# Box-Cox Transformations and Bias Reduction in Extreme Value Theory

Authors: Lígia Henriques-Rodrigues; M. Ivette Gomes
Journal: Computational and Mathematical Methods
 (2022)

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## Abstract

The Box-Cox transformations are used to make the data more suitable for statistical analysis. We know from the literature that this transformation of the data can increase the rate of convergence of the tail of the distribution to the generalized extreme value distribution, and as a byproduct, the bias of the estimation procedure is reduced. The reduction of bias of the Hill estimator has been widely addressed in the literature of extreme value theory. Several techniques have been used to achieve such reduction of bias, either by removing the main component of the bias of the Hill estimator of the extreme value index (EVI) or by constructing new estimators based on generalized means or norms that generalize the Hill estimator. We are going to study the Box-Cox Hill estimator introduced by Teugels and Vanroelen, in 2004, proving the consistency and asymptotic normality of the estimator and addressing the choice and estimation of the power and shift parameters of the Box-Cox transformation for the EVI estimation. The performance of the estimators under study will be illustrated for finite samples through small-scale Monte Carlo simulation studies.


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## Body

## 1. Introduction and Preliminaries

In extreme value theory, one is often interested in the estimation of the extreme value index (EVI),ξ, a parameter controlling the heaviness of the right tail F¯≔1−F, with F the distribution function (d.f.) of the process under study. In a univariate independent and identically distributed (i.i.d.) setup, let us assume that we have access to a random sample of size n and use the notation Xi:n to denote the i-th ascending order statistic, 1≤i≤n. Similar to the central limit theorem, the extremal types theorem states that if there exists a nondegenerate limit distribution for a suitable normalization of the maximum,
(1)PXn:n−bnan≤x⟶n⟶∞Gξx,being Gξ. an extreme value (EV) d.f. of the type:
(2)Gξx≡EVξx≔exp−1+ξx−1/ξ,1+ξx>0,ifξ=0,exp−exp−x,x∈R,ifξ=0.From Equations (1) and (2), the d.f. F is said to pertain to the domain of attraction for maxima of EVξ, with ξ∈R. Such a fact is denoted by F∈DMEVξ. The distribution in Equation (2) encompasses three types of distributions: if ξ<0, the right tail is light; i.e., F has a finite right endpoint (xF≔supx:Fx<1<+∞); if ξ>0, the right tail is heavy of a negative polynomial type and F has an infinite right endpoint; if ξ=0, the right tail is of an exponential type and the right endpoint can be either finite or infinite.In this paper, we are going to work with heavy-tailed models, i.e., models withξ>0 that belong to the so-called Fréchet domain of attraction, denoted by DM+≔DMEVξξ>0. These types of models are quite common in areas like biostatistics, computer science, finance, insurance, and statistical quality control, among others.


### 1.1. Box-Cox Transformations in Extreme Value Theory

LetX be a random variable (r.v.) with d.f. F≡FX∈DMEVξ, with probability density function fX. The Box-Cox (BC) transformation [1] of X is a function of the parameter σ∈R given by the following:
(3)Y=Xσ−1σ,σ≠0,lnX,σ=0,where σ=1 corresponds to a mere change in location, σ=1/2 to the square root transformation, and σ=−1 to the reciprocal transformation.The use of BC transformations in EV theory has been quite diverse. Bali [2] proposed the BC-GEV distribution to model the empirical distribution of returns as a generalization of the GEV distribution in Equation (2) and of the generalized Pareto distribution, with d.f. GPξx=1+lnGξx. The d.f. of the BC-GEV distribution is as follows:
(4)Fξx;σ=Gξxσ−1σ+1.Under a semiparametric framework, Teugels and Vanroelen [3] provided the theoretical framework through the use of regular variation (see Seneta [4] and Bingham et al. [5] for details on regular variation) to determine the optimal values of σ that maximize the rate of convergence of the second-order condition (to be detailed in Section 1.2) for values of ξ≥0. Another application of the BC transformation was presented in the work of Eastoe and Tawn [6], under a Bayesian framework. Wadsworth et al. [7] use the BC transformation in the inference procedure to account parametrically for the uncertainty surrounding the scale extrapolation in physical processes. Paulauskas and Vaiciulis [8, 9] used the BC transformation and considered estimators based on block scheme considering a family of functions of order statistics.In the next theorem, the link between the domain of attraction ofFX and of the transformed variable Y, with d.f. FY, is established.Theorem 1 (Wadsworth et al. [7]).
Under the conditions such that convergence in Equation (1) holds, with norming sequences an and bn, then with Gξ. the d.f. in Equation (2)
(5)PYn:n−bn∗an∗≤x⟶n⟶∞Gξ∗yholds for some finite ξ∗ where
(6)bn∗=bnσ−1σ,an∗=anbnσ−1,and we can say that FY∈DMEVξ∗. Furthermore, if FX is twice differentiable for sufficiently large x, then the limiting shape parameter ξ∗ takes the form:
(7)ξ∗=ξ+limx⟶xFhxxσ−1,with hx=1−Fx/fx the reciprocal hazard function of the parent distribution FX. For any such distribution which has ξ≤0, ξ∗=ξ, due to fact that hx/x tends then to zero, as x⟶xF.Remark 2.
Note that ifξ≤0, then ξ∗=ξ≤0 and FY is in the same domain of attraction of FX. If ξ>0, i.e., if we are in the Fréchet domain of attraction, ξ∗ can be different of ξ and FY can no longer belong to the Fréchet domain of attraction.


### 1.2. First- and Second-Order Frameworks

WithF∈DM+, let F⟵ denote the generalized inverse function of F, defined by F⟵t≔infx:Fx≥t, let U be the associated (reciprocal) quantile function, defined as Ut≔F⟵1−1/t,t≥1, and let RVα stand for the class of regularly varying functions at infinity with index α, i.e., positive measurable functions g. such that limt⟶∞gtx/gt=xα, for all x>0.In this setup, we have the validity of the first-order condition (de Haan [10]):
(8)F∈DM+⟺1−F∈RV−1/ξ⟺U∈RVξ.To develop the asymptotic nondegenerate properties of the estimators under study, we need to assume the existence of an auxiliary functionAt, with ∣A∣∈RVρ [11], where ρ≤0 is a shape second-order parameter, and for every x>0, we often work with a second-order condition (SOC) of the type:
(9)limt⟶∞lnUtx−lnUt−ξlnxAt=xρ−1ρ.Slightly more restrictively, we are going to work with models in the Fréchet domain of attraction (ξ>0) that belong to Hall-Welsh class of models [12], i.e., models with the second-order right tail expansion:
(10)1−Fx≔xC−1/ξ1+E1xC−−ρ/ξ+ox−−ρ/ξ+ε,x⟶∞,with E1≠0, ρ<0, and C>0, the first-order scale parameter. For these models, the second-order tail quantile function is as follows:
(11)Ut=Ctξ1+D1tρ+otρ,t⟶∞,with D1=ξE1. If the SOC in Equation (9) holds with At=ξβtρ, then D1=At/ρtρ. Most common heavy-tailed distributions, like the Fréchet, the generalized Pareto, the Burr and Student’s t, belong to the class in Equation (11). The log-gamma and the log-Pareto models (ρ=0 in Equation (9)) do not belong to the class in Equation (11).Remark 3.
Under a parametric framework, Wadsworth et al. [7] showed that for models in Hall-Welsh class, a BC transformation of the data increases the rate of convergence of the tail of the distribution to the GEV distribution if σ=−ρ/ξ, and as a byproduct, the bias of the estimation procedure is reduced.
Let us consider a BC transformation such thatX∗=Xσ+a and denote by X1∗,⋯,Xn∗ the transformed sample, with d.f. F∗, associated with the original sample X1,⋯,Xn, where Xi∗=TXi, with TX=Xσ+a. The first-order condition associated with the X⋆ is as follows:
(12)limt⟶∞U∗txU∗t=xξσ=xξ∗⟺U∗∈RVξ∗=ξσ.
Under the validity of Equation (12) and for some C⋆,C>0, we have U⋆t~C⋆tξ⋆=Ctξσ.
Teugels and Vanroelen [3] used the theory of extended regular variation (see de Haan and Ferreira [13]) to determine the optimal values of σ that maximize the rate of convergence of the second-order condition for values of ξ≥0 and proved in Theorems 4.1, 4.2, and 4.3 that when ξ>0 and ξ+ρ>0, a BC transformation has no effect on the SOC of U∗ unless σ=−ρ/ξ; if ξ>0 and ξ+ρ<0, the BC transformation can have a negative or positive effect on the SOC of U∗ and the positive effect happens when σ=−ρ/ξ. If ξ>0 and ξ+ρ=0, for any value other than σ=−ρ/ξ=1, there is no improvement in the speed of convergence of the SOC. Thus, the value σ that maximizes the rate of convergence of the SOC that will be considered in this work is as follows:
(13)σ=−ρξ.
This choice will enhance the bias reduction in the estimation procedure.


