Extremes DOI 10.1007/s10687-016-0261-5 Mean-of-order p reduced-bias extreme value index estimation under a third-order framework Frederico Caeiro1 · M. Ivette Gomes2 · Jan Beirlant3,4 · Tertius de Wet5 Received: 5 May 2015 / Revised: 6 June 2016 / Accepted: 8 June 2016 © Springer Science+Business Media New York 2016 Abstract Reduced-bias versions of a very simple generalization of the ‘classical’ Hill estimator of a positive extreme value index (EVI) are put forward. The Hill estimator can be regarded as the logarithm of the mean-of-order-0 of a certain set of statistics. Instead of such a geometric mean, it is sensible to consider the mean-oforder-p (MOP) of those statistics, with p real. Under a third-order framework, the asymptotic behaviour of the MOP, optimal MOP and associated reduced-bias classes of EVI-estimators is derived. Information on the dominant non-null asymptotic bias is also provided so that we can deal with an asymptotic comparison at optimal levels of some of those classes. Large-scale Monte-Carlo simulation experiments are undertaken to provide finite sample comparisons.  Frederico Caeiro fac@fct.unl.pt M. Ivette Gomes ivette.gomes@fc.ul.pt Jan Beirlant jan.beirlant@kuleuven.be Tertius de Wet tdewet@sun.ac.za 1 CMA and DM, FCT, Universidade Nova de Lisboa, Lisbon, Portugal 2 CEAUL and DEIO, FCUL, Universidade de Lisboa, Lisbon, Portugal 3 KU Leuven, Leuven, Belgium 4 University of the Free State, Bloemfontein, South Africa 5 Stellenbosch University, Stellenbosch, South Africa F. Caeiro et al. Keywords Bias estimation · Heavy tails · Optimal levels · Semi-parametric reduced-bias estimation · Statistics of extremes AMS 2000 Subject Classifications Primary 62G32 · Secondary 65C05 1 Introduction Given a sample of size n of independent and identically distributed random variables (RVs), X1 , . . . , Xn , with a common cumulative distribution function (CDF) F , let us denote the associated ascending order statistics by X1:n ≤ · · · ≤ Xn:n . Let us further assume that there exist sequences of real constants {an > 0} and {bn ∈ R} such that (Xn:n − bn ) /an converges in distribution to a non-degenerate RV. Then, as first proved in Gnedenko (1943), the limiting CDF is necessarily of the type of the general extreme value (EV) CDF, given by  exp(−(1 + ξ x)−1/ξ ), 1 + ξ x > 0, if ξ  = 0, EVξ (x) = (1.1) exp(− exp(−x)), x ∈ R, if ξ = 0. The CDF F is then  said to belong to the max-domain of attraction of EVξ and the notation F ∈ DM EVξ is used. The parameter ξ in Eq. 1.1 is the extreme value index (EVI) for maxima, the primary parameter of large extreme events. This EVI measures the heaviness of the right tail-function, F (x) := 1 − F (x), and the heavier the right tail, the larger ξ is. Let us further use the notation Ra for the class of regularly varying functions at infinity, with an index of regular variation a ∈ R, i.e. positive measurable functions g(·) such that for all x > 0, g(tx)/g(t) → x a , as t → ∞ (see Bingham et al. (1987), for details on regular variation). In this article we work with Pareto-type underlying parents, with a positive EVI, or equivalently, right tail-functions such that F (x) = x −α L(x), α = 1/ξ > 0, with L ∈ R0 . These heavy-tailed models are quite common in many areas of application, such as biostatistics, computer science, finance, insurance and social sciences, among others. For Pareto-type models, the Hill (H) estimator (Hill 1975) is the classical estimator, which is the average of the log-excesses of the k top data, V := {Vik := ln Xn−i+1:n − ln Xn−k:n , 1 ≤ i ≤ k < n} , (1.2) being thus the logarithm of the geometric mean (or mean-of-order-0) of U := exp(V) = {Uik := Xn−i+1:n /Xn−k:n , 1 ≤ i ≤ k < n} . (1.3) Brilhante et al. (2013) considered the mean-of-order-p (MOP) of U, in Eq. 1.3, with p ≥ 0. Here, we more generally consider p ∈ R, and the class of MOP EVI-estimators, ⎧ −1 k ⎪ ⎪ p 1 1 ⎪ ⎪ U , if p < 1/ξ, p  = 0, 1 − ⎨p ik k i=1 Hk (p) := (1.4) 1/k ⎪ k k  ⎪ 1 ⎪ ⎪ Vik =: Hk , if p = 0, =k Uik ⎩ ln i=1 i=1 Mean-of-order p reduced-bias extreme value index estimation... with Hk ≡ Hk (0), the Hill EVI-estimator. This class of MOP EVI-estimators depends on the tuning parameter p ∈ R, and can be consistent for any real p < 1/ξ . It is indeed a highly flexible class of EVI-estimators, but it is not asymptotically unbiased for the moderate k-values leading to minimum mean square error (MSE). For technical simplicity, let us consider the Hall-Welsh class of models (Hall and Welsh 1985), i.e. let us assume that, with U (t) := F ← (1 − 1/t), t ≥ 1, the tail quantile function, and using the notation F ← (t) := inf{x : F (x) ≥ t}, 0 ≤ t ≤ 1,   U (t) = C t ξ 1 + A(t)/ρ + o(t ρ ) , A(t) := ξβt ρ , (1.5) as t → ∞, C > 0, ξ > 0, ρ < 0 and β  = 0. Then, as noticed in Brilhante et al. (2014), there is an optimal value for p given by    √  pM := ϕρ /ξ, with ϕρ := 1−ρ/2− ρ 2 − 4ρ + 2 2 ∈ 0, 1 − 2/2 , (1.6) which maximizes the asymptotic efficiency of the class of estimators in Eq. 1.4 with respect to the Hill estimator, leading to an optimal MOP class, Hk := Hk (pM ). From Brilhante et al. (2013) (see also, de Haan and Peng (1998), √ for p = 0), it follows that under Eq. 1.5, for 0 ≤ p < 1/(2ξ ), λ(1) := lim k A(n/k) finite, n→∞ A   and the notation N μ, σ 2 indicating a normal RV with mean value μ and variance σ 2 , we have that,   √ 2 (1−pξ )2 d 1−pξ (1) 2 (1) 2 , bH(p) b , σ = 1−ρ−pξ k (Hk (p) − ξ ) −→ N λ(1) , σH(p) = ξ 1−2pξ . A H(p) H(p) n→∞ (1.7) It is easy to check that the result in Eq. 1.7 also holds for p < 0. Hence the MOP class in Eq. 1.4 has a dominant component of asymptotic bias given by A(n/k)(1 − pξ )/(1 − ρ − pξ ) = ξβ(n/k)ρ (1 − pξ )/(1 − ρ − pξ ). We thus find that RVs of the type RBk (p, β, ρ, φ) := Hk (p) 1 − β(1 − φ)  n ρ , 1−ρ−φ k (1.8) will lead to reduced bias (RB) estimators, replacing φ by estimators of pξ or of ϕρ = pM ξ . Gomes et al. (2015) introduced the partially RB MOP class of EVI-estimators,   (1.9) RBk (p) ≡ RBk (p, β̂, ρ̂) := RBk p, β̂, ρ̂, ϕρ̂ , also dependent on a tuning parameter p and on (β̂, ρ̂), an adequate estimator of (β, ρ) in Eq. 1.5. In this paper we further consider   (1.10) RBk (p) ≡ RBk (p, β̂, ρ̂) := RBk p, β̂, ρ̂, pHk (p) . Both in Eqs. 1.9 and 1.10, we can consider p replaced by an estimator of the optimal pM in Eq. 1.6. Note further that for p = 0 in Eq. 1.10 we get the simplest class of corrected-Hill (CH) EVI-estimators, provided in Caeiro et al. (2005), the pioneering F. Caeiro et al. article on minimum variance reduced-bias (MVRB) EVI-estimation, i.e. the class in Eq. 1.10 generalizes the class   (1.11) CHk ≡ CHk (β̂, ρ̂) := RBk 0, β̂, ρ̂, 0 . In Section 2, in addition to a few technical details in the field of extreme value theory, we make a brief reference to the asymptotic behavior of the H and the CH EVI-estimators and to the estimation of second-order parameters. In Section 3, and under a third-order framework, we deal with the asymptotic behaviour of the class of MOP EVI-estimators in Eq. 1.4, for any real p < 1/ξ , and consequently of the particular case when using pM , in Eq. 1.6. We further proceed with the study of the RB classes, in Eqs. 1.9 and 1.10, providing information on the dominant non-null asymptotic bias, so that we can deal, in Section 4.2, with the asymptotic comparison, at optimal levels, of the CH EVI-estimators in Eq. 1.11 and the optimal RB MOP classes of EVI-estimators in Eqs. 1.9 and 1.10. In Section 4.1 the asymptotic comparison at optimal levels of the MOP EVI-estimators, for a real p, is developed. Section 5 is dedicated to the finite sample properties of the classes of RB MOP EVIestimators, in comparison to the behaviour of CH EVI-estimators both at optimal simulated and at non-optimally estimated levels, done through a large-scale simulation study. Some overall conclusions are drawn in Section 6. Finally, in Section 7, the proofs of the theorems in Section 3 are given. 