Estimating 𝐿-Functionals for Heavy-Tailed Distributions and Application

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.

1. Introduction

1.1. 𝐿-Functionals

Let 𝑋 be a real random variable (rv) with continuous distribution function (df) 𝐹. The corresponding 𝐿-functionals are defined by

𝐿(𝐽)∶=¹₀𝐽(𝑠)𝑄(𝑠)𝑑𝑠, (1.1) 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

𝑚_𝑖∶=¹₀𝐽_𝑖(𝑠)𝑄(𝑠)𝑑𝑠,𝑖=1,2,3,4, (1.2) where

𝐽_𝑖(𝑠)∶=𝑠(1−𝑠)𝜙_𝑖(𝑠),0<𝑠<1, (1.3) with 𝜙_𝑖 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.

Indeed 𝐿-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

Π(𝑋)∶=^∞₀𝑔(1−𝐹(𝑥))𝑑𝑥, (1.4) 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

Π(𝑋)=¹₀𝑔^′(1−𝑠)𝑄(𝑠)𝑑𝑠, (1.5) 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

𝐻(𝑋)∶=⁰_−∞(𝑔(1−𝐹(𝑥))−1)𝑑𝑥+^∞₀𝑔(1−𝐹(𝑥))𝑑𝑥. (1.6) Likewise, by integration by parts it is shown that

𝐻(𝑋)=¹₀𝑔^′(1−𝑠)𝑄(𝑠)𝑑𝑠. (1.7) Wang [8] and Jones and Zitikis [9] have defined the risk two-sided deviation by

Δ_𝑟(𝑋)∶=¹₀𝐽_𝑟(𝑠)𝑄(𝑠)𝑑𝑠,0<𝑟<1, (1.8)

with

𝐽_𝑟𝑟(𝑠)∶=2𝑠^1−𝑟−(1−𝑠)^1−r𝑠^1−𝑟(1−𝑠)^1−𝑟,0<𝑠<1. (1.9) As we see, Π(𝑋),𝐻(𝑋), and Δ_𝑟(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 quantity 𝐿(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,…,𝑋_𝑛) be a sample of size 𝑛≥1 from an rv 𝑋 with df 𝐹, then the sample estimator of 𝐿(𝐽) is

𝐿_𝑛(𝐽)∶=¹₀𝐽(𝑠)𝑄_𝑛(𝑠)𝑑𝑠, (1.10) where 𝑄_𝑛(𝑠)∶=inf{𝑥∈ℝ∶𝐹_𝑛(𝑥)≥𝑠}, 0<𝑠≤1, is the empirical quantile function that corresponds to the empirical df 𝐹_𝑛(𝑥)∶=𝑛⁻¹∑^𝑛_𝑖=1𝕀{𝑋_𝑖≤𝑥} for 𝑥∈ℝ, pertaining to the sample (𝑋₁,…,𝑋_𝑛) with 𝕀(⋅) denoting the indicator function. It is clear that 𝐿_𝑛(𝐽) may be rewritten into

𝐿_𝑛(𝐽)=^𝑛_𝑖=1𝑎_𝑖,𝑛𝑋_𝑖,𝑛, (1.11) where 𝑎_𝑖,𝑛∶=∫^𝑖/𝑛_{(𝑖−1)/𝑛}𝐽(𝑠)𝑑𝑠, 𝑖=1,…,𝑛, and 𝑋_1,𝑛≤⋯≤𝑋_𝑛,𝑛 denote the order statistics based upon the sample (𝑋₁,…,𝑋_𝑛). The first general theorem on the asymptotic normality of 𝐿_𝑛(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

√𝑛(𝐿_𝑛(𝐽)−𝐿(𝐽))−−−^𝒟−→𝒩(0,𝜎²(𝐽)),as𝑛→∞, (1.12) provided that

𝜎²(𝐽)∶=¹₀¹₀(min(𝑠,𝑡)−𝑠𝑡)𝐽(𝑠)𝐽(𝑡)𝑑𝑄(𝑠)𝑄(𝑡)<∞. (1.13) 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 𝐿_𝑛(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 𝑚_𝑖 when 1/2<𝛼<2/3 and for the sample risk two-sided deviation Δ_𝑟(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. Section 2 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 of 𝛼-Stable Laws

A df is said to belong to the domain of attraction of a stable law with stability index 0<𝛼≤2, notation: 𝐹∈𝐷(𝛼), if there exist two real sequences 𝐴_𝑛>0 and 𝐶_𝑛 such that

𝐴_𝑛⁻¹(^𝑛_𝑖=1𝑋_𝑖−𝐶_𝑛)−−−^𝒟−→𝑆_𝛼(𝜎,𝛿,𝜇),as𝑛→∞, (2.1) where 𝑆_𝛼(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]. 𝑆_𝛼(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 tail 1−𝐺 is regularly varying at infinity with index −𝛼. That is lim_𝑡→∞(1−𝐺(𝑥𝑡))(1−𝐺(𝑡))=𝑥^−𝛼,forany𝑥>0. (2.2)(ii)There exists 0≤𝑝≤1 such that lim_𝑥→∞1−𝐹(𝑥)1−𝐺(𝑥)=𝑝,lim_𝑥→∞𝐹(−𝑥)1−𝐺(𝑥)=1−𝑝=∶𝑞. (2.3)

Let, for 0<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 lim_𝑠↓0𝐾(1−𝑥𝑠)𝐾(1−𝑠)=𝑥^−1/𝛼,forany𝑥>0. (2.4)(ii^′)There exists 0≤𝑝≤1 such that lim_𝑠↓0𝑄₁(1−𝑠)𝐾(1−𝑠)=𝑝^1/𝛼,lim_𝑠↓0𝑄₂(1−𝑠)𝐾(1−𝑠)=(1−𝑝)^1/𝛼=∶𝑞^1/𝛼. (2.5)

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

lim_𝑠↓0(𝐴(𝑠))⁻¹(𝐾(1−𝑥𝑠)𝐾(1−𝑠)−𝑥^−1/𝛼)=𝑥^−1/𝛼𝑥^𝜚−1𝜚,forany𝑥>0, (2.6) where 𝜚≤0 is the second-order parameter. If 𝜚=0, interpret (𝑥^𝜚−1)/𝜚 as log𝑥. The second-order condition for heavy-tailed distributions has been introduced by de Haan and Stadtmüller [37].

3. Estimating 𝐿-Functionals When 𝐹∈𝐷(𝛼)

3.1. Extreme Quantile Estimation

The right and left extreme quantiles of small enough level 𝑡 of df 𝐹, respectively, are two reals 𝑥_𝑅 and 𝑥_𝐿 defined by 1−𝐹(𝑥_𝑅)=𝑡 and 𝐹(𝑥_𝐿)=𝑡, that is, 𝑥_𝑅=𝑄(1−𝑡) and 𝑥_𝐿=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=𝑘_𝑛 and ℓ=ℓ_𝑛 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 𝑥_𝑅 and 𝑥_𝐿 are defined, respectively, by

𝑥_𝐿=𝑄_𝐿𝑘(𝑡)∶=(𝑛)^1/𝛼𝐿𝑋_𝑘,𝑛𝑡^{−1/𝛼𝐿},as𝑥𝑡↓0,_𝑅=𝑄_𝑅ℓ(1−𝑡)∶=(𝑛)^1/𝛼𝑅𝑋_{𝑛−ℓ,𝑛}𝑡^{−1/𝛼𝑅},as𝑡↓0, (3.1) where

𝛼_𝐿=𝛼_𝐿1(𝑘)∶=(𝑘^𝑘_𝑖=1log⁺(−𝑋_𝑖,𝑛)−log⁺(−𝑋_𝑘,𝑛))⁻¹,𝛼_𝑅=𝛼_𝑅1(ℓ)∶=(ℓ^ℓ_𝑖=1log⁺(𝑋_{𝑛−𝑖+1,𝑛})−log⁺(𝑋_{𝑛−ℓ,𝑛}))⁻¹ (3.2) are two forms of Hill's estimator [42] for the stability index 𝛼 which could also be estimated, using the order statistics 𝑍_1,𝑛≤⋯≤𝑍_𝑛,𝑛 associated to a sample (𝑍₁,…,𝑍_𝑛) from 𝑍, as follows:

1𝛼=𝛼(𝑚)∶=(𝑚^𝑚_𝑖=1log⁺(𝑍_{𝑛−𝑖+1,𝑛})−log⁺(𝑍_{𝑛−𝑚,𝑛}))⁻¹, (3.3) with log⁺𝑢∶=max(0,log𝑢) and 𝑚=𝑚_𝑛 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 Fractions 𝑘 and ℓ

