Estimating L-Functionals for Heavy-Tailed Distributions and Application

Volume 2010, Article ID 707146,34pages doi:10.1155/2010/707146

Research Article

Estimating L-Functionals for Heavy-Tailed Distributions and Application

Abdelhakim Necir and Djamel Meraghni

Laboratory of Applied Mathematics, Mohamed Khider University of Biskra, 07000 Biskra, Algeria

Correspondence should be addressed to Abdelhakim Necir,[email protected] Received 5 October 2009; Accepted 21 January 2010

Academic Editor: Riˇcardas Zitikis

Copyrightq2010 A. Necir and D. Meraghni. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

L-functionals summarize numerous statistical parameters and actuarial risk measures. Their sample estimators are linear combinations of order statisticsL-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 forL-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

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

LJ: ₁

0

JsQsds, 1.1

whereQs:inf{x∈R:Fx≥s},0< s≤1,is the quantile function pertaining to dfFand Jis a measurable function defined on0,1 see, e.g. Serfling,1. Several authors have used the quantity LJto solve some statistical problems. For example, in a work by Chernoff et al.2theL-functionals have a connection with optimal estimators of location and scale parameters in parametric families of distributions. Hosking3introduced theL-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 Seheult4have defined the trimmedL-moments to answer some questions related to heavy- tailed distributions for which means do not exist, and therefore theL-moment method cannot be applied. In the case where the trimming parameter equals one, the first four theoretical trimmedL-moments are

mi: ₁

0

JisQsds, i1,2,3,4, 1.2

where

Jis:s1−sφis, 0< s <1, 1.3 withφ_ipolynomials of orderi−1seeSection 4. A partial study of statistical estimation of trimmedL-moments was given recently by Hosking5.

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 Denneberg6and Wang7; see also Wang 8. The breakthrough in the area was offered by Jones and Zitikis9, 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 areasee, e.g., Jones and Zitikis10,11; Brazauskas et al.

12,13and Greselin et al.14. These works mainly discuss CLT-type results. We have been utilizing the aforementioned relationship between distortion risk measures andL-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

ΠX: _∞

0

g1−Fxdx, 1.4

whereg is a non decreasing concave function withg0 0 andg1 1.By a change of variables and integration by parts,ΠXmay be rewritten into

ΠX ₁

0

g1−sQsds, 1.5

wheregdenotes the Lebesgue derivative ofg.For heavy-tailed claim amounts, the empirical estimation with confidence bounds forΠXhas been discussed by Necir et al.17and Necir and Meraghni18. IfX∈Rrepresents financial data such as asset log-returns, the distortion risk measures are defined by

HX:

₀

−∞

g1−Fx−1 dx

_∞

0

g1−Fxdx. 1.6

Likewise, by integration by parts it is shown that

HX ₁

0

g1−sQsds. 1.7

Wang8and Jones and Zitikis9have defined the risk two-sided deviation by ΔrX:

₁

0

JrsQsds, 0< r <1, 1.8

with

J_rs: r 2

s^1−r−1−s^1−r

s^1−r1−s^1−r , 0< s <1. 1.9 As we see,ΠX, HX,andΔrXareL-functionals for specific weight functions. For more details about the distortion risk measures one refers to Wang8,16. A discussion on their empirical estimation is given by Jones and Zitikis9.

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 letN0, η²denote the normal distribution with mean 0 and varianceη².

The natural estimators of quantity LJ are linear combinations of order statistics calledL-statistics. For more details on this kind of statistics one refers to Shorack and Wellner 19, page 260. Indeed, letX1, . . . , Xnbe a sample of sizen≥1 from an rvXwith dfF, then the sample estimator ofLJis

L_nJ: ₁

0

JsQnsds, 1.10

whereQns : inf{x ∈ R : Fnx ≥ s}, 0 < s ≤ 1, is the empirical quantile function that corresponds to the empirical df F_nx : n⁻¹_n

i1I{Xi ≤ x} for x ∈ R, pertaining to the sample X1, . . . , X_nwith I·denoting the indicator function. It is clear thatL_nJmay be rewritten into

L_nJ ⁿ

i1

a_i,nX_i,n, 1.11

where ai,n : _i/n

i−1/nJsds,i 1, . . . , n, and X1,n ≤ · · · ≤ Xn,n denote the order statistics based upon the sampleX1, . . . , Xn.The first general theorem on the asymptotic normality of LnJis established by Chernoffet al.2. Since then, a large number of authors have studied the asymptotic behavior of L-statistics. A partial list consists of Bickel20, Shorack21,22,

Stigler23,24, Ruymgaart and Van Zuijlen25, Sen26, Boos27, Mason28, and Singh 29. Indeed, we have