2 Preliminary results In statistics of univariate extremes, whenever working with large values, i.e. with the right tail-function, F = 1 − F , of the model F underlying the data, F is usually said to be heavy-tailed whenever F ∈ R−1/ξ ,ξ > 0, or equivalently, the tail quantile + function U ∈ Rξ . Then F ∈ DM EVξ >0 =: DM . We thus assume the validity of any of the common first-order conditions: + F ∈ DM ⇐⇒ F ∈ R−1/ξ ⇐⇒ U ∈ Rξ . (2.1) 2.1 A brief review of higher-order conditions for a heavy right tail-function The second-order parameter ρ (≤ 0) rules the rate of convergence in the first-order condition, in Eq. 2.1, and can be defined as the non-positive parameter appearing in the limiting relation   ρ ln U (tx) − ln U (t) − ξ ln x x − 1 /ρ, if ρ < 0, (2.2) lim = ψρ (x) := ln x, if ρ = 0, t→∞ A(t) often assumed to hold for every x > 0, with A ultimately decreasing and where |A| must be of regular variation with index ρ. This second-order condition (SOC) has been widely accepted as an appropriate condition to specify a Pareto-type distribution and enables easily the derivation of the non-degenerate asymptotic bias of EVI-estimators, under a semi-parametric framework. For further details on the topic, see Beirlant et al. (2004) and de Haan and Ferreira (2006). Mean-of-order p reduced-bias extreme value index estimation... To obtain information on the normal asymptotic behaviour of estimators of second-order parameters and on the asymptotic bias of RB EVI-estimators, it is sensible to further assume a third-order condition (TOC), ruling the rate of convergence in Eq. 2.2, and which guarantees that lim t→∞ ln U (tx)−ln U (t)−ξ ln x A(t) B(t) − ψρ (x) = ψρ+ρ ′ (x), (2.3) where |B| ∈ Rρ ′ . Whenever dealing with bias reduction, it is usual to assume that Eq. 2.2 holds with A(t) = ξβt ρ , ρ < 0, as in Eq. 1.5. Models like the log-gamma (ρ = 0) and the standard Pareto (ρ = −∞) are excluded from this class. But most heavy-tailed models used in applications, such as the EVξ (x), in Eq. 1.1, the generalized Pareto, GPξ (x) = 1 + ln EVξ (x), x ≥ 0, the Fréchet, Fξ (x) = exp(−x −1/ξ ), x > 0, and the Student’s t CDFs, among others, belong to the Hall-Welsh class. It is often further assumed that the slightly more restrictive TOC,    (2.4) U (t) = C t ξ 1 + ξβt ρ /ρ + O t 2ρ , as t → ∞, holds. To assume that Eq. 2.4 holds is equivalent to saying that Eq. 2.3 holds with ρ = ρ ′ < 0 and A(t) = ξ β t ρ , B(t) = β ′ t ρ = β ′ A(t) βξ =: ζ A(t) ξ , ζ = β ′ /β  = 0, (2.5) where β and β ′ can possibly be arbitrary slowly varying functions. 2.2 Asymptotic behaviour of Hill and CH EVI-estimators + To have consistency of the H EVI-estimator over the whole DM , we need to work with intermediate values of k, i.e., a sequence of integers k = kn , 1 ≤ k < n, such that k = kn → ∞ and kn = o(n), as n → ∞. (2.6) As mentioned above, the Hill estimator usually reveals a high asymptotic bias. Indeed, it follows from Eq. 1.7 that under the general SOC in Eq. 2.2,   √ √  √ d k (Hk − ξ ) = N 0, ξ 2 + bH(1) kA(n/k) + op kA(n/k) , √ √ where the asymptotic bias bH(1) kA(n/k) = kA(n/k)/(1 − ρ) can be very large, moderate or small, i.e. goes to infinity, to a constant or to zero as n → ∞. Under the same conditions as before, and for CHk in Eq. 1.11, an adequate external estimator (β̂, ρ̂) of the vector of second-order parameters (β, ρ) using k1 upper order statistics, √ k1 > k, enables us to guarantee that k (CHk − ξ ) is asymptotically normal with variance also equal to ξ 2 but with a null mean value (see Remark 2 for possible choices of k1 ). Indeed, from the results in Caeiro et al. (2005), we know that it is possible to obtain   √  √ d k (CHk − ξ ) = N 0, ξ 2 + op kA(n/k) . F. Caeiro et al. More specifically, under the Pareto-type TOC in Eq. 2.4, we can adequately estimate the vector of second-order parameters, (β, ρ), and write (Caeiro et al. 2009; Caeiro and Gomes 2011),   √ √ 2    √ d k (CHk − ξ ) = N 0, ξ 2 + b(2) kA (n/k)+Op A(n/k) + op kA2 (n/k) , CH   ζ (2) 1 (2.7) − (1−ρ) = ξ1 1−2ρ bCH 2 , with ζ given in Eq. 2.5. Consequently, CHk outperforms Hk for all integer k. 2.3 Comments on the estimation of second-order parameters We recall the class of semi-parametric estimators of the second-order parameter ρ proposed by Fraga Alves et al. (2003). Under adequate general conditions, they are asymptotically normal estimators of ρ, whenever ρ < 0, showing, for a large variety of models and for a wide range of large k-values, highly stable sample paths as functions of k, the number of upper order statistics used. Such a class of estimators has been first parameterized by a tuning parameter τ ≥ 0, but more generally τ can be considered as a real number (Caeiro and Gomes 2006). It is defined as       ρ̂k (τ ) := −3 Vn (k; τ ) − 1 / Vn (k; τ ) − 3 , (2.8) (ℓ) Vn (k; τ ) :=   (1) τ Mk,n (2)   (2) τ/2 − Mk,n /2 τ/2  Mk,n /2 k ℓ i=1 Vik , 1 k where, with Vik given in Eq. 1.2, Mk,n := a bτ = b ln a, whenever τ = 0, (3) τ/3 , − Mk,n /6 ℓ ≥ 1, and the notation τ ∈ R. Consistency and asymptotic normality of the estimators in Eq. 2.8 were proved in Fraga Alves et al. (2003). Caeiro et al. (2009) made explicit both the asymptotic bias and variance of these ρ-estimators. Remark 1 (Choice of the tuning parameter τ in the ρ-estimation) The theoretical and simulated results in Fraga Alves et al. (2003), together with their use in RB estimation, lead us to advise in practice the use of τ = 0 for ρ ∈ [−1, 0) and τ = 1 for ρ ∈ (−∞, −1). However, practitioners should not choose blindly the value of τ in Eq. 2.8. It is sensible to draw a few sample paths of ρ̂k (τ ), as functions of k, choosing the value of τ which provides the highest stability for large k, by means of any stability criterion, such as the one suggested in Gomes and Pestana (2007). In practice, the choice of τ is more crucial than the choice of k. In the simulation study in Section 5 we are interested in models with ρ ∈ [−1, 0), the region where bias reduction is indeed strongly needed, and we thus always consider τ = 0. Remark 2 (Adequate choice of k for the ρ-estimation) As stated in Caeiro and Gomes (2008) the ideal situation for the estimation of ρ would be the choice of an ‘optimal’ Mean-of-order p reduced-bias extreme value index estimation... opt k-level, k1 , in the sense of minimal MSE. Denoting by ρ̂ = ρ̂k opt (τ ) any of the 1 opt ρ-estimators in Eq. 2.8 computed at such a k1 , ρ̂ − ρ = op (1/ ln n), as n → ∞, (2.9) a condition needed for an MVRB EVI-estimation. We stress that in practice, such opt a k1 , being of a high theoretical interest, is only of limited interest, at the current state-of-the-art. However, if we consider a level k of the order of n1−ǫ , for some small ǫ > 0, we can also guarantee Eq. 2.9 for a large class of models (see Caeiro et al. (2009), among others). This is the reason why, such as done in Caeiro et al. (2005), Gomes and Pestana (2007) and Gomes et al. (2007, 2008), the pioneering articles in MVRB-estimation, we advise in practice, as a compromise between theoretical and practical considerations, the use of any intermediate level such as k1 = k1,ǫ = ⌊n1−ǫ ⌋ for some ǫ > 0, small, with ⌊x⌋ denoting the integer part of x. The choice of ǫ is not crucial, and we often take ǫ = 0.01. Regarding the β-estimation, we shall consider the β-estimators introduced in Gomes and Martins (2002) and based on the scaled log-spacings Ui := i{ln Xn−i+1:n − ln Xn−i:n }, 1 ≤ i < n. On the basis of any consistent estimator ρ̂ of the second-order parameter ρ, we shall consider the β-estimator, β̂k (ρ̂), where, with ρ < 