EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 6024 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Class of Reduced Bias Estimators of Distortion Risk Measures under Dependence Serials with Heavy-Tailed Marginals Aminetou Agbrabatt1, El Hadji Deme2,∗, Sidiya Ahmedou2, Mamadou A. Barry2, Khalil El Waled1 1 FST, Université de Nouachott Al Aasriya, BP 2373, Nouakchott, Mauritanie 2 LERSTAD, UFR SAT, Université Gaston Berger, BP 234, Saint-Louis, Sénégal Abstract. In this paper, we introduce a class of semi-parametric estimators of the distortion risk premiums for dependent insurance losses with heavy-tailed marginals. Our approach is based on the kernel estimation of the tail index and extreme quantiles under the first and second orders regularly varying assumptions for stationary insured risks with heavy-tailed distribution under dependence serials. Moreover, we illustrate the behaviour of our proposed estimator and give a comparison between this estimator and the classical one in terms of the absolute bias and the root median squared error. 2020 Mathematics Subject Classifications: 62E20, 62H12 Key Words and Phrases: Risk Premiums, Insurance, Kernel Estimation, Heavy-Tailed, Bias reduction, Extreme value, Dependence serials 1. Introduction Risk measurement or premium calculation principles are used to quantify insurance losses and financial valuations. A number of risk measures have been proposed to manage these risks, and we refer to [1], [2], [3], [4] and the references therein. The most commonly used one is the net premium (mean) of a non-negative loss random variable X over the probability space (Ω,A, P ), with a tail distribution function F := 1− F and defined as π = E(X) = ∫ ∞ 0 F (x)dx. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6024 Email addresses: aminetou.aghrabatt@gmail.com (A. Aghrabatt), elhadji.deme@egb.edu.sn (E. Deme), h.sidiya84@gmail.com S. Ahmedou), barrymamadou-aliou@ugb.edu.sn (M. A. Barry), khalil.elwaled@gmail.com (K. El Waled) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Aghrabatt et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6024 2 of 25 Premiums are required to be greater than or equal to the mean E(X) in order to avoid that the insurer loses money on average. One way to achieve this goal consists in considering the following distortion risk premium introduced by [4] as: πg = ∫ ∞ 0 g(F (x))dx, (1) where g is a concave function defined from [0, 1] onto [0, 1], such that g(0) = 0 and g(1) = 1, and called the distortion function. The distortion function g is parameterized by a one-dimensional parameter β ≥ 1, called the distortion parameter and represents the risk aversion. It controls the amount of the risk loading included in the premium for a given riskiness of the loss variable X. Let Q be the quantile function corresponding to F and defined by Q(s) = inf{x : F (x) ≥ s}, for every s ∈ [0, 1). The quantile function Q plays a pivotal role in defining numerous risk measures, and is a well known risk measure itself, called the Value-at-Risk (VaR). By a change of variables, the distortion risk premium πg(R) can be rewritten in terms of the quantile function Q as follows: πg = − ∫ 1 0 g(s)dQ(1− s). (2) The risk measure πg, which can also be viewed as a premium calculation principle, has manifested in the econometric literature, particularly in dual theory of choice under risk, and has been introduced into actuarial literature by [4]. A number of risk measures of this form have been discussed by [5]. Important properties of the distortion risk measure, such as coherence and second order stochastic