# EstimatingL-Functionals for Heavy-Tailed Distributions and Application

Authors: Abdelhakim Necir; Djamel Meraghni
Journal: Journal of Probability and Statistics
 (2010)

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## Abstract

L-functionals summarize numerous statistical parameters and actuarial risk
measures. Their sample estimators are linear combinations of order statistics
(L-statistics). There exists a class of heavy-tailed distributions for which the
asymptotic normality of these estimators cannot be obtained by classical results.
In this paper we propose, by means of extreme value theory, alternative
estimators for L-functionals and establish their asymptotic normality. Our
results may be applied to estimate the trimmed L-moments and financial risk
measures for heavy-tailed distributions.


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## Body

## 1. Introduction

### 1.1.L-Functionals

LetX be a real random variable (rv) with continuous distribution function (df) F. The corresponding L-functionals are defined by(1.1)L(J):=∫01J(s)Q(s)ds,
where Q(s):=inf{x∈ℝ:F(x)≥s},0<s≤1, is the quantile function pertaining to df F and J is a measurable function defined on [0,1] (see, e.g. Serfling, [1]). Several authors have used the quantity L(J) to solve some statistical problems. For example, in a work by Chernoff et al. [2] the L-functionals have a connection with optimal estimators of location and scale parameters in parametric families of distributions. Hosking [3] introduced the L-moments as a new approach of statistical inference of location, dispersion, skewness, kurtosis, and other aspects of shape of probability distributions or data samples having finite means. Elamir and Seheult [4] have defined the trimmed L-moments to answer some questions related to heavy-tailed distributions for which means do not exist, and therefore the L-moment method cannot be applied. In the case where the trimming parameter equals one, the first four theoretical trimmed L-moments are(1.2)mi:=∫01Ji(s)Q(s)ds,i=1,2,3,4,
where(1.3)Ji(s):=s(1-s)ϕi(s),0<s<1,
with ϕi polynomials of order i-1 (see Section 4). A partial study of statistical estimation of trimmed L-moments was given recently by Hosking [5].Deriving asymptotics of complex statistics is a challenging problem, and this was indeed the case for a decade since the introduction of the distortion risk measure by Denneberg [6] and Wang [7]; see also Wang [8]. The breakthrough in the area was offered by Jones and Zitikis [9], who revealed a fundamental relationship between the distortion risk measure and the classical L-statistic, thus opening a broad gateway for developing statistical inferential results in the area (see, e.g., Jones and Zitikis [10, 11]; Brazauskas et al. [12, 13] and Greselin et al. [14]). These works mainly discuss CLT-type results. We have been utilizing the aforementioned relationship between distortion risk measures and L-statistics to develop a statistical inferential theory for distortion risk measures in the case of heavy-tailed distributions.IndeedL-functionals have many applications in actuarial risk measures (see, e.g., Wang [8, 15, 16]). For example, if X≥0 represents an insurance loss, the distortion risk premium is defined by(1.4)Π(X):=∫0∞g(1-F(x))dx,
where g is a non decreasing concave function with g(0)=0 and g(1)=1. By a change of variables and integration by parts, Π(X) may be rewritten into(1.5)Π(X)=∫01g′(1-s)Q(s)ds,
where g′ denotes the Lebesgue derivative of g. For heavy-tailed claim amounts, the empirical estimation with confidence bounds for Π(X) has been discussed by Necir et al. [17] and Necir and Meraghni [18]. If X∈ℝ represents financial data such as asset log-returns, the distortion risk measures are defined by(1.6)H(X):=∫-∞0(g(1-F(x))-1)dx+∫0∞g(1-F(x))dx.
Likewise, by integration by parts it is shown that(1.7)H(X)=∫01g′(1-s)Q(s)ds.
Wang [8] and Jones and Zitikis [9] have defined the risk  two-sided deviation by(1.8)Δr(X):=∫01Jr(s)Q(s)ds,0<r<1,with(1.9)Jr(s):=r2s1-r-(1-s)1-rs1-r(1-s)1-r,0<s<1.
As we see, Π(X),H(X), and Δr(X) are L-functionals for specific weight functions. For more details about the distortion risk measures one refers to Wang [8, 16]. A discussion on their empirical estimation is given by Jones and Zitikis [9].


### 1.2. Estimation of L-Functionals and Motivations

In the sequel let→p and →𝒟, respectively, stand for convergence in probability and convergence in distribution and let 𝒩(0,η2) denote the normal distribution with mean 0 and variance η2.The natural estimators of quantityL(J) are linear combinations of order statistics called L-statistics. For more details on this kind of statistics one refers to Shorack and Wellner [19, page 260]. Indeed, let (X1,…,Xn) be a sample of size n≥1 from an rv X with df F, then the sample estimator of L(J) is(1.10)L̂n(J):=∫01J(s)Qn(s)ds,
where Qn(s):=inf{x∈ℝ:Fn(x)≥s}, 0<s≤1, is the empirical quantile function that corresponds to the empirical df Fn(x):=n-1∑i=1n𝕀{Xi≤x} for x∈ℝ, pertaining to the sample (X1,…,Xn) with 𝕀(·) denoting the indicator function. It is clear that L̂n(J) may be rewritten into(1.11)L̂n(J)=∑i=1nai,nXi,n,
where ai,n:=∫(i-1)/ni/nJ(s)ds, i=1,…,n, and X1,n≤⋯≤Xn,n denote the order statistics based upon the sample (X1,…,Xn). The first general theorem on the asymptotic normality of L̂n(J) is established by Chernoff et al. [2]. Since then, a large number of authors have studied the asymptotic behavior of L-statistics. A partial list consists of Bickel [20], Shorack [21, 22], Stigler [23, 24], Ruymgaart and Van Zuijlen [25], Sen [26], Boos [27], Mason [28], and Singh [29]. Indeed, we have(1.12)n(L̂n(J)-L(J))→𝒟𝒩(0,σ2(J)),asn→∞,
provided that(1.13)σ2(J):=∫01∫01(min(s,t)-st)J(s)J(t)dQ(s)Q(t)<∞.
In other words, for a given function J, condition (1.13) excludes the class of distributions F for which σ2(J) is infinite. For example, if we take J=1, L(J) is equal to the expected value EX and hence the natural estimator of L̂n(J) is the sample mean X¯n. In this case, result (1.12) corresponds to the classical central limit theorem which is valid only when the variance of F is finite. How then can be construct confidence bounds for the mean of a df when its variance is infinite? This situation arises when df F belongs to the domain of attraction of α-stable laws (heavy-tailed) with characteristic exponent α∈(1,2); see Section 2. This question was answered by Peng [30, 31] who proposed an alternative asymptotically normal estimator for the mean. Remark 3.3 below shows that this situation also arises for the sample trimmed L-moments mi when 1/2<α<2/3 and for the sample risk two-sided deviation Δr(X) when 1/(r+1/2)<α<1/r for any 0<r<1.To solve this problem in a more general setting, we propose, by means of the extreme value theory, asymptotically normal estimators of L-functionals for heavy-tailed distributions for which σ2(J)=∞.The remainder of this paper is organized as follows. Section2 is devoted to a brief introduction on the domain of attraction of α-stable laws. In Section 3 we define, via the extreme value approach, a new asymptotically normal estimator of L-functionals and state our main results. Applications to trimmed L-moments, risk measures, and related quantities are given in Section 4. All proofs are deferred to Section 5.


## 1.1.L-Functionals

LetX be a real random variable (rv) with continuous distribution function (df) F. The corresponding L-functionals are defined by(1.1)L(J):=∫01J(s)Q(s)ds,
where Q(s):=inf{x∈ℝ:F(x)≥s},0<s≤1, is the quantile function pertaining to df F and J is a measurable function defined on [0,1] (see, e.g. Serfling, [1]). Several authors have used the quantity L(J) to solve some statistical problems. For example, in a work by Chernoff et al. [2] the L-functionals have a connection with optimal estimators of location and scale parameters in parametric families of distributions. Hosking [3] introduced the L-moments as a new approach of statistical inference of location, dispersion, skewness, kurtosis, and other aspects of shape of probability distributions or data samples having finite means. Elamir and Seheult [4] have defined the trimmed L-moments to answer some questions related to heavy-tailed distributions for which means do not exist, and therefore the L-moment method cannot be applied. In the case where the trimming parameter equals one, the first four theoretical trimmed L-moments are(1.2)mi:=∫01Ji(s)Q(s)ds,i=1,2,3,4,
where(1.3)Ji(s):=s(1-s)ϕi(s),0<s<1,
with ϕi polynomials of order i-1 (see Section 4). A partial study of statistical estimation of trimmed L-moments was given recently by Hosking [5].Deriving asymptotics of complex statistics is a challenging problem, and this was indeed the case for a decade since the introduction of the distortion risk measure by Denneberg [6] and Wang [7]; see also Wang [8]. The breakthrough in the area was offered by Jones and Zitikis [9], who revealed a fundamental relationship between the distortion risk measure and the classical L-statistic, thus opening a broad gateway for developing statistical inferential results in the area (see, e.g., Jones and Zitikis [10, 11]; Brazauskas et al. [12, 13] and Greselin et al. [14]). These works mainly discuss CLT-type results. We have been utilizing the aforementioned relationship between distortion risk measures and L-statistics to develop a statistical inferential theory for distortion risk measures in the case of heavy-tailed distributions.IndeedL-functionals have many applications in actuarial risk measures (see, e.g., Wang [8, 15, 16]). For example, if X≥0 represents an insurance loss, the distortion risk premium is defined by(1.4)Π(X):=∫0∞g(1-F(x))dx,
where g is a non decreasing concave function with g(0)=0 and g(1)=1. By a change of variables and integration by parts, Π(X) may be rewritten into(1.5)Π(X)=∫01g′(1-s)Q(s)ds,
where g′ denotes the Lebesgue derivative of g. For heavy-tailed claim amounts, the empirical estimation with confidence bounds for Π(X) has been discussed by Necir et al. [17] and Necir and Meraghni [18]. If X∈ℝ represents financial data such as asset log-returns, the distortion risk measures are defined by(1.6)H(X):=∫-∞0(g(1-F(x))-1)dx+∫0∞g(1-F(x))dx.
Likewise, by integration by parts it is shown that(1.7)H(X)=∫01g′(1-s)Q(s)ds.
Wang [8] and Jones and Zitikis [9] have defined the risk  two-sided deviation by(1.8)Δr(X):=∫01Jr(s)Q(s)ds,0<r<1,with(1.9)Jr(s):=r2s1-r-(1-s)1-rs1-r(1-s)1-r,0<s<1.
As we see, Π(X),H(X), and Δr(X) are L-functionals for specific weight functions. For more details about the distortion risk measures one refers to Wang [8, 16]. A discussion on their empirical estimation is given by Jones and Zitikis [9].


## 1.2. Estimation of L-Functionals and Motivations

In the sequel let→p and →𝒟, respectively, stand for convergence in probability and convergence in distribution and let 𝒩(0,η2) denote the normal distribution with mean 0 and variance η2.The natural estimators of quantityL(J) are linear combinations of order statistics called L-statistics. For more details on this kind of statistics one refers to Shorack and Wellner [19, page 260]. Indeed, let (X1,…,Xn) be a sample of size n≥1 from an rv X with df F, then the sample estimator of L(J) is(1.10)L̂n(J):=∫01J(s)Qn(s)ds,
where Qn(s):=inf{x∈ℝ:Fn(x)≥s}, 0<s≤1, is the empirical quantile function that corresponds to the empirical df Fn(x):=n-1∑i=1n𝕀{Xi≤x} for x∈ℝ, pertaining to the sample (X1,…,Xn) with 𝕀(·) denoting the indicator function. It is clear that L̂n(J) may be rewritten into(1.11)L̂n(J)=∑i=1nai,nXi,n,
where ai,n:=∫(i-1)/ni/nJ(s)ds, i=1,…,n, and X1,n≤⋯≤Xn,n denote the order statistics based upon the sample (X1,…,Xn). The first general theorem on the asymptotic normality of L̂n(J) is established by Chernoff et al. [2]. Since then, a large number of authors have studied the asymptotic behavior of L-statistics. A partial list consists of Bickel [20], Shorack [21, 22], Stigler [23, 24], Ruymgaart and Van Zuijlen [25], Sen [26], Boos [27], Mason [28], and Singh [29]. Indeed, we have(1.12)n(L̂n(J)-L(J))→𝒟𝒩(0,σ2(J)),asn→∞,
provided that(1.13)σ2(J):=∫01∫01(min(s,t)-st)J(s)J(t)dQ(s)Q(t)<∞.
In other words, for a given function J, condition (1.13) excludes the class of distributions F for which σ2(J) is infinite. For example, if we take J=1, L(J) is equal to the expected value EX and hence the natural estimator of L̂n(J) is the sample mean X¯n. In this case, result (1.12) corresponds to the classical central limit theorem which is valid only when the variance of F is finite. How then can be construct confidence bounds for the mean of a df when its variance is infinite? This situation arises when df F belongs to the domain of attraction of α-stable laws (heavy-tailed) with characteristic exponent α∈(1,2); see Section 2. This question was answered by Peng [30, 31] who proposed an alternative asymptotically normal estimator for the mean. Remark 3.3 below shows that this situation also arises for the sample trimmed L-moments mi when 1/2<α<2/3 and for the sample risk two-sided deviation Δr(X) when 1/(r+1/2)<α<1/r for any 0<r<1.To solve this problem in a more general setting, we propose, by means of the extreme value theory, asymptotically normal estimators of L-functionals for heavy-tailed distributions for which σ2(J)=∞.The remainder of this paper is organized as follows. Section2 is devoted to a brief introduction on the domain of attraction of α-stable laws. In Section 3 we define, via the extreme value approach, a new asymptotically normal estimator of L-functionals and state our main results. Applications to trimmed L-moments, risk measures, and related quantities are given in Section 4. All proofs are deferred to Section 5.