0,   k  −ρ   ρ  k  −ρ  k β̂k (ρ) := 1 i k i=1 k  k  −ρ  i 1 1 k i=1 k k k n 1 1 i Ui k i=1 Ui − k i=1 k      k  −ρ i 1 k i −2ρ U U − i i k k i=1 k i=1 . (2.10) Gomes and Martins (2002) obtained the asymptotic behaviour of β̂k (ρ) in Eq. 2.10, under the second-order framework in Eq. 2.2. The full derivation of the asymptotic behaviour of β̂k (ρ̂) under a third-order framework, was derived in Caeiro et al. (2009), as a generalization of a result in Gomes et al. (2008). If we replace, in Eq. 2.10, ρ by ρ̂k (τ ) given in Eq. 2.8, we obtain   p (2.11) β̂k (ρ̂k (τ )) − β ∼ −β ln(n/k) ρ̂k (τ ) − ρ . If we consider β̂ ≡ β̂k1 (ρ̂), with ρ̂ any of the estimators in Eq. 2.8, computed at any  of the  values k1 suggested in Remark 2, β̂ − β is, from Eq. 2.11, of the order of ρ̂ − ρ ln(n/k1 ). Consequently, Eq. 2.9 enables us to guarantee the consistency of β̂ ≡ β̂k1 (ρ̂). Remark 3 We have so far advised the use of the ρ-estimators in Fraga Alves et al. (2003) and the β-estimators in Gomes and Martins (2002). But recent classes of βestimators (Caeiro and Gomes 2006; Gomes et al. 2010; Henriques-Rodrigues et al. 2015) and ρ-estimators (Goegebeur et al. 2008, 2010; Ciuperca and Mercadier 2010; de Wet et al. 2012; Worms and Worms 2012; Deme et al 2013; Caeiro and Gomes 2014, 2015; Henriques-Rodrigues et al. 2014) are potential candidates for the (β, ρ)estimation. Overviews on reduced-bias estimation can be found in Chapter 6 of Reiss and Thomas (2007), Beirlant et al. (2012) and Gomes et al. (2015). F. Caeiro et al. 3 Asymptotic behaviour under a TOC Adaptations of the results in Caeiro et al. (2005) and Gomes et al. (2015) enable us to state the following theorem, still under a SOC only. √ Theorem 1 Under Eq. 1.5 and assuming that λ(1) := limn→∞ k A(n/k) is A 2 2 , given in Eq. 1.7, and whenever finite, we have for p < 1/(2ξ ), σRB(p) := σH(p) φ̂ = ϕρ̂ or pHk (p),    d √  (1) (1) 2 , bRB(p) k RBk (p, β̂, ρ̂, φ̂) − ξ −→ N λA(1) bRB(p) (φ) := (φ), σRB(p) n→∞ ρ(pξ −φ) (1−pξ −ρ)(1−ρ−φ) , (3.1) with φ the limit in probability of φ̂, and RBk (p, β, ρ, φ) given in Eq. 1.8. Itthus follows that the  asymptotic bias in Theorem 1 will be 0 in case of RBk (p) = RBk p, β̂, ρ̂, pHk (p) , for any p < 1/(2ξ ), including pM , in Eq. 1.6, but not in case   of RBk (p) = RBk p, β̂, ρ̂, ϕρ̂ when p  = pM . Remark 4 On the basis of Theorem 1 we can say that for models in Eq. 1.5, and just as CHk outperforms Hk , both RBk (pM ) and RBk (pM ) outperform Hk = Hk (pM ) for all k. Further details regarding a comparative behavior of RB and H can be found in Gomes et al. (2015). A generalization of Theorem 2 in Brilhante et al. (2013) enables us to further state: Theorem 2 Under the TOC in Eq. 2.4, or equivalently, if we assume that both Eq. 2.3 and Eq. 2.5 hold, and for levels k such that Eq. 2.6 holds, we have the validity of the following asymptotic distributional representation for any real number p < 1/(2ξ ): d σH(p) Vk (p) √ k (2) 2 Hk (p) = ξ + (1) + bH(p) A(n/k) + Op +bH(p) A (n/k) + op (A2 (n/k)), 2 bH(p) :=  A(n/k) √ k  (1−pξ )(ζ (1−pξ −ρ)2 +pξρ) , ξ(1−pξ −ρ)2 (1−pξ −2ρ)   (1) 2 , σH(p) given in Eq. 1.7, and Vk (p) asymptotically standard normal. with bH(p) (3.2) Generalizing Theorem 2 in Gomes et al. (2016a) and Theorems 2 and 3 in Gomes et al. (2015), we provide in Theorem 4 the asymptotic behaviour of the EVI-estimators in Eqs. 1.9 and 1.10. We first consider, in Theorem 3, the behaviour of RBk (p, β, ρ, φ) in Eq. 1.8, for φ = ϕρ , φ = pξ and the behaviour of RBk (p, β, ρ, φ̂) with φ̂ = pHk (p). Mean-of-order p reduced-bias extreme value index estimation... Theorem 3 Under the conditions of Theorem 2, we can write d σRB(p) Vk (p) √ k (1) + bRB(p) (φ)A(n/k)    (2) √ (1+op (1)),(3.3) (φ)A2 (n/k) + Op A(n/k) + bRB(p) RBk (p, β, ρ, φ) = ξ + k (1) (φ) was given in Eq. 3.1, and with ζ given in Eq. 2.5, where bRB(p)   1−pξ ζ (1−pξ −ρ)2 +pξρ 1−φ (2) bRB(p) (φ) := ξ(1−pξ −ρ) (1−pξ −ρ)(1−pξ −2ρ) − 1−φ−ρ , for both φ = ϕρ and√φ = pξ . Consequently, if k A(n/k) → λ(1) , finite, A √   d (1) 2 k (RBk (p, β, ρ, φ) − ξ ) −→ N λ(1) bRB(p) (φ), σRB(p) , A n→∞ (3.4) (3.5) with a null mean value if φ = pξ < 1/2 or {φ = ϕρ and p = p√M }, with pM given in Eq. 1.6. If such a mean value is null and we further assume that k A2 (n/k) → λ(2) , A finite, √   d (2) 2 bRB(p) (φ), σRB(p) k (RBk (p, β, ρ, φ) − ξ ) −→ N λ(2) . (3.6) A n→∞ Moreover, if we consider RBk (p, β, ρ, pHk (p)), Eqs. 3.3, 3.5 and 3.6 hold, with (1) (pH (p)) = 0 and bRB(p) k   1−pξ ζ (1−pξ −ρ)3 +pξρ 2 −(1−pξ )(1−pξ −ρ)(1−pξ −2ρ) (2) . (3.7) bRB(p) (pHk (p)) = ξ(1−pξ 3 1−pξ −2ρ −ρ) (2) (φ) in Eq. 3.4, whenever Remark 5 The value p = 0 in the expressions of bRB(p) (2) (pH (p)) in Eq. 3.7, leads to b(2) (0) = b(2) , already given in φ = pξ , and of bRB(p) k CH RB(0) Eq. 2.7, and in agreement with the results in Caeiro et al. (2009). Theorem 4 Under the conditions of Theorem 3, and for any of the estimators ρ̂ and β̂ in Eqs. 2.8 √ and  in 2.10, respectively,  both computed at a level k1 such that Eq. 2.9 holds, k RBk (p, β̂, ρ̂, φ̂) − ξ , φ̂ = ϕρ̂ or pHk (p), i.e. the normal- ized RB MOP EVI-estimators RBk (p) and RBk (p), respectively given in Eqs. 1.9 (1) (φ), given in Eq. 3.1, and 1.10, are asymptotically normal with a mean value, bRB(p) 2 2 , given in Eq. 1.7, φ the limit in probability of φ̂, and a variance σRB(p) ≡ σH(p) √ , finite, i.e. Eqs. 3.3 and 3.5 hold with RBk (p, β, ρ, φ) when k A(n/k) → λ(1) A replaced by RBk (p, β̂, ρ̂, φ̂). Consequently, RBk (pM ) and RBk (p) have a null dom(1) (φ), b(2) (φ) and inant component of asymptotic bias. More specifically, with bRB(p) RB(p)   (1) (2) bRB(p) (pHk (p)), bRB(p) (pHk (p)) respectively given in Eqs. 3.1, 3.4 and 3.7, we can write     (1) (2) b(1) , b(2) := bRB(p) (ϕρ ), bRB(p) (ϕρ ) , RB(p) RB(p)   (1) (2) b(1) , b(2) := bRB(p) (pξ ) = 0, bRB(p) (pHk (p)) . RB(p) RB(p) F. Caeiro et al. For RBk (pM ) and RBk (p), p √ < 1/(2ξ ), we can further get the limiting result in k = o(k1 ), as n → ∞, Eq. 3.6 for levels k such that k A(n/k) → ∞, provided that √ and we choose k1 optimal for the estimation of ρ, i.e., such that k1 A2 (n/k1 ) → λ(2) , A1 finite. √ validity of the condition, (ρ̂ − ρ) ln(n/k) = √  we can assume the  Alternatively, op 1/( kA(n/k)) , for k such that kA2 (n/k) → λ(2) , finite. A 4 Asymptotic comparison at optimal levels We next proceed to the comparison of some of the aforementioned EVI-estimators at their optimal levels. This is done in a way similar to the one used, for classical EVI-estimation, in de Haan and Peng (1998), among others, and in Gomes et al. (2007) and Caeiro and Gomes (2011), for specific sets of RB EVI-estimators. Let us assume that ξ̂k• denotes any arbitrary semi-parametric EVI-estimator, for which the asymptotic distributional representation ξ̂k• = ξ + σ• Zk• √ k + b•(1) A(n/k) + b•(2) A2 (n/k) + op (A2 (n/k)) (4.1) holds for any intermediate sequence of integers k = kn , and where Zk• is asymptoti√   d (1) cally standard normal. Then, k ξ̂k• − ξ → N (λ(1) b• , σ•2 ) provided that k is such A √ , finite, as n → ∞. If b•(1) = 0, and we consider levels k that k A(n/k) → λ(1) A √  • √ 2  d (2) , finite, as n → ∞, k ξ̂k − ξ → N (λ(2) b• , σ•2 ). such that k A (n/k) → λ(2) A A  • With Var∞ ξ̂k := σ•2 /k, the so-called asymptotic mean square error (AMSE) is       AMSE ξ̂k• := Var∞ ξ̂k• + Bias2∞ ξ̂k•   (1) 2 2 (1) σ•2 /k + b• A (n/k), if b•  = 0, =   2 (2) (1) A4 (n/k), if b• = 0, b•(2)  = 0. σ•2 /k + b• Regular variation theory (Bingham et al. 1987), enables us to show that there exists a function η(n) = η(n, ξ, ρ), such that ⎧ ⎫ 2 ⎪ ⎪  2 − 2ρ  (1)  1−2ρ ⎪ ⎪ (1) ⎬     ⎨ σ• 1−2ρ b• , if b• = 0, • =: LMSE ξ̂0• , lim η(n) AMSE ξ̂0 = 2   4ρ n→∞ ⎪ ⎪   ⎪ ⎩ σ 2 − 1−4ρ b•(2) 1−4ρ , if b•(1) = 0, b•(2) = 0 ⎪ ⎭ •   where ξ̂0• := ξ̂k•0|• and k0|• := arg min MSE ξ̂k• . k We consider the following: Definition 1 Given two biased estimators ξ̂k(1) and ξ̂k(2) , for each of which a distribu (1) (2)  tional representation of the type in Eq. 4.1 holds, with constants σ1 , b1 , b1 and   (1) (2) σ2 , b2 , b2 , respectively, both computed at their optimal levels, the asymptotic (2) (1) root efficiency (AREFF) of ξ̂0 relative to ξ̂0 is Mean-of-order p reduced-bias extreme value index estimation... 