### 1.3. Reduced Bias EVI Estimation

For heavy right tails, i.e., whenever Equation (8) holds, the classical extreme value index (EVI) estimator is the Hill (H) estimator [14], with functional form
(14)Hn,k≔1k∑i=1klnXn−i+1:nXn−k:n,the average of the log-excesses, and consequently also the average of the k scaled log-spacings Ui≔ilnXn−i+1:n−lnXn−i:n, 1≤i≤k.But the Hill EVI estimator, in Equation (14), exhibits the usual trade-off between bias and variance, and therefore, the suitable accommodation of the asymptotic bias has been widely discussed among EVT practitioners. The first papers are due to Peng [15], Beirlant et al. [16], Feuerverger and Hall [17], and Gomes et al. [18], among others. Such a bias reduction usually provided more stable sample trajectories and a lower mean square error at the optimal level, but at the expense of an increase in variance, which did not allow the estimators proposed in the aforementioned papers to surpass the classical estimators for small k values. Further details can be found in Gomes and Guillou [19], among others.For heavy-tailed models, Caeiro et al. [20, 21] and Gomes et al. [22, 23], among others, were able to overcome the bias-variance trade-off, keeping the variance of the proposed reduced bias EVI estimators equal to ξ2/k, the variance of the Hill EVI estimator. For that reason, these estimators were coined minimum variance reduced bias (MVRB) EVI estimators. These classes of estimators depend upon the adequate consistent estimation of the vector of second-order parameters, β,ρ∈R,R− in At=ξβtρ in a high level k1 of larger order than k, the number of top order statistics used in the EVI estimation, and surmount the classical EVI estimators for all k values.Practical issues related with the estimation ofβ,ρ and algorithms are presented by Gomes and Pestana [24], among others. Among several classes of MVRB EVI estimators, we refer to the class introduced in Caeiro et al. [20], the corrected Hill (CH) estimator:
(15)CHn,k≡CHn,kβ^,ρ^≔Hn,k1−β^1−ρ^nkρ∧,with β^,ρ^ a set of adequate estimators of the second-order parameters β,ρ (see Section 3). The estimator in Equation (15) will be used in Section 4 for comparison with the new class of MVRB EVI estimators introduced in this paper. Other techniques have been used in the literature to reduce the bias of the Hill estimator, in Equation (14), and new estimators based on generalized means or norms that generalize the Hill estimator have been proposed (see Brilhante et al. [25], Caeiro et al. [26–29], Paulauskas and Vaiciulis [8, 9], Beran et al. [30], and Beirlant et al. [31], among others).In Section2 of this paper, we present the BC Hill estimator and derive the consistency and asymptotic normality of the estimator under study assuming that the BC parameters are known. Next, in Section 3, we address the estimation of the BC parameters, and in Section 4, we present the finite sample behaviour of the BC Hill estimators and compare it with the corrected Hill estimator. Finally, in Section 5, some concluding remarks and future work are put forward.


## 1.1. Box-Cox Transformations in Extreme Value Theory

LetX be a random variable (r.v.) with d.f. F≡FX∈DMEVξ, with probability density function fX. The Box-Cox (BC) transformation [1] of X is a function of the parameter σ∈R given by the following:
(3)Y=Xσ−1σ,σ≠0,lnX,σ=0,where σ=1 corresponds to a mere change in location, σ=1/2 to the square root transformation, and σ=−1 to the reciprocal transformation.The use of BC transformations in EV theory has been quite diverse. Bali [2] proposed the BC-GEV distribution to model the empirical distribution of returns as a generalization of the GEV distribution in Equation (2) and of the generalized Pareto distribution, with d.f. GPξx=1+lnGξx. The d.f. of the BC-GEV distribution is as follows:
(4)Fξx;σ=Gξxσ−1σ+1.Under a semiparametric framework, Teugels and Vanroelen [3] provided the theoretical framework through the use of regular variation (see Seneta [4] and Bingham et al. [5] for details on regular variation) to determine the optimal values of σ that maximize the rate of convergence of the second-order condition (to be detailed in Section 1.2) for values of ξ≥0. Another application of the BC transformation was presented in the work of Eastoe and Tawn [6], under a Bayesian framework. Wadsworth et al. [7] use the BC transformation in the inference procedure to account parametrically for the uncertainty surrounding the scale extrapolation in physical processes. Paulauskas and Vaiciulis [8, 9] used the BC transformation and considered estimators based on block scheme considering a family of functions of order statistics.In the next theorem, the link between the domain of attraction ofFX and of the transformed variable Y, with d.f. FY, is established.Theorem 1 (Wadsworth et al. [7]).
Under the conditions such that convergence in Equation (1) holds, with norming sequences an and bn, then with Gξ. the d.f. in Equation (2)
(5)PYn:n−bn∗an∗≤x⟶n⟶∞Gξ∗yholds for some finite ξ∗ where
(6)bn∗=bnσ−1σ,an∗=anbnσ−1,and we can say that FY∈DMEVξ∗. Furthermore, if FX is twice differentiable for sufficiently large x, then the limiting shape parameter ξ∗ takes the form:
(7)ξ∗=ξ+limx⟶xFhxxσ−1,with hx=1−Fx/fx the reciprocal hazard function of the parent distribution FX. For any such distribution which has ξ≤0, ξ∗=ξ, due to fact that hx/x tends then to zero, as x⟶xF.Remark 2.
Note that ifξ≤0, then ξ∗=ξ≤0 and FY is in the same domain of attraction of FX. If ξ>0, i.e., if we are in the Fréchet domain of attraction, ξ∗ can be different of ξ and FY can no longer belong to the Fréchet domain of attraction.