Extreme value-based estimators rely essentially on the numbers 𝑘 and ℓ of lower- and upper-order statistics used in estimate computation. Estimators 𝛼_𝐿 and 𝛼_𝑅 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 index 𝛼; 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 𝛼_𝐿, 𝛼_𝑅, and 𝛼 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, 𝛼_𝑅 is much more suitable than 𝛼_𝐿 when estimating the stability index of a distribution which is skewed to the right (𝛿>0)whereas Figure 2 shows that 𝛼_𝐿 is much more reliable than 𝛼_𝑅 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 𝛼, 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, 𝛼 (solid line), 𝛼𝑅 (dashed line), and 𝛼𝐿 (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, 𝛼 (solid line), 𝛼𝑅 (dashed line), and 𝛼𝐿 (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, 𝛼 (solid line), 𝛼𝑅 (dashed line) and 𝛼𝐿 (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 sequences 𝑘 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 distribution 𝑆1.2(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 on 𝐽

For application needs, the following regularity assumptions on function 𝐽 are required:

(H1) 𝐽 is differentiable on (0,1),(H2)𝜆∶=lim_𝑠↓0𝐽(1−𝑠)𝐽(𝑠)<∞,(H3)both 𝐽(𝑠) and 𝐽(1−𝑠) are regularly varying at zero with common index 𝛽∈ℝ.(H4)there exists a function 𝑎(⋅) not changing sign near zero such that lim_𝑡↓0𝐽(𝑥𝑡)/𝐽(𝑡)−𝑥^𝛽𝑎(𝑡)=𝑥^𝛽𝑥^𝜔−1𝜔,forany𝑥>0, (3.4) where 𝜔≤0 is the second-order parameter.

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 (𝐽_𝑖)_𝑖=2,4 with (𝛽,𝜆)=(1,±1) (see Section 4.1) and by function 𝐽_𝑟 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(𝐽_𝑖)=∞,𝑖=1,…,4. Likewise, recall (1.9); whenever 1/(𝑟+1/2)<𝛼<1/𝑟, the two-sided deviation Δ_𝑟(𝑋) exists while 𝜎²(𝐽_𝑟)=∞.

3.4. Defining the Estimator and Main Results

We now have all the necessary tools to introduce our estimator of 𝐿(𝐽), given in (1.1), when 𝐹∈𝐷(𝛼) with 0<𝛼<2. Let 𝑘=𝑘_𝑛 and ℓ=ℓ_𝑛 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 𝛼_𝐿 and 𝛼_𝑅 are consistent estimators of 𝛼 (see, Mason [43]), then we have for all large 𝑛

1ℙ(1+𝛽−𝛼_𝐿1>0)=ℙ(1+𝛽−𝛼_𝑅>0)=1+𝑜(1). (3.5) Observe now that 𝐿(𝐽) defined in (1.1) may be split in three integrals as follows:

𝐿(𝐽)=₀^𝑘/𝑛𝐽(𝑡)𝑄(𝑡)𝑑𝑡+^1−ℓ/𝑛_𝑘/𝑛𝐽(𝑡)𝑄(𝑡)𝑑𝑡+¹_1−ℓ/𝑛𝐽(𝑡)𝑄(𝑡)𝑑𝑡=∶𝑇_𝐿,𝑛+𝑇_𝑀,𝑛+𝑇_𝑅,𝑛. (3.6) Substituting 𝑄_𝐿(𝑡) and 𝑄_𝑅(1−𝑡) for 𝑄(𝑡) and 𝑄(1−𝑡) in 𝑇_𝐿,𝑛 and 𝑇_𝑅,𝑛, respectively and making use of assumption (H3) and (3.5) yield that for all large 𝑛

𝐽(𝑡)𝑄₀^𝑘/𝑛_𝐿𝑘(𝑡)𝑑𝑡=(𝑛)^1/𝛼𝐿𝑋_𝑘,𝑛0^𝑘/𝑛𝑡^{−1/𝛼𝐿}𝐽(𝑡)𝑑𝑡=(1+𝑜(1))(𝑘/𝑛)𝐽(𝑘/𝑛)1+𝛽−1/𝛼_𝐿𝑋_𝑘,𝑛,_0ℓ/𝑛𝐽(1−𝑡)𝑄_𝑅ℓ(1−𝑡)𝑑𝑡=(𝑛)^1/𝛼𝑅𝑋_𝑛−ℓ,_𝑛0^ℓ/𝑛𝑡^{−1/𝛼𝑅}𝐽(1−𝑡)𝑑𝑡=(1+𝑜(1))(ℓ/𝑛)𝐽(1−ℓ/𝑛)1+𝛽−1/𝛼_𝑅𝑋_{𝑛−ℓ,𝑛}. (3.7) Hence we may estimate 𝑇_𝐿,𝑛 and 𝑇_𝑅,𝑛 by

𝑇_𝐿,𝑛∶=(𝑘/𝑛)𝐽(𝑘/𝑛)1+𝛽−1/𝛼_𝐿𝑋_𝑘,𝑛,𝑇_𝑅,𝑛∶=(ℓ/𝑛)𝐽(1−ℓ/𝑛)1+𝛽−1/𝛼_𝑅𝑋_{𝑛−ℓ,𝑛}, (3.8) respectively. As an estimator of 𝑇_𝑀,𝑛 we take the sample one that is

𝑇_𝑀,𝑛∶=^1−ℓ/𝑛_𝑘/𝑛𝐽(𝑡)𝑄_𝑛(𝑡)𝑑𝑡=^𝑛−ℓ_𝑖=𝑘+1𝑎_𝑖,𝑛𝑋_𝑖,𝑛, (3.9) with the same constants 𝑎_𝑖,𝑛 as those in (1.11). Thus, the final form of our estimator is

𝐿_𝑘,ℓ(𝐽)=(𝑘/𝑛)𝐽(𝑘/𝑛)1+𝛽−1/𝛼_𝐿𝑋_𝑘,𝑛+^𝑛−ℓ_𝑖=𝑘+1𝑎_𝑖,𝑛𝑋_𝑖,𝑛+(ℓ/𝑛)𝐽(1−ℓ/𝑛)1+𝛽−1/𝛼_𝑅𝑋_{𝑛−ℓ,𝑛}. (3.10) A universal estimator of 𝐿(𝐽) may be summarized by

𝐿^∗_𝑛(𝐽)=𝐿_𝑘,ℓ(𝐽)𝕀(𝜎²(𝐽)=∞)+𝐿_𝑛(𝐽)𝕀(𝜎²(𝐽)<∞), (3.11) where 𝐿_𝑛(𝐽) is as in (1.11). More precisely

𝐿^∗_𝑛(𝐽)=𝐿_𝑘,ℓ(𝐽)𝕀(𝐴(𝛼,𝛽))+𝐿_𝑛(𝐽)𝕀(𝐴(𝛼,𝛽)), (3.12) 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 case ℓ=𝑘 and 𝐽=1the asymptotic normality of the trimmed mean 𝑇_𝑀,𝑛 has been established in Theorem 1 of Csörgő et al. [56]. The following theorem gives the asymptotic normality of 𝑇_𝑀,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,

𝜎²(𝑥,𝑦;𝐽)∶=_𝑥^1−𝑦_𝑥^1−𝑦(min(𝑠,𝑡)−𝑠𝑡)𝐽(𝑠)𝐽(𝑡)𝑑𝑄(𝑠)𝑄(𝑡)<∞, (3.13) and let 𝜎²_𝑛(𝐽)∶=𝜎²(𝑘/𝑛,ℓ/𝑛;𝐽).

Theorem 3.4. Assume that 𝐹∈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 {𝐵_𝑛(𝑠),0≤𝑠≤1,𝑛=1,2,…} such that one has for all large 𝑛√𝑛(𝑇_𝑀,𝑛−𝑇_𝑀,𝑛)𝜎_𝑛∫(𝐽)=−^{1−ℓ/𝑛𝑘/𝑛}𝐽(𝑠)𝐵_𝑛(𝑠)𝑑𝑠𝜎_𝑛(𝐽)+𝑜_𝑝(1), (3.14) and therefore √𝑛(𝑇_𝑀,𝑛−𝑇_𝑀,𝑛)𝜎_𝑛(𝐽)^𝒟−−−−→𝒩(0,1)as𝑛→∞. (3.15)

The asymptotic normality of our estimator is established in the following theorem.

Theorem 3.5. Assume that 𝐹∈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 √𝑛(𝐿_𝑘,ℓ(𝐽)−𝐿(𝐽))𝜎_𝑛(𝐽)^𝒟−−−−→𝒩(0,𝜎²₀),as𝑛→∞, (3.16) where 𝜎²₀=𝜎²₀(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. (3.17)

The following corollary is more practical than Theorem 3.5 as it directly provides confidence bounds for 𝐿(𝐽).