1.005 9 0.97 0.9 0.8 1.01 0 .9 0.9 1.02 5 1.015 0.6 0.99 0.97 0.95 1 0.0 0.5 1.023 1.0 1.5 2.0 0.4 0.2 0.0 0.2 0.4 a Fig. 1 The indicator ARREF∗a|0 , in Eq. 4.3, as a function of (a, ρ) AREFF1|2 ≡ AREFFξ̂ (1) |ξ̂ (2) := 0 = 0      (1) (2) LMSE ξ̂0 LMSE ξ̂0 ⎧  −2ρ  (1)  1 ⎪ ⎪  b2  1−2ρ (1) (1) ⎪ ⎨ σσ21 , if b1  = 0 ∧ b2  = 0,  (1)  b 1 (4.2)  −4ρ  (2)  1 ⎪ b ⎪  (2) (2) (1) (1) σ 2 2  1−4ρ ⎪ , if b1 = b2 = 0 ∧ b1 , b2  = 0. ⎩ σ1  (2)  b1 Remark 6 Note that the AREFF indicator, in Eq. 4.2, is defined in such a way that the higher the AREFF indicator is, the better is the first estimator. 4.1 Asymptotic comparison of optimal MOP EVI-estimators Let us first consider the MOP EVI-estimators Hk (p) in Eq. 1.4. We have LMSE(p) =  ξ 2 (1−pξ )2 1−2pξ − 2ρ 1−2ρ  1−pξ 1−pξ −ρ  2 1−2ρ . To measure the performance of the optimal Hk (p), we have computed the AREFFindicator, in Eq. 4.2, now denoted AREFFp|0 . As was done in Brilhante et al. (2013) for p ≥ 0, we can reparameterize AREFFp|0 , so that we have dependence on two parameters only, ρ and a = pξ < 1/2, possibly negative. In Fig. 1, we picture the values of 1  −2ρ  √  1−a−ρ  1−2ρ 1−2a ∗ AREFFa|0 = . (4.3)  (1−ρ)(1−a)  1−a We always lose efficiency when p < 0, and for p > 0, the gain in efficiency is not terribly high, as already detected in Brilhante et al. (2013). But, at optimal levels, F. Caeiro et al. there is a wide region of the (a, ρ)-plane where the new class of MOP EVI-estimators performs better than the H EVI-estimator. Remark 7 Note that as detected in Brilhante et al. (2013), at optimal levels, the optimal class, H = HpM can beat the optimal Hill EVI-estimator, H00 := Hk0|H (0), k0|H := arg min MSE(Hk (0)), in the whole (ξ, ρ)-plane. See also, Paulaskas and Vaičiulis (2013). For a MOP location-invariant EVI-estimation, see Gomes et al. (2016b). Remark 8 For an asymptotic comparison at optimal levels of the MOP, with p ≥ 0, and the classical moment (Dekkers et al. 1989), generalized Hill (Beirlant et al. 1996, 2005) and mixed-moment (Fraga Alves et al. 2003) EVI-estimators, see Brilhante et al. (2013). Remark 9 The lack of efficiency of the MOP EVI-estimators with p < 0 together with the results in Stehlı́k et al. (2010) and Beran et al. (2014), related to the robustness of the MOP EVI-estimators associated with p = −1 (a = −ξ < 0), deserves a further discussion of the topic, ‘robustness versus efficiency’, as initiated in Dell’Aquila and Embrechts (2006), and the finding of indicators that simultaneously take both concepts into account. This is, however, a relevant topic beyond the scope of this article. 4.2 Asymptotic comparison at optimal levels of CH and optimal RB MOP classes We finally proceed to the comparison of CH, RB and RB, at their optimal levels. In Figs. 2 and 3, we respectively present in the (ζ, ρ)–plane, with ζ the parameter given in Eq. 2.5, the patterns of the contour plots of the AREFF-indicators of RB|CH and 0.0 1.01 1.02 1.04 2 1.05 1. 01.9 5 0.99 1.01 0.995 1 0.7 5 1.03 0.7 5 0.5 1.0 3 0 1. 1.5 02 1. 2.0 0.5 0.0 0.5 1.0 Fig. 2 Contour plot of the indicator AREFFRB|CH , as a function of(ζ, ρ) 1.5 2.0 Mean-of-order p reduced-bias extreme value index estimation... 0.0 0. 6 3 0.9 0.95 0.97 0.99 5 1.075 1.0 1 .0 25 04 1. 1.0 0.5 0.8 7 0.9 1.0 2 1.0 15 1.0 9 0.9 1.1 1.0 1 2 1.5 2.0 0.5 0.0 0.5 1.0 1.5 2.0 Fig. 3 Contour plot of the indicator AREFFRB|RB , as a function of (ζ, ρ) of RB|RB. We further present in Fig. 4 the contour plots of the AREFF-indicator of RB|CH. Despite the fact that asymptotically, at optimal levels, H = HpM beats H in the whole (ξ, ρ)-plane, as detected by simulation in Gomes et al. (2016a), neither RB nor RB beats CH in the whole (ζ, ρ)-plane, but both RB and RB beat CH in wide regions of the (ζ, ρ)-plane. Also, asymptotically, and at optimal levels, the new class RB outperforms the class RB, introduced in Gomes et al. (2015), in a wide region 0.0 0.99 0.975 0.95 .9 0.8 1.05 1.1 0 1 1.005 0.5 1.025 1.02 1.015 1.01 1.007 1.005 1.025 1.02 1 1.0 1.015 1.01 1.0 1.007 1.007 1.007 1 01 0.9 0.95 1.005 1.005 0.99 0.975 1.5 1.02 1. 0 15 0.8 1. 2.0 0.5 0.0 0.5 1.0 Fig. 4 Contour plot of the indicator AREFFRB|CH , as a function of (ζ, ρ) 1.5 2.0 F. Caeiro et al. of the (ζ, ρ)-plane. However, despite being only partially RB, RB is also able to moderately beat RB. 5 Finite sample properties of the EVI-estimators In Brilhante et al. (2013), multi-sample Monte-Carlo simulation experiments of size 5000 × 20 were implemented for the class of MOP EVI-estimators, in Eq. 1.4, compared to the MVRB CH EVI-estimators, in Eq. 1.11, for a large set of ξ -values and for several sample sizes n from Fréchetξ (ρ = −1), EVξ (ρ = −ξ ), GPξ (ρ = −ξ ), and the Student-tν , with ν = 1, 2, 4, i.e. for values of ξ = 1, 0.5, 0.25 (ξ = 1/ν, ρ = −2/ν). For the same models as before, further including, for a large set of (ξ, ρ)-values, the Burrξ,ρ model, F (x) = 1 − (1 + x −ρ/ξ )1/ρ , x ≥ 0, a similar comparative analysis of Hk = Hk (pM ) and RBk (pM ), with pM and RBk (p) respectively given in Eqs. 1.6 and 1.9, was carried out in Gomes et al. (2016a), where the notation H∗ and CH∗ was used. Similar, but smaller scale simulation experiments were performed in Gomes et al. (2015) for the class RBk (p) given in Eq. 1.9. For all the aforementioned models and a model out of the Hall-Welsh class in Eq. 1.5,  the Log-Gamma(a, ξ ) model, with density f (x) = (ln x)a−1 x −1/ξ −1 / ξ a Ŵ(a) , x ≥ 1 (a, ξ > 0), we have now further included in the simulation experiments RBk (p), given in Eq. 1.10, for different admissible values of p as well as for p = pM , either assumed known or estimated. Indeed, we shall now also estimate the optimal k-value for the H EVI-estimation, as given in Hall (1982), computing 2/(1−2ρ̂)    and HA := Hk̂0|H . k̂0|H0 = (1 − ρ̂)n−ρ̂ / β̂ −2ρ̂ (5.1) 0 With ϕρ given in Eq. 1.6 and p̂M = ϕρ̂ / HA , we further consider the EVI-estimators H∗k := Hk (p̂M ), ∗ RBk := RBk (p̂M ), ∗ RBk := RBk (p̂M ). (5.2) For adaptive EVI-estimation, and in the lines of Brilhante et al. (2013), we consider sensible the use of a double–bootstrap algorithm for the joint optimal choice of (k, p), in the sense of minimal bootstrap MSE, a topic beyond the scope of this article. Anyhow, to show the potential of the new EVI-estimators, we shall consider them all computed at k̂0|H0 , i.e. H∗A := H∗k̂ 0|H0 , ∗ ∗ RBA := RBk̂0|H , 0 ∗ ∗ RBA := RBk̂0|H , 0 (5.3) ∗  ∗ with k̂0|H0 and H∗k , RBk , RBk , respectively given in Eqs. 5.1, and 5.2. Note that levels of this type are not yet at all optimal for RB EVI-estimation. For details on multi-sample simulation, we refer Gomes and Oliveira (2001). Mean-of-order p reduced-bias extreme value index estimation... 5.1 Mean values and mean square error patterns For each value of n and for each of the aforementioned models, we have first simulated the mean values (E) and root MSEs (RMSEs) of the aforementioned EVIestimators, as functions of the k upper order statistics involved in the estimation, and on the basis of the first run of size 5000. As an illustration, and due to the difficulty of estimation in this case, since ρ = −0.25, we present Fig. 5, associated with an EV0.25 parent. Due to the closeness of the sample paths of the EVI-estimators in Eq. 5.2, we picture only the most efficient one at optimal k in the sense of minimal MSE, either ∗ ∗ RB or RB . We further represent H, H∗ , CH, and among the values of p = ℓ/(10ξ ), ℓ = 1, 2, 3, 4, the more efficient (RB(p), RB(p)) EVI-estimator, with Hk , RBk (p), RBk (p), CHk and H∗k , respectively defined in Eqs. 1.2, 1.9, 1.10, 1.11 and 5.2. Figure 6 is similar to Fig. 5, but for a Fréchetξ underlying parent, with ξ = 0.25 (ρ = −1). In this less problematic case, there is not a big difference among the RB EVI-estimators, but there is a strong reduction in bias. To provide some idea about the robustness of the new EVI-estimators for model misspecification, we present Fig. 7, similar to Fig. 5, but for a Log-Gamma(a, ξ ) parent, with a = 2 and ξ = 1. Conclusions are similar for underlying parents inside and outside Hall-Welsh class of models in Eq. 1.5. For all adequate k, there is a reduction in RMSE when we move from Hk to H∗k , next to CHk and finally to either RBk or RBk . A similar comment applies to bias reduction. Indeed, on the basis of the simulations, the use of p as a tuning parameter seems to be the most sensible way of using these estimators in practice. 5.1.1 Mean values of the EVI-estimators at optimal levels As an illustration of the bias reduction achieved with the RB MOP EVI-estimators in Eqs. 1.9 and 1.10 at simulated optimal levels (levels where RMSEs are minimal E[.] 0.5 RMSE[.] H 0.20 H* H CH H* CH RB * RB * 0.15 0.4 = 4(RB) = 4(RB) = 4(RB) 0.3 0.10 = 4(RB) = 0.25 0.2 k k 0 100 200 k 0.05 0 100 k 200 Fig. 5 Mean values (left) and RMSEs (right) for an EV0.25 underlying parent and n = 1000 F. Caeiro et al. E[.] RMSE[.] 0.04 0.26 H H* H = 2(RB) H* 0.03 CH = 0.25 0.25 CH = 2(RB) RB * RB * 0.02 = 2(RB) = 2(RB) k 0.24 k 0 200 400 600 800 k 0.01 0 1000 200 400 600 800 k 1000 Fig. 6 Mean values (left) and RMSEs (right) for an Frechet0.25 underlying parent and n = 1000 as functions of k), see Tables 1, 2 and 3, respectively related to EV0.25 , Fréchet and Log-Gamma(2,1) models. We present, for n = 100, 200, 500, 1000, 2000 and 5000, and in the first ∗ eleven entries, the simulated mean values at optimal levels of H, H∗ , CH, RB , ∗   RB , and RB(p), RB(p) , p = ℓ/(10ξ ), ℓ = 1, 2, 4, with Hk , RBk (p), RBk (p), ∗  ∗ CHk and H∗k , RBk , RBk , respectively defined in Eqs. 1.2, 1.9, 1.10, 1.11 and 5.2. In the last four lines, we further provide information on the simulated mean ∗  ∗ values of the non-optimal adaptive estimators HA and H∗A , RBA , RBA , respectively given in Eqs. 5.1 and 5.3. Information on 95 % confidence intervals, based on the 20 replicates with 5000 runs each, is also provided. The simulated mean value closest to the target ξ , in both classes of EVI-estimators, i.e. the ones related to simulated optimal levels and to non-optimally estimated levels, is written in E[.] RMSE[.] 0.35 1.3 H H H* H* = 4(RB) 1.2 CH 0.25 1.1 CH = 4(RB) = 4(RB) 1 =1 k 0.9 k 0 200 400 = 4(RB) 0.15 0 200 k k 400 Fig. 7 Mean values (left) and RMSEs (right) for a Log-Gamma(a, ξ ) parent, with (a, ξ ) = (2, 1), and n = 1000 ∗ n 100 200 500 1000 2000 5000 H 0.427 ± 0.0012 0.392 ± 0.0026 0.365 ± 0.0019 0.348 ± 0.0012 0.335 ± 0.0013 0.321 ± 0.0010 H∗ CH RB ∗ RB ∗ RB(p)(ℓ = 1) RB(p)(ℓ = 1) RB(p)(ℓ = 2) RB(p)(ℓ = 2) RB(p)(ℓ = 4) RB(p)(ℓ = 4) HA H∗A ∗ RBA ∗ RBA 0.382 ± 0.0027 0.407 ± 0.0012 0.372 ± 0.0021 0.371 ± 0.0029 0.353 ± 0.0014 0.351 ± 0.0017 0.342 ± 0.0017 0.339 ± 0.0012 0.330 ± 0.0008 0.327 ± 0.0010 0.317 ± 0.0008 0.316 ± 0.0009 0.367 ± 0.0022 0.359 ± 0.0018 0.344 ± 0.0014 0.334 ± 0.0016 0.324 ± 0.0008 0.313 ± 0.0008 0.368 ± 0.0025 0.360 ± 0.0018 0.346 ± 0.0014 0.335 ± 0.0014 0.326 ± 0.0007 0.314 ± 0.0008 0.368 ± 0.0026 0.365 ± 0.0021 0.353 ± 0.0024 0.346 ± 0.0029 0.359 ± 0.0019 0.359 ± 0.0018 0.345 ± 0.0017 0.343 ± 0.0017 0.345 ± 0.0014 0.345 ± 0.0014 0.335 ± 0.0013 0.336 ± 0.0014 0.334 ± 0.0015 0.335 ± 0.0015 0.327 ± 0.0014 0.327 ± 0.0012 0.324 ± 0.0007 0.325 ± 0.0007 0.319 ± 0.0007 0.320 ± 0.0008 0.313 ± 0.0007 0.314 ± 0.0008 0.310 ± 0.0009 0.310 ± 0.0009 0.283 ± 0.0011 0.291 ± 0.0009 0.291 ± 0.0004 0.291 ± 0.0007 0.290 ± 0.0005 0.290 ± 0.0002 0.527 ± 0.0017 0.486 ± 0.0013 0.447 ± 0.0007 0.423 ± 0.0004 0.403 ± 0.0004 0.382 ± 0.0002 0.376 ± 0.0032 0.382 ± 0.0017 0.376 ± 0.0008 0.369 ± 0.0005 0.361 ± 0.0003 0.300 ± 0.0020 0.504 ± 0.0017 0.375 ± 0.0032 0.305 ± 0.0013 0.468 ± 0.0013 0.382 ± 0.0017 0.305 ± 0.0013 0.433 ± 0.0007 0.376 ± 0.0008 0.302 ± 0.0010 0.412 ± 0.0004 0.368 ± 0.0005 0.300 ± 0.0009 0.394 ± 0.0003 0.361 ± 0.0003 0.295 ± 0.0010 0.374 ± 0.0002 0.351 ± 0.0002 0.351 ± 0.0002 Mean-of-order p reduced-bias extreme value index estimation... ∗ Table 1 Simulated mean values of Hk ≡ Hk (0), H∗k , CHk , RBk , RBk , RBk (p) and RBk (p), p = ℓ/(10ξ ), ℓ = 1, 2, 4, at simulated optimal levels, and of ∗  ∗ HA , H∗A , RBA , RBA , for EVξ underlying parents, ξ = 0.25, together with 95 % confidence intervals ∗ ∗ Table 2 Simulated mean values of Hk /ξ , H∗k /ξ , CHk /ξ , RBk /ξ , RBk /ξ , RBk (p)/ξ and RBk (p)/ξ , p = ℓ/(10ξ ), ℓ = 1, 2, 4, at simulated optimal levels, and of ∗   ∗ HA /ξ, H∗A /ξ, RBA /ξ, RBA /ξ , for Fréchet underlying parents, together with 95 % confidence intervals Fréchet parents n 100 200 500 1000 2000 5000 H 1.109 ± 0.0027 1.085 ± 0.0028 1.063 ± 0.0013 1.049 ± 0.0014 1.039 ± 0.0009 1.029 ± 0.0006 0.982 ± 0.0030 0.986 ± 0.0395 0.995 ± 0.0016 0.999 ± 0.0008 1.000 ± 0.0005 H∗ CH RB ∗ RB ∗ RB(p)(ℓ = 1) RB(p)(ℓ = 1) RB(p)(ℓ = 2) RB(p)(ℓ = 2) RB(p)(ℓ = 4) RB(p)(ℓ = 4) HA H∗A ∗ RBA 1.076 ± 0.0024 1.057 ± 0.0010 1.047 ± 0.0008 1.037 ± 0.0011 1.029 ± 0.0006 1.000 ± 0.0004 0.997 ± 0.0028 1.001 ± 0.0134 1.002 ± 0.0009 1.002 ± 0.0007 1.002 ± 0.0004 1.001 ± 0.0004 0.969 ± 0.0018 0.975 ± 0.0008 0.979 ± 0.0007 0.980 ± 0.0005 0.980 ± 0.0005 0.980 ± 0.0002 0.997 ± 0.0028 1.002 ± 0.0029 0.995 ± 0.0030 0.997 ± 0.0062 1.006 ± 0.0377 1.004 ± 0.0303 1.003 ± 0.0013 1.010 ± 0.0014 1.001 ± 0.0010 1.003 ± 0.0006 1.010 ± 0.0009 1.001 ± 0.0008 1.003 ± 0.0005 1.010 ± 0.0006 1.001 ± 0.0006 1.001 ± 0.0003 1.009 ± 0.0003 1.001 ± 0.0004 1.002 ± 0.0028 1.002 ± 0.0186 1.004 ± 0.0008 1.004 ± 0.0007 1.002 ± 0.0006 1.001 ± 0.0002 1.020 ± 0.0018 1.016 ± 0.0295 1.014 ± 0.0009 1.010 ± 0.0006 1.007 ± 0.0004 1.005 ± 0.0003 1.096 ± 0.0016 1.077 ± 0.0011 1.056 ± 0.0006 1.045 ± 0.0006 1.037 ± 0.0004 1.029 ± 0.0004 0.963 ± 0.0017 0.979 ± 0.0012 0.988 ± 0.0006 0.994 ± 0.0005 0.998 ± 0.0181 1.001 ± 0.0004 0.989 ± 0.0013 1.077 ± 0.0017 0.963 ± 0.0018 1.000 ± 0.0164 1.063 ± 0.0011 0.979 ± 0.0012 1.000 ± 0.0011 1.047 ± 0.0007 0.988 ± 0.0006 1.000 ± 0.0005 1.038 ± 0.0006 0.994 ± 0.0006 1.000 ± 0.0003 1.032 ± 0.0035 0.998 ± 0.0182 1.000 ± 0.0003 1.026 ± 0.0003 1.001 ± 0.0004 F. Caeiro et al. ∗ RBA 1.097 ± 0.0021 ∗ n 100 200 500 1000 2000 5000 H 1.271 ± 0.0035 1.242 ± 0.0033 1.211 ± 0.0026 1.189 ± 0.0022 1.174 ± 0.0018 1.156 ± 0.0019 H∗ CH RB ∗ RB ∗ RB(p)(ℓ = 1) RB(p)(ℓ = 1) RB(p)(ℓ = 2) RB(p)(ℓ = 2) RB(p)(ℓ = 4) RB(p)(ℓ = 4) HA H∗A ∗ RBA ∗ RBA 1.249 ± 0.0039 1.083 ± 0.0016 1.228 ± 0.0031 1.125 ± 0.0011 1.200 ± 0.0024 1.168 ± 0.0008 1.181 ± 0.0018 1.168 ± 0.0026 1.168 ± 0.0017 1.169 ± 0.0034 1.152 ± 0.0018 1.155 ± 0.0310 1.084 ± 0.0013 1.126 ± 0.0012 1.169 ± 0.0008 1.166 ± 0.0024 1.163 ± 0.0035 1.151 ± 0.0325 1.099 ± 0.0017 1.137 ± 0.0011 1.177 ± 0.0007 1.171 ± 0.0024 1.166 ± 0.0034 1.152 ± 0.0321 1.093 ± 0.0013 1.099 ± 0.0017 1.077 ± 0.0015 1.097 ± 0.0017 1.132 ± 0.0012 1.137 ± 0.0011 1.115 ± 0.0012 1.135 ± 0.0011 1.173 ± 0.0006 1.171 ± 0.0007 1.154 ± 0.0006 1.175 ± 0.0007 1.168 ± 0.0023 1.171 ± 0.0024 1.164 ± 0.0023 1.168 ± 0.0023 1.163 ± 0.0035 1.166 ± 0.0034 1.174 ± 0.0013 1.161 ± 0.0037 1.150 ± 0.0325 1.152 ± 0.0321 1.150 ± 0.0244 1.150 ± 0.0246 1.017 ± 0.0013 1.050 ± 0.0014 1.085 ± 0.0007 1.094 ± 0.0024 1.104 ± 0.0018 1.106 ± 0.0044 1.307 ± 0.0130 1.287 ± 0.0161 1.323 ± 0.0157 1.243 ± 