## 1.2. First- and Second-Order Frameworks

WithF∈DM+, let F⟵ denote the generalized inverse function of F, defined by F⟵t≔infx:Fx≥t, let U be the associated (reciprocal) quantile function, defined as Ut≔F⟵1−1/t,t≥1, and let RVα stand for the class of regularly varying functions at infinity with index α, i.e., positive measurable functions g. such that limt⟶∞gtx/gt=xα, for all x>0.In this setup, we have the validity of the first-order condition (de Haan [10]):
(8)F∈DM+⟺1−F∈RV−1/ξ⟺U∈RVξ.To develop the asymptotic nondegenerate properties of the estimators under study, we need to assume the existence of an auxiliary functionAt, with ∣A∣∈RVρ [11], where ρ≤0 is a shape second-order parameter, and for every x>0, we often work with a second-order condition (SOC) of the type:
(9)limt⟶∞lnUtx−lnUt−ξlnxAt=xρ−1ρ.Slightly more restrictively, we are going to work with models in the Fréchet domain of attraction (ξ>0) that belong to Hall-Welsh class of models [12], i.e., models with the second-order right tail expansion:
(10)1−Fx≔xC−1/ξ1+E1xC−−ρ/ξ+ox−−ρ/ξ+ε,x⟶∞,with E1≠0, ρ<0, and C>0, the first-order scale parameter. For these models, the second-order tail quantile function is as follows:
(11)Ut=Ctξ1+D1tρ+otρ,t⟶∞,with D1=ξE1. If the SOC in Equation (9) holds with At=ξβtρ, then D1=At/ρtρ. Most common heavy-tailed distributions, like the Fréchet, the generalized Pareto, the Burr and Student’s t, belong to the class in Equation (11). The log-gamma and the log-Pareto models (ρ=0 in Equation (9)) do not belong to the class in Equation (11).Remark 3.
Under a parametric framework, Wadsworth et al. [7] showed that for models in Hall-Welsh class, a BC transformation of the data increases the rate of convergence of the tail of the distribution to the GEV distribution if σ=−ρ/ξ, and as a byproduct, the bias of the estimation procedure is reduced.
Let us consider a BC transformation such thatX∗=Xσ+a and denote by X1∗,⋯,Xn∗ the transformed sample, with d.f. F∗, associated with the original sample X1,⋯,Xn, where Xi∗=TXi, with TX=Xσ+a. The first-order condition associated with the X⋆ is as follows:
(12)limt⟶∞U∗txU∗t=xξσ=xξ∗⟺U∗∈RVξ∗=ξσ.
Under the validity of Equation (12) and for some C⋆,C>0, we have U⋆t~C⋆tξ⋆=Ctξσ.
Teugels and Vanroelen [3] used the theory of extended regular variation (see de Haan and Ferreira [13]) to determine the optimal values of σ that maximize the rate of convergence of the second-order condition for values of ξ≥0 and proved in Theorems 4.1, 4.2, and 4.3 that when ξ>0 and ξ+ρ>0, a BC transformation has no effect on the SOC of U∗ unless σ=−ρ/ξ; if ξ>0 and ξ+ρ<0, the BC transformation can have a negative or positive effect on the SOC of U∗ and the positive effect happens when σ=−ρ/ξ. If ξ>0 and ξ+ρ=0, for any value other than σ=−ρ/ξ=1, there is no improvement in the speed of convergence of the SOC. Thus, the value σ that maximizes the rate of convergence of the SOC that will be considered in this work is as follows:
(13)σ=−ρξ.
This choice will enhance the bias reduction in the estimation procedure.


## 1.3. Reduced Bias EVI Estimation

For heavy right tails, i.e., whenever Equation (8) holds, the classical extreme value index (EVI) estimator is the Hill (H) estimator [14], with functional form
(14)Hn,k≔1k∑i=1klnXn−i+1:nXn−k:n,the average of the log-excesses, and consequently also the average of the k scaled log-spacings Ui≔ilnXn−i+1:n−lnXn−i:n, 1≤i≤k.But the Hill EVI estimator, in Equation (14), exhibits the usual trade-off between bias and variance, and therefore, the suitable accommodation of the asymptotic bias has been widely discussed among EVT practitioners. The first papers are due to Peng [15], Beirlant et al. [16], Feuerverger and Hall [17], and Gomes et al. [18], among others. Such a bias reduction usually provided more stable sample trajectories and a lower mean square error at the optimal level, but at the expense of an increase in variance, which did not allow the estimators proposed in the aforementioned papers to surpass the classical estimators for small k values. Further details can be found in Gomes and Guillou [19], among others.For heavy-tailed models, Caeiro et al. [20, 21] and Gomes et al. [22, 23], among others, were able to overcome the bias-variance trade-off, keeping the variance of the proposed reduced bias EVI estimators equal to ξ2/k, the variance of the Hill EVI estimator. For that reason, these estimators were coined minimum variance reduced bias (MVRB) EVI estimators. These classes of estimators depend upon the adequate consistent estimation of the vector of second-order parameters, β,ρ∈R,R− in At=ξβtρ in a high level k1 of larger order than k, the number of top order statistics used in the EVI estimation, and surmount the classical EVI estimators for all k values.Practical issues related with the estimation ofβ,ρ and algorithms are presented by Gomes and Pestana [24], among others. Among several classes of MVRB EVI estimators, we refer to the class introduced in Caeiro et al. [20], the corrected Hill (CH) estimator:
(15)CHn,k≡CHn,kβ^,ρ^≔Hn,k1−β^1−ρ^nkρ∧,with β^,ρ^ a set of adequate estimators of the second-order parameters β,ρ (see Section 3). The estimator in Equation (15) will be used in Section 4 for comparison with the new class of MVRB EVI estimators introduced in this paper. Other techniques have been used in the literature to reduce the bias of the Hill estimator, in Equation (14), and new estimators based on generalized means or norms that generalize the Hill estimator have been proposed (see Brilhante et al. [25], Caeiro et al. [26–29], Paulauskas and Vaiciulis [8, 9], Beran et al. [30], and Beirlant et al. [31], among others).In Section2 of this paper, we present the BC Hill estimator and derive the consistency and asymptotic normality of the estimator under study assuming that the BC parameters are known. Next, in Section 3, we address the estimation of the BC parameters, and in Section 4, we present the finite sample behaviour of the BC Hill estimators and compare it with the corrected Hill estimator. Finally, in Section 5, some concluding remarks and future work are put forward.


## 2. The Box-Cox Hill Estimator

The Box-Cox Hill (BC Hill) estimator introduced by Teugels and Vanroelen [3] has the functional form:
(16)Hn,kBCσ,a≔1kσ∑i=1klnXn−i+1:nσ+aXn−k:nσ+a,where a>0, so that Xiσ+ai=1n is strictly positive, σ=−ρ/ξ>0, as in Equation (13) and, Hn,kBCσ,0≡Hn,k, in Equation (14). The estimators Hn,k, in Equation (14), and Hn,kBCσ,a, in Equation (16), are both consistent for the estimation of ξ, for intermediate sequences of integers k, i.e., such that
(17)k=kn,1≤k<n,kn⟶∞,kn=on,asn⟶∞.Under the second-order framework in Equation (9), the asymptotical distributional representation
(18)Hn,k=dξ+ξkZk+11−ρAnk1+op1holds true [32], where Zk=k∑i=1kEi/k−1, with Ei a sequence of i.i.d. standard exponential r.v.’s, is an asymptotically standard normal r.v.Remark 4.
The BC Hill estimator is also a generalization of the EVI estimator introduced by Gomes and Oliveira [33],
(19)Hn,ka≡Hn,kBC1,a≔1k∑i=1klnXn−i+1:n+aXn−k:n+a,where the shift a imposed to the data is a tuning parameter. The authors studied the adequate choice of a that enabled the reduction of the main component of the bias of the Hill’s estimator, in Equation (14). The results obtained in the simulation studies for models with σ=1 are in line with the ones provided in Section 4.