## 2. Domain of Attraction ofα-Stable Laws

A df is said to belong to the domain of attraction of a stable law with stability index0<α≤2, notation: F∈D(α), if there exist two real sequences An>0 and Cn such that(2.1)An-1(∑i=1nXi-Cn)→𝒟Sα(σ,δ,μ),asn→∞,
where Sα(σ,δ,μ) is a stable distribution with parameters 0<α≤2, -1≤δ≤+1, σ>0 and -∞<μ<+∞ (see, e.g., Samorodnitsky and Taqqu [32]). This class of distributions was introduced by Lévy during his investigations of the behavior of sums of independent random variables in the early 1920s [33]. Sα(σ,δ,μ) is a rich class of probability distributions that allow skewness and thickness of tails and have many important mathematical properties. As shown in early work by Mandelbrot (1963) and Fama [34], it is a goodcandidate to accommodate heavy-tailed financial series and produces measures of risk based on the tails of distributions, such as the Value-at-Risk. They also have been proposed as models for many types of physical and economic systems, for more details see Weron [35]. This class of distributions have nice heavy-tail properties. More precisely, if we denote by G(x):=ℙ(|X|≤x)=F(x)-F(-x),x>0, the df of Z:=|X|, then the tail behavior of F∈D(α), for 0<α<2, may be described by the following(i)
The tail1-G is regularly varying at infinity with index -α. That is
(2.2)limt→∞(1-G(xt))(1-G(t))=x-α,for  anyx>0.(ii)
There exists0≤p≤1 such that
(2.3)limx→∞1-F(x)1-G(x)=p,limx→∞F(-x)1-G(x)=1-p=:q.Let, for0<s<1,K(s):=inf{x>0:G(x)≥s}be the quantile function pertaining to G and Q1(s):=max(-Q(1-s),0) and Q2(s):=max(Q(s),0). Then Proposition A.3 in a work by Csörgő et al. [36] says that the set of conditions above is equivalent to the following.(i′)
K(1-·) is regularly varying at 0 with index -1/α. That
(2.4)lims↓0K(1-xs)K(1-s)=x-1/α,for  anyx>0.(ii′)
There exists0≤p≤1 such that
(2.5)lims↓0Q1(1-s)K(1-s)=p1/α,lims↓0Q2(1-s)K(1-s)=(1-p)1/α=:q1/α.Our framework is a second-order condition that specifies the rate of convergence in statement(i′). There exists a function A, not changing sign near zero, such that(2.6)lims↓0(A(s))-1(K(1-xs)K(1-s)-x-1/α)=x-1/αxϱ-1ϱ,for  anyx>0,
where ϱ≤0 is the second-order parameter. If ϱ=0, interpret (xϱ-1)/ϱ as logx. The second-order condition for heavy-tailed distributions has been introduced by de Haan and Stadtmüller [37].


## 3. EstimatingL-Functionals When F∈D(α)

### 3.1. Extreme Quantile Estimation

The right and left extreme quantiles of small enough levelt of df F, respectively, are two reals xR and xL defined by 1-F(xR)=t and F(xL)=t, that is, xR=Q(1-t) and xL=Q(t). The estimation of extreme quantiles for heavy-tailed distributions has got a great deal of interest, see for instance Weissman [38], Dekkers and de Haan [39], Matthys and Beirlant [40] and Gomes et al. [41]. Next, we introduce one of the most popular quantile estimators. Let k=kn and ℓ=ℓn be sequences of integers (called trimming sequences) satisfying 1<k<n,1<ℓ<n, k→∞, ℓ→∞, k/n→0 and ℓ/n→0, as n→∞. Weissman's estimators of extreme quantiles xR and xL are defined, respectively, by(3.1)x̂L=Q̂L(t):=(kn)1/αLXk,nt-1/αL,ast↓0,x̂R=Q̂R(1-t):=(ℓn)1/αRXn-ℓ,nt-1/αR,ast↓0,
where(3.2)α̂L=α̂L(k):=(1k∑i=1klog+(-Xi,n)-log+(-Xk,n))-1,α̂R=α̂R(ℓ):=(1ℓ∑i=1ℓlog+(Xn-i+1,n)-log+(Xn-ℓ,n))-1
are two forms of Hill's estimator [42] for the stability index α which could also be estimated, using the order statistics Z1,n≤⋯≤Zn,n associated to a sample (Z1,…,Zn) from Z, as follows:(3.3)α̂=α̂(m):=(1m∑i=1mlog+(Zn-i+1,n)-log+(Zn-m,n))-1,
with log+u:=max(0,logu) and m=mn being an intermediate sequence fulfilling the same conditions as k and ℓ. Hill's estimator has been thoroughly studied, improved, and even generalized to any real-valued tail index. Its weak consistency was established by Mason [43] assuming only that the underlying distribution is regularly varying at infinity. The almost sure convergence was proved by Deheuvels et al. [44] and more recently by Necir [45]. The asymptotic normality has been investigated, under various conditions on the distribution tail, by numerous workers like, for instance, Csörgő and Mason [46], Beirlant and Teugels [47], and Dekkers et al. [48].


### 3.2. A Discussion on the Sample Fractionsk and ℓ

Extreme value-based estimators rely essentially on the numbersk and ℓ of lower- and upper-order statistics used in estimate computation. Estimators α̂L and α̂R have, in general, substantial variances for small values of k and ℓ and considerable biases for large values of k and ℓ. Therefore, one has to look for optimal values for k and ℓ, which balance between these two vices.Numerically, there exist several procedures for the thorny issue of selecting the numbers of order statistics appropriate for obtaining good estimates of the stability indexα; see, for example, Dekkers and de Haan [49], Drees and Kaufmann [50], Danielsson et al. [51], Cheng and Peng [52] and Neves and Alves [53]. Graphically, the behaviors of α̂L, α̂R, and α̂ as functions of k, ℓ and m, respectively, are illustrated by Figures 1, 2, and 3 drawn by means of the statistical software R [54]. According to Figure 1, α̂R is much more suitable than α̂L when estimating the stability index of a distribution which is skewed to the right (δ>0)whereas Figure 2 shows that α̂L is much more reliable than α̂R when the distribution is skewed to the left (δ<0). In the case where the distribution is symmetric (δ=0), both estimators seem to be equally good as seen in Figure 3. Finally, it is worth noting that, regardless of the distribution skewness, estimator α̂, based on the top statistics pertaining to the absolute value of X, works well and gives good estimates for the characteristic exponent α.Figure 1
Plots of Hill estimators, as functions of the number of extreme statistics,α̂ (solid line), α̂R (dashed line), and α̂L (dotted line) of the characteristic exponent α of a stable distribution skewed to the right, based on 1000 observations with 50 replications. The horizontal line represents the true value of α=1.2.Figure 2
Plots of Hill estimators, as functions of the number of extreme statistics,α̂ (solid line), α̂R (dashed line), and α̂L (dotted line) of the characteristic exponent α of a stable distribution skewed to the left, based on 1000 observations with 50 replications. The horizontal line represents the true value of α=1.2.Figure 3
Plots of Hill estimators, as functions of the number of extreme statistics,α̂ (solid line), α̂R (dashed line) and α̂L (dotted line) of the characteristic exponent α of a symmetric stable distribution, based on 1000 observations with 50 replications. The horizontal line represents the true value of α=1.2.It is clear that, in general, there is no reason for the trimming sequencesk and ℓ to be equal. We assume that there exists a positive real constant θ such that ℓ/k→θ as n→∞. If the distribution is symmetric, the value of θ is equal to 1; otherwise, it is less or greater than 1 depending on the sign of the distribution skewness. For an illustration, see Figure 4 where we plot the ratio θ for several increasing sample sizes.Figure 4
Plots of the ratios of the numbers of extreme statistics, as functions of the sample size, for a stable symmetric distributionS1.2(1,0,0) (solid line), a stable distribution skewed to the right S1.2(1,0.5,0) (dashed line) and a stable distribution skewed to the left S1.2(1,-0.5,0) (dotted line).


### 3.3. Some Regularity Assumptions onJ

For application needs, the following regularity assumptions on functionJ are required:(H1)
J is differentiable on (0,1),(H2)
λ:=lims↓0J(1-s)J(s)<∞,(H3)
bothJ(s) and J(1-s) are regularly varying at zero with common index β∈ℝ.(H4)
there exists a functiona(·) not changing sign near zero such that
(3.4)limt↓0J(xt)/J(t)-xβa(t)=xβxω-1ω,for  anyx>0,
where ω≤0 is the second-orderparameter.The three remarks below give more motivations to this paper.Remark 3.1.
Assumption (H3) has already been used by Mason and Shorack [55] to establish the asymptotic normality of trimmed L-statistics. Condition (H4) is just a refinement of (H3) called the second order condition that is required for quantile function K in (2.6).Remark 3.2.
Assumptions (H1)–(H4) are satisfied by all weight functions(Ji)i=2,4 with (β,λ)=(1,±1) (see Section 4.1) and by function Jr in (1.9) with (β,λ)=(r-1,-1). These two examples show that the constants β and λ may be positive or negative depending on application needs.Remark 3.3.
L-functionals L(J) exist for any 0<α<2 and β∈ℝ such that 1/α-β<1. However, Lemma 5.4 below shows that for 1/α-β>1/2 we have σ2(J)=∞. Then, recall (1.3); whenever 1/2<α<2/3, the trimmed L-moments exist however σ2(Ji)=∞,i=1,…,4. Likewise, recall (1.9); whenever 1/(r+1/2)<α<1/r, the two-sided deviation Δr(X) exists while σ2(Jr)=∞.


### 3.4. Defining the Estimator and Main Results

We now have all the necessary tools to introduce our estimator ofL(J), given in (1.1), when F∈D(α) with 0<α<2. Let k=kn and ℓ=ℓn be sequences of integers satisfying 1<k<n, 1<ℓ<n, k→∞, ℓ→∞, k/n→0, ℓ/n→0, and the additional condition ℓ/k→θ<∞ as n→∞. First, we must note that since 1+β-1/α>0 (see Remark 3.3) and since both α̂L and α̂R are consistent estimators of α (see, Mason [43]), then we have for all large n(3.5)ℙ(1+β-1α̂L>0)=ℙ(1+β-1α̂R>0)=1+o(1).
Observe now that L(J) defined in (1.1) may be split in three integrals as follows:(3.6)L(J)=∫0k/nJ(t)Q(t)dt+∫k/n1-ℓ/nJ(t)Q(t)dt+∫1-ℓ/n1J(t)Q(t)dt=:TL,n+TM,n+TR,n.
Substituting Q̂L(t) and Q̂R(1-t) for Q(t) and Q(1-t) in TL,n and TR,n, respectively and making use of assumption (H3) and (3.5) yield that for all large n(3.7)∫0k/nJ(t)Q̂L(t)dt=(kn)1/αLXk,n∫0k/nt-1/αLJ(t)dt=(1+o(1))(k/n)J(k/n)1+β-1/α̂LXk,n,∫0ℓ/nJ(1-t)Q̂R(1-t)dt=(ℓn)1/αRXn-ℓ,n∫0ℓ/nt-1/αRJ(1-t)dt=(1+o(1))(ℓ/n)J(1-ℓ/n)1+β-1/α̂RXn-ℓ,n.
Hence we may estimate TL,n and TR,n by(3.8)T̂L,n:=(k/n)J(k/n)1+β-1/α̂LXk,n,T̂R,n:=(ℓ/n)J(1-ℓ/n)1+β-1/α̂RXn-ℓ,n,
respectively. As an estimator of TM,n we take the sample one that is(3.9)T̂M,n:=∫k/n1-ℓ/nJ(t)Qn(t)dt=∑i=k+1n-ℓai,nXi,n,
with the same constants ai,n as those in (1.11). Thus, the final form of our estimator is(3.10)L̂k,ℓ(J)=(k/n)J(k/n)1+β-1/α̂LXk,n+∑i=k+1n-ℓai,nXi,n+(ℓ/n)J(1-ℓ/n)1+β-1/α̂RXn-ℓ,n.
A universal estimator of L(J) may be summarized by(3.11)L̂n*(J)=L̂k,ℓ(J)𝕀(σ2(J)=∞)+L̂n(J)𝕀(σ2(J)<∞),
where L̂n(J) is as in (1.11). More precisely(3.12)L̂n*(J)=L̂k,ℓ(J)𝕀(A(α,β))+L̂n(J)𝕀(A¯(α,β)),
where A(α,β):={(α,β)∈(0,2)×ℝ:1/2<1/α-β<1} and A¯(α,β) is its complementary in (0,2)×ℝ.Note that for the particular caseℓ=k and J=1the asymptotic normality of the trimmed mean T̂M,n has been established in Theorem 1 of Csörgő et al. [56]. The following theorem gives the asymptotic normality of T̂M,n for more general trimming sequences k and ℓ and weighting function J. For convenience, we set, for any 0<x<1/2 and 0<y<1/2,(3.13)σ2(x,y;J):=∫x1-y∫x1-y(min(s,t)-st)J(s)J(t)dQ(s)Q(t)<∞,
and let σn2(J):=σ2(k/n,ℓ/n;J).Theorem 3.4.
Assume thatF∈D(α) with 0<α<2. For any  measurable function J satisfying assumption (H3) with index β∈ℝ such that 0<1/α-β<1 and for any sequences  of integers k and ℓ such that1<k<n, 1<ℓ<n, k→∞, ℓ→∞, k/n→0, andℓ/n→0, as  n→∞, there exists a probability space (Ω,A,P) carrying the sequence X1,X2,… and a sequence of Brownian bridges {Bn(s),0≤s≤1,n=1,2,…} such that one has for all large  n(3.14)n(T̂M,n-TM,n)σn(J)=-∫k/n1-ℓ/nJ(s)Bn(s)dsσn(J)+op(1),
and therefore
(3.15)n(T̂M,n-TM,n)σn(J)→𝒟𝒩(0,1)asn→∞.The asymptotic normality of our estimator is established in the following theorem.Theorem 3.5.
Assume thatF∈D(α) with 0<α<2. For any  measurable function J satisfying assumptions (H1)–(H4) with index  β∈ℝ such that 1/2<1/α-β<1, and for any sequences of integers k and ℓ such that 1<k<n, 1<ℓ<n, k→∞, ℓ→∞, k/n→0, ℓ/n→0, ℓ/k→θ<∞, and ka(k/n)A(k/n)→0 as n→∞,  one has
(3.16)n(L̂k,ℓ(J)-L(J))σn(J)→𝒟𝒩(0,σ02),asn→∞,
where
(3.17)σ02=σ02(α,β):=(αβ+1)(2αβ+2-α)×(2α2+(βα-1)2+2α(βα-1)2((1+β)α-1)4+1(1+β)α-1)+1.The following corollary is more practical than Theorem3.5 as  it directly provides confidence bounds for L(J).Corollary 3.6.
Under the assumptions of Theorem3.5 one has
(3.18)n(L̂k,ℓ(J)-L(J))(ℓ/n)1/2J(1-ℓ/n)Xn-ℓ,n→𝒟𝒩(0,V2),asn→∞,
where
(3.19)V2=V2(α,β,λ,θ,p):=(1+λ-2(qp)-2/αθ-2β+2/α-1)×(2α2+(βα-1)2+2α(βα-1)2((1+β)α-1)4+1(1+β)α-1)+1,
with (p,q) as in statement (ii) of Section 2 and (λ,β) as in assumptions (H2)-(H3) of Section 3.