√n L_nJ−LJ _D

−−−→ N 0, σ²J

, asn−→ ∞, 1.12 provided that

σ²J: ₁

0

₁

0

mins, t−stJsJtdQsQt<∞. 1.13

In other words, for a given functionJ, condition1.13excludes the class of distributionsF for whichσ²Jis infinite. For example, if we takeJ 1,LJis equal to the expected value EXand hence the natural estimator ofL_nJis the sample meanX_n.In this case, result1.12 corresponds to the classical central limit theorem which is valid only when the variance ofF is finite. How then can be construct confidence bounds for the mean of a df when its variance is infinite? This situation arises when dfF belongs to the domain of attraction of α-stable lawsheavy-tailedwith characteristic exponentα∈1,2; seeSection 2. This question was answered by Peng30,31who proposed an alternative asymptotically normal estimator for the mean.Remark 3.3below shows that this situation also arises for the sample trimmedL- momentsm_iwhen 1/2 < α <2/3 and for the sample risk two-sided deviationΔrXwhen 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 ofL- functionals for heavy-tailed distributions for whichσ²J ∞.

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. InSection 3 we define, via the extreme value approach, a new asymptotically normal estimator ofL-functionals and state our main results. Applications to trimmed L-moments, risk measures, and related quantities are given inSection 4. All proofs are deferred toSection 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∈Dα,if there exist two real sequencesAn>0 andCnsuch that

A⁻¹_{n n}

i1

Xi−Cn

−−−→D Sα

σ, δ, μ

, asn−→ ∞, 2.1

where Sασ, δ, μis a stable distribution with parameters 0 < α ≤ 2, −1 ≤ δ ≤ 1, σ > 0 and−∞< μ < ∞see, e.g., Samorodnitsky and Taqqu32. This class of distributions was introduced by L´evy during his investigations of the behavior of sums of independent random variables in the early 1920s 33.Sασ, 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 good candidate 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 Weron35. This class of distributions have nice heavy-tail properties. More precisely, if we denote byGx:P|X| ≤ x Fx−F−x, x >0,the df ofZ:|X|,then the tail behavior ofF ∈Dα,for 0< α <2, may be described by the following

iThe tail 1−Gis regularly varying at infinity with index−α.That is

t→ ∞lim

1−Gxt

1−Gt x^−α, for anyx >0. 2.2

iiThere exists 0≤p≤1 such that

xlim→ ∞

1−Fx

1−Gx p, lim

x→ ∞

F−x

1−Gx 1−p:q. 2.3

Let, for 0< s <1, Ks:inf{x >0 :Gx≥s} be the quantile function pertaining toGand Q₁s : max−Q1−s,0and Q₂s : maxQs,0.Then Proposition A.3 in a work by Cs ¨org˝o et al.36says that the set of conditions above is equivalent to the following.

iK1− ·is regularly varying at 0 with index−1/α.That

lims↓0

K1−xs

K1−s x^−1/α, for any x >0. 2.4

iiThere exists 0≤p≤1 such that

lims↓0

Q₁1−s

K1−s p^1/α, lim

s↓0

Q₂1−s K1−s

1−p1/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 functionA,not changing sign near zero, such that

lims↓0As⁻¹

K1−xs K1−s −x^−1/α

x^−1/αx−1

, for anyx >0, 2.6 where ≤ 0 is the second-order parameter. If 0, interpret x −1/ as logx. The second-order condition for heavy-tailed distributions has been introduced by de Haan and Stadtm ¨uller37.

3. Estimating L-Functionals When F ∈ Dα

The right and left extreme quantiles of small enough leveltof dfF, respectively, are two reals x_R and x_L defined by 1−FxR t andFxL t, that is, x_R Q1−t andx_L Qt.

The estimation of extreme quantiles for heavy-tailed distributions has got a great deal of interest, see for instance Weissman 38, Dekkers and de Haan39, Matthys and Beirlant 40and Gomes et al.41. Next, we introduce one of the most popular quantile estimators.

Letkknand nbe sequences of integerscalled trimming sequencessatisfying 1< k <

n,1< < n,k → ∞, → ∞,k/n → 0 and/n → 0, asn → ∞. Weissman’s estimators of extreme quantilesx_Randx_Lare defined, respectively, by

xLQLt: k n

1/αL

Xk,nt^−1/αL, ast↓0,

x_RQ_R1−t:

n 1/αR

X_n−,nt^−1/αR, ast↓0,

3.1

where

α_Lα_Lk: 1

k k

i1

log −Xi,n−log −Xk,n ₋₁

,

αRαR: 1

i1

log Xn−i 1,n−log Xn−,n

₋₁ 3.2

are two forms of Hill’s estimator42for the stability indexαwhich could also be estimated, using the order statisticsZ_1,n ≤ · · · ≤ Z_n,n associated to a sample Z1, . . . , Z_n fromZ, as follows:

ααm : 1

m m

i1

log Zn−i 1,n−log Zn−m,n ₋₁

, 3.3

with log u:max0,loguandm m_n being an intermediate sequence fulfilling the same conditions as k and . Hill’s estimator has been thoroughly studied, improved, and even generalized to any real-valued tail index. Its weak consistency was established by Mason43 assuming only that the underlying distribution is regularly varying at infinity. The almost sure convergence was proved by Deheuvels et al.44and more recently by Necir45. The asymptotic normality has been investigated, under various conditions on the distribution tail, by numerous workers like, for instance, Cs ¨org˝o and Mason46, Beirlant and Teugels47, and Dekkers et al.48.