### 2.1. Asymptotic Behaviour of the Box-Cox Hill Estimator Whenσ and a Are Known

In this section, we start by deriving the asymptotic behaviour of the BC Hill estimator assuming that the BC parameters,σ and a, are known. We then get an optimal value for a that allows us to obtain a new class of MVRB EVI estimators.Theorem 5.
In Hall-Welsh class of models in Equation (11), for intermediate k, i.e., for k values such that Equation (17) holds, we have the validity of following distributional representation,
(20)Hn,kBCσ,a=dξ+ξkZk+An/k1−ρ1−aβCσ1+op1.
IfkAn/k⟶λ, possibly nonnull, we can guarantee that
(21)kHn,kBCσ,a−ξ⟶n⟶∞dξN0,1+λ1−ρ1−aβCσ,with N0,1 denoting a standard normal r.v. If we further choose
(22)a=βCσ,then
(23)kHn,kBCσ,a−ξ⟶n⟶∞dξN0,1.Proof.
The universal uniform transformation and from the definition of the reciprocal quantile functionU. in Equation (11), we get Xi:n=dUYi:n, with Y a unit Pareto r.v. In what follows, we shall use the following relations: for j>i, Yj:n/Yi:n=dYj−i:n−i, lnYi:n=dEi:n, where E denotes a standard exponential r.v., and Yn−k:n~pn/k, whenever Equation (8) holds.
(24)Hn,kBCσ,a=1kσ∑i=1klnXn−i+1:nσ+aXn−k:nσ+a=d1kσ∑i=1klnUσYn−i+1:n+aUσYn−k:n+a=d1kσ∑i=1klnUYn−i+1:nUYn−k:nσ+1kσ∑i=1kln1+a/UσYn−i+1:n1+a/UσYn−k:n=1k∑i=1klnUYn−i+1:nUYn−k:n⏟Hn,k+1kσ∑i=1kaUσYn−i+1:n−aUσYn−k:n=Hn,k+1kσ∑i=1kaUσYn−k:n−1+UYn−k:nUYn−i+1:nσ.
For the second term in the right hand side, we know that under the validity of Equations (8) and (11), we can write UYn−k:n~Un/k=Cn/kξ and UYn−i+1:n/UYn−k:n=dYk−i+1:k−ξ1+op1. Then,
(25)Hn,kBCσ,a=Hn,k+1σaCσnk−ξσ1k∑i=1k−1+Yi−ξσ.
SinceEYa=1/1−a, for a<1,
(26)Hn,kBCσ,a=dξ+ξkZk+11−ρAn/k1+op1+aCσnk−ξσ−ξ1+ξσ1+op1=dξ+ξkZk+An/k1−ρ+aCσnk−ξσ−ξ1+ξσ1+op1.
Using the parametrizationsAt=ξβtρ and σ=−ρ/ξ, the distributional representation in Equation (20) holds true. We then get an optimal a value, the value of a in Equation (22), such that that the dominant component of the bias of the BC Hill estimator is null, and Equation (23) follows.Corollary 6.
Under the first-order condition in Equation (12), and for the heavy-tailed models in Hall and Welsh class, in Equation (11), such that U∗t=c1tξ∗+c2, with c1 and c2 real constants, c1≠0, the BC Hill estimator in Equation (16) is asymptotically unbiased for any k=kn⟶∞, not necessarily intermediate.Proof.
This result comes straightforwardly from the definition of the Box-Cox Hill estimator in Equation (16) and from the fact that we are working with heavy-tailed models in Hall and Welsh class, with quantile function U∗t=c1tξ∗+c2, c1 and c2 real constants, c1≠0.Remark 7.
The heavy-tailed models in Hall and Welsh class that satisfy the property in Corollary6 are the Burrξ,ρ, the log-logisticθ,ξ, and the GPξ parents. This behaviour will be illustrated in Section 4.In Table1, we show for some heavy-tailed models the optimal values of σ and a, in Equations (13) and (22), respectively, where cν=νBν/2,1/21/ν, with B the complete beta function.Remark 8.
Teugels and Vanroelen [3] considered two examples to show the bias reduction of the BC Hill estimator. In the first one, a Burrξ=1,ρ=−0.5 model was chosen. The power of the BC transformation is σ=0.5, and the authors used a=1, which is, according to Equation (22) and Table 1, the optimal value for bias reduction. In the second example, the authors considered a Cauchy model (ξ=1, ρ=−2), with σ=2. It was noticed that the choice of a is very important and a value around 0.5 was suggested as it was the value that had the strongest effect of minimizing the bias of the BC Hill estimator. We now know, from Equation (22) and Table 1, that the optimal value is a=2/3.Table 1
First-order, second-order, and Box-Cox parameters for some heavy-tailed models.
ModelFirst-orderSecond-orderBox-CoxξCρβσaBurrξ,ρξ1ρ1−ρ/ξ1Fréchetξξ1-11/21/ξ1/2Log‐logisticθ,ξξθ−ξ-111/ξ1/θGeneralized Pareto,GPξξ1/ξ−ξ111/ξCauchy11/π-22π2/322/3Studenttν1/νν/cν−2/νν+1cν2/ν+22νν+1/ν+2


## 2.1. Asymptotic Behaviour of the Box-Cox Hill Estimator Whenσ and a Are Known

In this section, we start by deriving the asymptotic behaviour of the BC Hill estimator assuming that the BC parameters,σ and a, are known. We then get an optimal value for a that allows us to obtain a new class of MVRB EVI estimators.Theorem 5.
In Hall-Welsh class of models in Equation (11), for intermediate k, i.e., for k values such that Equation (17) holds, we have the validity of following distributional representation,
(20)Hn,kBCσ,a=dξ+ξkZk+An/k1−ρ1−aβCσ1+op1.
IfkAn/k⟶λ, possibly nonnull, we can guarantee that
(21)kHn,kBCσ,a−ξ⟶n⟶∞dξN0,1+λ1−ρ1−aβCσ,with N0,1 denoting a standard normal r.v. If we further choose
(22)a=βCσ,then
(23)kHn,kBCσ,a−ξ⟶n⟶∞dξN0,1.Proof.
The universal uniform transformation and from the definition of the reciprocal quantile functionU. in Equation (11), we get Xi:n=dUYi:n, with Y a unit Pareto r.v. In what follows, we shall use the following relations: for j>i, Yj:n/Yi:n=dYj−i:n−i, lnYi:n=dEi:n, where E denotes a standard exponential r.v., and Yn−k:n~pn/k, whenever Equation (8) holds.
(24)Hn,kBCσ,a=1kσ∑i=1klnXn−i+1:nσ+aXn−k:nσ+a=d1kσ∑i=1klnUσYn−i+1:n+aUσYn−k:n+a=d1kσ∑i=1klnUYn−i+1:nUYn−k:nσ+1kσ∑i=1kln1+a/UσYn−i+1:n1+a/UσYn−k:n=1k∑i=1klnUYn−i+1:nUYn−k:n⏟Hn,k+1kσ∑i=1kaUσYn−i+1:n−aUσYn−k:n=Hn,k+1kσ∑i=1kaUσYn−k:n−1+UYn−k:nUYn−i+1:nσ.
For the second term in the right hand side, we know that under the validity of Equations (8) and (11), we can write UYn−k:n~Un/k=Cn/kξ and UYn−i+1:n/UYn−k:n=dYk−i+1:k−ξ1+op1. Then,
(25)Hn,kBCσ,a=Hn,k+1σaCσnk−ξσ1k∑i=1k−1+Yi−ξσ.
SinceEYa=1/1−a, for a<1,
(26)Hn,kBCσ,a=dξ+ξkZk+11−ρAn/k1+op1+aCσnk−ξσ−ξ1+ξσ1+op1=dξ+ξkZk+An/k1−ρ+aCσnk−ξσ−ξ1+ξσ1+op1.
Using the parametrizationsAt=ξβtρ and σ=−ρ/ξ, the distributional representation in Equation (20) holds true. We then get an optimal a value, the value of a in Equation (22), such that that the dominant component of the bias of the BC Hill estimator is null, and Equation (23) follows.Corollary 6.
Under the first-order condition in Equation (12), and for the heavy-tailed models in Hall and Welsh class, in Equation (11), such that U∗t=c1tξ∗+c2, with c1 and c2 real constants, c1≠0, the BC Hill estimator in Equation (16) is asymptotically unbiased for any k=kn⟶∞, not necessarily intermediate.Proof.
This result comes straightforwardly from the definition of the Box-Cox Hill estimator in Equation (16) and from the fact that we are working with heavy-tailed models in Hall and Welsh class, with quantile function U∗t=c1tξ∗+c2, c1 and c2 real constants, c1≠0.Remark 7.
The heavy-tailed models in Hall and Welsh class that satisfy the property in Corollary6 are the Burrξ,ρ, the log-logisticθ,ξ, and the GPξ parents. This behaviour will be illustrated in Section 4.In Table1, we show for some heavy-tailed models the optimal values of σ and a, in Equations (13) and (22), respectively, where cν=νBν/2,1/21/ν, with B the complete beta function.Remark 8.
Teugels and Vanroelen [3] considered two examples to show the bias reduction of the BC Hill estimator. In the first one, a Burrξ=1,ρ=−0.5 model was chosen. The power of the BC transformation is σ=0.5, and the authors used a=1, which is, according to Equation (22) and Table 1, the optimal value for bias reduction. In the second example, the authors considered a Cauchy model (ξ=1, ρ=−2), with σ=2. It was noticed that the choice of a is very important and a value around 0.5 was suggested as it was the value that had the strongest effect of minimizing the bias of the BC Hill estimator. We now know, from Equation (22) and Table 1, that the optimal value is a=2/3.Table 1
First-order, second-order, and Box-Cox parameters for some heavy-tailed models.
ModelFirst-orderSecond-orderBox-CoxξCρβσaBurrξ,ρξ1ρ1−ρ/ξ1Fréchetξξ1-11/21/ξ1/2Log‐logisticθ,ξξθ−ξ-111/ξ1/θGeneralized Pareto,GPξξ1/ξ−ξ111/ξCauchy11/π-22π2/322/3Studenttν1/νν/cν−2/νν+1cν2/ν+22νν+1/ν+2