### 3.5. Computing Confidence Bounds forL(J)

The form of the asymptotic varianceV2 in (3.20) suggests that, in order to construct confidence intervals for L(J), an estimate of p is needed as well. Using the intermediate order statistic Zn-m,n, de Haan and Pereira [57] proposed the following consistent estimator for p:(3.20)p̂n=p̂n(m):=1m∑i=1n𝕀{Xi>Zn-m,n},
where m=mn is a sequence of integers satisfying 1<m<n, m→∞, and m/n→0, as n→∞ (the same as that used in (3.3)).LetJ be a given weight function satisfying (H1)–(H4) with fixed constants β and λ. Suppose that, for n large enough, we have a realization (x1,…,xn) of a sample (X1,…,Xn) from rv X with df F fulfilling all assumptions of Theorem 3.5. The (1-ς)-confidence intervals for L(J) will be obtained via the following steps.Step 1.
Select the optimal numbersk*, ℓ*, and m* of lower- and upper-order statistics used in (3.2) and (3.3).Step 2.
DetermineXk*,n,Xn-ℓ*,n,J(k*/n), J(1-ℓ*/n), and θ*:=ℓ*/k*.Step 3.
Compute, using (3.2), α̂L*:=α̂L(k*) and α̂R*:=α̂R(ℓ*). Then deduce, by (3.10), the estimate L̂k*,ℓ*(J).Step 4.
Use (3.3) and (3.20) to compute α̂*:=α̂(m*) and p̂n*:=p̂n(m*). Then deduce, by (3.19), the asymptotic standard deviation
(3.21)V*:=V2(α̂*,β,λ,θ*,p̂n*).
Finally, the lower and upper (1-ς)-confidence bounds for L(J), respectively, will be
(3.22)L̂k*,ℓ*(J)-zς/2ℓ*V*Xn-ℓ*,nJ(1-ℓ*/n)n,L̂k*,ℓ*(J)+zς/2ℓ*V*Xn-ℓ*,nJ(1-ℓ*/n)n,
where zς/2 is the (1-ς/2) quantile of the standard normal distribution 𝒩(0,1) with 0<ς<1.


## 3.1. Extreme Quantile Estimation

The right and left extreme quantiles of small enough levelt of df F, respectively, are two reals xR and xL defined by 1-F(xR)=t and F(xL)=t, that is, xR=Q(1-t) and xL=Q(t). The estimation of extreme quantiles for heavy-tailed distributions has got a great deal of interest, see for instance Weissman [38], Dekkers and de Haan [39], Matthys and Beirlant [40] and Gomes et al. [41]. Next, we introduce one of the most popular quantile estimators. Let k=kn and ℓ=ℓn be sequences of integers (called trimming sequences) satisfying 1<k<n,1<ℓ<n, k→∞, ℓ→∞, k/n→0 and ℓ/n→0, as n→∞. Weissman's estimators of extreme quantiles xR and xL are defined, respectively, by(3.1)x̂L=Q̂L(t):=(kn)1/αLXk,nt-1/αL,ast↓0,x̂R=Q̂R(1-t):=(ℓn)1/αRXn-ℓ,nt-1/αR,ast↓0,
where(3.2)α̂L=α̂L(k):=(1k∑i=1klog+(-Xi,n)-log+(-Xk,n))-1,α̂R=α̂R(ℓ):=(1ℓ∑i=1ℓlog+(Xn-i+1,n)-log+(Xn-ℓ,n))-1
are two forms of Hill's estimator [42] for the stability index α which could also be estimated, using the order statistics Z1,n≤⋯≤Zn,n associated to a sample (Z1,…,Zn) from Z, as follows:(3.3)α̂=α̂(m):=(1m∑i=1mlog+(Zn-i+1,n)-log+(Zn-m,n))-1,
with log+u:=max(0,logu) and m=mn being an intermediate sequence fulfilling the same conditions as k and ℓ. Hill's estimator has been thoroughly studied, improved, and even generalized to any real-valued tail index. Its weak consistency was established by Mason [43] assuming only that the underlying distribution is regularly varying at infinity. The almost sure convergence was proved by Deheuvels et al. [44] and more recently by Necir [45]. The asymptotic normality has been investigated, under various conditions on the distribution tail, by numerous workers like, for instance, Csörgő and Mason [46], Beirlant and Teugels [47], and Dekkers et al. [48].


## 3.2. A Discussion on the Sample Fractionsk and ℓ

Extreme value-based estimators rely essentially on the numbersk and ℓ of lower- and upper-order statistics used in estimate computation. Estimators α̂L and α̂R have, in general, substantial variances for small values of k and ℓ and considerable biases for large values of k and ℓ. Therefore, one has to look for optimal values for k and ℓ, which balance between these two vices.Numerically, there exist several procedures for the thorny issue of selecting the numbers of order statistics appropriate for obtaining good estimates of the stability indexα; see, for example, Dekkers and de Haan [49], Drees and Kaufmann [50], Danielsson et al. [51], Cheng and Peng [52] and Neves and Alves [53]. Graphically, the behaviors of α̂L, α̂R, and α̂ as functions of k, ℓ and m, respectively, are illustrated by Figures 1, 2, and 3 drawn by means of the statistical software R [54]. According to Figure 1, α̂R is much more suitable than α̂L when estimating the stability index of a distribution which is skewed to the right (δ>0)whereas Figure 2 shows that α̂L is much more reliable than α̂R when the distribution is skewed to the left (δ<0). In the case where the distribution is symmetric (δ=0), both estimators seem to be equally good as seen in Figure 3. Finally, it is worth noting that, regardless of the distribution skewness, estimator α̂, based on the top statistics pertaining to the absolute value of X, works well and gives good estimates for the characteristic exponent α.Figure 1
Plots of Hill estimators, as functions of the number of extreme statistics,α̂ (solid line), α̂R (dashed line), and α̂L (dotted line) of the characteristic exponent α of a stable distribution skewed to the right, based on 1000 observations with 50 replications. The horizontal line represents the true value of α=1.2.Figure 2
Plots of Hill estimators, as functions of the number of extreme statistics,α̂ (solid line), α̂R (dashed line), and α̂L (dotted line) of the characteristic exponent α of a stable distribution skewed to the left, based on 1000 observations with 50 replications. The horizontal line represents the true value of α=1.2.Figure 3
Plots of Hill estimators, as functions of the number of extreme statistics,α̂ (solid line), α̂R (dashed line) and α̂L (dotted line) of the characteristic exponent α of a symmetric stable distribution, based on 1000 observations with 50 replications. The horizontal line represents the true value of α=1.2.It is clear that, in general, there is no reason for the trimming sequencesk and ℓ to be equal. We assume that there exists a positive real constant θ such that ℓ/k→θ as n→∞. If the distribution is symmetric, the value of θ is equal to 1; otherwise, it is less or greater than 1 depending on the sign of the distribution skewness. For an illustration, see Figure 4 where we plot the ratio θ for several increasing sample sizes.Figure 4
Plots of the ratios of the numbers of extreme statistics, as functions of the sample size, for a stable symmetric distributionS1.2(1,0,0) (solid line), a stable distribution skewed to the right S1.2(1,0.5,0) (dashed line) and a stable distribution skewed to the left S1.2(1,-0.5,0) (dotted line).


## 3.3. Some Regularity Assumptions onJ

For application needs, the following regularity assumptions on functionJ are required:(H1)
J is differentiable on (0,1),(H2)
λ:=lims↓0J(1-s)J(s)<∞,(H3)
bothJ(s) and J(1-s) are regularly varying at zero with common index β∈ℝ.(H4)
there exists a functiona(·) not changing sign near zero such that
(3.4)limt↓0J(xt)/J(t)-xβa(t)=xβxω-1ω,for  anyx>0,
where ω≤0 is the second-orderparameter.The three remarks below give more motivations to this paper.Remark 3.1.
Assumption (H3) has already been used by Mason and Shorack [55] to establish the asymptotic normality of trimmed L-statistics. Condition (H4) is just a refinement of (H3) called the second order condition that is required for quantile function K in (2.6).Remark 3.2.
Assumptions (H1)–(H4) are satisfied by all weight functions(Ji)i=2,4 with (β,λ)=(1,±1) (see Section 4.1) and by function Jr in (1.9) with (β,λ)=(r-1,-1). These two examples show that the constants β and λ may be positive or negative depending on application needs.Remark 3.3.
L-functionals L(J) exist for any 0<α<2 and β∈ℝ such that 1/α-β<1. However, Lemma 5.4 below shows that for 1/α-β>1/2 we have σ2(J)=∞. Then, recall (1.3); whenever 1/2<α<2/3, the trimmed L-moments exist however σ2(Ji)=∞,i=1,…,4. Likewise, recall (1.9); whenever 1/(r+1/2)<α<1/r, the two-sided deviation Δr(X) exists while σ2(Jr)=∞.