3.2. A Discussion on the Sample Fractionsk and

Extreme value-based estimators rely essentially on the numbers k and of lower- and upper-order statistics used in estimate computation. Estimatorsα_Landα_Rhave, in general, substantial variances for small values ofkandand considerable biases for large values ofk and.Therefore, one has to look for optimal values forkand,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;

0 20 40 60 80 100 0

0.5 1 1.5 2 2.5 3

Figure 1: Plots of Hill estimators, as functions of the number of extreme statistics,αsolid line,αRdashed line, andαLdotted lineof the characteristic exponentαof a stable distribution skewed to the right, based on 1000 observations with 50 replications. The horizontal line represents the true value ofα1.2.

0 20 40 60 80 100

0 0.5 1 1.5 2 2.5 3

Figure 2: Plots of Hill estimators, as functions of the number of extreme statistics,αsolid line,αRdashed line, andαLdotted lineof the characteristic exponentαof a stable distribution skewed to the left, based on 1000 observations with 50 replications. The horizontal line represents the true value ofα1.2.

see, for example, Dekkers and de Haan49, Drees and Kaufmann50, Danielsson et al.51, Cheng and Peng52and Neves and Alves53. Graphically, the behaviors ofα_L,α_R, andαas functions ofk,andm, respectively, are illustrated by Figures1,2, and3drawn by means of the statistical software R54. According toFigure 1,α_Ris much more suitable thanα_Lwhen estimating the stability index of a distribution which is skewed to the rightδ >0whereas Figure 2shows thatαLis much more reliable thanαRwhen the distribution is skewed to the leftδ <0.In the case where the distribution is symmetricδ0,both estimators seem to be equally good as seen inFigure 3. Finally, it is worth noting that, regardless of the distribution skewness, estimatorα, based on the top statistics pertaining to the absolute value ofX,works well and gives good estimates for the characteristic exponentα.

0 20 40 60 80 100 0

0.5 1 1.5 2 2.5 3

Figure 3: Plots of Hill estimators, as functions of the number of extreme statistics,αsolid line,αRdashed lineandαLdotted lineof the characteristic exponentαof a symmetric stable distribution, based on 1000 observations with 50 replications. The horizontal line represents the true value ofα1.2.

1000 2000 3000 4000 5000

0 1 2 3 4

Figure 4: Plots of the ratios of the numbers of extreme statistics, as functions of the sample size, for a stable symmetric distributionS1.21,0,0 solid line, a stable distribution skewed to the rightS1.21,0.5,0 dashed lineand a stable distribution skewed to the leftS1.21,−0.5,0 dotted line.

It is clear that, in general, there is no reason for the trimming sequenceskandto be equal. We assume that there exists a positive real constantθsuch that/k → θasn → ∞.If the distribution is symmetric, the value ofθis equal to 1; otherwise, it is less or greater than 1 depending on the sign of the distribution skewness. For an illustration, seeFigure 4where we plot the ratioθfor several increasing sample sizes.

3.3. Some Regularity Assumptions onJ

For application needs, the following regularity assumptions on functionJare required:

H1Jis differentiable on0,1, H2λ:lim_s↓0J1−s

Js <∞,

H3bothJsandJ1−sare regularly varying at zero with common indexβ∈R.

H4there exists a functiona·not changing sign near zero such that

lim_t↓0Jxt/Jt−x^β

at x^βx^ω−1

ω , for anyx >0, 3.4

whereω≤0 is the second-order parameter.

The three remarks below give more motivations to this paper.

Remark 3.1. AssumptionH3has already been used by Mason and Shorack55to establish the asymptotic normality of trimmed L-statistics. ConditionH4is just a refinement ofH3 called the second order condition that is required for quantile functionKin2.6.

Remark 3.2. AssumptionsH1–H4are satisfied by all weight functionsJi_i2,4withβ, λ 1,±1 seeSection 4.1and by functionJ_rin1.9withβ, λ r−1,−1.These two examples show that the constantsβandλmay be positive or negative depending on application needs.

Remark 3.3. L-functionalsLJexist for any 0 < α < 2 andβ ∈ Rsuch that 1/α−β < 1.

However,Lemma 5.4below shows that for 1/α−β > 1/2 we haveσ²J ∞. Then, recall 1.3; whenever 1/2< α <2/3,the trimmed L-moments exist howeverσ²Ji ∞, i1, . . . ,4.