Corollary 3.6. Under the assumptions of Theorem 3.5 one has √𝑛(𝐿_𝑘,ℓ(𝐽)−𝐿(𝐽))(ℓ/𝑛)^1/2𝐽(1−ℓ/𝑛)𝑋^𝒟_{𝑛−ℓ,𝑛}−−−−→𝒩(0,𝑉²),as𝑛→∞, (3.18) where 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, (3.19) 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 for 𝐿(𝐽)

The form of the asymptotic variance 𝑉² 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 𝑍_{𝑛−𝑚,𝑛}, de Haan and Pereira [57] proposed the following consistent estimator for 𝑝∶

𝑝_𝑛=𝑝_𝑛1(𝑚)∶=𝑚^𝑛_𝑖=1𝕀{𝑋_𝑖>𝑍_{𝑛−𝑚,𝑛}}, (3.20) where 𝑚=𝑚_𝑛 is a sequence of integers satisfying 1<𝑚<𝑛, 𝑚→∞, and 𝑚/𝑛→0, as 𝑛→∞ (the same as that used in (3.3)).

Let 𝐽 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,…,𝑥_𝑛) of a sample (𝑋₁,…,𝑋_𝑛) 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 numbers 𝑘^∗, ℓ^∗, and 𝑚^∗ of lower- and upper-order statistics used in (3.2) and (3.3).

Step 2. Determine 𝑋_𝑘^∗_,𝑛,𝑋_𝑛−ℓ^∗_,𝑛,𝐽(𝑘^∗/𝑛), 𝐽(1−ℓ^∗/𝑛), and 𝜃^∗∶=ℓ^∗/𝑘^∗.

Step 3. Compute, using (3.2), 𝛼^∗_𝐿∶=𝛼_𝐿(𝑘^∗) and 𝛼^∗_𝑅∶=𝛼_𝑅(ℓ^∗). Then deduce, by (3.10), the estimate 𝐿_𝑘^∗_,ℓ^∗(𝐽).

Step 4. Use (3.3) and (3.20) to compute 𝛼^∗∶=𝛼(𝑚^∗) and 𝑝^∗_𝑛∶=𝑝_𝑛(𝑚^∗). Then deduce, by (3.19), the asymptotic standard deviation 𝑉^∗∶=𝑉²(𝛼^∗,𝛽,𝜆,𝜃^∗,𝑝^∗_𝑛). (3.21) Finally, the lower and upper (1−𝜍)-confidence bounds for 𝐿(𝐽), respectively, will be 𝐿_𝑘^∗_,ℓ^∗(𝐽)−𝑧_𝜍/2√ℓ^∗𝑉^∗𝑋_𝑛−ℓ^∗_,𝑛𝐽(1−ℓ^∗/𝑛)𝑛,𝐿_𝑘^∗_,ℓ^∗(𝐽)+𝑧_𝜍/2√ℓ^∗𝑉^∗𝑋_𝑛−ℓ^∗_,𝑛𝐽(1−ℓ^∗/𝑛)𝑛, (3.22) where 𝑧_𝜍/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 When 𝐹∈𝐷(𝛼)

When the distribution mean 𝐸𝑋 exists, the skewness and kurtosis coefficients are, respectively, defined by 𝐿₁∶=𝜇₃/𝜇₂^3/2 and 𝐿₂∶=𝜇₄/𝜇²₂ with 𝜇_𝑘∶=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

𝜐₁𝑚∶=³𝑚₂,𝜐₂𝑚∶=⁴𝑚₂, (4.1) where 𝑚_𝑖,𝑖=2,3,4, are the trimmed L-moments defined in Section 1. The corresponding weight functions of (1.3) are defined as follows:

𝐽₂𝐽(𝑠)∶=6𝑠(1−𝑠)(2𝑠−1),₃(𝑠)∶=203𝑠(1−𝑠)(5𝑠²𝐽−5𝑠+1),₄(𝑠)∶=152𝑠(1−𝑠)(14𝑠³−21𝑠²+9𝑠−1). (4.2) 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,

𝜐₁𝑚∶=³𝑚₂,𝜐₂𝑚∶=⁴𝑚₂, (4.3)

where

𝑚_𝑖=−6(𝑘/𝑛)²2−1/𝛼_𝐿𝑋_𝑘,𝑛+^𝑛−ℓ_𝑗=𝑘+1𝑎^(𝑖)_𝑗,𝑛𝑋_𝑗,𝑛+6(ℓ/𝑛)²2−1/𝛼_𝑅𝑋_{𝑛−ℓ,𝑛},for𝑖=2,20(𝑘/𝑛)²3(2−1/𝛼_𝐿)𝑋_𝑘,𝑛+^𝑛−ℓ_𝑗=𝑘+1𝑎^(𝑖)_𝑗,n𝑋_𝑗,𝑛+20(ℓ/𝑛)²3(2−1/𝛼_𝑅)𝑋_{𝑛−ℓ,𝑛},for𝑖=3,−15(𝑘/𝑛)²2(2−1/𝛼_𝐿)𝑋_𝑘,𝑛+^𝑛−ℓ_𝑗=𝑘+1𝑎^(𝑖)_𝑗,𝑛𝑋_𝑗,𝑛+15(ℓ/𝑛)²2(2−1/𝛼_𝑅)𝑋_{𝑛−ℓ,𝑛},for𝑖=4, (4.4) with 𝑎^(𝑖)_𝑗,𝑛∶=∫^𝑗/𝑛_{(𝑗−1)/𝑛}𝐽_𝑖(𝑠)𝑑𝑠,𝑖=2,3,4, and 𝑗=1,…,𝑛.

Theorem 4.1. Assume that 𝐹∈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→∞,√𝑛(𝜐₁−𝜐₁)(ℓ/𝑛)^3/2𝑋^𝒟_{𝑛−ℓ,𝑛}−−−−→𝒩(0,𝑉²₁√),𝑛(𝜐₂−𝜐₂)(ℓ/𝑛)^3/2𝑋^𝒟_{𝑛−ℓ,𝑛}−−−−→𝒩(0,𝑉²₂), (4.5) where 𝑉²₁∶=36𝑚²₂(1−9𝑚₃10𝑚₂)²𝜎^∗2,𝑉²₂∶=2254𝑚²₂(1−4𝑚₄5𝑚₂)²𝜎^∗2, (4.6) with 𝜎^∗2∶=(1+(𝑞/𝑝)^−2/𝛼𝜃^2/𝛼−3)×(2𝛼²+(𝛼−1)²+2𝛼(𝛼−1)2(2𝛼−1)⁴+12𝛼−1)+1. (4.7)

4.2. Risk Two-Sided Deviation When 𝐹∈𝐷(𝛼)

Recall that the risk two-sided deviation is defined by

Δ_𝑟(𝑋)∶=¹₀𝐽_𝑟(𝑠)𝑄(𝑠)𝑑𝑠,0<𝑟<1, (4.8) where

𝐽_𝑟𝑟(𝑠)∶=2𝑠^1−𝑟−(1−𝑠)^1−𝑟𝑠^1−𝑟(1−𝑠)^1−𝑟,0<𝑠<1. (4.9) An asymptotically normal estimator for Δ_𝑟(𝑋), when 1/(𝑟+1/2)<𝛼<1/𝑟, is

Δ_𝑟(𝑋)=−𝑟(𝑘/𝑛)^𝑟2𝑟−4/𝛼_𝐿𝑋_𝑘,𝑛+^𝑛−ℓ_𝑗=𝑘+1𝑎^(𝑟)_𝑗,𝑛𝑋_𝑗,𝑛+𝑟(ℓ/𝑛)^𝑟2𝑟−4/𝛼_𝑅𝑋_{𝑛−ℓ,𝑛}, (4.10) where

𝑎^(𝑟)_𝑗,𝑛=12𝑖[(1−𝑛)^−𝑟−(1−𝑖−1𝑛)^−𝑟𝑖−(𝑛)^−𝑟+(𝑖−1𝑛)^−𝑟], (4.11)

𝑗=1,…,𝑛.

Theorem 4.2. Assume that 𝐹∈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→∞,√𝑛(Δ_𝑟(𝑋)−Δ_𝑟(𝑋))(ℓ/𝑛)^𝑟−1/2𝑋^𝒟_{𝑛−ℓ,𝑛}−−−−→𝒩(0,𝑉²_𝑟), (4.12) where 𝑉²_𝑟𝑟∶=²4𝑞(1+(𝑝)^−2/𝛼𝜃^{2/𝛼−2𝑟+1})×(2𝛼²+(𝑟𝛼−𝛼−1)²+2𝛼(𝑟𝛼−𝛼−1)2(𝑟𝛼−1)⁴+1𝑟𝛼−1)+1. (4.13)

5. Proofs

First we begin by the following three technical lemmas.