## 3.4. Defining the Estimator and Main Results

We now have all the necessary tools to introduce our estimator ofL(J), given in (1.1), when F∈D(α) with 0<α<2. Let k=kn and ℓ=ℓn be sequences of integers satisfying 1<k<n, 1<ℓ<n, k→∞, ℓ→∞, k/n→0, ℓ/n→0, and the additional condition ℓ/k→θ<∞ as n→∞. First, we must note that since 1+β-1/α>0 (see Remark 3.3) and since both α̂L and α̂R are consistent estimators of α (see, Mason [43]), then we have for all large n(3.5)ℙ(1+β-1α̂L>0)=ℙ(1+β-1α̂R>0)=1+o(1).
Observe now that L(J) defined in (1.1) may be split in three integrals as follows:(3.6)L(J)=∫0k/nJ(t)Q(t)dt+∫k/n1-ℓ/nJ(t)Q(t)dt+∫1-ℓ/n1J(t)Q(t)dt=:TL,n+TM,n+TR,n.
Substituting Q̂L(t) and Q̂R(1-t) for Q(t) and Q(1-t) in TL,n and TR,n, respectively and making use of assumption (H3) and (3.5) yield that for all large n(3.7)∫0k/nJ(t)Q̂L(t)dt=(kn)1/αLXk,n∫0k/nt-1/αLJ(t)dt=(1+o(1))(k/n)J(k/n)1+β-1/α̂LXk,n,∫0ℓ/nJ(1-t)Q̂R(1-t)dt=(ℓn)1/αRXn-ℓ,n∫0ℓ/nt-1/αRJ(1-t)dt=(1+o(1))(ℓ/n)J(1-ℓ/n)1+β-1/α̂RXn-ℓ,n.
Hence we may estimate TL,n and TR,n by(3.8)T̂L,n:=(k/n)J(k/n)1+β-1/α̂LXk,n,T̂R,n:=(ℓ/n)J(1-ℓ/n)1+β-1/α̂RXn-ℓ,n,
respectively. As an estimator of TM,n we take the sample one that is(3.9)T̂M,n:=∫k/n1-ℓ/nJ(t)Qn(t)dt=∑i=k+1n-ℓai,nXi,n,
with the same constants ai,n as those in (1.11). Thus, the final form of our estimator is(3.10)L̂k,ℓ(J)=(k/n)J(k/n)1+β-1/α̂LXk,n+∑i=k+1n-ℓai,nXi,n+(ℓ/n)J(1-ℓ/n)1+β-1/α̂RXn-ℓ,n.
A universal estimator of L(J) may be summarized by(3.11)L̂n*(J)=L̂k,ℓ(J)𝕀(σ2(J)=∞)+L̂n(J)𝕀(σ2(J)<∞),
where L̂n(J) is as in (1.11). More precisely(3.12)L̂n*(J)=L̂k,ℓ(J)𝕀(A(α,β))+L̂n(J)𝕀(A¯(α,β)),
where A(α,β):={(α,β)∈(0,2)×ℝ:1/2<1/α-β<1} and A¯(α,β) is its complementary in (0,2)×ℝ.Note that for the particular caseℓ=k and J=1the asymptotic normality of the trimmed mean T̂M,n has been established in Theorem 1 of Csörgő et al. [56]. The following theorem gives the asymptotic normality of T̂M,n for more general trimming sequences k and ℓ and weighting function J. For convenience, we set, for any 0<x<1/2 and 0<y<1/2,(3.13)σ2(x,y;J):=∫x1-y∫x1-y(min(s,t)-st)J(s)J(t)dQ(s)Q(t)<∞,
and let σn2(J):=σ2(k/n,ℓ/n;J).Theorem 3.4.
Assume thatF∈D(α) with 0<α<2. For any  measurable function J satisfying assumption (H3) with index β∈ℝ such that 0<1/α-β<1 and for any sequences  of integers k and ℓ such that1<k<n, 1<ℓ<n, k→∞, ℓ→∞, k/n→0, andℓ/n→0, as  n→∞, there exists a probability space (Ω,A,P) carrying the sequence X1,X2,… and a sequence of Brownian bridges {Bn(s),0≤s≤1,n=1,2,…} such that one has for all large  n(3.14)n(T̂M,n-TM,n)σn(J)=-∫k/n1-ℓ/nJ(s)Bn(s)dsσn(J)+op(1),
and therefore
(3.15)n(T̂M,n-TM,n)σn(J)→𝒟𝒩(0,1)asn→∞.The asymptotic normality of our estimator is established in the following theorem.Theorem 3.5.
Assume thatF∈D(α) with 0<α<2. For any  measurable function J satisfying assumptions (H1)–(H4) with index  β∈ℝ such that 1/2<1/α-β<1, and for any sequences of integers k and ℓ such that 1<k<n, 1<ℓ<n, k→∞, ℓ→∞, k/n→0, ℓ/n→0, ℓ/k→θ<∞, and ka(k/n)A(k/n)→0 as n→∞,  one has
(3.16)n(L̂k,ℓ(J)-L(J))σn(J)→𝒟𝒩(0,σ02),asn→∞,
where
(3.17)σ02=σ02(α,β):=(αβ+1)(2αβ+2-α)×(2α2+(βα-1)2+2α(βα-1)2((1+β)α-1)4+1(1+β)α-1)+1.The following corollary is more practical than Theorem3.5 as  it directly provides confidence bounds for L(J).Corollary 3.6.
Under the assumptions of Theorem3.5 one has
(3.18)n(L̂k,ℓ(J)-L(J))(ℓ/n)1/2J(1-ℓ/n)Xn-ℓ,n→𝒟𝒩(0,V2),asn→∞,
where
(3.19)V2=V2(α,β,λ,θ,p):=(1+λ-2(qp)-2/αθ-2β+2/α-1)×(2α2+(βα-1)2+2α(βα-1)2((1+β)α-1)4+1(1+β)α-1)+1,
with (p,q) as in statement (ii) of Section 2 and (λ,β) as in assumptions (H2)-(H3) of Section 3.


## 3.5. Computing Confidence Bounds forL(J)

The form of the asymptotic varianceV2 in (3.20) suggests that, in order to construct confidence intervals for L(J), an estimate of p is needed as well. Using the intermediate order statistic Zn-m,n, de Haan and Pereira [57] proposed the following consistent estimator for p:(3.20)p̂n=p̂n(m):=1m∑i=1n𝕀{Xi>Zn-m,n},
where m=mn is a sequence of integers satisfying 1<m<n, m→∞, and m/n→0, as n→∞ (the same as that used in (3.3)).LetJ be a given weight function satisfying (H1)–(H4) with fixed constants β and λ. Suppose that, for n large enough, we have a realization (x1,…,xn) of a sample (X1,…,Xn) from rv X with df F fulfilling all assumptions of Theorem 3.5. The (1-ς)-confidence intervals for L(J) will be obtained via the following steps.Step 1.
Select the optimal numbersk*, ℓ*, and m* of lower- and upper-order statistics used in (3.2) and (3.3).Step 2.
DetermineXk*,n,Xn-ℓ*,n,J(k*/n), J(1-ℓ*/n), and θ*:=ℓ*/k*.Step 3.
Compute, using (3.2), α̂L*:=α̂L(k*) and α̂R*:=α̂R(ℓ*). Then deduce, by (3.10), the estimate L̂k*,ℓ*(J).Step 4.
Use (3.3) and (3.20) to compute α̂*:=α̂(m*) and p̂n*:=p̂n(m*). Then deduce, by (3.19), the asymptotic standard deviation
(3.21)V*:=V2(α̂*,β,λ,θ*,p̂n*).
Finally, the lower and upper (1-ς)-confidence bounds for L(J), respectively, will be
(3.22)L̂k*,ℓ*(J)-zς/2ℓ*V*Xn-ℓ*,nJ(1-ℓ*/n)n,L̂k*,ℓ*(J)+zς/2ℓ*V*Xn-ℓ*,nJ(1-ℓ*/n)n,
where zς/2 is the (1-ς/2) quantile of the standard normal distribution 𝒩(0,1) with 0<ς<1.


## 4. Applications

### 4.1. TL-Skewness and TL-Kurtosis WhenF∈D(α)

When the distribution meanEX exists, the skewness and kurtosis coefficients are, respectively, defined by L1:=μ3/μ23/2 and L2:=μ4/μ22 with μk:=E(X-EX)k,k=2, 3, and 4 being the centered moments of the distribution. They play an important role in distribution classification, fitting models, and parameter estimation, but they are sensitive to the behavior of the distribution extreme tails and may not exist for some distributions such as the Cauchy distribution. Alternative measures of skewness and kurtosis have been proposed; see, for instance, Groeneveld [58] and Hosking [3]. Recently, Elamir and Seheult [4] have used the trimmed L-moments to introduce new parameters called TL-skewness and TL-kurtosis that are more robust against extreme values. For example, when the trimming parameter equals one, the TL-skewness and TL-kurtosis measures are, respectively, defined by(4.1)υ1:=m3m2,υ2:=m4m2,
where mi,i=2,3,4, are the trimmed L-moments defined in Section 1. The corresponding weight functions of (1.3) are defined as follows:(4.2)J2(s):=6s(1-s)(2s-1),J3(s):=203s(1-s)(5s2-5s+1),J4(s):=152s(1-s)(14s3-21s2+9s-1).
If we suppose that F∈D(α) with 1/2<α<2/3, then, in view of the results above, asymptotically normal estimators for υ1 and υ2 will be, respectively,(4.3)υ̂1:=m̂3m̂2,υ̂2:=m̂4m̂2,where(4.4)m̂i={-6(k/n)22-1/α̂LXk,n+∑j=k+1n-ℓaj,n(i)Xj,n+6(ℓ/n)22-1/α̂RXn-ℓ,n,fori=2,20(k/n)23(2-1/α̂L)Xk,n+∑j=k+1n-ℓaj,n(i)Xj,n+20(ℓ/n)23(2-1/α̂R)Xn-ℓ,n,fori=3,-15(k/n)22(2-1/α̂L)Xk,n+∑j=k+1n-ℓaj,n(i)Xj,n+15(ℓ/n)22(2-1/α̂R)Xn-ℓ,n,fori=4,
with aj,n(i):=∫(j-1)/nj/nJi(s)ds,i=2,3,4, and j=1,…,n.Theorem 4.1.
Assume thatF∈D(α) with 1/2<α<2/3. For any sequencesof integers k and ℓ such that 1<k<n,1<ℓ<n, k→∞, ℓ→∞, k/n→0, ℓ/n→0, ℓ/k→θ<∞, and ka(k/n)A(k/n)→0 as n→∞,  one has, respectively, as n→∞,(4.5)n(υ̂1-υ1)(ℓ/n)3/2Xn-ℓ,n→𝒟𝒩(0,V12),n(υ̂2-υ2)(ℓ/n)3/2Xn-ℓ,n→𝒟𝒩(0,V22),
where
(4.6)V12:=36m22(1-9m310m2)2σ*2,V22:=2254m22(1-4m45m2)2σ*2,
with
(4.7)σ*2:=(1+(q/p)-2/αθ2/α-3)×(2α2+(α-1)2+2α(α-1)2(2α-1)4+12α-1)+1.


### 4.2. Risk Two-Sided Deviation WhenF∈D(α)

Recall that the risk two-sided deviation is defined by(4.8)Δr(X):=∫01Jr(s)Q(s)ds,0<r<1,
where(4.9)Jr(s):=r2s1-r-(1-s)1-rs1-r(1-s)1-r,0<s<1.
An asymptotically normal estimator for Δr(X), when 1/(r+1/2)<α<1/r, is(4.10)Δ̂r(X)=-r(k/n)r2r-4/α̂LXk,n+∑j=k+1n-ℓaj,n(r)Xj,n+r(ℓ/n)r2r-4/α̂RXn-ℓ,n,
where(4.11)aj,n(r)=12[(1-in)-r-(1-i-1n)-r-(in)-r+(i-1n)-r],j=1,…,n.Theorem 4.2.
Assume thatF∈D(α) with 0<α<2 such that 1/(r+1/2)<α<1/r, for any 0<r<1. Then, for any sequences  of integers k and ℓ such that1<k<n, 1<ℓ<n, k→∞,ℓ→∞, k/n→0, ℓ/n→0,ℓ/k→θ<∞, and ka(k/n)A(k/n)→0 as n→∞, one has, as n→∞,(4.12)n(Δ̂r(X)-Δr(X))(ℓ/n)r-1/2Xn-ℓ,n→𝒟𝒩(0,Vr2),
where
(4.13)Vr2:=r24(1+(qp)-2/αθ2/α-2r+1)×(2α2+(rα-α-1)2+2α(rα-α-1)2(rα-1)4+1rα-1)+1.


## 4.1. TL-Skewness and TL-Kurtosis WhenF∈D(α)

When the distribution meanEX exists, the skewness and kurtosis coefficients are, respectively, defined by L1:=μ3/μ23/2 and L2:=μ4/μ22 with μk:=E(X-EX)k,k=2, 3, and 4 being the centered moments of the distribution. They play an important role in distribution classification, fitting models, and parameter estimation, but they are sensitive to the behavior of the distribution extreme tails and may not exist for some distributions such as the Cauchy distribution. Alternative measures of skewness and kurtosis have been proposed; see, for instance, Groeneveld [58] and Hosking [3]. Recently, Elamir and Seheult [4] have used the trimmed L-moments to introduce new parameters called TL-skewness and TL-kurtosis that are more robust against extreme values. For example, when the trimming parameter equals one, the TL-skewness and TL-kurtosis measures are, respectively, defined by(4.1)υ1:=m3m2,υ2:=m4m2,
where mi,i=2,3,4, are the trimmed L-moments defined in Section 1. The corresponding weight functions of (1.3) are defined as follows:(4.2)J2(s):=6s(1-s)(2s-1),J3(s):=203s(1-s)(5s2-5s+1),J4(s):=152s(1-s)(14s3-21s2+9s-1).
If we suppose that F∈D(α) with 1/2<α<2/3, then, in view of the results above, asymptotically normal estimators for υ1 and υ2 will be, respectively,(4.3)υ̂1:=m̂3m̂2,υ̂2:=m̂4m̂2,where(4.4)m̂i={-6(k/n)22-1/α̂LXk,n+∑j=k+1n-ℓaj,n(i)Xj,n+6(ℓ/n)22-1/α̂RXn-ℓ,n,fori=2,20(k/n)23(2-1/α̂L)Xk,n+∑j=k+1n-ℓaj,n(i)Xj,n+20(ℓ/n)23(2-1/α̂R)Xn-ℓ,n,fori=3,-15(k/n)22(2-1/α̂L)Xk,n+∑j=k+1n-ℓaj,n(i)Xj,n+15(ℓ/n)22(2-1/α̂R)Xn-ℓ,n,fori=4,
with aj,n(i):=∫(j-1)/nj/nJi(s)ds,i=2,3,4, and j=1,…,n.Theorem 4.1.
Assume thatF∈D(α) with 1/2<α<2/3. For any sequencesof integers k and ℓ such that 1<k<n,1<ℓ<n, k→∞, ℓ→∞, k/n→0, ℓ/n→0, ℓ/k→θ<∞, and ka(k/n)A(k/n)→0 as n→∞,  one has, respectively, as n→∞,(4.5)n(υ̂1-υ1)(ℓ/n)3/2Xn-ℓ,n→𝒟𝒩(0,V12),n(υ̂2-υ2)(ℓ/n)3/2Xn-ℓ,n→𝒟𝒩(0,V22),
where
(4.6)V12:=36m22(1-9m310m2)2σ*2,V22:=2254m22(1-4m45m2)2σ*2,
with
(4.7)σ*2:=(1+(q/p)-2/αθ2/α-3)×(2α2+(α-1)2+2α(α-1)2(2α-1)4+12α-1)+1.