Likewise, recall1.9; whenever 1/r 1/2< α <1/r,the two-sided deviationΔrXexists whileσ²Jr ∞.

3.4. Defining the Estimator and Main Results

We now have all the necessary tools to introduce our estimator ofLJ,given in1.1, when F ∈Dαwith 0< α <2. Letkk_nand _nbe sequences of integers satisfying 1< k < n, 1< < n,k → ∞, → ∞,k/n → 0,/n → 0, and the additional condition/k → θ <∞ asn → ∞.First, we must note that since 1 β−1/α >0seeRemark 3.3and since bothαL

andα_Rare consistent estimators ofαsee, Mason43, then we have for all largen

P

1 β− 1 α_L >0

P

1 β− 1 α_R >0

1 o1. 3.5

Observe now thatLJdefined in1.1may be split in three integrals as follows:

LJ _k/n

0

JtQtdt

_1−/n

k/n

JtQtdt ₁

1−/nJtQtdt:T_L,n T_M,n T_R,n. 3.6

SubstitutingQLtandQR1−tforQtandQ1−tinTL,nandTR,n, respectively and making use of assumptionH3and3.5yield that for all largen

_k/n

0

JtQLtdt k

n 1/αL

Xk,n

_k/n

0

t^−1/αLJtdt 1 o1k/nJk/n 1 β−1/α_LXk,n, _/n

0

J1−tQ_R1−tdt

n _1/αR

X_n−,_{n /n}

0

t^−1/αRJ1−tdt

1 o1/nJ1−/n

1 β−1/α_R X_n−,n.

3.7

Hence we may estimateT_L,nandT_R,nby

TL,n: k/nJk/n

1 β−1/α_LXk,n, TR,n: /nJ1−/n

1 β−1/α_R X_n−,n, 3.8

respectively. As an estimator ofT_M,nwe take the sample one that is

TM,n: _1−/n

k/n

JtQntdt ⁿ⁻

ik 1

ai,nXi,n, 3.9

with the same constantsai,nas those in1.11. Thus, the final form of our estimator is

Lk,J k/nJk/n 1 β−1/α_LXk,n

n−

ik 1

ai,nXi,n /nJ1−/n

1 β−1/α_R X_n−,n. 3.10

A universal estimator ofLJmay be summarized by L^∗_nJ L_k,JI σ²J ∞

L_nJI σ²J<∞

, 3.11

whereLnJis as in1.11. More precisely L^∗_nJ L_k,JI

A

α, β L_nJI A α, β

, 3.12

whereAα, β:{α, β∈0,2×R: 1/2<1/α−β <1}andAα, βis its complementary in 0,2×R.

Note that for the particular case k and J 1 the asymptotic normality of the trimmed meanT_M,n has been established in Theorem 1 of Cs ¨org˝o et al.56. The following

theorem gives the asymptotic normality ofTM,nfor more general trimming sequenceskand and weighting functionJ. For convenience, we set, for any 0< x <1/2 and 0< y <1/2,

σ² x, y;J

: _1−y

x

_1−y

x

mins, t−stJsJtdQsQt<∞, 3.13

and letσ_n²J:σ²k/n, /n;J.

Theorem 3.4. Assume thatF ∈ Dαwith 0 < α < 2. For any measurable functionJ satisfying assumption (H3) with indexβ ∈Rsuch that 0 < 1/α−β < 1 and for any sequences of integersk andsuch that1< k < n, 1< < n,k → ∞, → ∞,k/n → 0, and/n → 0, asn → ∞, there exists a probability spaceΩ, A, Pcarrying the sequenceX₁, X₂, . . .and a sequence of Brownian bridges{Bns,0≤s≤1, n1,2, . . .}such that one has for all largen

√n TM,n−TM,n

σnJ −

_1−/n

k/n JsBnsds

σnJ o_p1, 3.14

and therefore

√n T_M,n−T_M,n σ_nJ

−−−→ N0,D 1 asn−→ ∞. 3.15

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

Theorem 3.5. Assume thatF ∈ Dαwith 0 < α < 2. For any measurable functionJ satisfying assumptions (H1)–(H4) with indexβ ∈Rsuch that 1/2 < 1/α−β < 1,and for any sequences of integers k and such that 1 < k < n, 1 < < n,k → ∞, → ∞, k/n → 0,/n → 0, /k → θ <∞,and√

kak/nAk/n → 0 asn → ∞, one has

√n L_k,J−LJ σ_nJ

−−−→ ND 0, σ₀²

, asn−→ ∞, 3.16

where

σ₀²σ₀² α, β

:

αβ 1

2αβ 2−α

×

2α²

βα−12

2α βα−1 2

1 β

α−1₄ 1

1 β α−1

1.

3.17

The following corollary is more practical than Theorem 3.5 as it directly provides confidence bounds forLJ.