Lemma 5.1. Let 𝑓₁ 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 lim_𝑥↓0∫_𝑥^1/2𝑓₁(𝑠)𝑑𝑓₂(𝑠)𝑓₁(𝑥)𝑓₂=𝜏(𝑥).𝜅−𝜏 (5.1)

Lemma 5.2. Under the assumptions of Theorem 3.5, one has lim_𝑛→∞∫^1−ℓ/𝑛_𝑘/𝑛(𝑠(1−𝑠))^1/2−𝜈𝐽(𝑠)𝑑𝑄(𝑠)(𝑘/𝑛)^1/2−𝜈=𝐽(𝑘/𝑛)𝑄(𝑘/𝑛)1+𝜆𝜃^{1/2−𝜈+𝛽−1/𝛼}(𝑞/𝑝)^1/𝛼,𝛼(1/2−𝜈+𝛽)−1 (5.2) for any 0<𝜈<1/4.

Lemma 5.3. For any 0<𝑥<1/2 and 0<𝑦<1/2 one has 𝜎²(𝑥,𝑦;𝐽)=𝑥𝑐²(𝑥)+𝑦𝑐²(1−𝑦)+_𝑥^1−𝑦𝑐²(𝑡)𝑑𝑡−(𝑥𝑐(𝑥)+𝑦𝑐(1−𝑦)+_𝑥^1−𝑦𝑐(𝑡)𝑑𝑡)², (5.3) where 𝑐(𝑠)∶=∫^𝑠_1/2𝐽(𝑡)𝑑𝑄(𝑡), 0<𝑠<1/2.

Lemma 5.4. Under the assumptions of Theorem 3.5, one has lim_𝑛→∞(𝑘/𝑛)𝐽²(𝑘/𝑛)𝑄²(𝑘/𝑛)𝜎²_𝑛(𝐽)=𝑤²,lim_𝑛→∞(ℓ/𝑛)𝐽²(1−ℓ/𝑛)𝑄²(1−ℓ/𝑛)𝜎²_𝑛(𝐽)=𝜆²(𝑞𝑝)^2/𝛼𝜃^{2𝛽−2/𝛼+1}𝑤², (5.4) where 𝑤²∶=(𝛼𝛽+1)(2𝛼𝛽+2−𝛼)2(1+𝜆²(𝑞/𝑝)^2/𝛼𝜃^{2𝛽−2/𝛼+1}). (5.5)

Proof of Lemma 5.1. Let 𝑓^′₁ denote the derivative of 𝑓₁. Applying integration by parts, we get, for any 0<𝑥<1/2,_𝑥^1/2𝑓₁(𝑠)𝑑𝑓₂(𝑠)=𝑓₁(12)𝑓₂(12)−𝑓₁(𝑥)𝑓₂(𝑥)−_𝑥^1/2𝑓^′₁(𝑠)𝑓₂(𝑠)𝑑𝑠. (5.6) Since the product 𝑓₁𝑓₂ is regularly varying at zero with index −𝜏+𝜅<0, then 𝑓₁(𝑥)𝑓₂(𝑥)→0 as 𝑥↓0. Therefore lim_𝑥↓0∫_𝑥^1/2𝑓₁(𝑠)𝑑𝑓₂(𝑠)𝑓₁(𝑥)𝑓₂(𝑥)=−1−lim_𝑥↓0∫_𝑥^1/2𝑓^′₁(𝑠)𝑓₂(𝑠)𝑑𝑠𝑓₁(𝑥)𝑓₂.(𝑥) (5.7) By using Karamata's representation (see, e.g., Seneta [59]), it is easy to show that 𝑥𝑓^′₁(𝑥)=𝜅(1+𝑜(1))𝑓₁(𝑥),as𝑥↓0. (5.8) Hence lim_𝑥↓0∫_𝑥^1/2𝑓₁(𝑠)𝑑𝑓₂(𝑠)𝑓₁(𝑥)𝑓₂(𝑥)=−1−𝜅lim_𝑥↓0∫_𝑥^1/2𝑓^′₁(𝑠)𝑓₂(𝑠)𝑑𝑠𝑥𝑓^′₁(𝑥)𝑓₂.(𝑥) (5.9) It is clear that (5.8) implies that 𝑓^′₁ is regularly varying at zero with index 𝜅−1; therefore 𝑓^′₁𝑓₂ is regularly varying with index −𝜏+𝜅−1<0. Then, Theorem 1.2.1 by de Haan [60, page 15] yields lim_𝑥↓0−∫_𝑥^1/2𝑓^′₁(𝑠)𝑓₂(𝑠)𝑑𝑠𝑥𝑓^′₁(𝑥)𝑓₂=1(𝑥).𝜅−𝜏 (5.10) This completes the proof of Lemma 5.1.

Proof of Lemma 5.2. We have 𝐼_𝑛∶=^1−ℓ/𝑛_𝑘/𝑛(𝑠(1−𝑠))^1/2−𝜈𝐽(𝑠)𝑑𝑄(𝑠)=^1/2_𝑘/𝑛(𝑠(1−𝑠))^1/2−𝜈−𝐽(𝑠)𝑑𝑄(𝑠)_{1/21−ℓ/𝑛}(𝑠(1−𝑠))^1/2−𝜈𝐽(1−𝑠)𝑑𝑄(1−𝑠)=∶𝐼_1𝑛−𝐼_2𝑛. (5.11) 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 lim_𝑛→∞𝐼_1𝑛(𝑘/𝑛)^1/2−𝜈=𝐽(𝑘/𝑛)𝑄(𝑘/𝑛)1/𝛼.1/2−𝜈+𝛽−1/𝛼 (5.12) 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 lim_𝑛→∞𝐼_2𝑛(ℓ/𝑛)^1/2−𝜈=𝐽(1−ℓ/𝑛)𝑄(1−ℓ/𝑛)1/𝛼.1/2−𝜈+𝛽−1/𝛼 (5.13) Note that statement (ii^′) of Section 2 implies that lim_𝑠↓0𝑞𝑄(1−𝑠)/𝑄(𝑠)=−(𝑝)^1/𝛼. (5.14) The last two relations, together with assumption (H2) and the regular variation of 𝑄(1−𝑠), imply that lim_𝑛→∞𝐼_2𝑛(𝑘/𝑛)^1/2−𝜈𝐽(𝑘/𝑛)𝑄(𝑘/𝑛)=−(𝑞/𝑝)^1/𝛼𝜆𝜃^{1/2−𝜈+𝛽−1/𝛼}/𝛼.1/2−𝜈+𝛽−1/𝛼 (5.15) This achieves the proof of Lemma 5.2.

Proof of Lemma 5.3. We will use similar techniques to those used by Csörgő et al. [36, Proposition A.2]. For any 0<𝑠<1/2, we set 𝑊_𝑥,𝑦(𝑡)∶=𝑐(1−𝑦)for1−𝑦≤𝑡<1,𝑐(𝑡)for𝑥<𝑡<1−𝑦,𝑐(𝑥)for0<𝑡≤𝑥. (5.16) Then 𝜎²(𝑥,𝑦;𝐽) may be rewritten into 𝜎²(𝑥,𝑦;𝐽)=¹₀𝑊²_𝑥,𝑦(𝑠)𝑑𝑠−(¹₀𝑊_𝑥,𝑦(𝑠)𝑑𝑠)², (5.17) and the result of Lemma 5.3 follows immediately.