## 4.2. Risk Two-Sided Deviation WhenF∈D(α)

Recall that the risk two-sided deviation is defined by(4.8)Δr(X):=∫01Jr(s)Q(s)ds,0<r<1,
where(4.9)Jr(s):=r2s1-r-(1-s)1-rs1-r(1-s)1-r,0<s<1.
An asymptotically normal estimator for Δr(X), when 1/(r+1/2)<α<1/r, is(4.10)Δ̂r(X)=-r(k/n)r2r-4/α̂LXk,n+∑j=k+1n-ℓaj,n(r)Xj,n+r(ℓ/n)r2r-4/α̂RXn-ℓ,n,
where(4.11)aj,n(r)=12[(1-in)-r-(1-i-1n)-r-(in)-r+(i-1n)-r],j=1,…,n.Theorem 4.2.
Assume thatF∈D(α) with 0<α<2 such that 1/(r+1/2)<α<1/r, for any 0<r<1. Then, for any sequences  of integers k and ℓ such that1<k<n, 1<ℓ<n, k→∞,ℓ→∞, k/n→0, ℓ/n→0,ℓ/k→θ<∞, and ka(k/n)A(k/n)→0 as n→∞, one has, as n→∞,(4.12)n(Δ̂r(X)-Δr(X))(ℓ/n)r-1/2Xn-ℓ,n→𝒟𝒩(0,Vr2),
where
(4.13)Vr2:=r24(1+(qp)-2/αθ2/α-2r+1)×(2α2+(rα-α-1)2+2α(rα-α-1)2(rα-1)4+1rα-1)+1.


## 5. Proofs

First we begin by the following three technical lemmas.Lemma 5.1.
Letf1 and f2 be two continuous functions defined on (0,1) and regularly varying at zero with respective indices κ>0 and -τ<0 such that κ<τ. Suppose that f1 is differentiable at zero, then
(5.1)limx↓0∫x1/2f1(s)df2(s)f1(x)f2(x)=τκ-τ.Lemma 5.2.
Under the assumptions of Theorem3.5, one has
(5.2)limn→∞∫k/n1-ℓ/n(s(1-s))1/2-νJ(s)dQ(s)(k/n)1/2-νJ(k/n)Q(k/n)=1+λθ1/2-ν+β-1/α(q/p)1/αα(1/2-ν+β)-1,
for any  0<ν<1/4.Lemma 5.3.
For any0<x<1/2 and 0<y<1/2 one has
(5.3)σ2(x,y;J)=xc2(x)+yc2(1-y)+∫x1-yc2(t)dt-(xc(x)+yc(1-y)+∫x1-yc(t)dt)2,
where c(s):=∫1/2sJ(t)dQ(t), 0<s<1/2.Lemma 5.4.
Under the assumptions of Theorem3.5, one has
(5.4)limn→∞(k/n)J2(k/n)Q2(k/n)σn2(J)=w2,limn→∞(ℓ/n)J2(1-ℓ/n)Q2(1-ℓ/n)σn2(J)=λ2(qp)2/αθ2β-2/α+1w2,
where
(5.5)w2:=(αβ+1)(2αβ+2-α)2(1+λ2(q/p)2/αθ2β-2/α+1).Proof of Lemma5.1.
Letf1′ denote the derivative of f1. Applying integration by parts, we get, for any 0<x<1/2,(5.6)∫x1/2f1(s)df2(s)=f1(12)f2(12)-f1(x)f2(x)-∫x1/2f1′(s)f2(s)ds.
Since the product f1f2 is regularly varying at zero with index -τ+κ<0, then f1(x)f2(x)→0 as x↓0. Therefore
(5.7)limx↓0∫x1/2f1(s)df2(s)f1(x)f2(x)=-1-limx↓0∫x1/2f1′(s)f2(s)dsf1(x)f2(x).
By using Karamata's representation (see, e.g., Seneta [59]), it is easy to show that
(5.8)xf1′(x)=κ(1+o(1))f1(x),asx↓0.
Hence
(5.9)limx↓0∫x1/2f1(s)df2(s)f1(x)f2(x)=-1-κlimx↓0∫x1/2f1′(s)f2(s)dsxf1′(x)f2(x).
It is clear that (5.8) implies that f1′ is regularly varying at zero with index κ-1; therefore f1′f2 is regularly varying with index -τ+κ-1<0. Then, Theorem 1.2.1 by de Haan [60, page 15] yields
(5.10)limx↓0-∫x1/2f1′(s)f2(s)dsxf1′(x)f2(x)=1κ-τ.
This completes the proof of Lemma 5.1.Proof of Lemma5.2.
We have(5.11)In:=∫k/n1-ℓ/n(s(1-s))1/2-νJ(s)dQ(s)=∫k/n1/2(s(1-s))1/2-νJ(s)dQ(s)-∫1-ℓ/n1/2(s(1-s))1/2-νJ(1-s)dQ(1-s)=:I1n-I2n.
By taking, in Lemma 5.1, f1(s)=(s(1-s))1/2-νJ(s) and f2(s)=Q(s) with κ=1/2-ν+β,τ=1/α, and x=k/n, we get
(5.12)limn→∞I1n(k/n)1/2-νJ(k/n)Q(k/n)=1/α1/2-ν+β-1/α.
Likewise if we take f1(s)=(s(1-s))1/2-νJ(1-s) and f2(s)=Q(1-s) with κ=1/2-ν+β,τ=1/α, and x=ℓ/n, we have
(5.13)limn→∞I2n(ℓ/n)1/2-νJ(1-ℓ/n)Q(1-ℓ/n)=1/α1/2-ν+β-1/α.
Note that statement (ii′) of Section 2 implies that
(5.14)lims↓0Q(1-s)/Q(s)=-(qp)1/α.
The last two relations, together with assumption (H2) and the regular variation of Q(1-s), imply that
(5.15)limn→∞I2n(k/n)1/2-νJ(k/n)Q(k/n)=-(q/p)1/αλθ1/2-ν+β-1/α/α1/2-ν+β-1/α.
This achieves the proof of Lemma 5.2.Proof of Lemma5.3.
We will use similar techniques to those used by Csörgő et al. [36, Proposition A.2]. For any 0<s<1/2, we set
(5.16)Wx,y(t):={c(1-y)for1-y≤t<1,c(t)forx<t<1-y,c(x)for0<t≤x.
Then σ2(x,y;J) may be rewritten into
(5.17)σ2(x,y;J)=∫01Wx,y2(s)ds-(∫01Wx,y(s)ds)2,
and the result of Lemma 5.3 follows immediately.Proof of Lemma5.4.
From Lemma5.3 we may write
(5.18)σn2(J)(k/n)J2(k/n)Q2(k/n)=Tn1+Tn2+Tn3+Tn4,
where
(5.19)Tn1:=(k/n)c2(k/n)(k/n)J2(k/n)Q2(k/n),Tn2:=(ℓ/n)c2(1-ℓ/n)(k/n)J2(k/n)Q2(k/n),Tn3:=∫k/n1-ℓ/nc2(t)dt(k/n)J2(k/n)Q2(k/n),Tn4:=((k/n)c(k/n)+(ℓ/n)c(1-ℓ/n)+∫k/n1-ℓ/nc(t)dt)2(k/n)J2(k/n)Q2(k/n).
By the same arguments as in the proof of Lemma 5.2, we infer that
(5.20)limn→∞c(k/n)J(k/n)Q(k/n)=1αβ-1,limn→∞c(1-ℓ/n)J(k/n)Q(k/n)=λ(q/p)1/αθβ-1/ααβ-1.
Therefore
(5.21)limn→∞Tn1=1(αβ-1)2,limn→∞Tn2=λ2(q/p)2/αθ2β-2/α+1(αβ-1)2.
Next, we consider the third term Tn3 which may be rewritten into
(5.22)Tn3=∫k/n1/2c2(t)dt(k/n)J2(k/n)Q2(k/n)+∫1/21-ℓ/nc2(t)dt(k/n)J2(k/n)Q2(k/n).
Observe that
(5.23)∫k/n1/2c2(t)dt(k/n)J2(k/n)Q2(k/n)=(c(k/n)J(k/n)Q(k/n))2∫k/n1/2c2(t)dt(k/n)c2(k/n).
It is easy to verify that function c2(·) is regularly varying at zero with index 2(β-1/α). Thus, by Theorem 1.2.1 by de Haan [60] we have
(5.24)limn→∞∫k/n1/2c2(t)dt(k/n)c2(k/n)=α2-2αβ-α.
Hence
(5.25)limn→∞∫k/n1/2c2(t)dt(k/n)J2(k/n)Q2(k/n)=α(αβ-1)2(2-2αβ-α).
By similar arguments we show that
(5.26)limn→∞∫1/21-ℓ/nc2(t)dt(k/n)J2(k/n)Q2(k/n)=α(q/p)2/αλ2θ2β-2/α+1(αβ-1)2(2-2αβ-α).
Therefore
(5.27)limn→∞Tn3=1+α(q/p)2/αλ2θ2β-2/α+1(αβ-1)2(2-2αβ-α).
By analogous techniques we show that Tn4→0 as n→∞; we omit details. Summing up the three limits of Tni,i=1,2,3, achieves the proof of the first part of Lemma 5.4. As for the second assertion of the lemma, we apply a similar procedure.