## 3. Adaptive Estimation of the Box-Cox Parameters

The BC parameters depend on the estimation of the first-order scale parameterC and on the estimation of the second-order shape, ρ, and scale, β, parameters. In this work, we are going to consider a misspecification of the first-order scale parameter, considering that C=1.For the estimation of the second-order shape parameter,ρ, we consider the class of estimators introduced in [34] defined by the following:
(27)ρ^k;τ≔−3Tnτk−1Tnτk−3,with τ a real tuning parameter [35]:
(28)Tnτk≔lnMn1k−1/2lnMn2k/21/2lnMn2k/2−1/3lnMn3k/6,ifτ=0,Mn1kτ−Mn2k/2τ/2Mn2k/2τ/2−Mn3k/6τ/3,ifτ=0,Mnαk≔1k∑i=1klnXn−i+1:n−lnXn−k:nα.Note thatMn1k≡Hn,k, in Equation (14), and the results obtained in Fraga Alves et al. [34] and in Gomes and Martins [36] advise to consider τ=0 for values of ρ≥−1 and τ=1 for values of ρ>−1.For the estimation of the second-order scale parameter,β, we are going to consider the estimator introduced in Gomes and Martins [36], given by the following:
(29)β^k;ρ^≔knρ∧dk−ρ^×Dk0−Dk−ρ^dkρ^×Dk−ρ^−Dk−2ρ^,where for any α≥0,
(30)Dkα≔1k∑i=1kikαUi,dkα≔1k∑i=1kikα,with Ui=ilnXn−i+1:n−lnXn−i:n, 1≤i≤k, the scaled log-spacings, and ρ^ the ρ estimator in Equation (27).Asymptotic considerations as well as simulated results led several authors to consider the estimation of the second-order estimators at a high levelk1, given by k1≔n1−ε where ε~0+ and x denotes, as usual, the integer part of x. In the simulation studies in Section 4, the value ε=0.001 was used to compute the estimates of the second-order parameters. Under adequate general conditions, the classes of estimators in Equations (27) and (29) are asymptotically normally distributed and show highly stable sample paths as functions of k, the number of top order statistics used, for a wide range of large k values.Theorem 9 (Fraga Alves et al. [34] and Gomes and Martins [36]).
If the SOC in Equation (9) holds, for models in Hall and Welsh class with At=ξβtρ, ρ<0, if k is intermediate, i.e., if Equation (17) holds and if kAn/k⟶n⟶∞∞, then ρ^k converges in probability towards ρ, as n⟶∞, and β^k;ρ converges in probability towards β, as n⟶∞. Moreover, with Wk•~aNormal0,1, •=R,B, σρ=ξ1−ρ32ρ2−2ρ+1/ρ, and σβ=∣β∣1−ρ1−2ρ/∣ρ∣,
(31)ρ^k;τ=dρ+σρkAn/kWkR+op1,β^k;ρ=dβ+σβkAn/kWkB+op1.As stated earlier, the estimates ofβ and ρ are computed at the high level k1 and the following results in Caeiro and Gomes [37] hold true.Theorem 10 (Caeiro and Gomes [37]).
Under the conditions of the previous theorem, withρ^=ρ^k1;τ for any τ and k1 such that
(32)ρ^−ρ=op1lnn,asn⟶∞,
β^k;ρ^ is consistent for the estimation of β. Moreover, for models in Hall and Welsh class and with β^≔β^k1,ρ^, we may further say that β^−β=Oplnn/k1ρ^−ρ, at most of the order of ρ^−ρlnn.In addition to the classes presented in Equations (27) and (29), we also refer the classes introduced in Goegebeur et al. [38], Ciuperca and Mercadier [39], Worms and Worms [40], de Wet et al. [41], and Henriques-Rodrigues et al. [42], among others, for the estimation of ρ and the classes introduced in Caeiro and Gomes [35] and Gomes et al. [43], for the estimation of β.In this paper, we are going to work with adaptive estimates of the BC parameters in Equations (13) and (22). So, for the estimation of σ, we consider the following:
(33)σ^=−ρ^k1Hn,k^0H,with ρ^ the ρ estimator in Equation (27), k1 the high level used in the estimation of the second-order parameters, and Hn,k^0H the Hill estimator computed at the level k^0H≔k0Hβ^,ρ^, with k0Hβ,ρ≔argminkAMSEHn,k [44], given by the following:
(34)k0Hβ,ρ≔1−ρ2n−2ρ−2ρβ21/1−2ρ.For the estimation ofa and with the misspecification C=1, we are going to consider:
(35)a^=β^k1,with β^ the β estimator in Equation (29). The associated estimators of σ, σ^k=ρ^k/ξ^k and a, considering C=1, i.e. a^k=β^k, which are consistent for the estimation of σ and a, respectively, for sequences of k values such that Equation (17) holds. The BC Hill estimator, in Equation (16), assuming that the BC parameters are adaptively estimated has the functional form:
(36)Hn,kBCH≡Hn,kBCσ^,a^≔1kσ^∑i=1klnXn−i+1:nσ^+a^Xn−k:nσ^+a^,with σ^ and a^ presented in Equations (33) and (35), respectively. The consistency of the BCH estimator in Equation (36) for the estimation of the EVI, ξ, is presented in the next theorem.Theorem 11.
In Hall-Welsh class of models in Equation (11), for intermediate k, i.e., for k values such that Equation (17) holds, with σ^ and a^ presented in Equations (33) and (35), consistent estimators of σ and a, respectively, then Hn,kBCσ^,a^⟶Pξ.Proof.
The delta method is not going to be directly applied since the partial derivatives of the estimatorHn,kBCσ^,a^, in Equation (36), in order to σ,a are complex. We first note that Xn−i+1:n>Xn−k:n, 1≤i≤k<n, with σ^ a consistent estimator of σ>0 and a^ a consistent estimator of a>0, and defining a∧∗≔ra^, with r>0 (a consistent estimator of a∗=ra), then
(37)Xn−i+1:n+a∧∗Xn−k:n+a∧∗σ∧≤Xn−i+1:nσ^+a^Xn−k:nσ^+a^<Xn−i+1:nXn−k:nσ∧.
Ifσ^≥1, we can choose r=1, and if 0<σ^<1, the value of r depends upon the underlying model in study. So, we can write the following:
(38)Hn,ka∧∗≔1k∑i=1klnXn−i+1:n+a∧∗Xn−k:n+a∧∗≤1kσ^∑i=1klnXn−i+1:nσ^+a^Xn−k:nσ^+a^<1k∑i=1klnXn−i+1:nXn−k:n=Hn,k.
The estimator in the upper bound is the Hill estimator,Hn,k, in Equation (14), consistent for the estimation of ξ. For the estimator in the lower bound, the delta method leads to the following:
(39)Hn,ka∧∗=dHn,ka∗+∂Hn,ka∗∂a∗a∧∗−a∗1+op1,with
(40)Hn,ka⋆=1k∑i=1klnXn−i+1:n+a⋆Xn−k:n+a⋆.
The estimatorHn,ka∗ is of the same type of the estimator Hn,ka, in Equation (19), and is therefore consistent for the estimation of ξ as proven in Section 2 of Gomes and Oliveira [33]. The term ∂Hn,ka∗/∂a∗ can be written as follows:
(41)∂Hn,ka∗∂a∗=1k∑i=1kXn−k:n−Xn−i+1:nXn−i+1:nXn−k:n=1k∑i=1kXn−k:n/Xn−i+1:n−1Xn−k:n.
Let us next use the same kind of arguments referred to in the proof of Theorem5: Xi:n=dUYi:n, with Y a unit Pareto r.v. and U. the quantile function in Equation (11), for j>i, Yj:n/Yi:n=dYj−i:n−i, lnYi:n=dEi:n, where E denotes a standard exponential r.v., and Yn−k:n~pn/k, whenever Equation (8) holds. Then, we have the following:
(42)∂Hn,ka∗∂a∗~p1kUn/k−ξ1+ξ, and we can write the following:
(43)Hn,ka∧∗=dHn,ka∗−a∧∗−a∗kUn/kξ1+ξ1+op1.
Working withk values such that Equation (17) holds and given that a∧∗ is a consistent estimator of a∗ and Hn,ka∗ is consistent for the estimation of ξ, the estimator Hn,ka∧∗ is also consistent for the estimation of ξ and the consistency of Hn,kBCσ^,a^ follows.Remark 12.
In the paper by Gomes and Oliveira [33], the tuning parameter a of the estimator Hn,ka presented in Remark 4 was estimated by the order statistic 1−X1:n. According to the authors, the estimator Hn,k1−X1:n is consistent for the estimation of ξ, with asymptotic distributional properties similar to those of the Hill estimator, for models in DM+ with finite left endpoint.Remark 13.
The estimatorHn,ka∧∗ has the same functional form of the shifted Hill estimator introduced in Aban and Meerschaert [45]. The shifted Hill estimator was obtained for shifted Pareto parents of the type Fx=1−cx+a−1/ξ and is a conditional maximum likelihood estimator of ξ. The estimator is consistent for the estimation of ξ even though a^ is not consistent for the estimation of a under a semiparametric framework (see Remark 2.1 of Gomes et al. [46]).


