EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 4, 2023, 2509-2543 ISSN 1307-5543 – ejpam.com Published by New York Business Global Kernel estimation of the Quintile Share Ratio index of inequality for heavy-tailed income distributions Modou Kebe1, El Hadji Deme1,∗, Tchilabalo Abozou Kpanzou2, Solym Mawaki Manou-Abi3, Ebrima Sisawo1 1 LERSTAD, UFR SAT, Université Gaston Berger, BP 234, Saint-Louis, Sénégal 2 LaMMASD, Département de Mathématiques, Université de Kara, Togo 3 Institut Montpelliérain Alexander Grothendieck,UMR CNRS 5149, Place Eugéne Bataillon, 34090,, Montpellier, France. Centre Universitaire de Formation et de Recherche, Mayotte, France Abstract. Evidence from micro-data shows that capital incomes are exceedingly volatile, which makes up a disproportionately high contribution to the overall inequality in populations with the heavy-tailed nature on the income distributions for many countries. The quintile share ratio (QSR) is a recently introduced measure of income inequality, also forming part of the European Laeken indicators and which cover four important dimensions of social inclusion (health, education, employment and financial poverty). In 2001, the European Council decided that income inequality in the European Union member states should be described using a number of indicators including the QSR. Non-parametric estimation has been developed on the QSR index for heavy-tailed capital incomes distributions. However, this method of estimation does not give satisfactory statistical performances, since it suffers badly from under coverage, and so we cannot rely on the non- parametric estimator. Hence, we need another estimator in the case of heavy-tailed populations. This is the reason why we introduce, in this paper, a class of semi-parametric estimators of the QSR index of economic inequality for heavy-tailed income distributions. Our methodology is based on the extreme value theory, which offers adequate statistical results for such distributions. We establish their asymptotic distribution, and through a simulation study, we illustrate their behavior in terms of the absolute bias and the median squared error. The simulation results clearly show that our estimators work well. 2020 Mathematics Subject Classifications: 62E20, 62G30, 62G32 Key Words and Phrases: Inequality Measures, QSR index, Kernel estimation, Heavy-tailed, Extreme value Theory. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i4.4765 Email addresses: modoukebe44@gmail.com (M. Kebe), elhadji.deme@egb.edu.sn (E. Deme) , kpanzout@gmail.com (T. A. Kpanzou), solym.manou-abi@univ-mayotte.fr (S. M.Manou-Abi), esso870@gmail.com (E. Sisawo) https://www.ejpam.com 2509 © 2023 EJPAM All rights reserved. M. Kebe et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2509-2543 2510 1. Introduction Inequality measurement is an attempt to make sense of comparisons of income distributions in terms of criteria that can be derived from ethical principles, appealing mathematical constructions or simple intuition. A serious approach to inequality measurement should begin with a consideration of the entities to which the tools of distributional judgement are applied. In the recent decades, the capital income is among the major incomes in a big number of countries, there are many important studies devoted to the capital income, mainly dealing with relationships between capital income taxation and welfare benefits, and in particular with the incidence and efficiency effects of taxes on incomes from capital in various scenarios of development sectors. For further details see, e.g., [1], [6], [20], [25], [28], [40]. Measuring and analyzing incomes, risks and other random outcomes has been an active and fruitful research field. Academics and governmental researchers have been developing measures that would aid them in understanding income and loss distributions, their differ- ences with respect to geographic regions and changes over time periods. It is a fascinating area due to a number of reasons, one of them being the fact that different measures or indices are needed to reveal different features of capital income distributions. The Gini index has been widely used by economists and sociologists to measure economic inequality. It has been also studied extensively and its properties documented in a number of papers (see e.g. the survey papers [31], [43], ...). Also, [44] suggested the Zenga index of inequal- ity measure, which aggregates the ratios of lower and upper conditional tail expectations, and by so doing