### 5.1. Proof of Theorem3.4

Csörgő et al. [36] have constructed a probability space (Ω,A,P) carrying an infinite sequence ξ1,ξ2,… of independent rv's uniformly distributed on (0,1) and a sequence of Brownian bridges {Bn(s),0≤s≤1,n=1,2,…} such that, for the empirical process,(5.28)φn(s):=n1/2{Γn(s)-s},0≤s≤1,
where Γn(·) is the uniform empirical df pertaining to the sample (ξ1,…,ξn); we have for any 0≤ν<1/4 and for all large n(5.29)sup1/n≤s≤1-1/n|φn(s)-Bn(s)|(s(1-s))1/2-ν=Op(n-ν).
For each n≥1, let ξ1,n≤⋯≤ξn,n denote the order statistics corresponding to (ξ1,…,ξn). Note that for each n, the random vector (Q(ξ1,n),…,Q(ξn,n)) has the same distribution as (X1,n,…,Xn,n). Therefore, for 1≤i≤n, we shall use the rv's Q(ξi,n) to represent the rv's Xi,n, and without loss of generality, we shall be working, in all the following proofs, on the probability space above. According to this convention, the term T̂M,n defined in (3.9) may be rewritten into(5.30)T̂M,n=∫ξk,nξn-ℓ,nQ(s)dΨ(Γn(s)),
where Ψ(s):=∫0sJ(t)dt. Integrating by parts yields(5.31)n1/2(T̂M,n-TM,n)σn(J)=Δ1,n+Δ2,n+Δ3,n,
where(5.32)Δ1,n:=-n1/2∫k/n1-ℓ/n{Ψ(Γn(s))-Ψ(s)}dQ(s)σn(J),Δ2,n:=n1/2∫k/nξk,n{Ψ(Γn(s))-Ψ(k/n)}dQ(s)σn(J),Δ3,n:=n1/2∫ξn-ℓ,n1-ℓ/n{Ψ(Γn(s))-Ψ(1-ℓ/n)}dQ(s)σn(J).
Next, we show that(5.33)Δ1,n→𝒟𝒩(0,1)asn→∞,(5.34)Δi,n→p0asn→∞fori=2,3.
Making use of the mean-value theorem, we have for each n(5.35)Ψ(Γn(s))-Ψ(s)=(Γn(s)-s)J(ϑn(s)),
where {ϑn(s)}n≥1 is a sequence of rv's with values in the open interval of endpoints s∈(0,1) and Γn(s). Therefore(5.36)Δ1,n=-∫k/n1-ℓ/nφn(s)J(ϑn(s))dQ(s)σn(J).
This may be rewritten into(5.37)Δ1,n=-∫k/n1-ℓ/nφn(s)J(s)dQ(s)σn(J)-∫k/n1-ℓ/nφn(s)J(s){J(ϑn(s))-J(s)}dQ(s)σn(J)=:Δ1,n*+Δ1,n**.
Note that(5.38)∫k/n1-ℓ/n|φn(s)-Bn(s)|J(s)dQ(s)σn(J)≤supk/n≤s≤1-ℓ/n|φn(s)-Bn(s)|(s(1-s))1/2-ν∫k/n1-ℓ/n(s(1-s))1/2-ν|J(s)|dQ(s)/σn(J),
for 0<ν<1/4, which by (5.29) is equal to(5.39)Op(n-ν)∫k/n1-ℓ/n(s(1-s))1/2-ν|J(s)|dQ(s)σn(J).
Since we have, from Lemmas 5.2 and 5.3,(5.40)(nk)ν∫k/n1-ℓ/n(s(1-s))1/2-ν|J(s)|dQ(s)/σn(J)=O(1),asn→∞,
then the right-hand side of the last inequality is equal to Op(k-ν) which in turn tends to zero as n→∞. This implies that as n→∞(5.41)Δ1,n*=-∫k/n1-ℓ/nBn(s)J(s)dQ(s)σn(J)+op(1).
Next, we show that Δ1,n**=op(1). Indeed, function J is differentiable on (0,1); then by the mean-value theorem, there exists a sequence {ϑn*(s)}n≥1 of rv's with values in the open interval of endpoints s∈(0,1) and ϑn(s) such that for each n we have(5.42)Δ1,n**=∫k/n1-ℓ/nφn(s)J(s){ϑn(s)-s}J′(ϑn*(s))dQ(s)σn(J).
From inequalities (3.9) and (3.10) by Mason and Shorack [55], we infer that, for any 0<ρ<1, there exists 0<Mρ<∞ such that for all large n we have(5.43)|J′(ϑn*(s))|≤Mρ|J(s)|(s(1-s)),
for any 0<s≤1/2. On the other hand, we have for any 0<s<1(5.44)|ϑn(s)-s|≤|Γn(s)-s|.
Therefore(5.45)|Δ1,n**|≤Mρn-1/2∫k/n1/2(|φn(s)|2|J(s)|/s(1-s))dQ(s)σn(J).
This implies, since, for each n≥1, E|φn(s)|2<s(1-s), that(5.46)E|Δ1,n**|≤Mρn-1/2∫k/n1/2|J(s)|dQ(s)σn(J),
which tends to zero as n→∞.Next, we consider the termΔ2,n which may be rewritten into(5.47)Δ2,n=n1/2∫k/nξk,n{Ψ(Γn(s))-Ψ(s)}dQ(s)σn(J)+n1/2∫k/nξk,n{Ψ(s)-Ψ(k/n)}dQ(s)σn(J).
Making use of the mean-value theorem, we get(5.48)Δ2,n=∫k/nξk,nφn(s)J(μn(s))dQ(s)σn(J)+n1/2∫k/nξk,n(s-k/n)J(sn*)dQ(s)σn(J),
where μn(s) is a sequence of rv's with values in the open interval of endpoints s∈(k/n,ξk,n) and Γn(s) and sn* a sequence of rv's with values in the open interval of endpoints s∈(k/n,ξk,n) and k/n. Again we may rewrite Δ2,n into(5.49)Δ2,n=∫k/nξk,nφn(s)(J(μn(s))-J(s))dQ(s)σn(J)+∫k/nξk,nφn(s)J(s)dQ(s)σn(J)+n1/2J(kn)∫k/nξk,n(s-kn)(J(sn*)J(k/n)-1)dQ(s)/σn(J)+n1/2J(kn)∫k/nξk,n(s-kn)dQ(s)/σn(J).
Recall that, as n→∞, both k and ℓ tend to infinity with k/n→0 and ℓ/n→0. This implies that(5.50)nk1/2(ξk,n-kn)→𝒟𝒩(0,1)asn→∞,(5.51)nl1/2(ξn-l,n-1+ln)→𝒟𝒩(0,1)asn→∞,
(see, e.g., Balkema and de Haan [61, page 18]). Next, we use similar arguments to those used in the proof of Theorem 1 by Csörgő et al. [56]. For any 0<c<∞ write(5.52)Δ2,n(1)(c):=∫k/n-c(k1/2/n)k/n+c(k1/2/n)|φn(s)||J(μn(s))-J(s)|dQ(s)σn(J)+∫k/n-c(k1/2/n)k/n+c(k1/2/n)|J(s)||φn(s)|dQ(s)σn(J),Δ2,n(2)(c):=∫k/n-c(k1/2/n)k/n+c(k1/2/n)n1/2|s-k/n|J(sn*)dQ(s)σn(J)+n1/2J(k/n)∫k/n-c(k1/2/n)k/n+c(k1/2/n)(s-k/n)(J(sn*)/J(k/n)-1)dQ(s)σn(J).
Notice that by (5.49)
(5.53)limc→∞lim infn→∞ℙ{|Δ2,n|≤Δ2,n(1)(c)+Δ2,n(2)(c)}≥limc→∞lim infn→∞ℙ{|ξk,n-kn|≤ck1/2n}.
In view of (5.50), this last quantity equals 1. Therefore to establish (5.34) for i=2, it suffices to show that for each 0<c<∞(5.54)Δ2,n(1)(c)→p0,Δ2,n(2)(c)→0asn→∞.
By the mean-value theorem, there exists {μn*(s)}n≥1 a sequence of rv's with values in the open interval of endpoints s and μn(s) such that for each n we have(5.55)J(μn(s))-J(s)=(μn(s)-s)J′(μn*(s)).Since|μn(s)-s|≤|Γn(s)-s|, then by inequality (5.43) we infer that, for any 0<ρ<1, there exists 0<Mρ′<∞ such that for all large n we have(5.56)|J(μn(s))-J(s)|≤Mρ′n-1/2|φn(s)||J(s)|(s(1-s)).
This implies that the first term in Δ2,n(1)(c) is less than or equal to(5.57)Mρ′n-1/2∫k/n-c(k1/2/n)k/n+c(k1/2/n)(φn2(s)|J(s)|(s(1-s)))dQ(s)/σn(J).
Since E(φn2(s))≤s(1-s), then the expectedvalue of the previous quantity is less than or equal to(5.58)Mρ′n-1/2∫(1-c)k/n(1+c)k/n|J(s)|dQ(s)σn(J).
Likewise the expected value of the second term in Δ2,n(1)(c) is less than or equal to(5.59)∫k/n-c(k1/2/n)k/n+c(k1/2/n)(s(1-s))1/2|J(s)|dQ(s)σn(J)≤(k/n+c(k1/2/n))1/2∫k/n-c(k1/2/n)k/n+c(k1/2/n)|J(s)|dQ(s)σn(J).
Fix 1<ɛ<∞. It is readily verified that, for all large n, the quantity (5.58) is less than(5.60)Mρ′n-1/2∫k/ɛnɛ(k/n)|J(s)|dQ(s)σn(J),
and the right-hand side of (5.59) is less than(5.61)2(kn)1/2∫k/ɛnɛ(k/n)|J(s)|dQ(s)/σn(J).
Therefore for all large n(5.62)E(Δ2,n(1)(c))≤(2+Mρ′k-1/2)(kn)1/2∫k/ɛnɛ(k/n)|J(s)|dQ(s)/σn(J).
By routine manipulations, as in the proofs of Lemmas 5.1 and 5.2 (we omit details), we easily show that(5.63)limn→∞(kn)1/2∫k/ɛnɛ(k/n)|J(s)|dQ(s)/σn(J)=wαβ-1(ɛ1/α-β-ɛβ-1/α).
Since k→∞, then for any 0<c<∞(5.64)lim supn→∞E(Δ2,n(1)(c))≤2(ɛ1/α-β-ɛβ-1/α)w,
for any fixed 1<ɛ<∞. This implies that for all 0<c<∞(5.65)limn→∞E(Δ2,n(1)(c))=0.
Therefore, by Markov inequality, we have the first result of (5.34).Now, consider the termΔ2,n(2)(c). Observe that ns*/k is a sequence of rv's with values in the open interval of endpoints nξk,n/k and 1. On the other hand,(5.49) implies that nξk,n/k→p1 as n→∞. Hence ns*/k→p1 as well. Then, it is readily checked that, in view of relation (3.9) by Mason and Shorack [55], we have(5.66)sups∈HnJ(sn*)J(k/n)=Op(1)asn→∞.
Therefore(5.67)Δ2,n(2)(c)=Op(1)J(kn)∫k/n-c(k1/2/n)k/n+c(k1/2/n)n1/2|s-kn|dQ(s)/σn(J).
Observe that(5.68)∫k/n-c(k1/2/n)k/n+c(k1/2/n)n1/2|s-kn|dQ(s)≤c(kn)1/2∫k/n-c(k1/2/n)k/n+c(k1/2/n)dQ(s).
Hence for all large n(5.69)Δ2,n(2)(c)=Op(1)J(kn)∫k/ɛnɛ(k/n)dQ(s)/σn(J),
for any fixed 1<ɛ<∞, we have(5.70)Δ2,n(2)(c)=Op(1)(ɛ1/α-β-ɛβ-1/α)w.
this means that Δ2,n(2)(c)→0 as n→∞.By the same arguments and making use of (5.51) we show that Δ3,n→p0 as n→∞ (we omit details), which achieves the proof of Theorem 3.4.


### 5.2. Proof of Theorem3.5

Recall (3.8), (3.9), and (3.10) and write(5.71)L̂k,ℓ(J)-L(J)=(T̂L,n-TL,n)+(T̂M,n-TM,n)+(T̂R,n-TR,n).
It is easy to verify that(5.72)T̂R,n-TR,n=(k/n)J(k/n)Xk,nα̂L(1+β)α̂L-1-∫0k/nJ(s)Q(t)dt=Sn1L+Sn2L+Sn3L,
where(5.73)Sn1L:=(kn)J(kn)Xk,n{α̂L(1+β)α̂L-1-α(1+β)α-1},Sn2L:=α(k/n)Q(k/n)J(k/n)(1+β)α-1{Xk,nQ(k/n)-1},Sn3L:=α(k/n)J(k/n)Q(k/n)(1+β)α-1-∫0k/nJ(s)Q(t)dt.
Likewise we have(5.74)T̂R,n-TR,n=(ℓ/n)J(1-ℓ/n)Xn-ℓ,nα̂R(1+β)α̂R-1-∫0ℓ/nJ(s)Q(1-t)dt=Sn1R+Sn2R+Sn3R,
where(5.75)Sn1R:=(k/n)J(k/n)Xk,n{α̂L(1+β)α̂L-1-α(1+β)α-1},Sn2R:=α(ℓ/n)Q(1-ℓ/n)J(1-ℓ/n)(1+β)α-1{Xn-ℓ,nQ(1-ℓ/n)-1},Sn3R:=α(ℓ/n)Q(1-ℓ/n)J(1-ℓ/n)(1+β)α-1-∫0ℓ/nJ(s)Q(1-t)dt.
It is readily checked that Sn1R may be rewritten into(5.76)Sn1L=α̂Lα(k/n)J(k/n)Xk,n((1+β)α̂L-1)((1+β)α-1)(1α̂L-1α).
Since α̂L is a consistent estimator of α, then for all large n(5.77)Sn1L=(1+op(1))α2(k/n)J(k/n)Xk,n((1+β)α-1)2(1α̂L-1α).
In view of Theorems 2.3 and 2.4 of Csörgő and Mason [46], Peng [30], and Necir et al. [17] has been shown that under the second-order condition (2.6) and for all large n(5.78)kα(1α̂L-1α)=-nkBn(kn)+nk∫0k/nBn(s)sds+op(1),k(Xk,nQ(k/n)-1)=-α-1nkBn(kn)+op(1),Xk,nQ(kn)=1+op(1),
where {Bn(s),0≤s≤1,n=1,2,…} is the sequence of Brownian bridges defined in Theorem 3.4. This implies that for all large n(5.79)Sn1L=(1+op(1))α(k1/2/n)J(k/n)Q(k/n)((1+β)α-1)2{-nkBn(kn)+nk∫0k/nBn(s)sds+op(1)},Sn2L=α(k1/2/n)Q(k/n)J(k/n)(1+β)α-1{-nkBn(kn)+op(1)}.
Then, in view of Lemma 5.3, we get for all large n(5.80)n(Sn1L+Sn2L)σn(J)=-αw((1+β)α-1)2{-nkBn(kn)+nk∫0k/nBn(s)sds}-w(1+β)α-1nkBn(kn)+op(1).
By the same arguments (we omit details), we show that for all large n(5.81)n(Sn1R+Sn2R)σn(J)=αwR((1+β)α-1)2{nℓBn(1-ℓn)-nℓ∫1-ℓ/n1Bn(s)1-sds}+wR(1+β)α-1{-nℓBn(1-ℓn)}+op(1),
where wR:=|λ|(q/p)1/αθβ-1/α+1/2w.Similar arguments as those used in the proof of Theorem 1 by  Necir et al. [17] yield that(5.82)nSn3Rσn(J)=nSn3Rσn(J)=o(1)asn→∞.
Then, by (5.80), (5.81), and (5.82) we get(5.83)n(L̂k,ℓ(J)-L(J))σn(J)=-αw((1+β)α-1)2{-nkBn(kn)+nk∫0k/nBn(s)sds}-w(1+β)α-1nkBn(kn)+op(1)-∫k/n1-ℓ/nJ(s)Bn(s)dsσn(J)+αwR((1+β)α-1)2{nℓBn(1-ℓn)-nℓ∫1-ℓ/n1Bn(s)1-sds}+wR(1+β)α-1{-nℓBn(1-ℓn)}+op(1).
The asymptotic variance of n(L̂k,ℓ(J)-L(J))/σn(J) will be computed by(5.84)σ02=limn→∞{w2α2((1+β)α-1)4nk∫0k/nds∫0k/nmin(s,t)-ststdt+w2(1-βα)2((1-β)α-1)4nkkn(1-kn)+∫k/n1-ℓ/ndc(s)∫k/n1-ℓ/n(min(s,t)-st)dc(s)σn(J)+wR2(1-βα)2((1-β)α-1)4nℓℓn(1-ℓn)+wR2α2((1+β)α-1)4nℓ∫1-ℓ/n1ds∫1-ℓ/n1min(s,t)-ststdt-2w2α(1-βα)((1-β)α-1)4nk∫0k/nt-(k/n)ttdt+2wα((1+β)α-1)2nk∫0k/nds∫k/n1-ℓ/ns-stsdc(t)/σn(J)-2wwRα(1-βα)((1+β)α-1)4nknℓ∫0k/ns-(1-ℓ/n)ssds+2wwRα2((1+β)α-1)4nknℓ∫0k/nds∫1-ℓ/n1min(s,t)-ststdt-2w((1-βα)((1+β)α-1)2)nk∫k/n1-ℓ/n(kn-s(kn))dc(t)/σn(J)-2w((1-βα)((1+β)α-1)2)nk∫k/n1-ℓ/n(kn-s(kn))dc(t)/σn(J)+2wwR(1-βα)2((1+β)α-1)4nknℓ(kn-kn(1-ℓn))-2wwRα(1-βα)((1+β)α-1)4∫1-ℓ/n1k/n-(k/n)s1-sds-2wR((1-βα)((1+β)α-1)2)nℓ∫k/n1-ℓ/n(s-s(1-ℓn))dc(t)/σn(J)+2wR(α((1+β)α-1)2)nℓ∫1-ℓ/n1ds∫k/n1-ℓ/n(t-st)(1-s)dc(t)/σn(J)-2wR2α(1-βα)((1+β)α-1)4(nℓ)∫1-ℓ/n1(1-ℓ/n)-(1-ℓ/n)s1-sds}.
After calculation we get(5.85)σ02=w22α2((1+β)α-1)4+w2(1-βα)2((1+β)α-1)4+1+wR2(1-βα)2((1+β)α-1)4+wR22α2((1+β)α-1)4-2w2α(1-βα)((1+β)α-1)4+2wα((1+β)α-1)2w-2w(1-βα)((1+β)α-1)2w+2wRα((1+β)α-1)2wR-2wR(1-βα)((1+β)α-1)2wR-wR22α((1+β)α-1)4=(w2+wR2)[2α2+(βα-1)2+2α(βα-1)((1+β)α-1)4+2(1+β)α-1]+1.
Finally, it is easy to verify that(5.86)w2+wR2=(αβ+1)(2αβ+2-α)2.
This completes the proof of Theorem 3.5.