## 4. Finite Sample Behaviour of the Estimators

To assess the finite sample behaviour of the BC Hill estimators, in Equations (16) and (36), we have implemented a multisample Monte Carlo simulation experiment of size 1000×10 (see Gomes and Oliveira [47], for further details on multisample simulation) for sample sizes n=200, 500, 1000, 2000, and 5000, from the following models:
(1)
The Burrξ,ρ model with d.f. Fx=1−1+x−ρ/ξ1/ρ, x>0, ξ=0.25,0.5,1, and ρ=−0.5,(2)
The generalized Pareto model, GPξ, with d.f. Fx=1−1+ξx−1/ξ, x≥−1/ξ, and ξ=0.25,0.5,1,(3)
The Cauchy model, with d.f.Fx=1/2+1/πarctanx, x∈R, and ξ=1,(4)
The Studenttv model, with d.f. Fx=Γν+1/2/Γν/2πν∫−∞x1+y2/ν−ν+1/2dy, x∈R, and ν>0, where Γ denotes the (complete) gamma function, ξ=0,5,0.25 (ξ=1/ν).In Figures1–3, we present the mean values, E., and mean squared errors, MSE., for the models under study provided by the Hill estimator, H, in Equation (14), the BC Hill r.v. in Equation (16), assuming that the BC parameters are known and denoted by BCH0 in the figures, the BC Hill estimator, BCH, in Equation (36), assuming that the BC parameters are adaptively estimated by Equations (33) and (35), and the MVRB EVI estimator, CH, in Equation (15) for comparison. From Figures 1–3, we can state that:
(i)
The BCH estimator beats the Hill estimator for all simulated models and for allk values, as it presents smaller absolute bias and the minimum MSE is smaller than the minimum MSE of the Hill estimator;(ii)
For the simulated Burrξ,ρ models, the BCH estimator underestimates the true value of ξ for k values larger than 400. The MSE exhibit the usual shape being the minimum values smaller than the ones of the CH estimator. Note that if X has a Burrξ,ρ distribution, then X−ρ/ξ+1 has a strict Pareto distribution and this could justify the excellent results of the BCH0 statistic;(iii)
For the GPξ models with ξ=0.5,1, the mean values of the CH and BCH estimators almost overlap for a wide region of k values (k<500, when ξ=0.5 and k<600, when ξ=1), and from that k value forward, the bias of the BCH estimator is smaller than the one of the CH estimator. When ξ=2, the BCH estimator underestimates ξ for almost all k values, whereas the CH estimator overestimates ξ for the same k values. Regarding the minimum MSE, both estimators (CH and BCH) present similar values;(iv)
For the Cauchy model, the behaviour of CH and the BCH estimator is similar to the behaviour observed for theGP2 parent;(v)
For the Studenttv models (ν=2,4), the sample paths of the mean values of the CH and BCH estimators overlap from small up to moderate values of k, and from that value forward, the bias of the BCH estimator is smaller. Regarding the minimum MSE, both estimators (CH and BCH) present similar values.Figure 1
Mean values (left) and MSEs (right) of the estimators under study for samples of sizen=1000 from a Burrξ,ρ=−0.5 parent with ξ=0.25 (top), ξ=0.5 (middle), and ξ=1 (bottom).Figure 2
Mean values (left) and MSEs (right) of the estimators under study for samples of sizen=1000 from GPξ parent with ξ=0.5 (top), ξ=1 (middle), and ξ=2 (bottom).Figure 3
Mean values (left) and MSEs (right) of the estimators under study for samples of sizen=2000 from a Cauchy parent (top), a Student t2 parent (middle), and a Student t4 parent (bottom).In Table2, we present the estimates of the BC parameters σ and a, as described in Section 3, for the aforementioned models. Information on the 95% confidence intervals (CIs), computed on the basis of 10 replicates with 1000 runs each, is also provided. The simulated data used to support the findings of this study are available from the corresponding author upon request.Table 2
Box-Cox estimates and associated 95% CIs based on 10 replicates with 1000 runs each.
200500100020005000Burr parent withξ=0.25 and ρ=−0.5σ=22.2759±0.00602.4373±0.00312.5529±0.00442.6605±0.00342.8112±0.0027a=11.0114±0.00011.0116±0.00011.0114±0.00001.0120±0.00011.0125±0.0000Burr parent withξ=0.5 and ρ=−0.5σ=11.1404±0.00301.2206±0.00351.2758±0.00261.3321±0.00251.4035±0.0018a=11.0113±0.00011.0115±0.00011.0115±0.00001.0121±0.00001.0124±0.0000Burr parent withξ=1 and ρ=−0.5σ=0.50.5696±0.00090.6113±0.00100.6375±0.00110.6658±0.00120.7034±0.0006a=11.0114±0.00011.0114±0.00011.0114±0.00001.0121±0.00001.0125±0.0000GP parent withξ=0.5σ=11.1426±0.00471.2199±0.00271.2760±0.00301.3318±0.00241.4045±0.0012a=21.0114±0.00011.0115±0.00011.0115±0.00011.0121±0.00011.0125±0.0000GP parent withξ=1σ=10.7774±0.00270.7998±0.00170.8166±0.00130.8347±0.00130.8611±0.0008a=10.9913±0.00010.9925±0.00010.9930±0.00010.9928±0.00010.9927±0.0000GP parent withξ=2σ=11.0880±0.00181.1165±0.00101.1331±0.00151.1409±0.00131.1519±0.0006a=0.51.0133±0.00051.0152±0.00031.0157±0.00011.0166±0.00021.0172±0.0001Cauchy parent withξ=1σ=21.9266±0.00432.0056±0.00422.0507±0.00392.0859±0.00202.1143±0.0019a=2/31.0524±0.00051.0536±0.00021.0551±0.00011.0554±0.00011.0577±0.0001Studentt2 parent