0.0127 1.238 ± 0.022 1.198 ± 0.0100 1.124 ± 0.0114 1.164 ± 0.0424 1.247 ± 0.0170 1.182 ± 0.0155 1.193 ± 0.0267 1.164 ± 0.0119 1.097 ± 0.0016 1.275 ± 0.0362 1.124 ± 0.0115 1.129 ± 0.0015 1.262 ± 0.0236 1.164 ± 0.0427 1.149 ± 0.0022 1.306 ± 0.0167 1.247 ± 0.0170 1.148 ± 0.0022 1.230 ± 0.0133 1.182 ± 0.0155 1.147 ± 0.0058 1.229 ± 0.018 1.193 ± 0.0267 1.141 ± 0.0160 1.191 ± 0.0106 1.164 ± 0.0119 Mean-of-order p reduced-bias extreme value index estimation... ∗ Table 3 Simulated mean values of Hk ≡ Hk (0), H∗k , CHk , RBk , RBk , RBk (p) and RBk (p), p = ℓ/(10ξ ), ℓ = 1, 2, 4, at simulated optimal levels, and of ∗  ∗ HA , H∗A , RBA , RBA , for Log-Gamma(a, ξ ) underlying parents, a = 2, ξ = 1, together with 95% confidence intervals ∗ ∗ Table 4 Simulated RMSE of Hs00 and HA , respectively denoted RMSE00 and RMSEA and REFF-indicators of H∗k , CHk , RBk , RBk , RBk (p) and RBk (p), p = ℓ/(10ξ ), ∗  ∗ ℓ = 1, 2, 4, at simulated optimal levels, and of H∗A , RBA , RBA , for EVξ underlying parents, ξ = 0.25, together with 95 % confidence intervals n 100 200 500 1000 2000 5000 RMSE00 0.246 ± 0.1698 0.200 ± 0.1669 0.157 ± 0.1591 0.133 ± 0.1527 0.113 ± 0.1462 0.092 ± 0.1379 H∗ CH RB ∗ RB ∗ RB(p)(ℓ = 1) RB(p)(ℓ = 1) RB(p)(ℓ = 2) RB(p)(ℓ = 2) RB(p)(ℓ = 4) RB(p)(ℓ = 4) RMSEA H∗A ∗ RBA ∗ RBA 1.103 ± 0.0011 1.328 ± 0.0108 1.097 ± 0.0019 1.237 ± 0.0056 1.082 ± 0.0012 1.171 ± 0.0042 1.073 ± 0.0015 1.130 ± 0.0021 1.065 ± 0.0010 1.101 ± 0.0021 1.058 ± 0.0010 1.072 ± 0.0020 1.386 ± 0.0084 1.307 ± 0.0050 1.240 ± 0.0034 1.195 ± 0.0028 1.161 ± 0.0020 1.127 ± 0.0013 1.420 ± 0.0083 1.319 ± 0.0051 1.236 ± 0.0034 1.185 ± 0.0027 1.148 ± 0.0190 1.112 ± 0.0013 1.388 ± 0.0085 1.411 ± 0.0080 1.624 ± 0.0091 1.543 ± 0.0054 1.309 ± 0.0051 1.319 ± 0.0051 1.478 ± 0.0057 1.435 ± 0.0052 1.241 ± 0.0035 1.238 ± 0.0035 1.351 ± 0.0034 1.327 ± 0.0031 1.196 ± 0.0028 1.188 ± 0.0028 1.277 ± 0.0033 1.263 ± 0.0032 1.162 ± 0.0021 1.151 ± 0.0020 1.222 ± 0.0027 1.213 ± 0.0026 1.127 ± 0.0013 1.114 ± 0.0014 1.171 ± 0.0011 1.166 ± 0.0011 3.707 ± 0.1013 3.981 ± 0.0705 3.637 ± 0.0362 3.183 ± 0.0568 2.782 ± 0.0349 2.293 ± 0.0148 0.313 ± 0.2327 0.259 ± 0.2173 0.209 ± 0.1999 0.181 ± 0.1883 0.159 ± 0.1779 0.135 ± 0.1655 1.749 ± 0.0095 1.609 ± 0.0041 1.492 ± 0.0028 1.419 ± 0.0015 1.359 ± 0.0010 1.296 ± 0.0009 2.080 ± 0.0074 1.082 ± 0.0017 1.076 ± 0.0004 1.612 ± 0.0042 1.682 ± 0.0043 1.070 ± 0.0002 1.494 ± 0.0028 1.556 ± 0.0058 1.067 ± 0.0004 1.420 ± 0.0015 1.453 ± 0.0053 1.064 ± 0.0004 1.360 ± 0.0010 1.355 ± 0.0036 1.061 ± 0.0004 1.297 ± 0.0009 F. Caeiro et al. 1.754 ± 0.0095 1.896 ± 0.0054 ∗ n 100 200 500 1000 2000 5000 RMSE00 0.212 ± 0.1547 0.163 ± 0.1520 0.117 ± 0.1432 0.091 ± 0.1345 0.071 ± 0.1977 0.052 ± 0.1764 H∗ CH RB ∗ RB ∗ RB(p)(ℓ = 1) RB(p)(ℓ = 1) RB(p)(ℓ = 2) RB(p)(ℓ = 2) RB(p)(ℓ = 4) RB(p)(ℓ = 4) RMSEA H∗A ∗ RBA ∗ RBA 1.059 ± 0.0015 1.257 ± 0.0072 1.049 ± 0.0016 1.237 ± 0.1591 1.040 ± 0.0013 1.337 ± 0.0080 1.034 ± 0.0013 1.460 ± 0.0123 1.028 ± 0.0016 1.574 ± 0.0123 1.026 ± 0.0014 1.795 ± 0.0097 1.230 ± 0.0068 1.216 ± 0.1519 1.311 ± 0.0076 1.425 ± 0.0072 1.534 ± 0.0117 1.741 ± 0.0092 1.323 ± 0.0076 1.306 ± 0.1659 1.413 ± 0.0088 1.539 ± 0.0078 1.655 ± 0.0112 1.883 ± 0.0099 1.308 ± 0.0086 1.290 ± 0.2260 1.405 ± 0.0090 1.541 ± 0.0084 1.386 ± 0.0118 1.526 ± 0.0154 1.296 ± 0.0073 1.269 ± 0.0071 1.268 ± 0.0072 1.306 ± 0.0104 1.265 ± 0.0083 1.277 ± 0.1609 1.251 ± 0.1554 1.249 ± 0.1611 1.278 ± 0.2822 1.214 ± 0.2133 1.380 ± 0.0084 1.352 ± 0.0079 1.343 ± 0.0081 1.251 ± 0.0085 1.502 ± 0.0078 1.471 ± 0.0079 1.616 ± 0.0113 1.585 ± 0.0117 1.666 ± 0.0122 1.839 ± 0.0095 1.804 ± 0.0095 1.899 ± 0.0094 1.455 ± 0.0073 1.565 ± 0.0105 1.778 ± 0.0094 1.319 ± 0.0090 1.394 ± 0.0118 1.556 ± 0.0095 1.655 ± 0.0137 1.884 ± 0.0106 0.062 ± 0.1148 0.050 ± 0.1026 0.032 ± 0.0842 0.024 ± 0.0733 0.018 ± 0.0644 0.013 ± 0.0549 1.052 ± 0.0033 1.052 ± 0.0081 1.073 ± 0.0035 1.097 ± 0.0043 1.124 ± 0.0045 1.161 ± 0.0041 1.033 ± 0.0007 1.051 ± 0.0033 1.033 ± 0.0072 1.052 ± 0.0081 1.027 ± 0.0008 1.073 ± 0.0035 1.025 ± 0.0009 1.096 ± 0.0043 1.024 ± 0.0009 1.124 ± 0.0045 1.025 ± 0.0008 1.161 ± 0.0041 Mean-of-order p reduced-bias extreme value index estimation... ∗ Table 5 Simulated RMSE of Hs00 /ξ and HA /ξ , respectively denoted RMSE00 and RMSEA and REFF-indicators of H∗k , CHk , RBk , RBk , RBk (p) and RBk (p), p = ℓ/(10ξ ), ∗  ∗ ℓ = 1, 2, 4, at simulated optimal levels, and of H∗A , RBA , RBA , for Fréchet parents, together with 95% confidence intervals ∗ ∗ Table 6 Simulated RMSE of Hs00 and HA , respectively denoted RMSE00 and RMSEA and REFF-indicators of H∗k , CHk , RBk , RBk , RBk (p) and RBk (p), p = ℓ/(10ξ ), ∗  ∗ ℓ = 1, 2, 4, at simulated optimal levels, and of H∗A , RBA , RBA , for Log-Gamma(a, ξ ) underlying parents, a = 2, ξ = 1, together with 95 % confidence intervals n 100 200 500 1000 2000 5000 RMSE00 0.365 ± 0.1588 0.315 ± 0.1735 0.263 ± 0.1432 0.233 ± 0.1231 0.209 ± 0.0987 0.183 ± 0.0886 H∗ CH RB ∗ RB ∗ RB(p)(ℓ = 1) RB(p)(ℓ = 1) RB(p)(ℓ = 2) RB(p)(ℓ = 2) RB(p)(ℓ = 4) RB(p)(ℓ = 4) RMSEA H∗A ∗ RBA ∗ RBA 1.059 ± 0.0012 1.446 ± 0.0353 1.048 ± 0.0011 1.358 ± 0.0068 1.038 ± 0.0009 1.197 ± 0.0047 1.033 ± 0.0007 1.130 ± 0.0029 1.029 ± 0.0006 1.032 ± 0.0029 1.024 ± 0.0011 1.015 ± 0.0136 1.443 ± 0.0156 1.356 ± 0.0065 1.197 ± 0.0043 1.128 ± 0.0036 1.060 ± 0.0030 1.038 ± 0.0098 1.453 ± 0.0364 1.353 ± 0.0064 1.180 ± 0.0042 1.115 ± 0.0035 1.049 ± 0.0030 1.030 ± 0.0098 1.466 ± 0.0076 1.453 ± 0.0364 1.489 ± 0.0402 1.422 ± 0.0362 1.355 ± 0.0066 1.353 ± 0.0064 1.417 ± 0.0071 1.339 ± 0.0062 1.610 ± 0.0488 1.623 ± 0.0088 1.373 ± 0.0068 0.376 ± 0.0219 1.364 ± 0.0294 1.460 ± 0.0272 1.059 ± 0.0517 1.180 ± 0.0042 1.271 ± 0.0051 1.178 ± 0.0040 1.115 ± 0.0050 1.189 ± 0.0058 1.122 ± 0.0040 1.060 ± 0.0030 1.049 ± 0.0030 1.106 ± 0.0062 1.066 ± 0.0040 1.038 ± 0.0104 1.030 ± 0.0098 1.044 ± 0.0148 1.043 ± 0.0139 1.234 ± 0.0044 1.612 ± 0.00110 1.172 ± 0.0088 1.619 ± 0.0145 1.556 ± 0.0447 0.352 ± 0.0208 0.369 ± 0.0214 0.258 ± 0.0182 0.252 ± 0.0179 0.212 ± 0.0165 1.229 ± 0.0248 1.173 ± 0.0142 1.251 ± 0.0196 1.198 ± 0.0267 1.161 ± 0.0046 1.048 ± 0.0126 1.229 ± 0.0248 1.606 ± 0.0101 1.123 ± 0.0036 1.040 ± 0.0220 1.173 ± 0.0142 1.043 ± 0.0361 1.251 ± 0.0197 1.111 ± 0.0143 1.037 ± 0.0018 1.198 ± 0.0267 1.058 ± 0.0347 1.031 ± 0.028 1.161 ± 0.0047 F. Caeiro et al. 1.364 ± 0.0298 1.186 ± 0.0043 Mean-of-order p reduced-bias extreme value index estimation... bold. Generally denoting by T any of the aforementioned EVI-estimators, note that for Fréchet underlying parents, T /ξ does not depend on ξ . For a proof, see Gomes et al. (2016a). 5.1.2 Mean square errors and relative efficiency indicators at optimal levels We have computed   the Hill estimator at the simulated value of k0|H0 := arg mink RMSE Hk , the simulated optimal k in the sense of minimum RMSE. Such ∗ ∗ an estimator is denoted by Hs00 . With RB denoting RB or RB or RB(p) or RB(p) we have also computed RBs00 , i.e. the EVI-estimator RBk computed at the simulated  value of k0|RB := arg min MSE RB . The simulated indicators are REFFRB|H := k k    RMSE Hs00 /RMSE RBs00 . Similar REFF-indicators, REFFH∗ |H and REFFCH|H , have also been simulated. A similar indicator was further computed for the adaptive EVI-estimators and comparatively to the simulated RMSE of HA , denoted RMSEA in Tables 4, 5 and 6, used here as an illustration of the obtained results. In the first row, we provide the RMSE of Hs00 , denoted by RMSE00 , so that we can easily recover the RMSE of all other estimators. The following rows provide the REFF-indicators of the different EVI-estimators under study. As before, a similar mark (bold) is used for the highest REFF-indicators in both classes. 