Corollary 3.6. Under the assumptions ofTheorem 3.5one has

√n L_k,J−LJ /n^1/2J1−/nXn−,n

−−−→ ND 0, V²

, as n−→ ∞, 3.18

where

V²V²

α, β, λ, θ, p :

1 λ⁻²

q p

_−2/α

θ^{−2β 2/α−1}

×

2α²

βα−12

2α βα−1 2

1 β α−14

1 1 β

α−1

1,

3.19

withp, qas in statement (ii) ofSection 2andλ, βas in assumptions (H2)-(H3) ofSection 3.

3.5. Computing Confidence Bounds forLJ

The form of the asymptotic variance V² in 3.20 suggests that, in order to construct confidence intervals forLJ,an estimate ofpis needed as well. Using the intermediate order statisticZ_n−m,n, de Haan and Pereira57proposed the following consistent estimator forp:

p_n p_nm: 1 m

n i1

I{Xi> Z_n−m,n}, 3.20

wherem m_nis a sequence of integers satisfying 1 < m < n,m → ∞,andm/n → 0, as n → ∞the same as that used in3.3.

LetJ be a given weight function satisfyingH1–H4with fixed constantsβand λ.

Suppose that, fornlarge enough, we have a realizationx1, . . . , x_nof a sampleX1, . . . , X_n from rvXwith dfFfulfilling all assumptions ofTheorem 3.5. The1−ς-confidence intervals forLJwill be obtained via the following steps.

Step 1. Select the optimal numbersk^∗,^∗, andm^∗of lower- and upper-order statistics used in 3.2and3.3.

Step 2. DetermineX_k^∗_,n, X_n−^∗_,n, Jk^∗/n,J1−^∗/n,andθ^∗:^∗/k^∗.

Step 3. Compute, using 3.2, α^∗_L : αLk^∗and α^∗_R : αR^∗. Then deduce, by 3.10, the estimateL_k^∗_,^∗J.

Step 4. Use3.3and3.20to computeα^∗:αm ^∗andp_n^∗:pnm^∗.Then deduce, by3.19, the asymptotic standard deviation

V^∗: V²

α^∗, β, λ, θ^∗,p^∗_n

. 3.21

Finally, the lower and upper1−ς-confidence bounds forLJ,respectively, will be

L_k∗,^∗J−z_ς/2

√^∗V^∗X_n−^∗,nJ1−^∗/n

n ,

Lk^∗,^∗J zς/2

√^∗V^∗X_n−^∗_,nJ1−^∗/n

n ,

3.22

wherezς/2is the1−ς/2quantile of the standard normal distributionN0,1with 0< ς <1.

4. Applications

4.1. TL-Skewness and TL-Kurtosis WhenF∈Dα

When the distribution mean EX exists, the skewness and kurtosis coefficients are, respectively, defined byL₁ :μ₃/μ^3/2₂ andL₂:μ₄/μ²₂withμ_k:EX−EX^k, k2, 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, Groeneveld58and Hosking3. Recently, Elamir and Seheult4have 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

υ₁ : m₃

m₂, υ₂: m₄

m₂, 4.1

where m_i, i 2,3,4,are the trimmed L-moments defined inSection 1. The corresponding weight functions of1.3are defined as follows:

J₂s:6s1−s2s−1, J₃s: 20

3 s1−s 5s²−5s 1 , J₄s: 15

2 s1−s 14s³−21s² 9s−1 .

4.2

If we suppose that F ∈ Dα with 1/2 < α < 2/3, then, in view of the results above, asymptotically normal estimators forυ₁andυ₂will be, respectively,

υ1 : m3

m₂, υ2: m4

m₂, 4.3

where

m_i

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

−6k/n² 2−1/α_LX_k,n

n−

jk 1

aⁱ_j,nX_j,n 6/n²

2−1/α_RX_n−,n, fori2, 20k/n²

32−1/α_LX_k,n n−

jk 1

aⁱ_j,nX_j,n 20/n²

32−1/α_RX_n−,n, fori3,

−15k/n² 22−1/α_LXk,n

n−

jk 1

aⁱ_j,nXj,n 15/n²

22−1/α_RX_n−,n, fori4,

4.4

withaⁱ_j,n:_j/n

j−1/nJisds, i2,3,4,andj1, . . . , n.