takes into account the relative nature of the poor and the rich for a given population. It has been also explored from various points of view, see, eg. [22], [23], [24] and [25]. Another trend, somewhat different from Zenga’s but equally interesting, which is based on the Palma index, is considered in, e.g. [7]. The quintile share ratio (QSR) is a recently introduced measure of income inequality, also forming part of the European Laeken indicators which cover four important dimensions of social inclusion (Financial poverty, employment, health and education). In 2001, the European Council decided that income inequality in the European Union (EU) member states should be described using a number of indicators including the Quintile share ratio index. Compared to the Gini index, relatively little research is available on the statistical inference of the QSR index. [32] investigated the QSR and established its variance in a complex sampling design framework. The authors upgraded on earlier work by [35] and [36]. As is to be expected from its definition, the influence function of the QSR is un- bounded. The form of that influence function has been also derived by [30]. Also, [29] derived the asymptotic distribution of a non-parametric plug-in estimator for the QSR index. However, this estimation suffers badly from under coverage, and so we cannot rely on the non-parametric estimator. Hence, we need another estimator in the case of heavy-tailed populations. This is the reason why we introduce, in this paper, a class of semi-parametric estimators of the QSR index of inequality measure for heavy-tailed in- M. Kebe et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2509-2543 2511 come distributions. Our consideration is based on the extreme value methodology, which offers adequate statistical tools for such distributions. This paper is organised as follows. In Section 2, we shall recall the definition of the QSR index by presenting it in terms of upper and lower integrals. Also, we shall briefly consider a non-parametric estimator of the QSR index which is obtained by replacing the underlying cumulative distribution function (cdf) of the population by its empirical counterpart. As this estimator does not exhibit, satisfactory performance in heavy-tailed populations, we introduce, in Section 3, a class of semi-parametric estimators of the QSR index of economic inequality for heavy-tailed income distributions. Through the extreme value methodology, we establish their asymptotic distribution, and derived their improved results in Section 4. Under a simulation study, we illustrate their behavior in terms of the absolute bias and the median squared error in Section 5. 2. Definitions and empirical estimation Suppose that we have at our disposal a sample (X1, · · · , Xn), n > 1 of independent and identically distributed in a population represented by a non-negative random variable X ≥ 0, with capital income distribution F (x) = P(X ≤ x) and finite mean µF := E[X]. We assume that F is continuous and strictly increasing. The (generalized) inverse Q : [0, 1) 7→ [0,∞) of the income distribution F , known in the literature as the quantile function, is defined for all s ∈ [0, 1) by the formula Q(s) = inf{x, F (x) ≥ s}. Considering two different levels α and β, such that 0 < α < β < 1 as illustrated in [29], the QSR index at levels α and β of the capital income X denoted by η(Q,α, β), is the ratio of an upper integral U(Q, β) to a lower integral L(Q,α). More precisely, we have: η(Q,α, β) := U(Q, β) L(Q,α) = ∫ 1 β Q(s)ds∫ α 0 Q(s)ds . (1) The QSR index is then given by: η(Q, 0.2, 0.8). In what follows, we will consider the more general QSR index η(Q,α, β) and the results will follow directly from that. The empirical estimator of the distribution F is defined by Fn(x) = n−1 ∑n i=1 I{Xi≤x} and its corresponding empirical quantile function is expressed by Qn(s) = inf{x;Fn(x) ≥ s}, where IS is the indicator function of the set S. Denote by X1,n ≤ ... ≤ Xn,n the order statistics associated with the sample (X1, . . . , Xn). Thus, Qn(s) is equal to the i-th order statistic Xi,n for all s ∈ ((i − 1)/n, i/n], and for all i = 1, ..., n. For this, one natural candidate for the empirical estimator of η(Q,α, β) is obtained by replacing in (1) the true quantile Q(·) with the sample quantiles Qn(·). We arrive at the following ’traditional’ QSR index estimator (see, e.g., [29]): η̂n(α, β) := η(Qn, α, β) = ∫ 1 β Qn(s)ds∫ α 0 Qn(s)ds . (2) M. Kebe et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2509-2543 2512 