dominance have been well studied see, for example [6], [7] and [5]. Note that the class of concave distortion risk measures is only a subset of the class of coherent risk measures. Many special cases that have arisen in the finance and insurance literature are such: • The Net Premium principle: g(x) = x • Value-at-Risk (VaRα): g(x) = 1(1 − α, 1), for some α ∈ (0, 1), where 1(.) is the indicator function. • Tail Value at Risk: g(x) = min(x/(1− α), 1), for some α ∈ (0, 1). • Proportional Hazard Transform: g(x) = x1/ϱ, for some ϱ ≥ 1. • Dual-Power Transform: g(x) = 1− (1− x)ϱ, for some ϱ ≥ 1. • Gini principle: g(x) = (1 + ϱ)x− ϱx2, with 0 < ϱ ≤ 1. • Lookback distortion: g(x) = xϱ(1− ϱ log(x)), with 0 < ϱ ≤ 1. • Beta-distortion risk premium (eg, [5]): g(x) = 1 B(a,b) ∫ x 0 sa−1(1− s)b−1ds, where B(a, b)= ∫ 1 0 s a−1(1− s)b−1ds, a ≤ 1 ≤ b.. • MINMAXVAR2 risk premium (see [8] and references there in): where g(t) = 1− (1− x 1 1+µ )1+ν , µ > 0, ν > 0. A. Aghrabatt et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6024 3 of 25 Note that these distortion functions g(·) are equal or can be approximated to a power func- tion gβ(y) = t1/β, β ≥ 1, since they are regularly varying at zero with index 1/β, that is: g(t) = t1/βℓg(t), where ℓg(·) is a slowly varying function at zero satisfying ℓg(λt)/ℓg(t) → 1 as t → 0, for λ > 0. This condition is used in [8] and [9] to estimate the reinsurance risk premiums in the context of independent extreme risks. A standard reinsurance product is an excess of loss reinsurance, which means that the reinsurer only compensates the cedant’s loss above a certain retention amount R > 0. Consider an excess-of-loss reinsurance policy in excess of a high retention level R > 0, the distorted reinsurance premium of the total claim amount max(X −R, 0) is defined as: πgβ (R) = ∫ ∞ R gβ ( F (x) ) dx. (3) As in (2), the distortion risk premium πgβ (R) can be rewritten in terms of Value-at- Risk Q as follows: πgβ (R) = − ∫ F (R) 0 gβ(s)dQ(1− s). (4) In the reinsurance context, the purpose of estimating the premium πg(R) is to esti- mate, for each insured, the expected under the distorted probability of the excess claim amounts for a given period. This evaluation is often done using statistical methods. For more details, see [10]. Thus, reinsurance companies must calculate the premiums to cover these excess claims, which are usually very high. The extreme value theory (EVT) has become one of the leading theories in the development of statistical models for high insurance losses. We refer to [11], for general accounts on extreme-value theory. Many authors studied the estimation of the premium of these high excess losses by using classical EVT models, mainly based on the independent and identically distributed (i.i.d) assumption of the insured risks with large tails. One can mention among others, [12], [13], [8], [14], [15], [16], [9], etc. The reinsurance is also a risk mitigating tool, constituting an important instrument in the management of risk of an insurance company where dependencies and the heavy-tailed nature should be taken into account. When transferring risk, the cedent seeks a trade-off between profit and safety, which is on the nature of the insured risk and on the reinsurance premium calculation principle. The heavy-tailed nature of insurance claims requires that special attention paid to be analyzing the tail distributions of a claims amounts. Such distributions are mainly char- acterized by their index which make the possibility to indicate the size and the frequency of some extreme phenomena within the framework of a