ). More precisely, we have, 1 = ( (1 2)(σ ) ) (+ ) ( 2 )β 2 (b 2 β ) 2 ( ) (1 2)(σ 1 )2 1 4 ( 2)β 2 (b 1 γβ+b 2 β ) 2 n 2(+ ) 1 2(+ )(1+o(1)), if <, as n. n1 4(1+o(1)), if = 6

7 σ CG σ FAGH Figure 1: Pattern of the asymptotic standard deviations σ CG and σ FAGH for 2 < Reduced-bias Estimators of : Optimal selection of the tuning parameter From Theorem 2.1 we can conclude that the tuning parameter τ only affects ˆ FAGH n () and ˆ CG n () asymptotic bias. Changing that parameter does not affect the asymptotic variance. Then, if =, the dominant component of bias of ˆ n() is b 1 A(n/)+b 2 B(n/). If <, the dominant component of bias is b 2 B(n/). Consequently, if =, and consequentially B(n/) = O(A(n/)), the asymptotic bias is b 1A(n/)+b 2B(n/) = (b 1γβ +b 2β )(n/) and we can always find a value τ 0 such that b 1 γβ+b 2 β = 0 and consequently remove the dominant component of bias. Such a value is independent of γ and, with ξ := β /β, it is given by and τ0 FAGH τ0 FAGH (,ξ) = 6[(1 2)(4(2 )(1 )2 1) 4ξ(1 ) 5 ] (1 2) 3, (2.4) (3 )(3 2) τ CG 0 τ CG 0 (,ξ) = 1 2(3 2)(2 )ξ (1 2)(3 4). (2.5) Notice that the estimation of the tuning parameter τ is not possible because the estimation of β is still an open topic in extreme value theory. Using the available values, β and β, from Remar 1.1, we have for the Fréchet model, = 1, ξ = 5/3 and we have τ CG 0 = 29/ and τ FAGH 0 = 217/ We next show, in 7

8 Figure 2, left, and as a function of, the values of τ 0 for models with ξ = 1, lie the Burr model. In the right of the same figure we show as a function of ν, the values of τ 0 for models with ξ 1, lie the Student s t ν and the Half-t ν models. In Figure 3, we show the values of τ 0 in the (ξ,)-plane. 2 τ 0 CG τ 0 FAGH 2 τ 0 CG τ 0 FAGH 1 1 τ 0 (, 1) 0 τ 0 (ν) ν Figure 2: Left: τ 0 as function of, for models with ξ = 1. Right: τ 0 t ν and Half-t ν models. as function of ν for Student s τ 0 CG τ 0 FAGH ξ ξ Figure 3: Left: τ CG 0 in the in the (ξ,)-plane. Right: τ FAGH 0 in the in the (ξ,)-plane. 8

9 3 Finite Sample Properties: Monte Carlo Simulation Study We have implemented Monte Carlo simulation experiments with 5000 samples of size n, with n = 100, 200, 500, 1000, 2000, 5000, and for the class of -estimators ˆ n FAGH(τ) () and ˆ CG(τ) n () with [ n] n 1, where [x] denotes the integer part of x. We will consider the values τ = 0 and τ = 1 for ˆ FAGH(τ) n () which are the ones suggested in Fraga Alves et al. (2003) and in other papers, as well as the heuristic choice τ FAGH 0 if 1 = 1 if > 1. For ˆ CG(τ) n () and based on the results from Section 2.2 we decided to use τ = 1, 0.5 and 0. To simplify the notation in the Figures and Tables we will denote CG(τ) ˆ CG(τ) n and FAGH(τ) ˆ n FAGH(τ). We have considered the following underlying models from Remar 1.1: Fréchet(γ) with γ = 0.5, Burr(γ,) with γ = 0.5, = 2, 1, 0.75, 0.5, 0.25 and Half-t ν with ν=2, 4. Despite of the fact that for any of the models we will show results for all aforementioned τ-values, we are led to suggest the following simple and heuristic choice, 1 if 0.5 τ CG = 0.5 if 0.5 < 1 0 if > Mean Values and Root Mean Square Errors For each value of n and for each of the above-mentioned models, we show in Figures 4-11 the simulated patterns of mean value (left), E, and root mean squared error (right), RMSE, of the -estimators FAGH(τ) and CG(τ) as function of. In Tables 1-8 we present, for the different sample sizes, the mean values (E) and root mean squared errors (RMSE) of the different estimators under study, at their optimal levels, for Burr (Tables 1-5), Fréchet (Table 6) and Half-t ν (Tables 7, 8) parents. For any value of n, the smallest bias and RMSE, at the simulated optimal level, is in bold. 3.2 Some comments We may draw the following comments: 9