6 Overall conclusions • • • Regarding the MOP EVI-estimation, we always lose efficiency when p < 0, and for p ≥ 0, the gain in efficiency is not terribly high, as already detected in Brilhante et al. (2013). But, at optimal levels, the optimal MOP class, H = H(pM ) beats the H=H(0) EVI-estimator in the whole (ξ, ρ)-plane, a very uncommon situation among classical EVI-estimators. Despite this last comment, that asymptotically at optimal levels H beats H in the whole (ξ, ρ)-plane, neither RB nor RB beats CH in the whole (ζ, ρ)-plane. But, again asymptotically and at optimal levels, both RB and RB beat CH, and also RB beats RB, in wide, interesting regions of the (ζ, ρ)-plane. For the large variety of simulated models considered, there is generally a reduction in RMSE, as well as in bias. We get estimates closer to the target ξ , when ∗ • ∗ we move from H to H∗ , next to CH and finally to either RB or to RB . And this happens both at simulated optimal levels and non-optimally estimated levels. Such a reduction is particularly high for values of ρ approaching zero. This happens even when we work with models beyond the scope of the stated theorems, as illustrated with the Log-Gamma model. However, and as already mentioned above, there is always a value of p that enables both RBk (p) and RBk (p) to outperform the corresponding estimated RB EVI-estimator, in Eq. 5.2. This provides a motivation for the use of an algorithm that adaptively chooses p among both RBk (p) and RBk (p), respectively defined in Eqs. 1.9 and 1.10. For the joint adaptive choice of (k, p), we suggest either F. Caeiro et al. • heuristic choices or the use of simple and/or double-bootstrap methods, a topic beyond the scope of this article. For all simulated models, excluding only the Fréchet, and among the values considered p = ℓ/(10ξ ), ℓ = 1, 2, 3, 4, the value of p leading to the best performance of the RB MOP EVI-estimators was the one associated with ℓ = 4. 7 Proofs Proof (Theorem 2) The consistency of the statistics in Eq. 1.4, for all 0 < p < 1/ξ , has been proved in Brilhante et al. (2013), and it more generally holds for all real p < 1/ξ . Indeed, if Eq. 2.6 holds, and with Yi , i ≥ 1, independent unit Pareto RVs (with CDF FY (y) = 1 − 1/y, y ≥ 1), we obtain, for any real p,  p   p ξp U (Yn−i+1:n ) p = Yk−i+1:k (1 + op (1)), 1 ≤ i ≤ k. = Uik := XXn−i+1:n U (Y ) n−k:n n−k:n The term op (1) above is uniform in i, 1 ≤ i ≤ k. This comes from the results in Drees (1998) (see Theorem B.2.18 in de Haan and Ferreira, 2006), jointly with the fact that for uniform order statistics Ui:n , 1 ≤ i ≤ n, we have that 1/Ui:n can be uniformly bounded in probability by C[i/(n + 1)]−1 (for some constant C larger, resp. smaller, than 1 for an upper, resp. lower, bound). Since E(Y a ) = 1/(1 − a) if a < 1, the law of large numbers enables us to say that if p < 1/ξ , not necessarily positive, d 1 p Hk (p) = 1− k   1 k i=1 −1 ξp Yi (1 + op (1)) , converges weakly to ξ , as n → ∞. Let us now deal with the asymptotic nondegenerate behaviour of Hk (p), in Eq. 1.4, under the third-order framework in Eq. 2.3, but with A(·) and B(·) chosen as in Eq. 2.5. Note first that we can then write (2.3) as  ρ 2  2ρ   ρ  U (tx) = x ξ 1 + A(t) x ρ−1 + A2 (t) 21 x ρ−1 + ζξ x 2ρ−1 (1 + o(1)) . U (t) Since 1 k 1 k k i=1 k   X   Xn−i+1:n p Xn−k:n n−i+1:n Xn−k:n i=1 p = 1 k = 1 k k  k i=1 pξ Yi i=1    U (Yn−i+1:n ) p , U (Yn−k:n ) ρ 1 + pA(n/k) (Yi −1)/ρ + p 2ρ +pζ A2 (n/k))(Yi =: 1 k k  i=1 ×  pξ Yi we can thus write 2 A2 (n/k) 2  − 1)/(2ξρ) + op (A2 (n/k))  + pA(n/k) Uk (ξ, p, ρ) + pA2 (n/k) pVk (ξ,p,ρ) 2 + ζ Wk (ξ,p,ρ) ξ  + op (A2 (n/k)), ρ (Yi − 1)/ρ 2 Mean-of-order p reduced-bias extreme value index estimation... with Uk (ξ, p, ρ) = Wk (ξ, p, ρ) = 1 k k   pξ Yi i=1 1 k k  ρ Yi −1 ρ  , Vk (ξ, p, ρ) = 2ρ Yi −1 pξ Yi k  1 k pξ Yi i=1   ρ Yi −1 2 , ρ . 2ρ i=1 2 a For a < 1/2, we =  have Var(Y ) = 1/(1 − 2a) − (1/(1 − a)) 2 2 a ( (1 − a) (1 − 2a) . We thus know that for p < 1/(2ξ ), k √ √ pξ (p) 1 Yi − 1−pξ /(pξ ) =: Vk is asymptotically k(1 − pξ ) 1 − 2pξ k1 i=1 standard normal, and we can write 1 k k  pξ Yi i=1 = 1 1−pξ + pξ Vk (p) √ √ . k(1−pξ ) 1−2pξ   Also, and now for p < 1/ξ , and for any a ≤ 0, E Y pξ (Y a − 1) =   a 2a 2 pξ a 2 = (1−pξ )(1−pξ −a) and E Y (Y − 1) (1−pξ )(1−pξ −a)(1−pξ −2a) . We can thus write 1 k k   X n−i+1:n Xn−k:n i=1 p = 1 1−pξ 1+ 2 √pξ Vk (p) k(1−2pξ ) pA (n/k) + 1−pξ −2ρ  + pA(n/k) 1−pξ −ρ pξ +ζ (1−pξ −ρ) ξ(1−pξ −ρ) + Op   A(n/k) √ k  + op (A2 (n/k)) . Let’s go back to the EVI-estimator in Eq. 1.4, writing Hk (p) = 1 p 1 − (1 − pξ )/ 1 + + √pξ Vk (p) k(1−2pξ ) pA2 (n/k) 1−pξ −2ρ  + pA(n/k) 1−pξ −ρ + Op pξ +ζ (1−pξ −ρ) ξ(1−pξ −ρ)   A(n/k) √ k  + op (A2 (n/k)) . We can thus further write Hk (p) = ξ + − ξ(1−pξ √ √ )Vk (p) k 1−2pξ + p(1−pξ )A2 (n/k) (1−pξ −ρ)2 (1−pξ )A(n/k) 1−pξ −ρ + + Op  (1−pξ )A2 (n/k) 1−pξ −2ρ  A(n/k) √ k  pξ +ζ (1−pξ −ρ) ξ(1−pξ −ρ)  + op (A2 (n/k)). √ k √Op After some √  trivial computations, Eq. 3.2 follows. Indeed, since    A(n/k)/ k = Op (A(n/k)) → 0, as n → ∞, the summands Op A(n/k)/ k are totally irrelevant for the asymptotic bias, that follows in a straightforward fashion from the representations above. F. Caeiro et al. Proof (Theorem 3) Considering either φ = ϕρ or φ = pξ ,   β(1−φ)  n ρ √ √ )Vk (p) + (1−pξ )A(n/k) = ξ 1+ (1−pξ RBk (p, β, ρ, φ) := Hk (p) 1− 1−ρ−ϕ k ξ(1−pξ −ρ) ρ k 1−2pξ + (1−pξ )(ζ (1−pξ −ρ)2 +pξρ)A2 (n/k) ξ 2 (1−pξ −ρ)2 (1−pξ −2ρ)  + op (A2 (n/k)) 1− and we furthermore can write (1−pξ √ √ )Vk (p) k 1−2pξ RBk (p, β, ρ, φ) = ξ 1 + − A2 (n/k)(1−pξ )(1−φ) ξ 2 (1−pξ −ρ)(1−φ−ρ) + + β(1−φ) 1−ρ−φ  n ρ  k , ρ(pξ −φ)A(n/k) ξ(1−pξ −ρ)(1−φ−ρ) (1−pξ )(ζ (1−pξ −ρ)2 +pξρ)A2 (n/k) ξ 2 (1−pξ −ρ)2 (1−pξ −2ρ) + op (A2 (n/k)) . Consequently Eq. 3.3 holds with (1) ≡ b(1) bRB RB(p) = ρ(pξ −φ) (1−pξ −ρ)(1−φ−ρ)  ζ (1−pξ −ρ)2 +pξρ (1−pξ −ρ)(1−pξ −2ρ) and (2) bRB ≡ b(2) RB(p) = 1−pξ ξ(1−pξ −ρ) On the basis of Eq. 3.2, we can write  1 − ρ − p Hk (p) = (1 − pξ − ρ) 1 − − − √ ξp(1−pξ )V√k (p) k(1−pξ −ρ) 1−2pξ 1−φ 1−φ−ρ − p(1−pξ )(ζ (1−pξ −ρ)2 +pξρ)A2 (n/k) ξ(1−pξ −ρ)3 (1−pξ −2ρ) Moreover,  . p(1−pξ )A(n/k) (1−pξ −ρ)2  + op (A2 (n/k)) .  pA(n/k) (p) √k − 1−pξ 1 − p Hk (p) = (1 − pξ ) 1 − √ξpV −ρ k 1−2pξ    2 2 p ζ (1−pξ −ρ) +pξρ A (n/k) − ξ(1−pξ −ρ)2 (1−pξ −2ρ) + op (A2 (n/k)) . Consequently,   β(1−p Hk (p))  n ρ RBk (p, β, ρ, pHk (p)) := Hk (p) 1 − 1−ρ−p Hk (p) k   Hk (p)) = Hk (p) 1 − A(n/k)(1−p ξ(1−ρ−p Hk (p))  A(n/k)(1−pξ ) )A2 (n/k) = Hk (p) 1 − ξ(1−pξ −ρ) − pρ(1−pξ ξ(1−pξ −ρ)3    √ + op (A2 (n/k)) +Op A(n/k) k    (1−pξ √ + op (A2 (n/k)) = ξ 1 + √ √ )Vk (p) + Op A(n/k) k 1−2pξ 2 )2 − Aξ 2(n/k)(1−pξ (1−pξ −ρ)2 − k pρ(1−pξ )A2 (n/k) ξ(1−pξ −ρ)3  (1−pξ −ρ)2 +pξρ)A2 (n/k) , + (1−pξξ 2)(ζ 2 (1−pξ −ρ) (1−pξ −2ρ) Mean-of-order p reduced-bias extreme value index estimation... (1) ≡ b(1) i.e. Eq. 3.3 holds with bRB(p) RB(p) (2) bRB(p) ≡ b(2) =0 2 RB(p) 2 (1−pξ ) pρ(1−pξ ) )(ζ (1−pξ −ρ) +pξρ) = − ξ(1−pξ − (1−pξ + (1−pξ −ρ)2 −ρ)3 ξ(1−pξ −ρ)2 (1−pξ −2ρ)   1−pξ ζ (1−pξ −ρ)3 +pξρ 2 −(1−pξ )(1−pξ −ρ)(1−pξ −2ρ) . = (1−pξ ξ(1−pξ −ρ)(1−pξ −2ρ) −ρ)2 Proof (Theorem 4) If we estimate consistently β and ρ through the estimators β̂ and ρ̂, we can use Cramer’s delta-method, and obtain for any of the RB MOP classes of EVI-estimators, either in Eq. 1.9 or in Eq. 1.10, generally denoted RBk (p, β̂, ρ̂),      p ∗ ∗ . + ρ̂ −ρ ln(n/k) −bRB(p) A(n/k) β̂−β RBk (p, β̂, ρ̂)−RBk (p, β, ρ) ∼ aRB(p) β (7.1) ∗ ∗ ∗ ∗ for suitable functions aRB(p) ≡ aRB(p) (ξ, ρ) and bRB(p) ≡ bRB(p) (ξ, ρ), The first part of √ , finite, thus follows in the theorem, related to levels k such that k A(n/k) → λ(1) A a straight forward way from Eq. 7.1.   p Next, since Eq. 2.11 holds, i.e., (β̂ − β)/β ∼ − ln(n/k1 ) ρ̂ − ρ , we have     p ∗ ρ̂ −ρ A(n/k) ln(k/k1 )+bRB(p) =: Wk,k1 . RBk (p, β̂, ρ̂)−RBk (p, β, ρ) ∼ −aRB(p) Under the in the theorem, i.e., with k1 optimal for the ρ-estimation, ρ̂ −  conditions √ ρ = Op 1/ k1 A(n/k1 ) , √ k Wk,k1 = Op  √ √ k A(n/k) k1 A(n/k1 ) ln   k k1 = Op   1 −ρ k k1 2 ln   k k1 = op (1)if k/k1 → 0, and the second part of the theorem follows.   √ If we further assume that (ρ̂ − ρ) ln(n/k) = op 1/( kA(n/k)) , we are able to prove the final results in the theorem. Acknowledgments Research partially supported by National Funds through FCT — Fundação para a Ciência e a Tecnologia, projects UID/MAT/00006/2013 (CEA/UL), UID/MAT/00297/2013 (CMA/UNL), COST Action IC1408, the KU Leuven project HSTRT/14/001 and National Research Foundation of South Africa. The authors would like to thank the associate editor and the referees for their constructive comments that led to substantial improvement of a first version of this paper. References Beirlant, J., Vynckier, P., Teugels, J.: Excess functions and estimation of the extreme-value index. Bernoulli 2, 293–318 (1996) Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.: Statistics of Extremes. Theory and Applications. Wiley (2004) Beirlant, J., Dierckx, G., Guillou, A.: Estimation of the extreme-value index and generalised quantile plots. Bernoulli 11(6), 949–970 (2005) Beirlant, J., Caeiro, F., Gomes, M.I.: An overview and open research topics in statistics of univariate extremes. Revstat 10(1), 1–31 (2012) F. Caeiro et al. Beran, J., Schell, D., Stehlı́k, M.: The harmonic moment tail index estimator: asymptotic distribution and robustness. Ann. Inst. Statist. Math. 66, 193–220 (2014) Bingham, N., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1987) Brilhante, M.F., Gomes, M.I., Pestana, D.: A simple generalization of the Hill estimator. Comput. Stat. Data Anal. 1, 518–535 (2013) Brilhante, M.F., Gomes, M.I., Pestana, D.: The mean-of-order p extreme value index estimator revisited. In: Pacheco, A., et al (eds.) New Advances in Statistical Modeling and Application, pp. 163–175. Springer, Berlin (2014) Caeiro, F., Gomes, M.I.: A new class of estimators of a scale second order parameter. Extremes 9, 193– 211 (2006) Caeiro, F., Gomes, M.I.: Minimum-variance reduced-bias tail index and high quantile estimation. Revstat 6(1), 1–20 (2008) Caeiro, F., Gomes, M.I.: Asymptotic comparison at optimal levels of reduced-bias extreme value index estimators. Statistica Neerlandica 65(4), 462–488 (2011) Caeiro, F., Gomes, M.I.: A semi-parametric estimator of a shape second order parameter. In: Pacheco, A., et al (eds.) New Advances in Statistical Modeling and Application, pp. 163–175. Springer, Berlin (2014) Caeiro, F., Gomes, M.I.: Bias reduction in the estimation of a shape second order parameter of a heavy tailed model. J. Statist. Comput. and Simul. 85(17), 3405–3419 (2015) Caeiro, F., Gomes, M.I., Pestana, D.D.: Direct reduction of bias of the classical Hill estimator. Revstat 3(2), 111–136 (2005) Caeiro, F., Gomes, M.I., Henriques-Rodrigues, L.: Reduced-bias tail index estimators under a third order framework. Commun. Stat. Theory and Methods 38(7), 1019–1040 (2009) Ciuperca, G., Mercadier, C.: Semi-parametric estimation for heavy tailed distributions. Extremes 13(1), 55–87 (2010) Dell’Aquila, R., Embrechts, P.: Extremes and robustness: a contradiction? Fin. Mkts. Portfolio Mgmt 20, 103–118 (2006) Dekkers, A., Einmahl, J., de Haan, L.: A moment estimator for the index of an extreme-value distribution. Ann. Statist. 17, 1833–1855 (1989) Deme, E.H., Gardes, L., Girard, S.: On the estimation of the second order parameter for heavy-tailed distributions. Revstat 11(3), 277–299 (2013) Drees, H.: On smooth statistical tail function. Scand. J. Statist. 25, 187–210 (1998) Fraga Alves, M.I., Gomes, M.I., de Haan, L.: A new class of semi-parametric estimators of the second order parameter. Portugaliae Mathematica 60(1), 193–213 (2003) Fraga Alves, M.I., Gomes, M.I., de Haan, L., Neves, C.: The mixed moment estimator and location invariant alternatives. Extremes 12, 149–185 (2003) Gnedenko, B.: Sur la distribution limite du terme maximum d’une série aléatoire. Ann. Math. 44(6), 423– 453 (1943) Goegebeur, Y., Beirlant, J., de Wet, T.: Linking Pareto-tail kernel goodness-of-fit statistics with tail index at optimal threshold and second order estimation. Revstat 6(1), 51–69 (2008) Goegebeur, Y., Beirlant, J., de Wet, T.: Kernel estimators for the second order parameter in extreme value statistics. J. Statist. Planning and Inference 140(9), 2632–2654 (2010) Gomes, M.I., Guillou, A., et al: Extreme value theory and statistics of univariate extremes: a review. Int. Stat. Rev. 83(2), 263–292 (2015) Gomes, M.I., Martins, M.J.: Asymptotically unbiased estimators of the extreme value index based on external estimation of the second order parameter. Extremes 5(1), 5–31 (2002) Gomes, M.I., Oliveira, O.: The bootstrap methodology in Statistics of Extremes: choice of the optimal sample fraction. Extremes 4(4), 331–358 (2001) Gomes, M.I., Pestana, D.: A sturdy reduced-bias extreme quantile (VaR) estimator. J. Am. Stat. Assoc. 102(477), 280–292 (2007) Gomes, M.I., Martins, M.J., Neves, M.M.: Improving second order reduced-bias extreme value index estimation. Revstat 5(2), 177–207 (2007) Gomes, M.I., de Haan, L., Henriques-Rodrigues, L.: Tail index estimation for heavy-tailed models: accommodation of bias in weighted log-excesses. J. Royal Stat. Soc. B 70(1), 31–52 (2008) Mean-of-order p reduced-bias extreme value index estimation... Gomes, M.I., Henriques-Rodrigues, L., Pereira, H., Pestana, D.: Tail index and second order parameters’ semi-parametric estimation based on the log-excesses. J. Statist. Comput. and Simul. 80(6), 653–666 (2010) Gomes, M.I., Brilhante, M.F., Caeiro, F., Pestana, D.: A new partially reduced-bias mean-of-order p class of extreme value index estimators. Comput. Statist. Data Anal. 82, 223–237 (2015) Gomes, M.I., Brilhante, M.F., Pestana, D.: New reduced-bias estimators of a positive extreme value index. Commun. Stat. Simul. Comput. 45, 1–30 (2016a) Gomes, M.I., Henriques-Rodrigues, L., Manjunath, B.G.: Mean-of-order-p location-invariant extreme value index estimation. Revstat 14(3), 273–296 (2016b). Available at: https://www.ine.pt/revstat/pdf/ PORT-MOP-REVSTAT.pdf de Haan, L., Ferreira, A.: Extreme Value Theory: an Introduction Springer Science+Business Media. LLC, New York (2006) de Haan, L., Peng, L.: Comparison of extreme value index estimators. Statistica Neerlandica 52, 60–70 (1998) Hall, P.: On some simple estimates of an exponent of regular variation. J. Royal Stat. Soc. B 44, 37–42 (1982) Hall, P., Welsh, A.W.: Adaptive estimates of parameters of regular variation. Ann. Statist. 13, 331–341 (1985) Henriques-Rodrigues, L., Gomes, M.I., Fraga Alves, M.I., Neves, C.: PORT-Estimation of a shape secondorder parameter. Revstat 12(3), 299–328 (2014) Henriques-Rodrigues, L., Gomes, M.I., Manjunath, B.G.: Estimation of a scale second-order parameter related to the PORT methodology. J. of Stat Theory and Practice 9(3), 571–599 (2015) Hill, B.: A simple general approach to inference about the tail of a distribution. Ann. Statist. 3, 1163–1174 (1975) Paulaskas, V., Vaičiulis, M.: On the improvement of Hill and some other estimators. Lith. Math. J 53, 336– 355 (2013) Reiss, R.-D., Thomas, M.: Statistical Analysis of Extreme Values, with Application to Insurance, Finance, Hydrology and Other Fields, 2nd edition; 3rd edition, Birkhäuser Verlag (2007) Stehlı́k, M., Potocký, R., Waldl, H., Fabián, Z.: On the favourable estimation of fitting heavy tailed data. Comput. Stat. 25, 485–503 (2010) de Wet, T., Goegebeur, Y., Munch, M.: Asymptotically unbiased estimation of the second order tail parameter in extreme value statistics. Stat. Probab. Lett. 82, 565–573 (2012) Worms, J., Worms, R.: Estimation of second order parameters using probability weighted moments. ESAIM: Probab. Stat. 16, 97–113 (2012)