given probability distribution (See eg, [9]). The extreme value theory (EVT) offers satisfactory statistical results such heavy tailed distributions. Semiparametric estimators of reinsurance premiums for independent A. Aghrabatt et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6024 4 of 25 and identically distributed (i.i.d) from a heavy tailed losses have been largely studied in the literature. However, only recently, dependencies among risks have been considered (see, [17]) Also, in the dependence context of financial extreme losses with heavy-tailed marginals, [18], [19] and [20] investigated the estimation of the Value-at-Risk for extreme losses (high quantile). In high excess reinsurance losses, [17] introduced a semiparametric estimator of the risk premiums from heavy tailed dependent insured risks over an optimal retention level. This semiparametric estimator surfers from a bias problem due to the fact it depends on a classical estimator of the Value-at-Risk estimator (the Weissman’s estimator (see, [21]), which have the same problem. The aim of this paper is to generalize the estimator of reinsurance risk premiums proposed in [17]. As it exhibits a potential bias, we introduce its bias reduction approach under under dependent insured risks with heavy tailed marginals. Our consideration is based on the bias reduction approach proposed by [19] in the estimation of the Value-at Risk under dependence serials. The rest of the paper is organized as follows. In Section 2, we propose a statistical estimation of the distortion risk premiums under dependent serials. In Section 3, we establish the asymptotic properties of the proposed estimator. Then in Section 4, we match our theoretical results with a simulation assessment in order to highlight the efficiency of our methods. Finally, Sections 5 and 6 are respectively devoted to the conclusion and the proofs of our main results. 2. Estimating the distortion risk premiums 2.1. Extreme value theory under dependence serials Extreme value statistics are based on the fact that under rather mild conditions for large samples, a class of distribution functions can be considered to fit the distribution of the largest observation in a sample. From this limit theorem, it follows that the tail behav- ior of a distribution function can be characterized mainly by a single shape parameter, called the tail index or extreme value index. Based on the sign of extreme value index, the domain of attraction of the extreme value distribution can be divided into three subclasses. To this end, let’s consider Xi, i ∈ N a copies from a non negative stationary insured risk X defined over some probability space (Ω,A,P), with ncommon marginal distribution function (df) F (x) = P(X ≤ x). Indeed, under mild conditions on the dependence struc- ture, If the Xi, i ∈ N are weakly dependent then, the corresponding main result of the extreme value theory (See, [22], Section 3.7), under some mild dependence conditions, is based on the following weak convergence of the distribution function from the standardized A. Aghrabatt et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6024 5 of 25 maximum of n observations (X1, .., Xn), n > 1: L ( a−1 n ( max 1≤i≤n Xi − bn )) → Gθ γ weakly, (5) for some θ ∈ [0, 1], where an > 0, bn ∈ R are standardized sequences and Gγ(x) = exp ( −(1 + γx) −1/γ + ) , with y+ = max(y, 0) and Gγ(x) = exp(e−x), for γ = 0. Here, the real-valued parameter γ is referred to as the extreme value index of F , which in turn is said to belong to the maximum domain of attraction of Gγ , denoted by F ∈ DM(Gγ). Throughout this