Clearly, the estimator η̂n(α, β) can be rewritten as: η̂n(α, β) = n−1 n∑ j=[nβ]+1 Xj,n  /n−1 [nα]∑ j=1 Xj,n  , (3) where [x] is the integer part of x. Note that the QSR estimator η̂n(α, β) is the ratio of a U-statistic to a L-statistic. According to [29]), for a given capital income distribution F with finite variance, the following result holds: √ n ( η̂n(α, β) − η(Q,α, β) ) d→ N ( 0, σ2 η(α, β) ) , as n → ∞, where σ2 η(α, β) := σ2(β, 1) + (η(Q,α, β))2 σ2(0, α) − 2η(Q,α, β) ∫ α 0 sdQ(s) ∫ 1 β (1 − t)dQ(t), with for 0 ≤ s < t ≤ 1 and σ2(s, t) = ∫ t s ∫ t s (min(u; v) − uv)dQ(u)dQ(v). This result is violated when capital incomes have heavy-tailed distributions with infinite variance, since the asymptotic variance σ2 η(α, β) is also infinite. For more detail; refer to [29]). However, micro-data show that capital incomes account for a large part of disparity in populations. Furthermore, in some countries, capital incomes have been making up a disproportionately high contribution to the overall inequality (see, e.g., [17]). The present research has been motivated by the need for better understanding the distribution and inequality of capital incomes, which in many countries appear to be heavy-tailed (see, eg. [25]). To this aim, we assume that the income distribution F is heavy-tailed. This is equivalent to the fact that the survival function F := 1 − F associated to F is regularly varying at infinity with index −1/γ < 0. More precisely, F (x) = x−1/γℓF (x), x > 0, (4) where ℓF is a slowly varying function at infinity, i.e for all x > 0, ℓF (tx)/ℓF (t) → 1, as t → ∞. The relation (4) is also equivalent to Q(1 − s) = s−γℓQ(s), s ∈ (0, 1), where ℓQ(zs)/ℓQ(s) → 1, as s → 0, for all z ∈ (0, 1). From (4), one can easily see that for all x > 0 and z ∈ (0, 1): lim t→∞ F (tx) F (t) = x−1/γ and lim s→0 Q(1 − zs) Q(1 − s) = z−γ . (5) M. Kebe et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2509-2543 2513 The relation in (5) is namely called the first order regularly varying condition. The param- eter γ is called the tail index (or the extreme value index) and governs the tail behavior, with larger values indicating heavier tails. Its estimation has received a great attention in the extreme value literature (see, e.g., [10]). This kind of models and its unidentified parameters have been previously used by various authors such as [15], [25], [29] to assess inequality measure of capital incomes. Next, we also note that: • When γ > 1, the QSR index η(Q,α, β) and thus its estimator η̂n(α, β) are not defined. • When 0 < γ ≤ 1/2 (the lower half of the unit interval), then E[X2+ϵ] < ∞ for some ϵ > 0, and so we can use the available asymptotically normal estimator η̂n(α, β), (see, [29]). • 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 estimator η̂n(α, β) is violated (see, [29]). The last situation motivate the need of a specific estimator of the QSR index for heavy- tailed income distributions with infinite second moments. The class of heavy-tailed dis- tributions (the so-called Pareto-type 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, incomes of countries with very high economic inequality, etc. (see, e.g., [2]; [3]; [4]; [11] ; [13]; [14]; [15]; [25]; [34]; [39]; [41]). To better understand the heavy-tailed distribution and the inequality of capital incomes, which are governed by the unknown extreme value index γ, we make use, in this paper, of the extreme value methodology and propose asymptotically normal estimators of the QSR index η(Q,α, β). The following section concerns a class of semi-parametric estimators of the QSR index η(Q,α, β) for heavy-tailed income distributions with infinite second order moments. 3. Kernel estimation of the Quintile Share Ratio index In the rest of this paper, we shall be concerned with heavy-tailed capital income distribu- tions with index in the upper half of the unit interval. More precisely, we will deal with the case where F satisfies F (x) = x−1/γℓF (x), x > 0, 1/2 < γ < 1. (6) We have mentioned above that the tail index γ controls the behavior of income distri- bution F and its finite variance. In this spirit, we shall take into account the estimation M. Kebe et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2509-2543 2514 of γ in the construction of our class of semi-parametric estimators for QSR index η(Q,α, β). Now, Let k = k(n) be an intermediate sequence of integers, i.e., a sequence such that: 1 < k < n, k → ∞ and k/n → 0 as n → ∞. (7) Using the same arguments as in [34], the QSR index can be decomposed for 0 < α < β < 1 − k/n