10 E(^()) RMSE(^()) Figure 4: Simulated mean values (left) and RMSEs (right) of the estimators under study for n = 2000, from an Burr(0.5, 2) model. E(^()) RMSE(^()) Figure 5: Simulated mean values (left) and RMSEs (right) of the estimators under study for n = 2000, from an Burr(0.5, 1) model. As expected, since consistency of the -estimators is achieved for -values such that A(n/), as n, we do not have consistent estimates for small values of. For all the heavy-tailed models considered in this paper, the -estimator F AGH required much larger -values in order to have consistent estimates. This is more evident for the Fréchet model (see Figure 9). With an adequate choice of τ and for large -values, both -estimators have a stable simulated 10

11 E(^()) RMSE(^()) Figure 6: Simulated mean values (left) and RMSEs (right) of the estimators under study for n = 2000, from an Burr(0.5, 0.75) model. E(^()) RMSE(^()) Figure 7: Simulated mean values (left) and RMSEs (right) of the estimators under study for n = 2000, from an Burr(0.5, 0.5) model. sample path of the mean value, near the true value of. The adequate τ is usually a value close to the one we get from the asymptotic results given in Eqs. (2.4) and (2.5). Regarding the RMSE, the estimator CG has the smallest value for almost all (large) -values. For large values of, the simulated patterns of the mean value and root mean squared error of and are very close. Regarding mean values at optimal levels, CG -estimator performs better, beating always the 11

12 E(^()) RMSE(^()) Figure 8: Simulated mean values (left) and RMSEs (right) of the estimators under study for n = 2000, from an Burr(0.5, 0.25) model. E(^()) RMSE(^()) Figure 9: Simulated mean values (left) and RMSEs (right) of the estimators under study for n = 2000, from an Fréchet(0.5) model. FAGH -estimator, unless we wor with a Burr(0.5, 1) model and samples of size n Regarding the minimum RMSE, at optimal level, CG -estimator performs usually better. 12

13 E(^()) RMSE(^()) Figure 10: Simulated mean values (left) and RMSEs (right) of the estimators under study for n = 2000, from an Half t 2 model. E(^()) RMSE(^()) Figure 11: Simulated mean values (left) and RMSEs (right) of the estimators under study for n = 2000, from an Half t 4 model. 13

14 Table 1: Simulated mean values (E) and RMSE, at simulated optimal levels, of the -estimators under study, for Burr(0.5, 2) parents. n : E CG(-1) CG(-0.5) RMSE CG(-1) CG(-0.5) Table 2: Simulated mean values (E) and RMSE, at simulated optimal levels, of the -estimators under study, for Burr(0.5, 1) parents. n : E CG(-1) CG(-0.5) RMSE CG(-1) CG(-0.5)

15 Table 3: Simulated mean values (E) and RMSE, at simulated optimal levels, of the -estimators under study, for Burr(0.5, 0.75) parents. n : E CG(-1) CG(-0.5) RMSE CG(-1) CG(-0.5) Table 4: Simulated mean values (E) and RMSE, at simulated optimal levels, of the -estimators under study, for Burr(0.5, 0.5) parents. n : E CG(-1) CG(-0.5) RMSE CG(-1) CG(-0.5)

16 Table 5: Simulated mean values (E) and RMSE, at simulated optimal levels, of the -estimators under study, for Burr(0.5, 0.25) parents. n : E CG(-1) CG(-0.5) RMSE CG(-1) CG(-0.5) Table 6: Simulated mean values (E) and RMSE, at simulated optimal levels, of the -estimators under study, for Fréchet(0.5) parents. n : E CG(-1) CG(-0.5) RMSE CG(-1) CG(-0.5)

17 Table 7: Simulated mean values (E) and RMSE, at simulated optimal levels, of the -estimators under study, for Half-t 2 parents. n : E CG(-1) CG(-0.5) RMSE CG(-1) CG(-0.5) Table 8: Simulated mean values (E) and RMSE, at simulated optimal levels, of the -estimators under study, for Half-t 4 parents. n : E CG(-1) CG(-0.5) RMSE CG(-1) CG(-0.5)