paper, we assume that the non negative stationary insured risksXi, i ∈ N satisfies the following β-mixing dependence structure condition: β(m) := sup p≥1 E { sup C∈B∞p+m+1 |P(C|Bp 1)− P(C)| } → 0, (6) as m → ∞, where Bj i denotes the σ-algebra generated by (Xi, ..., Xj). Without loss of generality, β(m) measures the total variation distance between the unconditional distri- bution of the future of the time series and the conditional distribution of the future given the past of the time series when both are detached by m time points. Also; it is assumed that the common marginal distribution function F of the β-mixing insured risks Xi, i ∈ N, is heavy-tailed (belonging to the the Fréchet domain of attraction that is F ∈ M(Gγ), γ > 0). This is equivalent to the fact that its associated tail distri- bution function 1−F is regularly varying at infinity with index −1/γ < 0. More precisely, that is F (x) := 1− F (x) = x−1/γℓF (x), x > 0, (7) where ℓF is a slowly varying function at infinity, i.e for all x > 0, ℓF (tx)/ℓF (t) → 1, as t → ∞. The relation (7) is also equivalent to U(z) = Q(1− z−1) = zγℓU (z), z > 1, where ℓU (tz)/ℓU (t) → 1, as t → ∞, for all z > 1, where Q(1 − s) = inf{x, F (x) ≥ s} is the quantile function associated to F , namely called the Value-at-Risk. The class of heavy- tailed distributions includes distributions such as Pareto, Burr, Student, Lévy-stable, and log-gamma which are known to be appropriate models in Extreme value theory for fitting large insurance claims, large fluctuations of prices, log-returns,etc. (see, e.g., [23], [24], [12], [13], [8], [14], [25], [26], [15], [9], etc.). From (7), one can easily see that for all x > 0 and z > 1: lim t→∞ F (tx) F (t) = x−1/γ ⇔ lim t→∞ U(tz) U(t) = zγ . (8) The relation in (8) is namely called the first order regularly varying condition. The pa- rameter γ is the tail index (or the extreme value index) and governs the tail behavior, A. Aghrabatt et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6024 6 of 25 with larger values indicating heavier tails. Its estimation has received a great attention in the extreme value literature, especially in the case of i.i.d. random variables (cf. [11]). Although only few papers consider that for the case of time series with serial dependence features. We can mention among others, [27], [28]. And very recently [18], [19] and [20]. Next, we note that: • When γ > 1, the first moment E(X1) of the dependence insured losses is not defined. Thus, their associated reinsurance distorted risk premiums πgβ (R) is also not defined and it is not possible to do its statistical estimation. • When 0 < γ ≤ 1/2 (the lower half of the unit interval), then the second moments of the of the dependence insured losses, E[X2+ϵ 1 ] < ∞, for some ϵ > 0, and so a nonparametric estimator for the reinsurance distorted risk premiums πgβ (R) can be obtain by substituting in (3) the unknown distribution function F with its empirical component Fn defined as Fn(x) = n−1 ∑n i=1 1(Xi ≤ x), this estimator is asymptoti- cally normal. • When 1/2 < γ < 1 (the upper half of the unit interval), then the second moment is infinite, and so the asymptotic normality of the nonparametric estimator of πgβ (R). is violated. The last situation motivate the need of a specific estimator of the reinsurance distorted risk premiums for dependence insured loses with heavy-tailed distributions and infinite second moments, that is with index in he upper half of the unit interval (1/2 < γ < 1). By making use the extreme value theory which offers satisfactory statistical results for such distributions, we