with ξ=0.5σ=21.3044±0.00461.3636±0.00481.3944±0.00281.4171±0.00181.4397±0.0022a=1.51.0133±0.00021.0143±0.00011.0146±0.00011.0148±0.00001.0155±0.0000Studentt4 parent with ξ=0.25σ=21.7947±0.00821.9771±0.00782.0919±0.00562.1934±0.00312.3023±0.0040a=10/31.0236±0.00021.0242±0.00011.0245±0.00011.0247±0.00011.0257±0.0000Next, and based on the 1000 runs, we computed the simulated optimum level, i.e., the levelk^0≔argminkMSEξ^k, with ξ^=H,CH,BCH, and present in Tables 3–5 the simulated values of the optimal sample fraction (the optimal level divided by the number of positive elements in the sample), the mean value of each estimator computed at k^0, and the MSE of each estimator also computed at k^0. For each model, the smallest absolute bias and the smallest MSE are written in bold. Information on the 95% CIs, computed on the basis of 10 replicates with 1000 runs each, is also provided.Table 3
Optimal sample fraction and associated 95% CIs based on 10 replicates with 1000 runs each.
200500100020005000Burr parent withξ=0.25 and ρ=−0.5H0.0645±0.08870.0446±0.05550.0357±0.09910.0319±0.03420.0290±0.0342CH0.1225±0.13110.0740±0.12670.0635±0.12610.0501±0.08340.0427±0.0847BCH0.4730±0.19290.4242±0.31350.4001±0.15530.3971±0.13520.3917±0.2153Burr parent withξ=0.5 and ρ=−0.5H0.0670±0.07510.0448±0.06430.0360±0.02610.0308±0.07480.0291±0.0449CH0.1335±0.13710.0748±0.11990.0594±0.07960.0497±0.07820.0409±0.0951BCH0.5010±0.13200.4238±0.29080.4030±0.16530.3903±0.17000.3922±0.1698Burr parent withξ=1 and ρ=−0.5H0.0645±0.07290.0452±0.04560.0352±0.05530.0325±0.06450.0294±0.0514CH0.1265±0.15470.0840±0.12940.0614±0.16790.0490±0.11670.0426±0.0862BCH0.4830±0.25570.4194±0.33870.3977±0.20990.3946±0.23140.3939±0.1347GP parent withξ=0.5H0.0620±0.05920.0414±0.05230.0382±0.07080.0304±0.06090.0288±0.0353CH0.1295±0.17830.0844±0.12990.0615±0.14470.0473±0.09360.0407±0.0871BCH0.1470±0.09730.0844±0.12990.0579±0.08590.0445±0.10820.0366±0.0935GP parent withξ=1H0.1745±0.12470.1352±0.08390.1182±0.11790.1076±0.15550.0993±0.1192CH0.6810±0.14580.6810±0.11480.6839±0.10140.6799±0.06620.6777±0.1206BCH0.7815±0.11700.7802±0.12360.7839±0.15990.7895±0.12180.7932±0.1745GP parent withξ=2H0.3825±0.14650.3384±0.23550.3048±0.32790.2863±0.41160.2681±0.3300CH0.5695±0.13300.5244±0.15280.4578±0.39130.4362±0.40300.3929±0.4868BCH0.4260±0.13670.3756±0.19370.3342±0.27680.3050±0.27260.2606±0.4894Cauchy parent withξ=1H0.4540±0.0110.3418±0.01510.2971±0.00730.2510±0.00940.2032±0.0060CH0.5637±0.01800.4399±0.01210.3909±0.00940.3260±0.01000.2693±0.0110BCH0.5820±0.01970.4473±0.01170.3947±0.00830.3416±0.00590.2897±0.0053Studentt2 parent with ξ=0.5H0.2345±0.01250.1554±0.00640.1139±0.00720.0956±0.00480.0709±0.0036CH0.5449±0.01710.4729±0.01330.4211±0.01130.4009±0.00610.3779±0.0060BCH0.6799±0.03490.6077±0.01230.5371±0.01250.4912±0.00810.4479±0.0065Studentt4 parent with ξ=0.25H0.0943±0.00890.0597±0.00580.0457±0.00300.0312±0.00180.0208±0.0007CH0.1633±0.01530.1020±0.01090.0670±0.00580.0461±0.00320.0301±0.0010BCH0.2126±0.01500.1176±0.00900.0713±0.00480.0460±0.00330.0266±0.0024Table 4
Simulated mean values at optimal levels and associated 95% CIs based on 10 replicates with 1000 runs each.
200500100020005000Burr parent withξ=0.25 and ρ=−0.5H0.2872±0.00420.2702±0.00230.2597±0.00320.2563±0.00100.2519±0.0006CH0.2827±0.00200.2681±0.00220.2640±0.00170.2579±0.00110.2534±0.0007BCH0.2438±0.00050.2459±0.00080.2478±0.00030.2487±0.00020.2495±0.0002Burr parent withξ=0.5 and ρ=−0.5H0.5775±0.00680.5404±0.00470.5235±0.00210.5096±0.00350.5054±0.0018CH0.5687±0.00530.5363±0.00380.5254±0.00310.5154±0.00200.5051±0.0014BCH0.4870±0.00110.4921±0.00110.4953±0.00070.4979±0.00040.4991±0.0003Burr parent withξ=1 and ρ=−0.5H1.1476±0.01351.0805±0.00761.0446±0.00851.0285±0.00471.0125±0.0029CH1.1324±0.00971.0830±0.00671.0544±0.00871.0273±0.00641.0099±0.0034BCH0.9751±0.00270.9854±0.00200.9918±0.00140.9952±0.00130.9980±0.0008GP parent withξ=0.5H0.5705±0.00620.5431±0.00370.5265±0.00470.5126±0.00230.5045±0.0014CH0.5662±0.00390.5344±0.00420.5265±0.00390.5081±0.00240.5044±0.0015BCH0.5688±0.00340.5458±0.00400.5286±0.00310.5161±0.00280.5073±0.0020GP parent withξ=1H1.0686±0.00831.0384±0.00331.0224±0.00381.0151±0.00311.0070±0.0015CH1.0036±0.00461.0009±0.00201.0003±0.00091.0001±0.00100.9999±0.0007BCH0.9974±0.00270.9981±0.00130.9989±0.00080.9993±0.00060.9998±0.0004GP parent withξ=2H2.0798±0.00572.0474±0.00722.0294±0.00462.0197±0.00532.0089±0.0022CH2.0626±0.00452.0449±0.00292.0268±0.00342.0177±0.00302.0084±0.0014BCH1.8812±0.00341.9084±0.00371.9254±0.00281.9327±0.00191.9406±0.0008Cauchy parent withξ=1H1.0794±0.00501.0547±0.00451.0411±0.00271.0307±0.00241.0212±0.0014CH1.0656±0.00531.0487±0.00281.0386±0.00311.0282±0.00231.0204±0.0018BCH0.9236±0.00310.9524±0.00250.9627±0.00180.9731±0.00120.9823±0.0008Studentt2 parent with ξ=0.5H0.5858±0.00350.5556±0.00350.5414±0.00230.5359±0.00210.5267±0.0018CH0.5176±0.00170.5101±0.00190.5059±0.00140.5049±0.00070.5020±0.0006BCH0.5060±0.00350.5067±0.00160.5052±0.00080.5040±0.00060.5019±0.0006Studentt4 parent with ξ=0.25H0.3374±0.00400.3163±0.00360.3058±0.00260.2957±0.00120.2861±0.0008CH0.3230±0.00370.3069±0.00350.2961±0.00160.2889±0.00140.2818±0.0011BCH0.3161±0.00290.3054±0.00190.2958±0.00170.2894±0.00160.2814±0.0016Table 5