### 5.3. Proof of the Corollary

Straightforward by combining Theorem3.5 and Lemma 5.4, we omit details.


### 5.4. Proof of Theorem4.1

We will only present details for the proof concerning the first part of Theorem4.1. The proof for the second part is very similar. For convenience we set(5.87)Δm2:=m̂2-m2,Δm3:=m̂3-m3.
Then we have(5.88)υ̂1-υ1=m̂3m̂2-m3m2=m2Δm3-m3Δm2m̂2m2.
Since m̂2 is consistent estimator of m2, then for all large n(5.89)υ̂1-υ1=(1+oP(1))(Δm3m2-m3Δm2m22),
and therefore(5.90)n(υ̂1-υ1)σn(J3)=1m2(C1n+C2n)+op(1),
where(5.91)C1n:=nΔm3σn(J3),C2n:=-m3nΔm2m2σn(J3).
In view of (5.83) we may write that for all large n(5.92)C1n=-αw1(2α-1)2{-nkBn(kn)+nk∫0k/nBn(s)sds}-w12α-1nkBn(kn)+op(1)-∫k/n1-ℓ/nJ(s)Bn(s)dsσn(J)+αwR,1(2α-1)2{nℓBn(1-ℓn)-nℓ∫1-ℓ/n1Bn(s)1-sds}+wR,12α-1{-nℓBn(1-ℓn)}+op(1),and, by Lemma5.4, we infer that σn(J3)/σn(J2)→10/9 as n→∞,(5.93)-10m29m3C2n=-αw1(2α-1)2{-nkBn(kn)+nk∫0k/nBn(s)sds}-w12α-1nkBn(kn)+op(1)-∫k/n1-ℓ/nJ(s)Bn(s)dsσn(J)+αwR,1((1+β)α-1)2{nℓBn(1-ℓn)-nℓ∫1-ℓ/n1Bn(s)1-sds}+wR,12α-1{-nℓBn(1-ℓn)}+op(1),
with
(5.94)w1:=(α+1)(2α+2-α)2(1+(q/p)2/αθ3-2/α),wR,1:=(q/p)1/αθ3/2-1/αw1.
By the same arguments as the proof of Theorem 3.5 we infer that(5.95)n(υ̂1-υ1)(ℓ/n)3/2Xn-ℓ,n→𝒟𝒩(0,V12)asn→∞.This achieves the proof of Theorem4.1.


### 5.5. Proof of Theorem4.2

Theorem4.2 is just an application of Theorem 3.5, we omit details.


## 5.1. Proof of Theorem3.4

Csörgő et al. [36] have constructed a probability space (Ω,A,P) carrying an infinite sequence ξ1,ξ2,… of independent rv's uniformly distributed on (0,1) and a sequence of Brownian bridges {Bn(s),0≤s≤1,n=1,2,…} such that, for the empirical process,(5.28)φn(s):=n1/2{Γn(s)-s},0≤s≤1,
where Γn(·) is the uniform empirical df pertaining to the sample (ξ1,…,ξn); we have for any 0≤ν<1/4 and for all large n(5.29)sup1/n≤s≤1-1/n|φn(s)-Bn(s)|(s(1-s))1/2-ν=Op(n-ν).
For each n≥1, let ξ1,n≤⋯≤ξn,n denote the order statistics corresponding to (ξ1,…,ξn). Note that for each n, the random vector (Q(ξ1,n),…,Q(ξn,n)) has the same distribution as (X1,n,…,Xn,n). Therefore, for 1≤i≤n, we shall use the rv's Q(ξi,n) to represent the rv's Xi,n, and without loss of generality, we shall be working, in all the following proofs, on the probability space above. According to this convention, the term T̂M,n defined in (3.9) may be rewritten into(5.30)T̂M,n=∫ξk,nξn-ℓ,nQ(s)dΨ(Γn(s)),
where Ψ(s):=∫0sJ(t)dt. Integrating by parts yields(5.31)n1/2(T̂M,n-TM,n)σn(J)=Δ1,n+Δ2,n+Δ3,n,
where(5.32)Δ1,n:=-n1/2∫k/n1-ℓ/n{Ψ(Γn(s))-Ψ(s)}dQ(s)σn(J),Δ2,n:=n1/2∫k/nξk,n{Ψ(Γn(s))-Ψ(k/n)}dQ(s)σn(J),Δ3,n:=n1/2∫ξn-ℓ,n1-ℓ/n{Ψ(Γn(s))-Ψ(1-ℓ/n)}dQ(s)σn(J).
Next, we show that(5.33)Δ1,n→𝒟𝒩(0,1)asn→∞,(5.34)Δi,n→p0asn→∞fori=2,3.
Making use of the mean-value theorem, we have for each n(5.35)Ψ(Γn(s))-Ψ(s)=(Γn(s)-s)J(ϑn(s)),
where {ϑn(s)}n≥1 is a sequence of rv's with values in the open interval of endpoints s∈(0,1) and Γn(s). Therefore(5.36)Δ1,n=-∫k/n1-ℓ/nφn(s)J(ϑn(s))dQ(s)σn(J).
This may be rewritten into(5.37)Δ1,n=-∫k/n1-ℓ/nφn(s)J(s)dQ(s)σn(J)-∫k/n1-ℓ/nφn(s)J(s){J(ϑn(s))-J(s)}dQ(s)σn(J)=:Δ1,n*+Δ1,n**.
Note that(5.38)∫k/n1-ℓ/n|φn(s)-Bn(s)|J(s)dQ(s)σn(J)≤supk/n≤s≤1-ℓ/n|φn(s)-Bn(s)|(s(1-s))1/2-ν∫k/n1-ℓ/n(s(1-s))1/2-ν|J(s)|dQ(s)/σn(J),
for 0<ν<1/4, which by (5.29) is equal to(5.39)Op(n-ν)∫k/n1-ℓ/n(s(1-s))1/2-ν|J(s)|dQ(s)σn(J).
Since we have, from Lemmas 5.2 and 5.3,(5.40)(nk)ν∫k/n1-ℓ/n(s(1-s))1/2-ν|J(s)|dQ(s)/σn(J)=O(1),asn→∞,
then the right-hand side of the last inequality is equal to Op(k-ν) which in turn tends to zero as n→∞. This implies that as n→∞(5.41)Δ1,n*=-∫k/n1-ℓ/nBn(s)J(s)dQ(s)σn(J)+op(1).
Next, we show that Δ1,n**=op(1). Indeed, function J is differentiable on (0,1); then by the mean-value theorem, there exists a sequence {ϑn*(s)}n≥1 of rv's with values in the open interval of endpoints s∈(0,1) and ϑn(s) such that for each n we have(5.42)Δ1,n**=∫k/n1-ℓ/nφn(s)J(s){ϑn(s)-s}J′(ϑn*(s))dQ(s)σn(J).
From inequalities (3.9) and (3.10) by Mason and Shorack [55], we infer that, for any 0<ρ<1, there exists 0<Mρ<∞ such that for all large n we have(5.43)|J′(ϑn*(s))|≤Mρ|J(s)|(s(1-s)),
for any 0<s≤1/2. On the other hand, we have for any 0<s<1(5.44)|ϑn(s)-s|≤|Γn(s)-s|.
Therefore(5.45)|Δ1,n**|≤Mρn-1/2∫k/n1/2(|φn(s)|2|J(s)|/s(1-s))dQ(s)σn(J).
This implies, since, for each n≥1, E|φn(s)|2<s(1-s), that(5.46)E|Δ1,n**|≤Mρn-1/2∫k/n1/2|J(s)|dQ(s)σn(J),
which tends to zero as n→∞.Next, we consider the termΔ2,n which may be rewritten into(5.47)Δ2,n=n1/2∫k/nξk,n{Ψ(Γn(s))-Ψ(s)}dQ(s)σn(J)+n1/2∫k/nξk,n{Ψ(s)-Ψ(k/n)}dQ(s)σn(J).
Making use of the mean-value theorem, we get(5.48)Δ2,n=∫k/nξk,nφn(s)J(μn(s))dQ(s)σn(J)+n1/2∫k/nξk,n(s-k/n)J(sn*)dQ(s)σn(J),
where μn(s) is a sequence of rv's with values in the open interval of endpoints s∈(k/n,ξk,n) and Γn(s) and sn* a sequence of rv's with values in the open interval of endpoints s∈(k/n,ξk,n) and k/n. Again we may rewrite Δ2,n into(5.49)Δ2,n=∫k/nξk,nφn(s)(J(μn(s))-J(s))dQ(s)σn(J)+∫k/nξk,nφn(s)J(s)dQ(s)σn(J)+n1/2J(kn)∫k/nξk,n(s-kn)(J(sn*)J(k/n)-1)dQ(s)/σn(J)+n1/2J(kn)∫k/nξk,n(s-kn)dQ(s)/σn(J).
Recall that, as n→∞, both k and ℓ tend to infinity with k/n→0 and ℓ/n→0. This implies that(5.50)nk1/2(ξk,n-kn)→𝒟𝒩(0,1)asn→∞,(5.51)nl1/2(ξn-l,n-1+ln)→𝒟𝒩(0,1)asn→∞,
(see, e.g., Balkema and de Haan [61, page 18]). Next, we use similar arguments to those used in the proof of Theorem 1 by Csörgő et al. [56]. For any 0<c<∞ write(5.52)Δ2,n(1)(c):=∫k/n-c(k1/2/n)k/n+c(k1/2/n)|φn(s)||J(μn(s))-J(s)|dQ(s)σn(J)+∫k/n-c(k1/2/n)k/n+c(k1/2/n)|J(s)||φn(s)|dQ(s)σn(J),Δ2,n(2)(c):=∫k/n-c(k1/2/n)k/n+c(k1/2/n)n1/2|s-k/n|J(sn*)dQ(s)σn(J)+n1/2J(k/n)∫k/n-c(k1/2/n)k/n+c(k1/2/n)(s-k/n)(J(sn*)/J(k/n)-1)dQ(s)σn(J).
Notice that by (5.49)
(5.53)limc→∞lim infn→∞ℙ{|Δ2,n|≤Δ2,n(1)(c)+Δ2,n(2)(c)}≥limc→∞lim infn→∞ℙ{|ξk,n-kn|≤ck1/2n}.
In view of (5.50), this last quantity equals 1. Therefore to establish (5.34) for i=2, it suffices to show that for each 0<c<∞(5.54)Δ2,n(1)(c)→p0,Δ2,n(2)(c)→0asn→∞.
By the mean-value theorem, there exists {μn*(s)}n≥1 a sequence of rv's with values in the open interval of endpoints s and μn(s) such that for each n we have(5.55)J(μn(s))-J(s)=(μn(s)-s)J′(μn*(s)).Since|μn(s)-s|≤|Γn(s)-s|, then by inequality (5.43) we infer that, for any 0<ρ<1, there exists 0<Mρ′<∞ such that for all large n we have(5.56)|J(μn(s))-J(s)|≤Mρ′n-1/2|φn(s)||J(s)|(s(1-s)).
This implies that the first term in Δ2,n(1)(c) is less than or equal to(5.57)Mρ′n-1/2∫k/n-c(k1/2/n)k/n+c(k1/2/n)(φn2(s)|J(s)|(s(1-s)))dQ(s)/σn(J).
Since E(φn2(s))≤s(1-s), then the expectedvalue of the previous quantity is less than or equal to(5.58)Mρ′n-1/2∫(1-c)k/n(1+c)k/n|J(s)|dQ(s)σn(J).
Likewise the expected value of the second term in Δ2,n(1)(c) is less than or equal to(5.59)∫k/n-c(k1/2/n)k/n+c(k1/2/n)(s(1-s))1/2|J(s)|dQ(s)σn(J)≤(k/n+c(k1/2/n))1/2∫k/n-c(k1/2/n)k/n+c(k1/2/n)|J(s)|dQ(s)σn(J).
Fix 1<ɛ<∞. It is readily verified that, for all large n, the quantity (5.58) is less than(5.60)Mρ′n-1/2∫k/ɛnɛ(k/n)|J(s)|dQ(s)σn(J),
and the right-hand side of (5.59) is less than(5.61)2(kn)1/2∫k/ɛnɛ(k/n)|J(s)|dQ(s)/σn(J).
Therefore for all large n(5.62)E(Δ2,n(1)(c))≤(2+Mρ′k-1/2)(kn)1/2∫k/ɛnɛ(k/n)|J(s)|dQ(s)/σn(J).
By routine manipulations, as in the proofs of Lemmas 5.1 and 5.2 (we omit details), we easily show that(5.63)limn→∞(kn)1/2∫k/ɛnɛ(k/n)|J(s)|dQ(s)/σn(J)=wαβ-1(ɛ1/α-β-ɛβ-1/α).
Since k→∞, then for any 0<c<∞(5.64)lim supn→∞E(Δ2,n(1)(c))≤2(ɛ1/α-β-ɛβ-1/α)w,
for any fixed 1<ɛ<∞. This implies that for all 0<c<∞(5.65)limn→∞E(Δ2,n(1)(c))=0.
Therefore, by Markov inequality, we have the first result of (5.34).Now, consider the termΔ2,n(2)(c). Observe that ns*/k is a sequence of rv's with values in the open interval of endpoints nξk,n/k and 1. On the other hand,(5.49) implies that nξk,n/k→p1 as n→∞. Hence ns*/k→p1 as well. Then, it is readily checked that, in view of relation (3.9) by Mason and Shorack [55], we have(5.66)sups∈HnJ(sn*)J(k/n)=Op(1)asn→∞.
Therefore(5.67)Δ2,n(2)(c)=Op(1)J(kn)∫k/n-c(k1/2/n)k/n+c(k1/2/n)n1/2|s-kn|dQ(s)/σn(J).
Observe that(5.68)∫k/n-c(k1/2/n)k/n+c(k1/2/n)n1/2|s-kn|dQ(s)≤c(kn)1/2∫k/n-c(k1/2/n)k/n+c(k1/2/n)dQ(s).
Hence for all large n(5.69)Δ2,n(2)(c)=Op(1)J(kn)∫k/ɛnɛ(k/n)dQ(s)/σn(J),
for any fixed 1<ɛ<∞, we have(5.70)Δ2,n(2)(c)=Op(1)(ɛ1/α-β-ɛβ-1/α)w.
this means that Δ2,n(2)(c)→0 as n→∞.By the same arguments and making use of (5.51) we show that Δ3,n→p0 as n→∞ (we omit details), which achieves the proof of Theorem 3.4.