need to estimate the tail index γ and establish a class of semi- parametric estimators of the distorted risk premium pg(R) in the case of dependence insured risks. The most popular positive tail index estimators in the framework of extreme value theory is the original Hill’s estimator [29], defined as: γ̂ (H) k := 1 k k∑ i=1 logXn−i+1,n − logXn−k,n, (9) where X1,n ≤ · · · ≤ Xn,n stands for the order statistics and k = k(n) represents an intermediate sequence, that is, a sequence such that k → ∞ and k/n → 0, as n → ∞. Using the tail quantile function Xn−[kt],n, 0 < t < n/k, [19] proposed the following Kernel- type estimator of the tail index γ under β-mixing series: γ̂ (K) k = ∫ 1 0 ( logXn−[kt],n − logXn−k,n ) d(tK(t)), (10) where K is a kernel function integrated to one. Note that in the particular case where K = K := I(0,1), the estimator γ̂ K k corresponds to the well-known Hill’s estimator (9) of A. Aghrabatt et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6024 7 of 25 positive tail index γ. Also, it is easy to see that the kernel estimator γ̂ (K) k can be rewritten as follows: γ̂ (K) k := 1 k k∑ i=1 { i k K ( i k ) − i− 1 k K ( i− 1 k )} (logXn−i+1,n − logXn−k,n) . 2.2. Estimation of distortion risk under depedence serials To better understand the heavier of tail distribution of insured risks, which is governed by the unknown tail index, many authors used the extreme value methodology and inves- tigated semiparametric estimators of the distortion risk premiums in the case of β-mixing random variables, we use the class of kernel estimator defined in (11) and investigate semi- parametric estimators of the distortion risk premiums with optimal retention levels. Note that, the optimal retention level corresponds to the Value-at-Risk Q(1−k/n), which is the minimum amount that a company must have to cover the risk X with an uncer- tain ruin probability. Since F (Q(1 − k/n)) = k/n, from (4), the optimal risk distorted reinsurance premiums is defined as: πβ,n := πgβ (Q(1− k/n)) = − ∫ k/n 0 gβ(s)dQ(1− s). (11) The distorted reinsurance premium πβ,n is unknown since it depends on the unknown high quantile Q(1−s), s → 0. A Weissman-type estimator [21] of high quantiles for heavy-tailed distributions, based on the class of kernel estimators in (10), is defined as: Q̂ (K) k (1− s) = (ns/k)−γ̂ (K) k Xn−k,n, s → 0, (12) where γ̂ (K) k is the above class of kernel estimator for the extreme value index γ and the quantity Xn−k,n is a moderate quantile and assigned to be the empirical estimator of the optimal retention level Q(1− k/n). Substituting in (11) the extreme quantileQ(1−s), s → 0 with its Weisman’s type estimator Q (K) k (1− s), we introduce the following class of semiparametric estimator for πβ,n: π̂ (K) β,k,n = − ∫ k/n 0 gβ(s)dQ̂ (K) k (1− s), (13) which leads to: π̂ (K) β,k,n = γ̂ (K) k 1/β − γ̂ (K) k gβ(k/n)Xn−k,n, (14) provided that P(γ̂(K) k > 1/β) = o(1), for large values of n. In the particular case where K = K := I(0,1), π̂ (K) β,k,n corresponds exactly to the distortion risk premiums estimator introduced in [17] under dependence serials. From the second order regularly varying condition (CSO) and the regularity conditions on the β-mixing coefficients (CR), [17] established, the asymptotic normality of the estimator π̂ (K) β,k,n. A. Aghrabatt et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6024 8 of 25 3. Main Results 3.1. Asymptotic distribution of class of estimators π̂ (K) β,k,n In this section, we investigate the asymptotic distribution of the class of distortion premi- ums estimators π̂ (K) β,k,n introduced in (14). Clearly, this