Theorem 4.1. Assume thatF ∈Dαwith 1/2 < α < 2/3.For any sequences of integerskand such that 1 < k < n, 1 < < n,k → ∞, → ∞,k/n → 0,/n → 0,/k → θ < ∞,and

√kak/nAk/n → 0 asn → ∞, one has, respectively, asn → ∞,

√nυ1−υ1 /n^3/2X_n−,n

−−−→ ND 0, V₁² ,

√nυ2−υ2 /n^3/2X_n−,n

−−−→ ND 0, V₂² ,

4.5

where

V₁²: 36 m²₂

1− 9m₃ 10m2

₂

σ^∗2, V₂²: 225 4m²₂

1−4m₄

5m2

₂

σ^∗2, 4.6

with

σ^∗2: 1

q/p_−2/α θ^2/α−3

×

2α² α−1² 2αα−1 22α−1⁴

1 2α−1

1. 4.7

4.2. Risk Two-Sided Deviation WhenF ∈Dα Recall that the risk two-sided deviation is defined by

ΔrX: ₁

0

J_rsQsds, 0< r <1, 4.8

where

Jrs: r 2

s^1−r−1−s^1−r

s^1−r1−s^1−r , 0< s <1. 4.9

An asymptotically normal estimator forΔrX,when 1/r 1/2< α <1/r, is ΔrX − rk/n^r

2r−4/αL

X_k,n n−

jk 1

a^r_j,nX_j,n r/n^r 2r−4/αR

X_n−,n, 4.10

where

a^r_j,n 1 2

1− i

n _−r

−

1−i−1 n

_−r

− i

n

_−r i−1

n _−r

, 4.11

j1, . . . , n.

Theorem 4.2. Assume thatF ∈ Dαwith 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 such that1 < k < n, 1 < < n, k → ∞, → ∞,k/n → 0,/n → 0, /k → θ <∞,and√

kak/nAk/n → 0 asn → ∞, one has, asn → ∞,

√n ΔrX−ΔrX /n^r−1/2X_n−,n

−−−→ ND 0, V_r²

, 4.12

where

V_r²: r² 4

1

q p

_−2/α

θ^{2/α−2r 1}

×

2α² rα−α−1² 2αrα−α−1 2rα−1⁴

1 rα−1

1.

4.13

5. Proofs

First we begin by the following three technical lemmas.

Lemma 5.1. Letf1 and f2 be two continuous functions defined on0,1and regularly varying at zero with respective indicesκ >0 and−τ <0 such thatκ < τ. Suppose thatf₁is differentiable at zero, then

limx↓0

_1/2

x f₁sdf2s f1xf2x τ

κ−τ. 5.1

Lemma 5.2. Under the assumptions ofTheorem 3.5, one has

nlim→ ∞

_1−/n

k/n s1−s^1/2−νJsdQs

k/n^1/2−νJk/nQk/n 1 λθ1/2−ν β−1/α

q/p1/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

xc²x yc²

1−y _1−y

x

c²tdt−

xcx yc

1−y _1−y

x

ctdt ₂

, 5.3

wherecs:_s

1/2JtdQt, 0< s <1/2.

Lemma 5.4. Under the assumptions ofTheorem 3.5, one has

nlim→ ∞

k/nJ²k/nQ²k/n σ_n²J w²,

nlim→ ∞

/nJ²1−/nQ²1−/n

σ_n²J λ²

q p

_2/α

θ^{2β−2/α 1}w²,

5.4

where

w²:

αβ 1

2αβ 2−α 2 1 λ²

q/p_2/α

θ^{2β−2/α 1}. 5.5

Proof ofLemma 5.1. Letf₁ denote the derivative off₁.Applying integration by parts, we get, for any 0< x <1/2,

_1/2

x

f₁sdf2s f₁ 1

2

f₂ 1

2

−f₁xf2x− _1/2

x

f₁sf2sds. 5.6

Since the productf₁f₂is regularly varying at zero with index−τ κ <0,thenf₁xf2x → 0 asx↓0.Therefore

limx↓0

_1/2

x f₁sdf2s

f₁xf2x −1−lim

x↓0

_1/2

x f₁sf2sds

f₁xf2x . 5.7

By using Karamata’s representationsee, e.g., Seneta59, it is easy to show that

xf₁x κ1 o1f1x, asx↓0. 5.8

Hence

limx↓0

_1/2

x f1sdf2s

f₁xf2x −1−κlim

x↓0

_1/2

x f₁sf2sds

xf₁xf2x . 5.9

It is clear that5.8implies thatf₁ is regularly varying at zero with indexκ−1; thereforef₁f2

is regularly varying with index−τ κ−1<0. Then, Theorem 1.2.1 by de Haan60, page 15 yields

limx↓0 − _1/2

x f₁sf2sds xf₁xf2x 1

κ−τ. 5.10

This completes the proof ofLemma 5.1.

Proof ofLemma 5.2. We have

In: _1−/n

k/n

s1−s^1/2−νJsdQs

_1/2

k/n

s1−s^1/2−νJsdQs

− _1/2

1−/ns1−s^1/2−νJ1−sdQ1−s :I1n−I2n.