## 5.2. Proof of Theorem3.5

Recall (3.8), (3.9), and (3.10) and write(5.71)L̂k,ℓ(J)-L(J)=(T̂L,n-TL,n)+(T̂M,n-TM,n)+(T̂R,n-TR,n).
It is easy to verify that(5.72)T̂R,n-TR,n=(k/n)J(k/n)Xk,nα̂L(1+β)α̂L-1-∫0k/nJ(s)Q(t)dt=Sn1L+Sn2L+Sn3L,
where(5.73)Sn1L:=(kn)J(kn)Xk,n{α̂L(1+β)α̂L-1-α(1+β)α-1},Sn2L:=α(k/n)Q(k/n)J(k/n)(1+β)α-1{Xk,nQ(k/n)-1},Sn3L:=α(k/n)J(k/n)Q(k/n)(1+β)α-1-∫0k/nJ(s)Q(t)dt.
Likewise we have(5.74)T̂R,n-TR,n=(ℓ/n)J(1-ℓ/n)Xn-ℓ,nα̂R(1+β)α̂R-1-∫0ℓ/nJ(s)Q(1-t)dt=Sn1R+Sn2R+Sn3R,
where(5.75)Sn1R:=(k/n)J(k/n)Xk,n{α̂L(1+β)α̂L-1-α(1+β)α-1},Sn2R:=α(ℓ/n)Q(1-ℓ/n)J(1-ℓ/n)(1+β)α-1{Xn-ℓ,nQ(1-ℓ/n)-1},Sn3R:=α(ℓ/n)Q(1-ℓ/n)J(1-ℓ/n)(1+β)α-1-∫0ℓ/nJ(s)Q(1-t)dt.
It is readily checked that Sn1R may be rewritten into(5.76)Sn1L=α̂Lα(k/n)J(k/n)Xk,n((1+β)α̂L-1)((1+β)α-1)(1α̂L-1α).
Since α̂L is a consistent estimator of α, then for all large n(5.77)Sn1L=(1+op(1))α2(k/n)J(k/n)Xk,n((1+β)α-1)2(1α̂L-1α).
In view of Theorems 2.3 and 2.4 of Csörgő and Mason [46], Peng [30], and Necir et al. [17] has been shown that under the second-order condition (2.6) and for all large n(5.78)kα(1α̂L-1α)=-nkBn(kn)+nk∫0k/nBn(s)sds+op(1),k(Xk,nQ(k/n)-1)=-α-1nkBn(kn)+op(1),Xk,nQ(kn)=1+op(1),
where {Bn(s),0≤s≤1,n=1,2,…} is the sequence of Brownian bridges defined in Theorem 3.4. This implies that for all large n(5.79)Sn1L=(1+op(1))α(k1/2/n)J(k/n)Q(k/n)((1+β)α-1)2{-nkBn(kn)+nk∫0k/nBn(s)sds+op(1)},Sn2L=α(k1/2/n)Q(k/n)J(k/n)(1+β)α-1{-nkBn(kn)+op(1)}.
Then, in view of Lemma 5.3, we get for all large n(5.80)n(Sn1L+Sn2L)σn(J)=-αw((1+β)α-1)2{-nkBn(kn)+nk∫0k/nBn(s)sds}-w(1+β)α-1nkBn(kn)+op(1).
By the same arguments (we omit details), we show that for all large n(5.81)n(Sn1R+Sn2R)σn(J)=αwR((1+β)α-1)2{nℓBn(1-ℓn)-nℓ∫1-ℓ/n1Bn(s)1-sds}+wR(1+β)α-1{-nℓBn(1-ℓn)}+op(1),
where wR:=|λ|(q/p)1/αθβ-1/α+1/2w.Similar arguments as those used in the proof of Theorem 1 by  Necir et al. [17] yield that(5.82)nSn3Rσn(J)=nSn3Rσn(J)=o(1)asn→∞.
Then, by (5.80), (5.81), and (5.82) we get(5.83)n(L̂k,ℓ(J)-L(J))σn(J)=-αw((1+β)α-1)2{-nkBn(kn)+nk∫0k/nBn(s)sds}-w(1+β)α-1nkBn(kn)+op(1)-∫k/n1-ℓ/nJ(s)Bn(s)dsσn(J)+αwR((1+β)α-1)2{nℓBn(1-ℓn)-nℓ∫1-ℓ/n1Bn(s)1-sds}+wR(1+β)α-1{-nℓBn(1-ℓn)}+op(1).
The asymptotic variance of n(L̂k,ℓ(J)-L(J))/σn(J) will be computed by(5.84)σ02=limn→∞{w2α2((1+β)α-1)4nk∫0k/nds∫0k/nmin(s,t)-ststdt+w2(1-βα)2((1-β)α-1)4nkkn(1-kn)+∫k/n1-ℓ/ndc(s)∫k/n1-ℓ/n(min(s,t)-st)dc(s)σn(J)+wR2(1-βα)2((1-β)α-1)4nℓℓn(1-ℓn)+wR2α2((1+β)α-1)4nℓ∫1-ℓ/n1ds∫1-ℓ/n1min(s,t)-ststdt-2w2α(1-βα)((1-β)α-1)4nk∫0k/nt-(k/n)ttdt+2wα((1+β)α-1)2nk∫0k/nds∫k/n1-ℓ/ns-stsdc(t)/σn(J)-2wwRα(1-βα)((1+β)α-1)4nknℓ∫0k/ns-(1-ℓ/n)ssds+2wwRα2((1+β)α-1)4nknℓ∫0k/nds∫1-ℓ/n1min(s,t)-ststdt-2w((1-βα)((1+β)α-1)2)nk∫k/n1-ℓ/n(kn-s(kn))dc(t)/σn(J)-2w((1-βα)((1+β)α-1)2)nk∫k/n1-ℓ/n(kn-s(kn))dc(t)/σn(J)+2wwR(1-βα)2((1+β)α-1)4nknℓ(kn-kn(1-ℓn))-2wwRα(1-βα)((1+β)α-1)4∫1-ℓ/n1k/n-(k/n)s1-sds-2wR((1-βα)((1+β)α-1)2)nℓ∫k/n1-ℓ/n(s-s(1-ℓn))dc(t)/σn(J)+2wR(α((1+β)α-1)2)nℓ∫1-ℓ/n1ds∫k/n1-ℓ/n(t-st)(1-s)dc(t)/σn(J)-2wR2α(1-βα)((1+β)α-1)4(nℓ)∫1-ℓ/n1(1-ℓ/n)-(1-ℓ/n)s1-sds}.
After calculation we get(5.85)σ02=w22α2((1+β)α-1)4+w2(1-βα)2((1+β)α-1)4+1+wR2(1-βα)2((1+β)α-1)4+wR22α2((1+β)α-1)4-2w2α(1-βα)((1+β)α-1)4+2wα((1+β)α-1)2w-2w(1-βα)((1+β)α-1)2w+2wRα((1+β)α-1)2wR-2wR(1-βα)((1+β)α-1)2wR-wR22α((1+β)α-1)4=(w2+wR2)[2α2+(βα-1)2+2α(βα-1)((1+β)α-1)4+2(1+β)α-1]+1.
Finally, it is easy to verify that(5.86)w2+wR2=(αβ+1)(2αβ+2-α)2.
This completes the proof of Theorem 3.5.


## 5.3. Proof of the Corollary

Straightforward by combining Theorem3.5 and Lemma 5.4, we omit details.


## 5.4. Proof of Theorem4.1

We will only present details for the proof concerning the first part of Theorem4.1. The proof for the second part is very similar. For convenience we set(5.87)Δm2:=m̂2-m2,Δm3:=m̂3-m3.
Then we have(5.88)υ̂1-υ1=m̂3m̂2-m3m2=m2Δm3-m3Δm2m̂2m2.
Since m̂2 is consistent estimator of m2, then for all large n(5.89)υ̂1-υ1=(1+oP(1))(Δm3m2-m3Δm2m22),
and therefore(5.90)n(υ̂1-υ1)σn(J3)=1m2(C1n+C2n)+op(1),
where(5.91)C1n:=nΔm3σn(J3),C2n:=-m3nΔm2m2σn(J3).
In view of (5.83) we may write that for all large n(5.92)C1n=-αw1(2α-1)2{-nkBn(kn)+nk∫0k/nBn(s)sds}-w12α-1nkBn(kn)+op(1)-∫k/n1-ℓ/nJ(s)Bn(s)dsσn(J)+αwR,1(2α-1)2{nℓBn(1-ℓn)-nℓ∫1-ℓ/n1Bn(s)1-sds}+wR,12α-1{-nℓBn(1-ℓn)}+op(1),and, by Lemma5.4, we infer that σn(J3)/σn(J2)→10/9 as n→∞,(5.93)-10m29m3C2n=-αw1(2α-1)2{-nkBn(kn)+nk∫0k/nBn(s)sds}-w12α-1nkBn(kn)+op(1)-∫k/n1-ℓ/nJ(s)Bn(s)dsσn(J)+αwR,1((1+β)α-1)2{nℓBn(1-ℓn)-nℓ∫1-ℓ/n1Bn(s)1-sds}+wR,12α-1{-nℓBn(1-ℓn)}+op(1),
with
(5.94)w1:=(α+1)(2α+2-α)2(1+(q/p)2/αθ3-2/α),wR,1:=(q/p)1/αθ3/2-1/αw1.
By the same arguments as the proof of Theorem 3.5 we infer that(5.95)n(υ̂1-υ1)(ℓ/n)3/2Xn-ℓ,n→𝒟𝒩(0,V12)asn→∞.This achieves the proof of Theorem4.1.


## 5.5. Proof of Theorem4.2

Theorem4.2 is just an application of Theorem 3.5, we omit details.



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*Source: 707146-2010-03-23.xml*