Proof of Lemma 5.4. From Lemma 5.3 we may write 𝜎²_𝑛(𝐽)(𝑘/𝑛)𝐽²(𝑘/𝑛)𝑄²(𝑘/𝑛)=𝑇_𝑛1+𝑇_𝑛2+𝑇_𝑛3+𝑇_𝑛4, (5.18) where 𝑇_𝑛1∶=(𝑘/𝑛)𝑐²(𝑘/𝑛)(𝑘/𝑛)𝐽²(𝑘/𝑛)𝑄²(𝑘/𝑛),𝑇_𝑛2∶=(ℓ/𝑛)𝑐²(1−ℓ/𝑛)(𝑘/𝑛)𝐽²(𝑘/𝑛)𝑄²,𝑇(𝑘/𝑛)_𝑛3∫∶=^{1−ℓ/𝑛𝑘/𝑛}𝑐²(𝑡)𝑑𝑡(𝑘/𝑛)𝐽²(𝑘/𝑛)𝑄²,𝑇(𝑘/𝑛)_𝑛4∶=((𝑘/𝑛)𝑐(𝑘/𝑛)+(ℓ/𝑛)𝑐(1−ℓ/𝑛)+∫^1−ℓ/𝑛_𝑘/𝑛𝑐(𝑡)𝑑𝑡)²(𝑘/𝑛)𝐽²(𝑘/𝑛)𝑄²(k./𝑛) (5.19) By the same arguments as in the proof of Lemma 5.2, we infer that lim_𝑛→∞𝑐(𝑘/𝑛)=1𝐽(𝑘/𝑛)𝑄(𝑘/𝑛),𝛼𝛽−1lim_𝑛→∞𝑐(1−ℓ/𝑛)=𝐽(𝑘/𝑛)𝑄(𝑘/𝑛)𝜆(𝑞/𝑝)^1/𝛼𝜃^{𝛽−1/𝛼}.𝛼𝛽−1 (5.20) Therefore lim_𝑛→∞𝑇_𝑛1=1(𝛼𝛽−1)²,lim_𝑛→∞𝑇_𝑛2=𝜆²(𝑞/𝑝)^2/𝛼𝜃^{2𝛽−2/𝛼+1}(𝛼𝛽−1)². (5.21) Next, we consider the third term 𝑇_𝑛3 which may be rewritten into 𝑇_𝑛3=∫^1/2_𝑘/𝑛𝑐²(𝑡)𝑑𝑡(𝑘/𝑛)𝐽²(𝑘/𝑛)𝑄²+∫(𝑘/𝑛)^{1−ℓ/𝑛1/2}𝑐²(𝑡)𝑑𝑡(𝑘/𝑛)𝐽²(𝑘/𝑛)𝑄².(𝑘/𝑛) (5.22) Observe that ∫^1/2_𝑘/𝑛𝑐²(𝑡)𝑑𝑡(𝑘/𝑛)𝐽²(𝑘/𝑛)𝑄²(𝑘/𝑛)=(𝑐(𝑘/𝑛))𝐽(𝑘/𝑛)𝑄(𝑘/𝑛)²∫^1/2_𝑘/𝑛𝑐²(𝑡)𝑑𝑡(𝑘/𝑛)𝑐².(𝑘/𝑛) (5.23) 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 lim_𝑛→∞∫^1/2_𝑘/𝑛𝑐²(𝑡)𝑑𝑡(𝑘/𝑛)𝑐²=𝛼(𝑘/𝑛).2−2𝛼𝛽−𝛼 (5.24) Hence lim_𝑛→∞∫^1/2_𝑘/𝑛𝑐²(𝑡)𝑑𝑡(𝑘/𝑛)𝐽²(𝑘/𝑛)𝑄²=𝛼(𝑘/𝑛)(𝛼𝛽−1)².(2−2𝛼𝛽−𝛼) (5.25) By similar arguments we show that lim_𝑛→∞∫^1−ℓ/𝑛_1/2𝑐²(𝑡)𝑑𝑡(𝑘/𝑛)𝐽²(𝑘/𝑛)𝑄²=(𝑘/𝑛)𝛼(𝑞/𝑝)^2/𝛼𝜆²𝜃^{2𝛽−2/𝛼+1}(𝛼𝛽−1)².(2−2𝛼𝛽−𝛼) (5.26) Therefore lim_𝑛→∞𝑇_𝑛3=1+𝛼(𝑞/𝑝)^2/𝛼𝜆²𝜃^{2𝛽−2/𝛼+1}(𝛼𝛽−1)².(2−2𝛼𝛽−𝛼) (5.27) By analogous techniques we show that 𝑇_𝑛4→0 as 𝑛→∞; we omit details. Summing up the three limits of 𝑇_𝑛𝑖,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 Theorem 3.4

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 {𝐵_𝑛(𝑠),0≤𝑠≤1,𝑛=1,2,…} such that, for the empirical process,

𝜑_𝑛(𝑠)∶=𝑛^1/2{Γ_𝑛(𝑠)−𝑠},0≤𝑠≤1, (5.28) where Γ_𝑛(⋅) is the uniform empirical df pertaining to the sample (𝜉₁,…,𝜉_𝑛); we have for any 0≤𝜈<1/4 and for all large 𝑛

sup_{1/𝑛≤𝑠≤1−1/𝑛}|𝜑_𝑛(𝑠)−𝐵_𝑛(𝑠)|(𝑠(1−𝑠))^1/2−𝜈=𝑂_𝑝(𝑛^−𝜈). (5.29) For each 𝑛≥1, let 𝜉_1,𝑛≤⋯≤𝜉_𝑛,𝑛 denote the order statistics corresponding to (𝜉₁,…,𝜉_𝑛). Note that for each 𝑛, the random vector (𝑄(𝜉_1,𝑛),…,𝑄(𝜉_𝑛,𝑛)) has the same distribution as (𝑋_1,𝑛,…,𝑋_𝑛,𝑛). Therefore, for 1≤𝑖≤𝑛, we shall use the rv's 𝑄(𝜉_𝑖,𝑛) to represent the rv's 𝑋_𝑖,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 𝑇_𝑀,𝑛 defined in (3.9) may be rewritten into

𝑇_𝑀,𝑛=^{𝜉𝑛−ℓ,𝑛}_{𝜉𝑘,𝑛}𝑄(𝑠)𝑑Ψ(Γ_𝑛(𝑠)), (5.30) where Ψ(𝑠)∶=∫^𝑠₀𝐽(𝑡)𝑑𝑡. Integrating by parts yields

𝑛^1/2(𝑇_𝑀,𝑛−𝑇_𝑀,𝑛)𝜎_𝑛(𝐽)=Δ_1,𝑛+Δ_2,𝑛+Δ_3,𝑛, (5.31) where

Δ_1,𝑛𝑛∶=−^1/2∫^1−ℓ/𝑛_𝑘/𝑛{Ψ(Γ_𝑛(𝑠))−Ψ(𝑠)}𝑑𝑄(𝑠)𝜎_𝑛,Δ(𝐽)_2,𝑛𝑛∶=^1/2∫^{𝜉𝑘,𝑛}_𝑘/𝑛{Ψ(Γ_𝑛(𝑠))−Ψ(𝑘/𝑛)}𝑑𝑄(𝑠)𝜎_𝑛,Δ(𝐽)_3,𝑛𝑛∶=^1/2∫_𝜉^1−ℓ/𝑛_{𝑛−ℓ,𝑛}{Ψ(Γ_𝑛(𝑠))−Ψ(1−ℓ/𝑛)}𝑑𝑄(𝑠)𝜎_𝑛.(𝐽) (5.32) Next, we show that

Δ^𝒟_1,𝑛−−−−→𝒩(0,1)as𝑛→∞,Δ^𝑝_𝑖,𝑛−−−→0as𝑛→∞for𝑖=2,3. (5.33)(5.34) (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

Δ⁽¹⁾_2,𝑛∫(𝑐)∶=^{𝑘/𝑛+𝑐(𝑘1/2/𝑛)𝑘/𝑛−𝑐(𝑘1/2/𝑛)}|𝜑_𝑛(𝑠)||𝐽(𝜇_𝑛(𝑠))−𝐽(𝑠)|𝑑𝑄(𝑠)𝜎_𝑛+∫(𝐽)_{𝑘/𝑛+𝑐(𝑘1/2/𝑛)𝑘/𝑛−𝑐(𝑘1/2/𝑛)}|𝐽(𝑠)||𝜑_𝑛(𝑠)|𝑑𝑄(𝑠)𝜎_𝑛,Δ(𝐽)_(2)2,𝑛∫(𝑐)∶=^{𝑘/𝑛+𝑐(𝑘1/2/𝑛)𝑘/𝑛−𝑐(𝑘1/2/𝑛)}𝑛^1/2|𝑠−𝑘/𝑛|𝐽(𝑠^∗_𝑛)𝑑𝑄(𝑠)𝜎_𝑛+𝑛(𝐽)_1/2𝐽(𝑘/𝑛)∫^{𝑘/𝑛+𝑐(𝑘1/2/𝑛)}_{𝑘/𝑛−𝑐(𝑘1/2/𝑛)}(𝑠−𝑘/𝑛)(𝐽(𝑠^∗_𝑛)/𝐽(𝑘/𝑛)−1)𝑑𝑄(𝑠)𝜎_𝑛.(𝐽) (5.52) Notice that by (5.49) lim_𝑐→∞liminf_𝑛→∞ℙ{|Δ_2,𝑛|≤Δ⁽¹⁾_2,𝑛(𝑐)+Δ⁽²⁾_2,𝑛(𝑐)}≥lim_𝑐→∞liminf_𝑛→∞ℙ{|𝜉_𝑘,𝑛−𝑘𝑛𝑘|≤𝑐^1/2𝑛}. (5.53) 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<∞