5.11

By taking, inLemma 5.1,f1s s1−s^1/2−νJsandf2s Qswithκ1/2−ν β, τ 1/α,andxk/n,we get

nlim→ ∞

I1n

k/n^1/2−νJk/nQk/n 1/α

1/2−ν β−1/α. 5.12

Likewise if we takef1s s1−s^1/2−νJ1−sandf2s Q1−swithκ1/2−ν β, τ 1/α,andx/n,we have

nlim→ ∞

I2n

/n^1/2−νJ1−/nQ1−/n 1/α

1/2−ν β−1/α. 5.13

Note that statementiiofSection 2implies that

lims↓0 Q1−s/Qs − q

p 1/α

. 5.14

The last two relations, together with assumptionH2and the regular variation ofQ1−s, imply that

nlim→ ∞

I_2n

k/n^1/2−νJk/nQk/n −

q/p_1/α

λθ1/2−ν β−1/α/α

1/2−ν β−1/α . 5.15

This achieves the proof ofLemma 5.2.

Proof ofLemma 5.3. We will use similar techniques to those used by Cs ¨org˝o et al. 36, Proposition A.2. For any 0< s <1/2,we set

Wx,yt:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ c

1−y

for 1−y≤t <1, ct forx < t <1−y, cx for 0< t≤x.

5.16

Thenσ²x, y;Jmay be rewritten into

σ² x, y;J

₁

0

W_x,y² sds− ₁

0

W_x,ysds ₂

, 5.17

and the result ofLemma 5.3follows immediately.

Proof ofLemma 5.4. FromLemma 5.3we may write σ_n²J

k/nJ²k/nQ²k/n Tn1 Tn2 Tn3 Tn4, 5.18 where

T_n1: k/nc²k/n

k/nJ²k/nQ²k/n, T_n2: /nc²1−/n k/nJ²k/nQ²k/n, Tn3:

_1−/n

k/n c²tdt k/nJ²k/nQ²k/n,

Tn4: k/nck/n /nc1−/n _1−/n

k/n ctdt₂ k/nJ²k/nQ²k/n .

5.19

By the same arguments as in the proof ofLemma 5.2, we infer that

n→ ∞lim

ck/n

Jk/nQk/n 1

αβ−1,

nlim→ ∞

c1−/n

Jk/nQk/n λ q/p1/α

θ^β−1/α αβ−1 .

5.20

Therefore

nlim→ ∞T_n1 1

αβ−1₂, lim

n→ ∞T_n2 λ² q/p_2/α

θ^{2β−2/α 1}

αβ−1₂ . 5.21

Next, we consider the third termTn3which may be rewritten into

T_n3

_1/2

k/nc²tdt k/nJ²k/nQ²k/n

_1−/n

1/2 c²tdt

k/nJ²k/nQ²k/n. 5.22 Observe that

_1/2

k/nc²tdt

k/nJ²k/nQ²k/n

ck/n Jk/nQk/n

_{2 1/2}

k/nc²tdt

k/nc²k/n. 5.23 It is easy to verify that functionc²·is regularly varying at zero with index 2β−1/α.Thus, by Theorem 1.2.1 by de Haan60we have

nlim→ ∞

_1/2

k/nc²tdt

k/nc²k/n α

2−2αβ−α. 5.24

Hence

nlim→ ∞

_1/2

k/nc²tdt

k/nJ²k/nQ²k/n α αβ−12

2−2αβ−α. 5.25

By similar arguments we show that

nlim→ ∞

_1−/n

1/2 c²tdt

k/nJ²k/nQ²k/n α q/p_2/α

λ²θ^{2β−2/α 1} αβ−1₂

2−2αβ−α. 5.26

Therefore

nlim→ ∞Tn3 1 α q/p2/α

λ²θ^{2β−2/α 1} αβ−12

2−2αβ−α . 5.27

By analogous techniques we show thatTn4 → 0 asn → ∞; we omit details. Summing up the three limits ofT_ni, i1,2,3, achieves the proof of the first part ofLemma 5.4. As for the second assertion of the lemma, we apply a similar procedure.

5.1. Proof ofTheorem 3.4

Cs ¨org˝o et al.36have constructed a probability spaceΩ, A, Pcarrying an infinite sequence ξ1, ξ2, . . . of independent rv’s uniformly distributed on 0,1 and a sequence of Brownian bridges{Bns,0≤s≤1, n1,2, . . .}such that, for the empirical process,

ϕns:n^1/2{Γns−s}, 0≤s≤1, 5.28

whereΓn·is the uniform empirical df pertaining to the sampleξ1, . . . , ξn; we have for any 0≤ν <1/4 and for all largen

sup

1/n≤s≤1−1/n

ϕ_ns−B_ns s1−s^1/2−ν Op

n^−ν

. 5.29

For eachn ≥ 1, letξ1,n ≤ · · · ≤ ξn,n denote the order statistics corresponding toξ1, . . . , ξn. Note that for eachn,the random vector Qξ1,n, . . . , Qξn,nhas the same distribution as X1,n, . . . , Xn,n. Therefore, for 1 ≤ i ≤ n,we shall use the rv’s Qξi,nto 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 termTM,ndefined in3.9may be rewritten into