Simulated MSE at optimal levels and associated 95% CIs based on 10 replicates with 1000 runs each.
200500100020005000Burr parent withξ=0.25 and ρ=−0.5H0.0062±0.00030.0028±0.00010.0015±0.00000.0008±0.00000.0003±0.0000CH0.0037±0.00010.0018±0.00010.0010±0.00000.0006±0.00000.0002±0.0000BCH0.0007±0.00000.0003±0.00000.0001±0.00000.0001±0.00000.0000±0.0000Burr parent withξ=0.5 and ρ=−0.5H0.0251±0.00090.0115±0.00030.0063±0.00020.0032±0.00010.0013±0.0000CH0.0145±0.00050.0074±0.00020.0043±0.00010.0023±0.00000.0010±0.0000BCH0.0028±0.00010.0012±0.00000.0006±0.00000.0003±0.00000.0001±0.0000Burr parent withξ=1 and ρ=−0.5H0.1000±0.00310.0462±0.00140.0250±0.00070.0124±0.00040.0051±0.0001CH0.0580±0.00170.0289±0.00070.0171±0.00050.0088±0.00030.0038±0.0001BCH0.0116±0.00020.0050±0.00010.0025±0.00010.0013±0.00000.0005±0.0000GP parent withξ=0.5H0.0251±0.00070.0114±0.00040.0061±0.00010.0032±0.00010.0013±0.0000CH0.014±0.00060.0073±0.00020.0041±0.00010.0022±0.00010.0009±0.0000BCH0.0108±0.00040.0064±0.00020.0039±0.00010.0022±0.00010.0009±0.0000GP parent withξ=1H0.0314±0.00090.0144±0.00030.0077±0.00010.0041±0.00010.0018±0.0001CH0.0076±0.00020.0030±0.00010.0015±0.00010.0008±0.00000.0003±0.0000BCH0.0058±0.00020.0024±0.00000.0012±0.00000.0006±0.00000.0002±0.0000GP parent withξ=2H0.0551±0.00160.0243±0.00050.0128±0.00030.0067±0.00030.0027±0.0001CH0.0373±0.00090.0165±0.00020.0089±0.00020.0047±0.00020.0020±0.0000BCH0.0469±0.00060.0237±0.00040.0144±0.00050.0095±0.00030.0059±0.0001Cauchy parent withξ=1H0.0365±0.00110.0168±0.00050.0094±0.00030.0053±0.00010.0025±0.0001CH0.0283±0.00080.0130±0.00040.0073±0.00020.0041±0.00010.0020±0.0001BCH0.0227±0.00050.0099±0.00030.0057±0.00020.0031±0.00010.0015±0.0000Studentt2 parent with ξ=0.5H0.0236±0.00050.0117±0.00030.0069±0.00020.0043±0.00010.0022±0.0000CH0.0064±0.00020.0025±0.00010.0013±0.00000.0007±0.00000.0003±0.0000BCH0.0041±0.00010.0017±0.00000.0009±0.00000.0005±0.00000.0002±0.0000Studentt4 parent with ξ=0.25H0.0210±0.00060.0114±0.00030.0071±0.00020.0048±0.00010.0028±0.0001CH0.0124±0.00040.0070±0.00010.0046±0.00010.0032±0.00010.0020±0.0001BCH0.0080±0.00020.0055±0.00010.0041±0.00010.0031±0.00010.0020±0.0001Table2 shows that the estimation of a works well for models with first-order scale parameter, C=1, such as the Burr and GP1, as expected and that the estimation of σ could be improved in some parents. However, from Tables 4 and 5, we noticed that the precise estimation of σ has more impact on the behaviour of the BCH estimator in some models, like the GP0.25 and GP0.5 than on other models.At optimal levels, the BCH estimator is always better than the CH estimator for all simulated Burr parents, and in this case, the misspecificationC=1 has no impact on the BCH estimator since C=1 (see Table 1). The simulated mean values, at optimal levels, of the CH estimator are better than the ones of the BCH estimator for all GP parents, but the BCH estimator has competitive MSEs for the GP0.5 and GP1 parents. For the Cauchy and Student t2 and t4 parents, the BCH estimator is better than the CH estimator for almost all sample sizes, even with the misspecification C=1.


## 5. Concluding Remarks and Future Work

In this paper, we have studied the BC Hill estimator introduced by Teugels and Vanroelen, in 2004. We proved the consistency and asymptotic normality of the estimators and with the appropriate choice of the BC parameters we were able to obtain a new class of MVRB EVI estimators. We considered the adaptive estimation of the BC parameters, and the finite sample behaviour of the estimators was illustrated by a Monte Carlo simulation study. From the simulation study, we conclude that the BC Hill estimator performs better than the corrected Hill estimator for several heavy-tailed models in terms of simulated mean values and simulated MSEs. Furthermore, the estimation procedure presents some robustness to the estimation of the BC parameters, as can be depicted by the misspecification of the first-order scale parameter,C=1. Note that several heavy-tailed models satisfy this criterion. The estimation of the BC parameters could be improved by deriving the maximum likelihood estimators, a topic under development and out of the scope of this paper. Other topics of interest for future research, out of the scope of this paper, are the determination of the optimal level k used to compute the BC EVI estimates and the use of the BC Hill estimator to derive new classes of estimators for other parameters of extreme events such as high quantiles and the second-order shape parameter, ρ.



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*Source: 3854763-2022-03-10.xml*