Δ⁽¹⁾_2,𝑛(𝑐)−−−^𝑝→0,Δ⁽²⁾_2,𝑛(𝑐)→0as𝑛→∞. (5.54) By the mean-value theorem, there exists {𝜇^∗_𝑛(𝑠)}_𝑛≥1 a sequence of rv's with values in the open interval of endpoints 𝑠 and 𝜇_𝑛(𝑠) such that for each 𝑛 we have

𝐽(𝜇_𝑛(𝑠))−𝐽(𝑠)=(𝜇_𝑛(𝑠)−𝑠)𝐽^′(𝜇^∗_𝑛(𝑠)). (5.55)

Since |𝜇_𝑛(𝑠)−𝑠|≤|Γ_𝑛(𝑠)−𝑠|, then by inequality (5.43) we infer that, for any 0<𝜌<1, there exists 0<𝑀^′_𝜌<∞ such that for all large 𝑛 we have

|𝐽(𝜇_𝑛𝑀(𝑠))−𝐽(𝑠)|≤^′𝜌𝑛^−1/2|𝜑_𝑛(𝑠)||𝐽(𝑠)|.(𝑠(1−𝑠)) (5.56) This implies that the first term in Δ⁽¹⁾_2,𝑛(𝑐) is less than or equal to

𝑀^′_𝜌𝑛^{−1/2𝑘/𝑛+𝑐(𝑘1/2/𝑛)}_{𝑘/𝑛−𝑐(𝑘1/2/𝑛)}(𝜑²_𝑛(𝑠)|𝐽(𝑠)|(𝑠(1−𝑠)))𝑑𝑄(𝑠)/𝜎_𝑛(𝐽). (5.57) Since 𝐸(𝜑²_𝑛(𝑠))≤𝑠(1−𝑠), then the expectedvalue of the previous quantity is less than or equal to

𝑀^′_𝜌𝑛^−1/2∫^{(1+𝑐)𝑘/𝑛}_{(1−𝑐)𝑘/𝑛}|𝐽(𝑠)|𝑑𝑄(𝑠)𝜎_𝑛.(𝐽) (5.58) Likewise the expected value of the second term in Δ⁽¹⁾_2,n(𝑐) is less than or equal to

∫^{𝑘/𝑛+𝑐(𝑘1/2/𝑛)}_{𝑘/𝑛−𝑐(𝑘1/2/𝑛)}(𝑠(1−𝑠))^1/2|𝐽(𝑠)|𝑑𝑄(𝑠)𝜎_𝑛≤(𝐽)(𝑘/𝑛+𝑐(𝑘^1/2/𝑛))^1/2∫^{𝑘/𝑛+𝑐(𝑘1/2/𝑛)}_{𝑘/𝑛−𝑐(𝑘1/2/𝑛)}|𝐽(𝑠)|𝑑𝑄(𝑠)𝜎_𝑛.(𝐽) (5.59) Fix 1<𝜀<∞. It is readily verified that, for all large 𝑛, the quantity (5.58) is less than

𝑀^′_𝜌𝑛^−1/2∫^{𝜀(𝑘/𝑛)}_{𝑘/𝜀𝑛}|𝐽(𝑠)|𝑑𝑄(𝑠)𝜎_𝑛,(𝐽) (5.60) and the right-hand side of (5.59) is less than

𝑘2(𝑛)^{1/2𝜀(𝑘/𝑛)}_{𝑘/𝜀𝑛}|𝐽(𝑠)|𝑑𝑄(𝑠)/𝜎_𝑛(𝐽). (5.61) Therefore for all large 𝑛

𝐸(Δ⁽¹⁾_2,𝑛(𝑐))≤(2+𝑀^′_𝜌𝑘^−1/2𝑘)(𝑛)^{1/2𝜀(𝑘/𝑛)}_{𝑘/𝜀𝑛}|𝐽(𝑠)|𝑑𝑄(𝑠)/𝜎_𝑛(𝐽). (5.62) By routine manipulations, as in the proofs of Lemmas 5.1 and 5.2 (we omit details), we easily show that

lim_𝑛→∞(𝑘𝑛)^{1/2𝜀(𝑘/𝑛)}_{𝑘/𝜀𝑛}|𝐽(𝑠)|𝑑𝑄(𝑠)/𝜎_𝑛𝑤(𝐽)=𝛼𝛽−1(𝜀^{1/𝛼−𝛽}−𝜀^{𝛽−1/𝛼}). (5.63) Since 𝑘→∞, then for any 0<𝑐<∞

limsup_𝑛→∞𝐸(Δ⁽¹⁾_2,𝑛(𝑐))≤2(𝜀^{1/𝛼−𝛽}−𝜀^{𝛽−1/𝛼})𝑤, (5.64) for any fixed 1<𝜀<∞. This implies that for all 0<𝑐<∞

lim_𝑛→∞𝐸(Δ⁽¹⁾_2,𝑛(𝑐))=0. (5.65) Therefore, by Markov inequality, we have the first result of (5.34).

Now, consider the term Δ⁽²⁾_2,𝑛(𝑐). Observe that 𝑛𝑠^∗/𝑘 is a sequence of rv's with values in the open interval of endpoints 𝑛𝜉_𝑘,𝑛/𝑘 and 1. On the other hand, (5.49) implies that 𝑛𝜉_𝑘,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

sup_{𝑠∈𝐻𝑛}𝐽(𝑠^∗_𝑛)𝐽(𝑘/𝑛)=𝑂_𝑝(1)as𝑛→∞. (5.66) Therefore

Δ⁽²⁾_2,𝑛(𝑐)=𝑂_𝑝𝑘(1)𝐽(𝑛)^{𝑘/𝑛+𝑐(𝑘1/2/𝑛)}_{𝑘/𝑛−𝑐(𝑘1/2/𝑛)}𝑛^1/2𝑘|𝑠−𝑛|𝑑𝑄(𝑠)/𝜎_𝑛(𝐽). (5.67) Observe that

^{𝑘/𝑛+𝑐(𝑘1/2/𝑛)}_{𝑘/𝑛−𝑐(𝑘1/2/𝑛)}𝑛^1/2𝑘|𝑠−𝑛𝑘|𝑑𝑄(𝑠)≤𝑐(𝑛)^{1/2𝑘/𝑛+𝑐(𝑘1/2/𝑛)}_{𝑘/𝑛−𝑐(𝑘1/2/𝑛)}𝑑𝑄(𝑠). (5.68) Hence for all large 𝑛

Δ⁽²⁾_2,𝑛(𝑐)=𝑂_𝑝𝑘(1)𝐽(𝑛)^{𝜀(𝑘/𝑛)}_{𝑘/𝜀𝑛}𝑑𝑄(𝑠)/𝜎_𝑛(𝐽), (5.69) for any fixed 1<𝜀<∞, we have

Δ⁽²⁾_2,𝑛(𝑐)=𝑂_𝑝(1)(𝜀^{1/𝛼−𝛽}−𝜀^{𝛽−1/𝛼})𝑤. (5.70) this means that Δ⁽²⁾_2,𝑛(𝑐)→0 as 𝑛→∞.By the same arguments and making use of (5.51) we show that Δ^𝑝_3,𝑛−→0 as 𝑛→∞ (we omit details), which achieves the proof of Theorem 3.4.

5.2. Proof of Theorem 3.5

Recall (3.8), (3.9), and (3.10) and write

𝐿_𝑘,ℓ(𝐽)−𝐿(𝐽)=(𝑇_𝐿,𝑛−𝑇_𝐿,𝑛)+(𝑇_𝑀,𝑛−𝑇_𝑀,𝑛)+(𝑇_𝑅,𝑛−𝑇_𝑅,𝑛). (5.71) It is easy to verify that

𝑇_𝑅,𝑛−𝑇_𝑅,𝑛=(𝑘/𝑛)𝐽(𝑘/𝑛)𝑋_𝑘,_𝑛𝛼_𝐿(1+𝛽)𝛼_𝐿−−1₀^𝑘/𝑛𝐽(𝑠)𝑄(𝑡)𝑑𝑡=𝑆^𝐿_𝑛1+𝑆^𝐿_𝑛2+𝑆^𝐿_𝑛3, (5.72) where

𝑆^𝐿_𝑛1𝑘∶=(𝑛𝑘)𝐽(𝑛)𝑋_𝑘,_𝑛{𝛼_𝐿(1+𝛽)𝛼_𝐿−𝛼−1𝑆(1+𝛽)𝛼−1},_𝐿𝑛2∶=𝛼(𝑘/𝑛)𝑄(𝑘/𝑛)𝐽(𝑘/𝑛){𝑋(1+𝛽)𝛼−1^𝑘,_𝑛𝑆𝑄(𝑘/𝑛)−1},_𝐿𝑛3∶=𝛼(𝑘/𝑛)𝐽(k/𝑛)𝑄(𝑘/𝑛)−(1+𝛽)𝛼−1₀^𝑘/𝑛𝐽(𝑠)𝑄(𝑡)𝑑𝑡. (5.73) Likewise we have