class of estimator is directly related to the kernel estimator γ̂ (K) k of the tail index γ. Note that in extreme value theory (EVT), to prove the asymptotic distribution of the tail index estimators such as the Hill’s estimator or the kernel-type one, we need a second order condition which specifies the rate of convergence for the left-hand side of the equations in (8) to their limits (See, eg. [30], [11] and [31]). This condition can be formulated in different ways as shown below. We will use the formulation later-on. Second order regularly varying condition (CSO). Suppose that there exists a positive or negative function A with lim t→∞ A(t) = 0 and a real number ρ ≤ 0 such that lim t→∞ 1 A(t) ( U(tx) U(t) − xγ ) = xγ xρ − 1 ρ , ∀x > 0. (15) The rate of the convergence for the function A to 0 is essential since it helps to exhibit the bias term of the tail index estimators. The asymptotic normality of the original Hill’s estimator has been established for β-mixing sequences in [27] and [28]. Also, from the assumption that the intermediate sequence k is such that k1/2A(n/k) → λ ∈ R, as n → ∞ and assuming the following regularity condi- tions on the β-mixing coefficients: Regularity conditions (CR). There exist ϵ > 0, a bivariate function r and a sequence ℓn such that, as n → ∞, (a) β(ℓ) ℓ n+ ℓ log2 k√ k → 0; (b) n ℓ k Cov ( ℓ∑ i=1 I{Xi>F←(1−kx/n)}, ℓ∑ i=1 I{Xi>F←(1−ky/n)} ) → r(x, y), ∀ 0 ≤x, y≤1 + ϵ; (c) For some constant C: n ℓ k E ( ℓ∑ i=1 I{F←(1−ky/n) 0. The consistency and the asymptotic normality of ρ̂kρ have been established in [20] in the case of β-mixing serials under the second order condition CSO and the assumptions kρ → ∞, kρ/n → 0 and k 1/2 ρ A(n/kρ) → ∞, as n → ∞. Our next goal is to establish, under suitable assumptions, the asymptotic normality of π̃ (K ∆̂∗opt ) β,k,n,ρ̂ . This is done in the following theorem. Theorem 2. Under the assumptions of Theorem 1, if ρ̂ is either a canonical negative value ρ̂ = ρ = ρ0 or an external estimator ρ̂ = ρ̂kρ, consistent in probability to ρ, with kρ := kρ(n), an intermediate sequence of integers greater than k, satisfying kρ → ∞ and kρ/n → 0, as n → ∞, then we have: √ k ( π̃ (K ∆̂∗opt ) β,k,n ρ̂ − πβ,n ) gβ(k/n)Q(1− k/n) d→ N ( 0, ÃV(γ, ρ, α) ) , where ÃV(γ, ρ, α) = (a0) 2r(1, 1) + (a1) 2 ∫ ∫ [0,1]2 [ r(t, s) ts − r(t, 1) t − r(1, s) s + r(1, 1) ] d { sK∆̂∗opt (s) } d { tK∆̂∗opt (t) } + (a3) 2 ∫ ∫ [0,1]2 [ r(t, s) ts − r(t, 1) t − r(1, s) s + r(1, 1) ] d { s (1−K2,ρ(s)) } d { t (1−K2,ρ(t)) } + 2a0 a1 ∫ 1 0 [ r(t, 1) t − r[1, 1] ] d { tK∆̂∗opt (t) } + 2a0 a3 ∫ 1 0 [ r(t, 1) t − r[1, 1] ] d { t (1−K2,ρ(t)) } + 2a1 a3 ∫ ∫ [0,1]2 [ r(t, s) ts − r(t, 1) t − r(1, s) s + r(1, 1) ] d { sK∆̂∗opt (s) } d { t (1−K2,ρ(t)) } , with a0 = βγ2 1− βγ , a1 = βγ (1− βγ)2 and a3 = − βγ(1− ρ)(1− 2ρ) ρ2(1− βγ)(1− β γ − β ρ) . A. Aghrabatt et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6024 14 of 25 4. Simulation Study In this section, the class of biased estimator π̂ (K) β,k,n and the reduced-bias estimator π̃ (K ∆̂∗opt ) β,k,n, ρ̂ of the distortion risk measure πβ,n with optimal retention level Q(1− k/n) are compared in a simulation study. To this end, we consider the following classical stationary models, which satisfy the regularity (CR) assumptions: • (Autoregressive (AR) model): Consider first the stationary solution of the AR(1) equation: Xi = θXi−1 + Zi, i = 1, ..., n, (27) for some θ ∈ (0, 1) and i.i.d. random variables Zi. The distribution function of the innova- tions Zi is denoted by FZ . Assume that FZ admits a positive Lebesgue density which is L1 Lipschiz-continuous; see [27] eq. (42). Suppose