as follows: η(Q), α, β) := 1 L(Q,α) {∫ 1−k/n β Q(s)ds + ∫ 1 1−k/n Q(s)ds } . (8) As mentioned above, one can estimate the moderate quantile Q(s), β ≤ s ≤ 1 − k/n by its empirical estimator Qn(s). But the case where 1 − k/n < s < 1 corresponds to high quantiles i.e Q(s), s → 1 and it is not possible to use the empirical estimation Qn(s). Under the first order regularly varying condition (5), we have Q(1−zs) ≈ z−γQ(1−s), s → 0. By setting zs = 1 − u and s = k/n, we obtain the following approximation: Q(u) ≈ (n(1 − u)/k)−γQ(1 − k/n), u → 1. (9) This leads to the following Weissman’s type estimator ([42]) of high quantile Q(u), u → 1: Q (K) n,k (u) = (n k (1 − u) )−γ̂ (K) n,k Xn−k,n, (10) with γ̂Kn,k, the kernel class of estimators for the tail index γ, introduced in [9] and given by: γ̂ (K) n,k = 1 k k∑ j=1 jK ( j k + 1 ) log ( Xn−j+1,n Xn−j,n ) , (11) where K is a kernel integrating to one. Note that in the particular case where K = K := I(0,1), the estimator γ̂ K n,k corresponds to the well-known Hill’s estimator ([27]) of the tail index γ defined by: γ̂n,k := γ̂ (K) n,k = 1 k k∑ j=1 j log ( Xn−j+1,n Xn−j,n ) . (12) The estimator γ̂n,k is the most popular estimator of the tail index γ in the framework of heavy-tailed distributions. The Weissman’s estimator ([42]) of high quantile is thus defined as Q (K) n,k (u). Next, replacing in (8), Q(s) by its empirical quantile estimator Qn(s), for β ≤ s ≤ 1−k/n and by its high quantiles estimator Q (K) n,k (s), for 1−k/n < s < 1, we arrive at the following kernel-type estimators of η(Q,α, β), 0 < α < β < 1 − k/n: η̂ (K) n,k (α, β) = 1 Ln(α) {∫ 1−k/n β Qn(s)ds + ∫ 1 1−k/n Q (K) n,k (s)ds } , (13) M. Kebe et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2509-2543 2515 where Ln(α) = n−1 [nα]∑ j=1 Xj,n is the above empirical estimator of the lower integral L(α), which can be rewritten as: η̂ (K) n,k (α, β) = 1 Ln(α) { n−k∑ j=1 [( j n − β ) + − ( j − 1 n − β ) + ] Xj,n + (k/n) 1 − γ̂ (K) n,k Xn−k,n } , (14) where (s − t)+ is the classical notation for the positive part of (s − t). The estimator η̂ (K) n,k (α, β) generalizes the one proposed in [15], when we use a general kernel instead of K. In the above definitions and in what follows we indicate by Q(·) and Qn(·) the quantile function and its empirical counterpart (both functions are left-continuous). Let us now state an asymptotic normality result for η̂ (K) n,k (α, β). As it exhibit a bias, we will introduce a bias reduction method to estimate the QSR index. 4. Main results In extreme value analysis, one can easily achieve asymptotic normality results by impos- ing a second order regularly varying condition (RU), (see, e.g., [10], Page 48), which is necessary to quantify the speed of convergence in (5). This condition can be formulated in different ways, below we state it in terms of the tail quantile functions U(x) = Q(1−1/x): (RU): There exist a function a(x) → 0 as x → ∞ of constant sign for large values of x and a second order parameter ρ < 0 such that, for any x > 0, lim t→∞ logU(tx) − logU(t) − γ log x a(t) = xρ − 1 ρ . (15) Note that the condition (RU) implies that |a| is regularly varying with index ρ (see, e.g., [10];[18]; [33]). As an example of heavy-tailed distributions satisfying the second order regularly varying condition (RU), we have the so called and frequently used Hall’s model which is a class of cdf’s, such that U(t) = ctγ(1 + dρ−1a(t) + o(tρ)), as t → ∞, where γ > 0, ρ ≤ 0, c > 0, and d ∈ R∗. This sub-class of heavy-tailed models contains the distri- butions such as Pareto, Burr, Fréchet and Student-t. For statistical inference concerning the second-order parameter ρ, we refer, for example, to [12] and [38]. Section 4.1 below gives the asymptotic normality of our proposed estimator η̂ (K) n,k (α, β). 4.1. Asymptotic normality of the kernel estimator η̂ (K) n,k (α, β). To establish the asymptotic normality of the kernel-type estimator η̂ (K) n,k (α, β), some clas- sical assumptions about the kernel K are needed. M. Kebe et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2509-2543 2516 Condition (K). Let K be a function defined on (0, 1] such that (i) K(s) ≥ 0, whenever, 0 < s ≤ 1 and K(1) = 0; (ii) K(·) is differentiable, non increas- ing and right continuous on (0, 1]; (iii) K and K ′ are bounded; (iv) ∫ 1 0 K(u)du = 1; (v) ∫ 1 0 u−1/2K(u)du < 1. These conditions are not restrictive but are satisfied by the usual weight functions used in the literature, including the power kernel K(s) = (1 + τ)sτ I{0 0 sup λ1/n≤ s≤1−λ1/n |βn(s) − Bn(s)| (s(1 − s))1/2−ν = OP(n−ν), (17) where βn is the resulting empirical quantile function denoted by: βn(t) = √ n (t− Vn(t)) with Vn(s) = ξj,n, j−1 n < s ≤ j n , j = 1, ..., n and Vn(0) = 0. Before we establish the asymptotic results in Theorem ‘1, let’s introduce the following notations. Next, from (8), (13), (14) and (42), the QSR index η(Q,α, β) and its biased estimator η̂ (K) n,k (α, β), 0 < α < β < 1 − k/n, can be respectively rewritten as: η(Q,α, β) := Un,k,1(Q, β) L(Q,α) + Un,k,2(Q) L(Q,α) , and η̂ (K) n,k (α, β) := Un,k,1(Qn, β) Ln(α) + Un,k,2 ( Q (K) n,k ) Ln(α) , where Ln(α) := ∫ α 0 Qn(s)ds = n−1 ∑[nα] j=1 Xj,n is the empirical estimator of the lower integral L(Q,α) = ∫ α 0 Q(s)ds and the U -functional integrals are defined as: Un,k,1(Q, t) := ∫ 1−k/n t Q(s)ds, for 0 ≤ t < 1 − k/n Un,k,2(Q) := ∫ 1 1−k/n Q(s)ds, Un,k,1(Qn, t) := ∫ 1−k/n t Qn(s)ds, for 0 ≤ t < 1 − k/n, Un,k,2 ( Q (K) n,k ) := ∫ 1 1−k/n Q (K) n,k (s)ds = (k/n)Xn−k,n 1 − γ̂ (K) n,k . with Qn(·) (respectively, Q (K) n,k is the empirical estimator (respectively, the Weissman’s type estimator) of the quantile function Q(·) and k = k(n) is a sequence of integers satis- fying k → ∞, k/n → 0 and as n → ∞. To simplify the proof, we need the following preliminary results whose proofs are given after this one. Lemma 1. Assume that the distribution F satisfies the regularly varying condition (5) with γ ∈ (1/2, 1). If k = k(n) is a sequence of integers satisfying k → ∞, k/n → 0, as M. Kebe et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2509-2543 2518 n → ∞, then for 0 < α < 1 − k/n, we have: √ n ( Ln(α) − L(Q,α) ) (k/n)1/2Xn−k,n = oP(1), as n → ∞. (18) Lemma 2. Assume that the distribution F satisfies the regularly varying condition (5) with γ ∈ (1/2, 1). If k = k(n) is a sequence of integers satisfying k → ∞, k/n → 0, as n → ∞, then for 0 < α < β < 1 − k/n, we have: √ n { Un,k,1(Qn, β) Ln(α) − Un,k,1(Q, β) L(Q,α) } (k/n)1/2Xn−k,n d = Wn,α,1 + oP(1), (19) as n → ∞, where Wn,α,1 := − ∫ 1−k/n 0 Bn(s)dQ(s) L(Q,α) (k/n)1/2Q(1 − k/n) . Lemma 3. Under the assumptions of Theorem 1, we have for 0 < α < 1 − k/n: √ n Un,k,2 ( Q (K) n,k ) Ln(α) − Un,k,2(Q) L(Q,α)  (k/n)1/2Xn−k,n d = λAB(η) K (γ, ρ, α) + Wn,α,2 + Wn,α,3 + oP(1), (20) as n → ∞, where AB(η) K (γ, ρ, α) is defined in Theorem 1 and Wn,α,2 := − γ (1 − γ) L(Q,α) √ n k Bn(1 − k/n), Wn,α,3 := γ (1 − γ)2 L(Q,α) √ n k ∫ 1 0 s−1Bn(1 − sk/n)d(sK(s)). Now, coming back to the proof of the theorem, under assumptions, we have: η̂ (K) n,k (α, β) − η(Q,α, β) = { Un,k,1(Qn, β) Ln(α) − Un,k,1(Q, β) L(Q,α) } + Un,k,2 ( Q (K) n,k ) Ln(α) − Un,k,2(Q) L(Q,α)  := An,1 + An,2. (21) For all values of n large enough, we get respectively from Lemma 2 and Lemma 3: √ n An,1 (k/n)1/2Xn−k,n d = Wn,α,1 + oP(1), M. Kebe et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2509-2543 2519 and √ nAn,2 (k/n)1/2Xn−k,n d = λAB(η) K (γ, ρ, α) + Wn,α,2 + Wn,α,3 + oP(1). This leads to √ n ( η̂ (K) n,k (α, β) − η(Q,α, β) ) (k/n)1/2Xn−k,n d = λAB(η) K (γ, ρ, α) + Wn,α,1 +Wn,α,2 + Wn,α,3 + oP(1). (22) Now, our next step is to compute the asymptotic variance of the process Wn,α,1+ Wn,α,2+ Wn,α,3. The computations quite direct and we give below the main arguments, i.e. EW 2 n,α,1 = ∫ 1−k/n 0 (1 − t) (∫ t 0 sdQ(s) ) dQ(t) L2(Q,α) k/nQ2(1 − k/n) + ∫ 1−k/n 0 t (∫ 1−k/n t (1 − s)dQ(s) ) dQ(t) L2(Q,α) k/nQ2(1 − k/n) = ∫ 1 k/n u (∫ 1 u dQ(1 − v) ) dQ(1 − u) L2(Q,α) k/nQ2(1 − k/n) − ∫ 1 k/n u (∫ 1 u vdQ(1 − v) ) dQ(1 − u) L2(Q,α) k/nQ2(1 − k/n) + ∫ 1 k/n (∫ u k/n vdQ(1 − v) ) dQ(1 − u) L2(Q,α) k/nQ2(1 − k/n) − ∫ 1 k/n u (∫ u k/n vdQ(1 − v) ) dQ(1 − u) L2(Q,α) k/nQ2(1 − k/n) =: Qn,α,1 + Qn,α,2 + Qn,α,3 + Qn,α,4. Recall now that Q(1−s) = s−γℓQ(s) with ℓQ a slowly varying function at 0. By integration by parts and using Lemma 6 in [13] Qn,α,1 = 1 2L2(Q,α) [ 1 + ∫ 1 k/nQ 2(1 − u)du k/nQ2(1 − k/n) ] −→ γ (2γ − 1)L2(Q,α) . Remark that d (∫ 1 u vdQ(1 − v) ) = −u dQ(1 − u) which implies that Qn,α,2 = − 1 2L2(Q,α) k n [∫ 1 k/n vdQ(1 − v) k/nQ(1 − k/n) ]2 = o(1) (23) this last result coming from the fact that, according to Proposition 1.3.6 in [5] for all ε > 0, x−εℓ(x) −→ ∞ as x → 0. Thus, choosing 0 < ε < γ − 1 2 entails 0 ≤ s (∫ 1 s td(Q(1 − t)) sQ(1 − s) )2 = s ( 1 + ∫ 1 s t−γℓQ(t)dt s1−γℓQ(s) )2 ≤ s ( 1 + Csγ−1−ε )2 = O ( s1+2[γ−1−ε] ) = o(1) where C is a suitable constant. Consequently, Qn,α,2 −→ 0. The two others terms, Qn,α,3 and Qn,α,4, can be treated similarly, leading to Qn,α,3 = Qn,α,1 −→ γ (2γ − 1)L2(Q,α) M. Kebe et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2509-2543 2520 Qn,α,4 = Qn,α,2 −→ 0. Finally, EW 2 n,α,1 −→ 2γ (2γ − 1)L2(Q,α) , and direct computations now lead to E(W 2 n,α,2) −→ γ2 (1 − γ)2 L2(Q,α) EW 2 n,α,3 −→ γ2 (1 − γ)4 L2Q,α) ∫ 1 0 K2(s)ds by Corollary 1 in [13] E(Wn,α,1Wn,α,2) −→ γ (1 − γ)L2(Q,α) by (23), E(Wn,α,1Wn,α,3) = 0 and E(Wn,α,2Wn,α,3) = 0. Combining all these results, Theorem 1 follows. Proof of Lemma 1. Let t ∈ |0, 1 − k/n), we have: √ n ( Un,k,1(Qn, t) − Un,k,1(Q, t) ) (k/n)1/2Xn−k,n = ∫ 1−k/n t √ n ( Qn(s) −Q(s) ) ds (k/n)1/2Xn−k,n . Since Q(1 − ·) is a regularly varying function at zero with index −γ, then from Theorem 2.4.1 in [10], Xn−k,n = Qn(1 − k/n) = Q(1 − k/n)(1 + oP(1)), as n → ∞. Using the approach in (17) and the Vervaat process (see [45]), [34] showed in Statement 4.3, p. 8, for all t ∈ (0; 1), that∫ 1−k/n t √ n(Qn(s) −Q(s))ds (k/n)1/2Xn−k,n d = − ∫ 1−k/n 0 Bn(s)dQ(s) (k/n)1/2Q(1 − k/n) + oP(1). (24) Also note that this result in (24) [when t = 0] is equivalent to that of [37]. More precisely, we have ∫ 1−k/n 0 √ n(Qn(s) −Q(s))ds (k/n)1/2Xn−k,n d = − ∫ 1−k/n 0 Bn(s)dQ(s) (k/n)1/2Q(1 − k/n) + oP(1). (25) This concludes that for 0 ≤ t < 1 − k/n, we have, as n → ∞ : √ n ( Un,k,1(Qn, t) − Un,k,1(Q, t) ) (k/n)1/2Xn−k,n d = − ∫ 1−k/n 0 Bn(s)dQ(s) (k/n)1/2Q(1 − k/n) + oP(1). (26) Next, we remark also that, L(Q,α) := Un,k,1(Q, 0)−Un,k,1(Q,α) and Ln(α) := Un,k,1(Qn, 0)− Un,k,1(Q,α). This leads to: √ n ( Ln(α) − L(Q,α) ) (k/n)1/2Xn−k,n = √ n ( Un,k,1(Qn, 0) − Un,k,1(Q, 0) ) (k/n)1/2Xn−k,n M. Kebe et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2509-2543 2521 − √ n ( Un,k,1(Qn, α) − Un,k,1(Q,α) ) (k/n)1/2Xn−k,n . From (26), we get for all large values of n: √ n ( Ln(α) − L(Q,α) ) (k/n)1/2Xn−k,n = oP(1). (27) The Lemma 1 follows. Proof of Lemma 2. We first have: Un,k,1(Qn, β) Ln(α) − Un,k,1(Q, β) L(Q,α) = Un,k,1(Qn, β) Ln(α) − Un,k,1(Qn, β) L(Q,α) + Un,k,1(Qn, β) L(Q,α) − Un,k,1(Q, β) L(Q,α) = 1 L(Q,α) ( Un,k,1(Qn, β) − Un,k,1(Q, β) ) − Un,k,1(Qn, β) Ln(α)L(Q,α) ( Ln(α) − L(Q,α) ) . This leads to: √ n { Un,k,1(Qn, β) Ln(α) − Un,k,1(Q, β) L(Q,α) } (k/n)1/2Xn−k,n = 1 L(Q,α) × √ n (Un,k,1(Qn, β) − Un,k,1(Q, β)) (k/n)1/2Xn−k,n − Un,k,1(Qn, β) Ln(α)L(Q,α) × √ n (Ln(α) − L(Q,α)) (k/n)1/2Xn−k,n . Next, from (26), we have for all large values of n, √ n (Un,k,1(Qn, β) − Un,k,1(Q, β)) L(Q,α)(k/n)1/2Xn−k,n d = ∫ 1−k/n 0 Bn(s)dQ(s) L(Q,α)(k/n)1/2Q(1 − k/n) + oP(1). (28) Since the right term in (28) is bounded in probability, we get for all large values of n, Un,k,1(Qn, β) = Un,k,1(Q, β) + oP(1). Remarking that Un,k,1(Q, β) = ∫ 1−k/n β Q(s)ds and k/n → 0, as n → ∞, we have Un,k,1(Qn, β) = ∫ 1 β Q(s)ds { 1 + oP(1) } . M. Kebe et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2509-2543 2522 In the other hand, from Lemma 1, we have Ln(α) = L(Q,α)+oP(1), as n → ∞. Therefore, using again the Lemma 1 and the fact that the lower integral L(Q,α) and the upper integral∫ 1 β Q(s)ds are finite, we get for all n large enough: Un,k,1(Qn, β) Ln(α)L(Q,α) × √ n (Ln(α) − L(Q,α)) (k/n)1/2Xn−k,n = oP(1). (29) Finally, combining (28) and (29), the Lemma 2 follows. Proof of Lemma 3. We use the following decomposition: Un,k,2 ( Q (K) n,k ) Ln(α) − Un,k,2(Q) L(Q,α) = Un,k,2 ( Q (K) n,k ) Ln(α) − Un,k,2 ( Q (K) n,k ) L(Q,α) + Un,k,2 ( Q (K) n,k ) L(Q,α) − Un,k,2(Q) L(Q,α) = 1 L(Q,α) ( Un,k,2 ( Q (K) n,k ) − Un,k,2(Q) ) − Un,k,2 ( Q (K) n,k ) Ln(α)L(Q,α) ( Ln(α) − L(Q,α) ) . This implies that: √ n Un,k,2 ( Q (K) n,k ) Ln(α) − Un,k,2(Q) L(Q,α)  (k/n)1/2Xn−k,n = 1 L(Q,α) × √ n ( Un,k,2 ( Q (K) n,k ) − Un,k,2(Q) ) (k/n)1/2Xn−k,n − Un,k,2 ( Q (K) n,k ) Ln(α)L(Q,α) × √ n ( Ln(α) − L(Q,α) ) (k/n)1/2Xn−k,n .(30) Recall that Un,k,2 ( Q (K) n,k ) = k/n 1 − γ̂ (K) n,k Xn−k,n. According to Theorem 1 in [13], we have as n → ∞: √ k ( γ̂ (K) n,k − γ ) d = √ k a (n/k) ∫ 1 0 s−ρK(s)ds+γ √ n k ∫ 1 0 s−1Bn ( 1 − s k n ) d (sK(s))+oP(1) (31) This leads to the weak consistency of γ̂ (K) n,k to γ. Since Q(1−·) is a regularly varying function at zero with index −γ, then from Theorem 2.4.1 in [10], Xn−k,n = Q(1 − k/n)(1 + oP(1)), M. Kebe et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2509-2543 2523 as n → ∞ and (k/n)Q(1 − k/n) = (k/n)1−γℓQ(k/n). Since γ ∈ (1/2, 1), we have from Proposition 1.3.6 in [5], (k/n)1−γℓQ(k/n) → 0, as n → ∞. Therefore Un,k,2 ( Q (K) n,k ) P→ 0, as n → ∞. Finally, according to the Lemma 1 in this paper, the second right term of the Equation 30 is equal to oP(1). Now, it allows us to look at the first right term of the Equation 30. Clearly Xn−k,n d = U(Yn−k,n) with U(x) = Q(1 − 1/x) and Un,k,2 ( Q (K) n,k ) d = k/n 1 − γ̂ (K) n,k U (Yn−k,n) . By remarking that Xn−k,n = U(n/k)(1 + oP(1)) with