𝑇_𝑅,𝑛−𝑇_𝑅,𝑛=(ℓ/𝑛)𝐽(1−ℓ/𝑛)𝑋_𝑛−ℓ,_𝑛𝛼_𝑅(1+𝛽)𝛼_𝑅−−1₀^ℓ/𝑛𝐽(𝑠)𝑄(1−𝑡)𝑑𝑡=𝑆^𝑅_𝑛1+𝑆^𝑅_𝑛2+𝑆^𝑅_𝑛3, (5.74) where

𝑆^𝑅_𝑛1∶=(𝑘/𝑛)𝐽(𝑘/𝑛)𝑋_𝑘,_𝑛{𝛼_𝐿(1+𝛽)𝛼_𝐿−𝛼−1𝑆(1+𝛽)𝛼−1},_𝑅𝑛2∶=𝛼(ℓ/𝑛)𝑄(1−ℓ/𝑛)𝐽(1−ℓ/𝑛){𝑋(1+𝛽)𝛼−1^𝑛−ℓ,_𝑛𝑆𝑄(1−ℓ/𝑛)−1},_𝑅𝑛3∶=𝛼(ℓ/𝑛)𝑄(1−ℓ/𝑛)𝐽(1−ℓ/𝑛)−(1+𝛽)𝛼−1₀^ℓ/𝑛𝐽(𝑠)𝑄(1−𝑡)𝑑𝑡. (5.75) It is readily checked that 𝑆^𝑅_𝑛1 may be rewritten into

𝑆^𝐿_𝑛1=𝛼_𝐿𝛼(𝑘/𝑛)𝐽(𝑘/𝑛)𝑋_𝑘,_𝑛((1+𝛽)𝛼_𝐿(1−1)((1+𝛽)𝛼−1)𝛼_𝐿−1𝛼). (5.76) Since 𝛼_𝐿 is a consistent estimator of 𝛼, then for all large 𝑛

𝑆^𝐿_𝑛1=(1+𝑜_𝑝𝛼(1))²(𝑘/𝑛)𝐽(𝑘/𝑛)𝑋_𝑘,_𝑛((1+𝛽)𝛼−1)²(1𝛼_𝐿−1𝛼). (5.77) 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

√𝑘𝛼(1𝛼_𝐿−1𝛼)=−𝑛𝑘𝐵_𝑛(𝑘𝑛)+𝑛𝑘₀^𝑘/𝑛𝐵_𝑛(𝑠)𝑠𝑑𝑠+𝑜_𝑝√(1),𝑋𝑘(^𝑘,_𝑛𝑄(𝑘/𝑛)−1)=−𝛼⁻¹𝑛𝑘𝐵_𝑛(𝑘𝑛)+𝑜_𝑝𝑋(1),_𝑘,_𝑛𝑄(𝑘𝑛)=1+𝑜_𝑝(1), (5.78) where {𝐵_𝑛(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

𝑆^𝐿_𝑛1=(1+𝑜_𝑝(1))𝛼(𝑘^1/2/𝑛)𝐽(𝑘/𝑛)𝑄(𝑘/𝑛)((1+𝛽)𝛼−1)²{−𝑛𝑘𝐵_𝑛(𝑘𝑛)+𝑛𝑘₀^𝑘/𝑛𝐵_𝑛(𝑠)𝑠𝑑𝑠+𝑜_𝑝𝑆(1)},_𝐿𝑛2=𝛼(𝑘^1/2/𝑛)𝑄(𝑘/𝑛)𝐽(𝑘/𝑛)(1+𝛽)𝛼−1{−𝑛𝑘𝐵_𝑛(𝑘𝑛)+𝑜_𝑝(1)}. (5.79) Then, in view of Lemma 5.3, we get for all large 𝑛

√𝑛(𝑆^𝐿_𝑛1+𝑆^𝐿_𝑛2)𝜎_𝑛(𝐽)=−𝛼𝑤((1+𝛽)𝛼−1)²{−𝑛𝑘𝐵_𝑛(𝑘𝑛)+𝑛𝑘₀^𝑘/𝑛𝐵_𝑛(𝑠)𝑠−𝑤𝑑𝑠}(1+𝛽)𝛼−1𝑛𝑘𝐵_𝑛(𝑘𝑛)+𝑜_𝑝(1). (5.80) By the same arguments (we omit details), we show that for all large 𝑛

√𝑛(𝑆^𝑅_𝑛1+𝑆^𝑅_𝑛2)𝜎_𝑛=(𝐽)𝛼𝑤_𝑅((1+𝛽)𝛼−1)²{𝑛ℓ𝐵_𝑛ℓ(1−𝑛)−𝑛ℓ¹_1−ℓ/𝑛𝐵_𝑛(𝑠)+𝑤1−𝑠𝑑𝑠}_𝑅(1+𝛽)𝛼−1{−𝑛ℓ𝐵_𝑛ℓ(1−𝑛)}+𝑜_𝑝(1), (5.81) where 𝑤_𝑅∶=|𝜆|(𝑞/𝑝)^1/𝛼𝜃^{𝛽−1/𝛼+1/2}𝑤.Similar arguments as those used in the proof of Theorem 1 by Necir et al. [17] yield that

√𝑛𝑆^𝑅_𝑛3𝜎_𝑛=√(𝐽)𝑛𝑆^𝑅_𝑛3𝜎_𝑛(𝐽)=𝑜(1)as𝑛→∞. (5.82) Then, by (5.80), (5.81), and (5.82) we get

√𝑛(𝐿_𝑘,ℓ(𝐽)−𝐿(𝐽))𝜎_𝑛(𝐽)=−𝛼𝑤((1+𝛽)𝛼−1)²{−𝑛𝑘𝐵_𝑛(𝑘𝑛)+𝑛𝑘₀^𝑘/𝑛𝐵_𝑛(𝑠)𝑠−𝑤𝑑𝑠}(1+𝛽)𝛼−1𝑛𝑘𝐵_𝑛(𝑘𝑛)+𝑜_𝑝∫(1)−^{1−ℓ/𝑛𝑘/𝑛}𝐽(𝑠)𝐵_𝑛(𝑠)𝑑𝑠𝜎_𝑛+(𝐽)𝛼𝑤_𝑅((1+𝛽)𝛼−1)²{𝑛ℓ𝐵_𝑛ℓ(1−𝑛)−𝑛ℓ¹_1−ℓ/𝑛𝐵_𝑛(𝑠)+𝑤1−𝑠𝑑𝑠}_𝑅(1+𝛽)𝛼−1{−𝑛ℓ𝐵_𝑛ℓ(1−𝑛)}+𝑜_𝑝(1). (5.83) The asymptotic variance of √𝑛(𝐿_𝑘,ℓ(𝐽)−𝐿(𝐽))/𝜎_𝑛(𝐽) will be computed by