that as x → ∞, 1 − FZ(x) ∼ px−1/γℓ(x) and FZ(−x) ∼ qx−1/γℓ(x), for some slowly varying function ℓ and p = 1 − q ∈ (0, 1). Then from Sect. 3.2 of [27], we get that 1 − F (x) ∼ dθ(1 − FZ(x)), as x −→ ∞, where dθ = (1 − θ1/γ)−1. Furthermore, the regularity conditions hold with/ r(x, y) = x ∧ y + ∑∞ m=1 (cm(x, y) + cm(y, x)) , where cm(x, y) = x ∧ yθm/γ . • (Moving average (MA) model): Consider the stationary solution of MA(1) equation: Xi = θZi−1 + Zi, i = 1, ..., n; (28) where the innovation Zi satisfies the same conditions as in the above AR(1) model. And from Sect. 3.2 of [27], we obtain 1−F (x) ∼ dθ(1−FZ(x)) as x → ∞, where dθ = 1+θ1/γ . One can also compute the covariance structure as : r(x, y) = x ∧ y + (1 + θ1/γ)−1(x ∧ yθ1/γ + y ∧ xθ1/γ). Now, we proceed by generating the data for the three (03) models. This involves an independent model and the two models mentioned above. We first generate the i.i.d in- novations (Z1, ..., Zn), such that: FZ(z) = { (1− q)(1− F̃ (−z)) if z < 0, 1− q + qF̃ (z)) if z > 0, where F̃ stands for the Fréchet distribution function F̃ (z) = exp((−z)−1/γ) for z > 0, and p = 0.75. Then FZ belongs to the domain of attraction with extreme value index γ > 0. In the following table, we generate the three (03) time series models under simulation with their tail distribution, which are needed to compute the true distortion risk premiums: For each generating model, we simulate N = 1000 samples with size n = 1000. To evaluate of the true value of distortion risk premiums πβ,n , we use the approximation of the tail distribution F given in Table 1. Also, we apply to each sample both estimators π̂ (K) β,k,n and π̃ (K∆̂∗ opt ) β,k,n,ρ̂ , for different integers of top order statistic k = 1, ...,m, where m is the number of positive values of the simulated A. Aghrabatt et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6024 15 of 25 Description Independence AR(1) MA(1) of Models Xi = Zi Xi = θXi−1 + Zi Xi = θZi−1 + Zi Tail-distribution F (x) = FZ(x) F (x) ∼ (1− θ1/γ)−1FZ(x) F (x) ∼ (1 + θ1/γ)FZ(x) coefficients θ = 0 and γ = 0.6 θ = 0.3 and γ = 0.6 θ = 0.3 and γ = 0.6 Table 1: Description of models under simulation with their associated tail distribution functions. samples. For computation and the comparison of the estimators, we adopt the following steps: • The class of estimators π̂ (K) β,k,n is computed with the tail index estimators γ̂ (K) k , k = 1, ...,mn and two different kernel function K satisfying the assumption (K). The first kernel is the power function , defined as K(s) = (1 + τ)sτ I{0 0, then gβ(k/nt)U(k/nt) → 0, as t → ∞. Thus, an integration by parts yields to: πβ,n = gβ(k/n) 1 β ∫ ∞ 1 t−1/β−1(U(nt/k)− U(n/k))dt. Therefore, by using again U(n/k) = Q(1− k/n), we get: √ kA3 gβ(k/n)Q(1− k/n) = √ k [ βγ 1− βγ − 1 β ∫ ∞ 1 t−1/β−1 ( U(nt/k) U(n/k) − 1 ) dt ] = − 1 β √ k ∫ ∞ 1 t−1−1/β ( U(nt/k) U(n/k) − tγ ) dt. (33) Assume that the second order condition (CSO) holds. From Theorem B.2.18 in [11], we have we have for a possibly different function Ã, with Ã(z) ∼ A(z), z → ∞, and for all ε, δ > 0, there exists some positive number z0 = z0(ε, δ) such that for tz ≥ z0:∣∣∣∣ U(tz) U(z) − tγ Ã(z) − tγ tρ − 1 ρ ∣∣∣∣ ≤ εtρ+γ max(tδ, t−δ). (34) A. Aghrabatt et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6024 22 of 25 Since k1/2A(n/k) → λ ∈ R, then from (33) and (34), we have for 0 < γ < 1, γ < 1/β and for all values of n large enough: √ kA3 gβ(k/n)Q(1− k/n) = −λ β ∫ ∞ 1 tγ−1/β−1 t ρ − 1 ρ dt(1 + o(1)) = − λβ (1− βγ − βρ)(1− βγ) (1 + o(1)). (35) Finlay, form (31), (32) and (35), we get as n → ∞: √ k ( π̂ (K) β,k,n − πβ,n ) gβ(k/n)Q(1− k/n) d→λ { β (1− βγ)2 ∫ 1 0 t−ρK(t)dt− β (1− βγ − βρ)(1− βγ) } + γβ (1− β γ)2 ∫ 1 0 ( t−1W (t)−W (1) ) d(tK(t)) + βγ2 1− β γ W (1). (36) Proof of Corollary 1. Computing the variance of the Gaussian terms appeared in the right side of (36) with respect to the covariance structure r(., .), the proof of Corollary 1 holds. Proof of Theorem 2. Recall that π̃ (K∆̂∗ opt ) β,k,nρ̂ : =  γ̂ (K∆̂∗ opt ) n,k 1 β − γ̂ (K∆̂∗ opt ) n,k + Ân,k(ρ̂ )( 1 β − γ̂ (K∆̂∗ opt ) n,k )( 1− βγ̂ (K∆̂∗ opt ) n,k − βρ̂ )  gβ(k/n) Xn−k,n. where Ân,k(ρ̂ ) := − (1− ρ̂)(1− 2ρ̂) ρ̂ 2 { γ̂ (K) n,k − γ̂ (K2,ρ̂) n,k } is the estimator of A(n/k). Next, as in the proof of Theorem 1, we have π̂ (K∆̂∗ opt ) β,R − πβ,n = H1 +H2 +H3 with H1 =  π̂ (K∆̂∗ opt ) 1 β − π̂ (K∆̂∗ opt ) − γ 1 β − γ  gβ(k/n)Xn−k,n H2 = γ 1 β − γ ( Xn−k,n Q(1− k/n) − 1 ) gβ(k/n)Q(1− k/n) H3 = γ 1 β − γ gβ(k/n)Q(1− k/n)− ∫ ∞ Q(1−k/n) gβ(F (x))dx H4 = Ân,k(ρ̂ )( 1 β − γ̂ (K∆̂∗ opt ) n,k )( 1− βγ̂ (K∆̂∗ opt ) n,k − βρ̂ )gβ(k/n) Xn−k,n. Under assumptions and for all n large enough, we have from (23), √ k ( γ̂ (K∆̂∗ opt ) k − γ ) d = γ ∫ 1 0 ( t−1W (t)−W (1) ) d(tK∆∗ opt (t)) {1 + oP(1)} . (37) A. Aghrabatt et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6024 23 of 25 Therefore, using the Delta-method procedure, we get for all n large enough: H1 d = (1 + oP(1)) β (1− β γ)2 k−1/2gβ(k/n)Xn−k,n { γ ∫ 1 0 ( t−1W (t)−W (1) ) d(tK∆̂∗ opt (t))) } , Next, using (30), we have implies Xn−k,n/Q(1− k/n) = 1 + oP(1), as n → ∞ and √ kH1 gβ(k/n)Q(1− k/n) d = (1 + oP(1)) γβ (1− β γ)2 ∫ 1 0 ( t−1W (t)−W (1) ) d(tK∆̂∗ opt (t)), (38) as n → ∞. Similarly, we have for all n large enough: √ kH2 gβ(k/n)Q(1− k/n) d = (1 + oP(1)) γβ 1− β γ √ k ( Xn−k,n Q(1− k/n) − 1 ) . Using again (30), we get: √ kH2 gβ(k/n)Q(1− k/n) d = (1 + oP(1)) βγ2 1− β γ W (1). (39) For the term H3, we note that it is exactly equal to the term A3 in the proof of the Theorem 1. Therefore, using the statement in (35), we have for all values of n large enough, √ kH3 gβ(k/n)Q(1− k/n) = − √ kA(n/k) β (1− βγ)(1− βγ − βρ) (1 + o(1)). (40) Now, for the term H4, using the fact that Xn−k,n/Q(1− k/n) = 1 + oP(1), we have √ kH4 gβ(k/n)Q(1− k/n) = √ k Ân,k(ρ̂) β( 1− β γ̂ (K∆̂∗ opt ) n,k )( 1− β γ̂ (K∆̂∗ opt ) n,k − β ρ̂ ) (1 + oP(1)). Since γ̂ (K∆̂∗ opt ) k and ρ̂ := ρ̂kρ are respectively consistent to γ and ρ, we obtain: √ kH4 gβ(k/n)Q(1− k/n) d = √ k Ân,k(ρ̂) β (1− β γ)(1− β γ − β ρ) (1 + oP(1)). (41) Hence, from (40) and (41), we get for all large n, √ k(H3 +H4) gβ(k/n)Q(1− k/n) = √ k ( Ân,k(ρ̂)−A(n/k) ) β (1− βγ)(1− β γ − β ρ) (1 + oP(1)). Recall from that Ân,k(ρ̂ ) := − (1− ρ̂)(1− 2ρ̂) ρ̂ 2 { γ̂ (K) n,k − γ̂ (K2,ρ̂) n,k } . Using again the consistency of ρ̂ := ρ̂kρ to ρ and the expansion in (16), we get for all large values of n: √ k ( γ̂ (K) n,k − γ̂ (K2,ρ̂) n,k ) d = √ k ( γ̂ (K) n,k − γ ) − √ k ( γ̂ (K2,ρ̂) n,k − γ ) + oP(1) d = − √ kA(n/k) ρ2 (1− ρ)(1− 2ρ) + γ ∫ 1 0 ( t−1W (t)−W (1) ) d { t (1−K2,ρ(t)) } + oP(1). A. Aghrabatt et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6024 24 of 25 With the consistency of ρ̂ := ρ̂kρ to ρ, this leads to Ân,k(ρ̂ ) d = A(n/k)− γ(1− ρ)(1− 2ρ) ρ2k1/2 ∫ 1 0 ( t−1W (t)−W (1) ) d { t (K −K2,ρ(t)) } + oP(k −1/2). This implies that for all large values of n: √ k(H3 +H4) gβ(k/n)Q(1− k/n) d = − βγ(1− ρ)(1− 2ρ) ρ2(1− βγ)(1− β γ − β ρ) ∫ 1 0 ( t−1W (t)−W (1) ) d { t (1−K2,ρ(t)) } +oP(1). (42) Finlay, form (38), (39) and (42), we get as n → ∞: √ k ( π̃ (K∆̂∗ opt ) β,k,nρ̂ − πβ,n ) gβ(k/n)Q(1− k/n) d→ γβ (1− β γ)2 ∫ 1 0 ( t−1W (t)−W (1) ) d(tK(t)) + βγ2 1− β γ W (1). Proof of Corollary 1. 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