U (n/k) = Q(1 − k/n), we have: √ n ( Un,k,2 ( Q (K) n,k ) − Un,k,2(Q) ) L(Q,α)(k/n)1/2Xn−k,n d = √ n ( Un,k,2 ( Q (K) n,k ) − Un,k,2(Q) ) L(Q,α)(k/n)1/2U (n/k) { 1 + oP(1) } . As a consequence, the following expansion holds: √ n ( Un,k,2 ( Q (K) n,k ) − Un,k,2(Q) ) L(Q,α)(k/n)1/2U (n/k) d = 4∑ j=1 Tn,j , where Tn,1 := √ k L(Q,α) (1 − γ̂ (K) n,k ) [ U (Yn−k,n) U(n/k) − ( k n Yn−k,n )γ] , Tn,2 := √ k L(Q,α)(1 − γ̂ (K) n,k ) [( k n Yn−k,n )γ − 1 ] , Tn,3 := 1 L(Q,α) (1 − γ̂ (K) n,k )(1 − γ) √ k ( γ̂ (K) n,k − γ ) , Tn,4 := √ n L(Q,α) (k/n)1/2U(n/k) [ k/n 1 − γ U(n/k) − Un,k,2(Q, β) ] . We study each term separately. Term Tn,1. According to the Theorem 2.3.9 in [10], for any δ > 0, we have √ k ( U (Yn−k,n) U(n/k) − ( k n Yn−k,n )γ) = √ k a (n k ){(k n Yn−k,n )γ ( k nYn−k,n )ρ − 1 ρ + oP(1) ( k n Yn−k,n )γ+ρ±δ } , We study each term separately. Term Tn,1. According to [10] Theorem 2.3.9) , for any δ > 0, we have √ k ( U (Yn−k,n) U(n/k) − ( k n Yn−k,n )γ) = √ k a (n k ){(k n Yn−k,n )γ ( k nYn−k,n )ρ − 1 ρ + oP(1) ( k n Yn−k,n )γ+ρ±δ } , M. Kebe et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2509-2543 2524 Thus, since kYn−k,n/n → 1, √ k a(n/k) → λ ∈ R and γ̂ (K) n,k P→ γ, as n → ∞, it readily follows that Tn,1 = oP(1). (32) Term Tn,2. The equality Yn−k,n d = (1 − ξn−k,n)−1 yields √ k [( k n Yn−k,n )γ − 1 ] d = √ k ((n k (1 − ξn−k,n) )−γ − 1 ) = −γ √ k (n k (1 − ξn−k,n) − 1 ) (1 + oP(1)) by a Taylor expansion = −γ √ n k βn ( 1 − k n ) (1 + oP(1)) = −γ √ n k ( Bn ( 1 − k n ) + OP(n−ν) ( k n )1/2−ν ) (1 + oP(1)), for 0 ≤ ν < 1/2, by [8]. Thus, using again the fact that γ̂ (K) n,k P→ γ, it follows that Tn,2 d = − γ L(Q,α)(1 − γ) √ n k Bn ( 1 − k n ) (1 + oP(1)) = Wn,α,2 + oP(1). (33) Term Tn,3. By using again the weak consistency of γ̂ (K) n,k to γ and the equation 31, we get Tn,3 d = 1 L(Q,α)(1 − γ)2 {√ k a (n/k) ∫ 1 0 s−1K(s)ds + γ √ n k ∫ 1 0 s−1Bn ( 1 − s k n ) d (sK(s)) } + oP(1) = 1 L(Q,α) (1 − γ)2 √ k a (n/k) ∫ 1 0 s−1K(s)ds + Wn,α,3 + oP(1). (34) Term Tn,4. A change of variables and an integration by parts yield Tn,4 = √ k L(Q,α) { 1 1 − γ − ∫ ∞ 1 x−2U(nx/k) U(n/k) dx } = − √ k L(Q,α) ∫ ∞ 1 x−2 ( U(nx/k) U(n/k) − xγ ) dx. Theorem 2.3.9 in [10] entails that, for γ ∈ (1/2, 1), Tn,4 = − √ ka ( n k ) L(Q,α) ∫ ∞ 1 xγ−2 x ρ − 1 ρ dx (1 + oP(1)) = √ ka ( n k ) L(Q,α) 1 (1 − γ)(γ + ρ− 1) (1 + oP(1)). (35) Combining (32)-(35), Lemma 3 follows. M. Kebe et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2509-2543 2525 Remark 1. Since Q(1 − ·) is a regularly varying function at zero with index −γ, then from Theorem 2.4.1 in [10], Xn−k,n = Q(1−k/n)(1+oP(1)), as n → ∞ and (k/n)1/2Q(1− k/n) = (k/n)1/2−γℓQ(k/n), for k → ∞, k/n → 0, as n → ∞. Remarking that n1/2/(k/n)1/2Q(1− k/n) is equal to k1/2/(k/n)1−γℓQ(k/n) and since γ ∈ (1/2, 1), we have from Proposition 1.3.6 in [5], (k/n)1−γℓQ(k/n) → 0, as n → ∞. and the rate of convergence in Theorem 1 tends to infinity as n goes to infinity. From Theorem 1, it is clear that the estimator η̂ (K) n,k (α, β) exhibits a bias due to the fact that we use in its construction a symptomatic estimator of Q(·) derived from the Weissman’s type estimator Q (K) n,k (·), which is known to have such a problem. To solve this issue, we propose in the next section to use a bias reduction method and to introduce an improved estimator of the QSR index η(Q,α, β). 4.2. Reduced bias estimation of the QSR index In this section, we propose to substitute in (13), the Weissman’s estimator Q (K) n,k with an asymptotically unbiased estimator of the extreme quantile. Our approach is similar to the bias reduction procedure introduced in [19] and [26]. In order to find an asymptotically unbiased estimator of the extreme quantile, we use the second order condition (RU), for which the following approximation holds: Q(u) ≈ (n k (1 − u) )−γ Q(1 − k/n) { 1 − a(n/k) ρ [ 1 − (n k (1 − u) )−ρ ]} , u → 1, (36) where γ, a(·) and ρ are unknown. The first part ( n k (1 − u) )−γ Q(1 − k/n) in the right side of (36) is exactly estimated by the Weissman’s type estimator Q (K) n,k (u) and defined in (9). Clearly, the estimator Q (K) n,k exhibits a potential bias because it depends on the Kernel type estimator γ̂ (K) n,k of the tail index γ, which from (16) has such problem. The expression 1 − ρ−1a(n/kn)[1 − ( n k (1 − u) )−ρ ] can be viewed as a correcting term since a(n/kn) tends to 0. This leads to the necessity to find good estimators for γ, a(n/kn) and ρ. We first propose to introduce an asymptotically unbiased estimator for γ by following an approach similar to [19]. To this