18 4 Adaptive Estimation of the Shape Second-order Parameter 4.1 An Algorithm for the Heuristic Choice of τ and Estimation of We next provide a heuristic algorithm similar in spirit to the ones in Figueiredo et al. (2012) for value-at-ris (VaR)-estimation and in Gomes et al. (2011) for estimation of the EVI. 1. Given an observed sample (x 1,x 2,...,x n ), compute, for = 1,2,...,n 1, the observed values of ˆ (τ) n (), with τ = 1(0.1) Define ˆ (τ) n (;j) = round(ˆ n (),j), = [n/2],...,n 1, the rounded values of ˆ (τ) n () to j decimal places. 3. Consider the sets of values associated to equal consecutive values of a ˆ (τ) n (; j), obtained in Step 2. Set (τ) min and (τ) max the minimum and maximum values, respectively, of the set with the largest range. The largest run size is then l (τ) := max (τ) (τ) min The adaptive estimate is the median of ˆ (τ) n (), (τ) min (τ) max. 5. Tae τ A as the value of τ that maximises l(τ). 6. The best estimate, ˆ A, is the value of the the median of ˆ (τ) n (), (τ) min (τ) max that corresponds to the maximum run size l (τ), computed in Step 3. Remar 4.1. More generally, we can consider in STEP 1, τ {τ 1 (0.1)τ 2 }, τ 1 τ Application to a simulated Fréchet sample We have simulated a sample of pseudo random numbers of size n = 2000, from a Fréchet model with γ = The sample was obtained from R software (R Development Core Team, 2012) through the code: set.seed(0) x<-(-log(runif(2000)))^(-.25) For this simulated sample, we have = 1. Woring again with the class of -estimators ˆ FAGH(τ) n () and ˆ CG(τ) n (), the algorithm presented here led us to ˆτ A FAGH and the adaptive -estimates ˆ FAGH A = and ˆ CG A = 0.7, ˆτ CG A = 1.1 = We conclude that the algorithm 18

19 in Section 4.1 provides similar -estimates for both estimators and those estimates are also close to the true value. 4.3 Monte Carlo Simulation Study We have implemented a small-scale Monte Carlo simulation study of the algorithm in Section 4.1 with the class of -estimators ˆ FAGH(τ) n () and ˆ CG(τ) (). Since the evaluation of the algorithm is n very time consuming we wored with 1000 samples from the of size n, with n = 100, 200, 500, 1000, 2000, 5000, and from the Burr(0.5,-0.5) model. In Table 9, we present the simulated mean value and RMSE of the adaptive simulated estimates. Regarding mean values and RMSE, the CG -estimator performs almost always better than the FAGH -estimator. If we compare these results with the ones provided in Table 4, we conclude that the algorithm for the adaptive estimation of lead us to estimates with smaller bias (n 2000) but larger variance. Table 9: Simulated mean values (E) and RMSE of the adaptive -estimates for Burr(0.5, -0.5) parents. n : E FAGH CG RMSE FAGH CG Conclusion In this wor we have made a detailed study of a recent class of -estimators, given by Eq. (1.8), and compare it with the class of -estimators in Fraga Alves et al. (2003), in Eq. (1.6). The theoretic and simulated results allow us to conclude that the recent class of -estimators may provide an interesting alternative to the available class of -estimators in Fraga Alves et al. (2003). For large sample sizes, the algorithm presented in section 4.1 seems to perform very well in the choice of the 19

20 tuning parameter τ and in the estimation of the second-order parameter. Acnowledgments: Research partially supported by National Funds through FCT Fundação para a Ciência e a Tecnologia, projects PEst-OE/MAT/UI0006/2011 (CEAUL), PEst-OE/MAT/UI0297/2011 (CMA/UNL) and PTDC/FEDER. References [1] Araújo Santos, P., Fraga Alves, M.I., Gomes, M.I. (2006). Peas over random threshold methododlogy for tail index and high quantile estimation. Revstat 4:3, [2] Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J. (2004). Statistics of Extremes. Theory and Applications. Wiley. [3] Beirlant, J., Caeiro, F., Gomes, M.I. (2012). An Overview and Open Research Topics in the Field of Statistics of Univariate Extremes. Revstat 10(1):1-31. [4] Caeiro, F., Gomes, M.I. (2006). A new class of estimators of the scale second order parameter. Extremes 9: [5] Caeiro, F., Gomes, M.I. (2008). Minimum-variance reduced-bias tail index and high quantile estimation. Revstat 6(1):1-20. [6] Caeiro, F., Gomes, M.I. (2012). A Semi-Parameter Estimator of a Shape Second Order Parameter, Notas e Comunicações CEAUL 07/2012. [7] Deers, A., Einmahl, J., de Haan, L. (1989). A moment estimator for the index of an extreme value distribution. Ann. Statist. 17: [8] Figueiredo, F., Gomes, M.I., Henriques-Rodrigues, L., Miranda, C. (2012). A computational study of a quasi-port methodology for VaR based on second-order reduced-bias estimation. J. Statist. Comput. and Simul. 82(4): [9] Fraga Alves, M.I., Gomes, M.I., de Haan, L. (2003). A new class of semiparametric estimators of the second order parameter, Portugaliae Math., 60(2):

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