𝜎²₀=lim_𝑛→∞{𝑤²𝛼²((1+𝛽)𝛼−1)⁴𝑛𝑘₀^𝑘/𝑛𝑑𝑠₀^𝑘/𝑛min(𝑠,𝑡)−𝑠𝑡𝑠𝑡𝑑𝑡+𝑤²(1−𝛽𝛼)²((1−𝛽)𝛼−1)⁴𝑛𝑘𝑘𝑛𝑘(1−𝑛∫)+^{1−ℓ/𝑛𝑘/𝑛}𝑑𝑐(𝑠)∫^1−ℓ/𝑛_𝑘/𝑛(min(𝑠,𝑡)−𝑠𝑡)𝑑𝑐(𝑠)𝜎_𝑛(𝐽)+𝑤²_𝑅(1−𝛽𝛼)²((1−𝛽)𝛼−1)⁴𝑛ℓℓ𝑛ℓ(1−𝑛)+𝑤²_𝑅𝛼²((1+𝛽)𝛼−1)⁴𝑛ℓ¹_1−ℓ/𝑛𝑑𝑠¹_1−ℓ/𝑛min(𝑠,𝑡)−𝑠𝑡𝑠𝑡𝑑𝑡−2𝑤²𝛼(1−𝛽𝛼)((1−𝛽)𝛼−1)⁴𝑛𝑘₀^𝑘/𝑛𝑡−(𝑘/𝑛)𝑡𝑡𝑑𝑡𝛼+2𝑤((1+𝛽)𝛼−1)²𝑛𝑘₀^𝑘/𝑛𝑑𝑠^1−ℓ/𝑛_𝑘/𝑛𝑠−𝑠𝑡𝑠𝑑𝑐(𝑡)/𝜎_𝑛(𝐽)−2𝑤𝑤_𝑅𝛼(1−𝛽𝛼)((1+𝛽)𝛼−1)⁴𝑛𝑘𝑛ℓ₀^𝑘/𝑛𝑠−(1−ℓ/𝑛)𝑠𝑠𝑑𝑠+2𝑤𝑤_𝑅𝛼²((1+𝛽)𝛼−1)⁴𝑛𝑘𝑛ℓ₀^𝑘/𝑛𝑑𝑠¹_1−ℓ/𝑛min(𝑠,𝑡)−𝑠𝑡𝑠𝑡𝑑𝑡−2𝑤((1−𝛽𝛼)((1+𝛽)𝛼−1)²)𝑛𝑘^1−ℓ/𝑛_𝑘/𝑛(𝑘𝑛𝑘−𝑠(𝑛))𝑑𝑐(𝑡)/𝜎_𝑛(𝐽)−2𝑤((1−𝛽𝛼)((1+𝛽)𝛼−1)²)𝑛𝑘^1−ℓ/𝑛_𝑘/𝑛(𝑘𝑛𝑘−𝑠(𝑛))𝑑𝑐(𝑡)/𝜎_𝑛(𝐽)+2𝑤𝑤_𝑅(1−𝛽𝛼)²((1+𝛽)𝛼−1)⁴𝑛𝑘𝑛ℓ(𝑘𝑛−𝑘𝑛ℓ(1−𝑛))−2𝑤𝑤_𝑅𝛼(1−𝛽𝛼)((1+𝛽)𝛼−1)^{411−ℓ/𝑛}𝑘/𝑛−(𝑘/𝑛)𝑠1−𝑠𝑑𝑠−2𝑤_𝑅((1−𝛽𝛼)((1+𝛽)𝛼−1)²)𝑛ℓ^1−ℓ/𝑛_𝑘/𝑛ℓ(𝑠−𝑠(1−𝑛))𝑑𝑐(𝑡)/𝜎_𝑛(𝐽)+2𝑤_𝑅(𝛼((1+𝛽)𝛼−1)²)𝑛ℓ¹_1−ℓ/𝑛𝑑𝑠^1−ℓ/𝑛_𝑘/𝑛(𝑡−𝑠𝑡)(1−𝑠)𝑑𝑐(𝑡)/𝜎_𝑛(𝐽)−2𝑤²_𝑅𝛼(1−𝛽𝛼)((1+𝛽)𝛼−1)⁴(𝑛ℓ)¹_1−ℓ/𝑛(1−ℓ/𝑛)−(1−ℓ/𝑛)𝑠1−𝑠𝑑𝑠}. (5.84)(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) After calculation we get

𝜎²₀=𝑤²2𝛼²((1+𝛽)𝛼−1)⁴+𝑤²(1−𝛽𝛼)²((1+𝛽)𝛼−1)⁴+1+𝑤²_𝑅(1−𝛽𝛼)²((1+𝛽)𝛼−1)⁴+𝑤²_𝑅2𝛼²((1+𝛽)𝛼−1)⁴−2𝑤²𝛼(1−𝛽𝛼)((1+𝛽)𝛼−1)⁴𝛼+2𝑤((1+𝛽)𝛼−1)²𝑤−2𝑤(1−𝛽𝛼)((1+𝛽)𝛼−1)²𝑤+2𝑤_𝑅𝛼((1+𝛽)𝛼−1)²𝑤_𝑅−2𝑤_𝑅(1−𝛽𝛼)((1+𝛽)𝛼−1)²𝑤_𝑅−𝑤²_𝑅2𝛼((1+𝛽)𝛼−1)⁴=(𝑤²+𝑤²_𝑅)[2𝛼²+(𝛽𝛼−1)²+2𝛼(𝛽𝛼−1)((1+𝛽)𝛼−1)⁴+2(1+𝛽)𝛼−1]+1. (5.85) Finally, it is easy to verify that

𝑤²+𝑤²_𝑅=(𝛼𝛽+1)(2𝛼𝛽+2−𝛼)2. (5.86) This completes the proof of Theorem 3.5.

5.3. Proof of the Corollary

Straightforward by combining Theorem 3.5 and Lemma 5.4, we omit details.

5.4. Proof of Theorem 4.1

We will only present details for the proof concerning the first part of Theorem 4.1. The proof for the second part is very similar. For convenience we set

Δ𝑚₂∶=𝑚₂−𝑚₂,Δ𝑚₃∶=𝑚₃−𝑚₃. (5.87) Then we have

𝜐₁−𝜐₁=𝑚₃𝑚₂−𝑚₃𝑚₂=𝑚₂Δ𝑚₃−𝑚₃Δ𝑚₂𝑚₂𝑚₂. (5.88) Since 𝑚₂ is consistent estimator of 𝑚₂, then for all large 𝑛

𝜐₁−𝜐₁=(1+𝑜_𝑃(1))(Δ𝑚₃𝑚₂−𝑚₃Δ𝑚₂𝑚²₂), (5.89) and therefore

√𝑛(𝜐₁−𝜐₁)𝜎_𝑛(𝐽₃)=1𝑚₂(𝐶_1𝑛+𝐶_2𝑛)+𝑜_𝑝(1), (5.90) where

𝐶_1𝑛√∶=𝑛Δ𝑚₃𝜎_𝑛(𝐽₃),𝐶_2𝑛𝑚∶=−³√𝑛Δ𝑚₂𝑚₂𝜎_𝑛(𝐽₃). (5.91) In view of (5.83) we may write that for all large 𝑛

𝐶_1𝑛=−𝛼𝑤₁(2𝛼−1)²{−𝑛𝑘𝐵_𝑛(𝑘𝑛)+𝑛𝑘₀^𝑘/𝑛𝐵_𝑛(𝑠)𝑠−𝑤𝑑𝑠}₁2𝛼−1𝑛𝑘𝐵_𝑛(𝑘𝑛)+𝑜_𝑝∫(1)−^{1−ℓ/𝑛𝑘/𝑛}𝐽(𝑠)𝐵_𝑛(𝑠)𝑑𝑠𝜎_𝑛+(𝐽)𝛼𝑤_𝑅,1(2𝛼−1)²{𝑛ℓ𝐵_𝑛ℓ(1−𝑛)−𝑛ℓ¹_1−ℓ/𝑛𝐵_𝑛(𝑠)+𝑤1−𝑠𝑑𝑠}_𝑅,12𝛼−1{−𝑛ℓ𝐵_𝑛ℓ(1−𝑛)}+𝑜_𝑝(1), (5.92)

and, by Lemma 5.4, we infer that 𝜎_𝑛(𝐽₃)/𝜎_𝑛(𝐽₂)→10/9 as 𝑛→∞,

−10𝑚₂9𝑚₃𝐶_2𝑛=−𝛼𝑤₁(2𝛼−1)²{−𝑛𝑘𝐵_𝑛(𝑘𝑛)+𝑛𝑘₀^𝑘/𝑛𝐵_𝑛(𝑠)𝑠−𝑤𝑑𝑠}₁2𝛼−1𝑛𝑘𝐵_𝑛(𝑘𝑛)+𝑜_𝑝∫(1)−^{1−ℓ/𝑛𝑘/𝑛}𝐽(𝑠)𝐵_𝑛(𝑠)𝑑𝑠𝜎_𝑛+(𝐽)𝛼𝑤_𝑅,1((1+𝛽)𝛼−1)²{𝑛ℓ𝐵_𝑛ℓ(1−𝑛)−𝑛ℓ¹_1−ℓ/𝑛𝐵_𝑛(𝑠)+𝑤1−𝑠𝑑𝑠}_𝑅,12𝛼−1{−𝑛ℓ𝐵_𝑛ℓ(1−𝑛)}+𝑜_𝑝(1), (5.93) with 𝑤₁∶=(𝛼+1)(2𝛼+2−𝛼)2(1+(𝑞/𝑝)^2/𝛼𝜃^3−2/𝛼),𝑤_𝑅,1∶=(𝑞/𝑝)^1/𝛼𝜃^3/2−1/𝛼𝑤₁. (5.94) By the same arguments as the proof of Theorem 3.5 we infer that

√𝑛(𝜐₁−𝜐₁)(ℓ/𝑛)^3/2𝑋^𝒟_{𝑛−ℓ,𝑛}−−−−→𝒩(0,𝑉²₁)as𝑛→∞. (5.95)

This achieves the proof of Theorem 4.1.

5.5. Proof of Theorem 4.2

Theorem 4.2 is just an application of Theorem 3.5, we omit details.

Acknowledgments

The authors thank anonymous referees for their insightful comments and valuable suggestions. The first author is indebted to Ric̆‌ardas Zitikis for his constructive advice and also for having introduced him (the first author) to the Actuarial Research Group (ARG) in London, Ontario.