end, consider two kernel functions K1 and K2 satisfying (K) and define a mixture of them in the form K∆ (s) = ∆K1 (s) + (1 − ∆)K2 (s), for ∆ ∈ R. Clearly K∆ also satisfies the condition (K) and hence by the result given in (16), the asymptotic bias λ ∫ 1 0 s−ρK∆(s)ds of γ̂ (K∆) n,k is such that λ ∫ 1 0 s−ρK∆(s)ds = λ∆ ∫ 1 0 s−ρK1(s)ds + λ(1 − ∆) ∫ 1 0 s−ρK2(s)ds. Equating the right-hand side of the above equation to zero leads to the value of eliminating M. Kebe et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2509-2543 2526 the asymptotic bias ∆∗ = ∫ 1 0 s−ρK2(s)ds∫ 1 0 s−ρ { K2(s) −K1(s) } ds , (37) provided ∫ 1 0 s−ρ { K2(s)−K1(s) } ds ̸= 0. Clearly, the tail index estimator γ̂ (K∆∗ ) n,k is shown to be asymptotically unbiased in the sense that the mean of its limiting distribution is zero, whatever the value of λ. More precisely, we have from (16): k1/2 ( γ̂ (K∆∗ ) n,k − γ ) d→ N ( 0, γ2 ∫ 1 0 K2 ∆∗(s)ds ) . (38) An open problem is to determine whether among this class of unbiased estimators γ̂ (K∆∗ ) n,k , we can find the asymptotically unbiased estimator with minimum variance. Clearly, the asymptotic variance γ2 ∫ 1 0 K2 ∆∗(s)ds is minimal for a minimum value of ∫ 1 0 K2 ∆∗(s)ds. Ac- cording to [19] and [13], the minimum of ∫ 1 0 K2 ∆∗(s)ds is obtained at the “optimal” function given by: K∆∗ opt (s) = ( 1 − ρ ρ )2 − (1 − ρ) (1 − 2ρ) ρ2 s−ρ, for s ∈ (0, 1) , (39) and K∆∗ opt (s) = 0 otherwise. Note that this unction can be viewed as a mixture between two power kernels: K1(s) := K(s) = I(0 0. The consistency of ρ̂ (∗) kρ to ρ have been established in [21] and [12]) under the second order condition (RU) 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 η̃ (K ∆̂∗ opt ) n,k,ρ̂ (α, β). 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: √ n ( η̃ (K ∆̂∗ opt ) n,k, ρ̂ (α, β) − η(Q,α, β) ) (k/n)1/2Xn−k,n d→ N ( 0, ÃV(γ, ρ, α) ) , where ÃV(γ, ρ, α) = γ4(γ − ρ)2 (2γ − 1)(1 − γ)4(γ + ρ− 1)2 L2(Q,α) . Proof of Theorem 2 For simplify the proof, we introduce the following Lemmas, whose proofs are given just after this one. Lemma 4. Suppose that the distribution F satisfies the second order condition (RU). If k → ∞, k/n → 0 and √ k a(n/k) → λ ∈ R, as n → ∞ and 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 γ̂ ( K ∆̂∗ opt ) n,k − γ  d = γ √ n/k ∫ 1 0 s−1Bn(1 − sk/n)d(sK∆∗ opt (s)) + oP (1) and √ k (ân,k(ρ̂ ) − a(n/k)) d = γ (1 − ρ) √ n/k ∫ 1 0 s−1Bn(1−sk/n)d(s(K1(s)−K∆∗ opt (s)))+oP (1) . M. Kebe et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2509-2543 2529 Lemma 5. Under the assumptions of Theorem 2, we have for 0 < α < 1 − k/n: √ n  Un,k,2 ( Q (K ∆̂∗ opt ) n,k,ρ̂ ) Ln(α) − Un,k,2(Q) L(Q,α)  (k/n)1/2Xn−k,n d = Wn,α,1 + Wn,α,2 + Wn,α,4 + Wn,α,5 + oP(1),(44) as n → ∞, where Wn,α,4 := ργ2 (γ + ρ− 1)(1 − γ)2 L(Q,α) √ n/k ∫ 1 0 s−1Bn(1 − sk/n)d(sK∆∗ opt (s)) Wn,α,5 := −(1 − γ)(1 − ρ) (γ + ρ− 1) Wn,α,3. Now coming back to this proof, under the assumptions, we have: η̂ (K ∆̂∗ opt ) n,k (α, β) − η(Q,α, β) = { Un,k,1(Qn, β) Ln(α) − Un,k,1(Q, β) L(Q,α) } +  Un,k,2 ( Q K ∆̂∗ opt ) n,k ) Ln(α) − Un,k,2(Q) L(Q,α)  := Bn,1 + Bn,2. (45) For all values of n large enough, we get respectively from Lemma 4 and Lemma 5: √ n Bn,1 (k/n)1/2Xn−k,n d = Wn,α,1 + oP(1), and √ nBn,2 (k/n)1/2Xn−k,n d = Wn,α,2 + Wn,α,4 + Wn,α,5 + oP(1). This leads to √ n ( η̂ (K) n,k (α, β) − η(Q,α, β) ) (k/n)1/2Xn−k,n d = Wn,α,1 + Wn,α,2 + Wn,α,4 + Wn,α,5 + oP(1). (46) We only have to compute the asymptotic variance of the sum of process in the right term of (46). As in Theorem 1, the computations are quite direct and the desired asymptotic variance can be obtained by noticing that EW 2 n,α,5 −→ γ2(1 − ρ)2 (1 − γ)2(γ + ρ− 1)2 L2(Q,α) M. Kebe et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2509-2543 2530 E(Wn,α,1Wn,α,4) = 0 E(Wn,α,5) = 0 EW 2 n,α,4 = γ4(1 − ρ)2 (1 − γ)4(γ + ρ− 1)2 L2(Q,α) E(Wn,α,2Wn,α,5) = 0 E(Wn,α,Wn,α,4) = 0 E(Wn,α,4Wn,α,5) = − ργ3(1 − ρ) (1 − γ)3(γ + ρ− 1)2 L2(Q,α) . Combining all these results, Theorem 2 follows. Proof of Lemma 4. Note that the first quantity of interest can be expanded as √ k γ̂ ( K ∆̂∗ opt ) n,k − γ  = √ k γ̂ ( K ∆̂∗ opt ) n,k − γ̂ ( K∆∗ opt ) n,k + √ k γ̂ ( K∆∗ opt ) n,k − γ  = √ k 1 k k∑ j=1 { K ∆̂∗ opt ( j k + 1 ) −K∆∗ opt ( j k + 1 )} j log ( Xn−j+1,n Xn−j,n ) + √ k γ̂ ( K∆∗ opt ) n,k − γ  , where K∆∗ opt (s) = ( (1 − ρ)2 ρ2 )2 − (1 − ρ) (1 − 2ρ) ρ2 s−ρ, t ∈ (0, 1), and K∆∗ opt (s) = 0 otherwise, K ∆̂∗ opt is defined as K∆∗ opt with ρ replaced by ρ̂. We have all ready mentioned that the function K∆∗ opt is viewed as a mixture between two power kernels: K1(s) := I(0