TM,n _ξn−,n

ξk,n

QsdΨΓns, 5.30

whereΨs:_s

0Jtdt.Integrating by parts yields n^1/2 TM,n−TM,n

σnJ Δ1,n Δ2,n Δ3,n, 5.31

where

Δ1,n:−n^1/2_1−/n

k/n {ΨΓns−Ψs}dQs

σnJ ,

Δ2,n: n^1/2_ξk,n

k/n{ΨΓns−Ψk/n}dQs

σnJ ,

Δ3,n: n^1/2_1−/n

ξn−,n {ΨΓns−Ψ1−/n}dQs

σ_nJ .

5.32

Next, we show that

Δ1,n−−−→ N0,D 1 asn−→ ∞, 5.33 Δi,n

−−−→p 0 asn−→ ∞fori2,3. 5.34

Making use of the mean-value theorem, we have for eachn

ΨΓns−Ψs Γns−sJϑns, 5.35

where{ϑns}_n≥1is a sequence of rv’s with values in the open interval of endpointss∈0,1 andΓns.Therefore

Δ1,n −_1−/n

k/n ϕ_nsJϑnsdQs

σnJ . 5.36

This may be rewritten into

Δ1,n− _1−/n

k/n ϕ_nsJsdQs

σnJ −

_1−/n

k/n ϕ_nsJs{Jϑns−Js}dQs σnJ

:Δ^∗_1,n Δ^∗∗_1,n.

5.37

Note that _1−/n

k/n ϕns−BnsJsdQs σ_nJ

≤ sup

k/n≤s≤1−/n

ϕns−Bns s1−s^1/2−ν

_1−/n

k/n

s1−s^1/2−ν|Js|dQs/σnJ,

5.38

for 0< ν <1/4,which by5.29is equal to O_pn^−ν_1−/n

k/n s1−s^1/2−ν|Js|dQs

σnJ . 5.39

Since we have, from Lemmas5.2and5.3, n

k

ν_1−/n

k/n

s1−s^1/2−ν|Js|dQs/σnJ O1, asn−→ ∞, 5.40

then the right-hand side of the last inequality is equal toO_pk^−νwhich in turn tends to zero asn → ∞. This implies that asn → ∞

Δ^∗_1,n− _1−/n

k/n BnsJsdQs

σ_nJ op1. 5.41

Next, we show that Δ^∗∗_1,n op1. Indeed, function J is differentiable on 0,1; then by the mean-value theorem, there exists a sequence{ϑ^∗_ns}_n≥1 of rv’s with values in the open interval of endpointss∈0,1andϑnssuch that for eachnwe have

Δ^∗∗_{1,n 1−/n}

k/n ϕnsJs{ϑns−s}Jϑ_n^∗sdQs

σ_nJ . 5.42

From inequalities3.9and3.10by Mason and Shorack55, we infer that, for any 0< ρ <1, there exists 0< M_ρ<∞such that for all largenwe have

Jϑ_n^∗s≤ M_ρ|Js|

s1−s, 5.43 for any 0< s≤1/2.On the other hand, we have for any 0< s <1

|ϑns−s| ≤ |Γns−s|. 5.44

Therefore

Δ^∗∗_1,n≤ M_ρn^−1/2_1/2

k/n ϕ_ns²|Js|/s1−s dQs

σ_nJ . 5.45

This implies, since, for eachn≥1,E|ϕns|² < s1−s,that

EΔ^∗∗_1,n≤ M_ρn^−1/2_1/2

k/n|Js|dQs

σnJ , 5.46

which tends to zero asn → ∞.

Next, we consider the termΔ2,nwhich may be rewritten into

Δ2,n n^1/2_ξk,n

k/n{ΨΓns−Ψs}dQs σnJ

n^1/2_ξk,n

k/n{Ψs−Ψk/n}dQs

σnJ . 5.47

Making use of the mean-value theorem, we get

Δ2,n _ξk,n

k/nϕnsJ μns

dQs σ_nJ

n^1/2_ξk,n

k/ns−k/nJs^∗_ndQs

σ_nJ , 5.48

whereμnsis a sequence of rv’s with values in the open interval of endpointss∈k/n, ξk,n and Γns and s^∗_n a sequence of rv’s with values in the open interval of endpoints s ∈ k/n, ξk,nandk/n.Again we may rewriteΔ2,ninto

Δ2,n _ξk,n

k/nϕns J

μns

−Js dQs σ_nJ

_ξk,n

k/nϕnsJsdQs σ_nJ n^1/2J

k n

_ξk,n

k/n

s− k

n

Js^∗_n Jk/n−1

dQs/σnJ

n^1/2J k

n _ξk,n

k/n

s− k

n

dQs/σnJ.

5.49