#### Abstract

-functionals summarize numerous statistical parameters and actuarial risk measures. Their sample estimators are linear combinations of order statistics (-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 -functionals and establish their asymptotic normality. Our results may be applied to estimate the trimmed -moments and financial risk measures for heavy-tailed distributions.

#### 1. Introduction

##### 1.1. -Functionals

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

where is the quantile function pertaining to df and is a measurable function defined on (see, e.g. Serfling, ). Several authors have used the quantity to solve some statistical problems. For example, in a work by Chernoff et al.  the -functionals have a connection with optimal estimators of location and scale parameters in parametric families of distributions. Hosking  introduced the -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  have defined the trimmed -moments to answer some questions related to heavy-tailed distributions for which means do not exist, and therefore the -moment method cannot be applied. In the case where the trimming parameter equals one, the first four theoretical trimmed -moments are

where

with polynomials of order (see Section 4). A partial study of statistical estimation of trimmed -moments was given recently by Hosking .

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  and Wang ; see also Wang . The breakthrough in the area was offered by Jones and Zitikis , who revealed a fundamental relationship between the distortion risk measure and the classical -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. ). These works mainly discuss CLT-type results. We have been utilizing the aforementioned relationship between distortion risk measures and -statistics to develop a statistical inferential theory for distortion risk measures in the case of heavy-tailed distributions.

Indeed -functionals have many applications in actuarial risk measures (see, e.g., Wang [8, 15, 16]). For example, if represents an insurance loss, the distortion risk premium is defined by

where is a non decreasing concave function with and By a change of variables and integration by parts, may be rewritten into

where denotes the Lebesgue derivative of For heavy-tailed claim amounts, the empirical estimation with confidence bounds for has been discussed by Necir et al.  and Necir and Meraghni . If represents financial data such as asset log-returns, the distortion risk measures are defined by

Likewise, by integration by parts it is shown that

Wang  and Jones and Zitikis  have defined the risk two-sided deviation by

with

As we see, and are -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 .

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

In the sequel let and , respectively, stand for convergence in probability and convergence in distribution and let denote the normal distribution with mean 0 and variance

The natural estimators of quantity are linear combinations of order statistics called -statistics. For more details on this kind of statistics one refers to Shorack and Wellner [19, page 260]. Indeed, let be a sample of size from an rv with df , then the sample estimator of is

where , , is the empirical quantile function that corresponds to the empirical df for pertaining to the sample with denoting the indicator function. It is clear that may be rewritten into

where , , and denote the order statistics based upon the sample The first general theorem on the asymptotic normality of is established by Chernoff et al. . Since then, a large number of authors have studied the asymptotic behavior of L-statistics. A partial list consists of Bickel , Shorack [21, 22], Stigler [23, 24], Ruymgaart and Van Zuijlen , Sen , Boos , Mason , and Singh . Indeed, we have

provided that

In other words, for a given function , condition (1.13) excludes the class of distributions for which is infinite. For example, if we take , is equal to the expected value and hence the natural estimator of is the sample mean In this case, result (1.12) corresponds to the classical central limit theorem which is valid only when the variance of 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 belongs to the domain of attraction of -stable laws (heavy-tailed) with characteristic exponent 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 -moments when and for the sample risk two-sided deviation when for any To solve this problem in a more general setting, we propose, by means of the extreme value theory, asymptotically normal estimators of -functionals for heavy-tailed distributions for which

The remainder of this paper is organized as follows. Section 2 is devoted to a brief introduction on the domain of attraction of -stable laws. In Section 3 we define, via the extreme value approach, a new asymptotically normal estimator of -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 -Stable Laws

A df is said to belong to the domain of attraction of a stable law with stability index notation: if there exist two real sequences and such that

where is a stable distribution with parameters , , and (see, e.g., Samorodnitsky and Taqqu ). This class of distributions was introduced by Lévy during his investigations of the behavior of sums of independent random variables in the early 1920s . 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 , 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 . This class of distributions have nice heavy-tail properties. More precisely, if we denote by the df of then the tail behavior of for may be described by the following

(i)The tail is regularly varying at infinity with index That is (ii)There exists such that

Let, for be the quantile function pertaining to and and Then Proposition in a work by Csörgő et al.  says that the set of conditions above is equivalent to the following.

is regularly varying at with index That There exists such that

Our framework is a second-order condition that specifies the rate of convergence in statement There exists a function not changing sign near zero, such that

where is the second-order parameter. If interpret as The second-order condition for heavy-tailed distributions has been introduced by de Haan and Stadtmüller .

#### 3. Estimating -Functionals When

##### 3.1. Extreme Quantile Estimation

The right and left extreme quantiles of small enough level of df , respectively, are two reals and defined by and , that is, and The estimation of extreme quantiles for heavy-tailed distributions has got a great deal of interest, see for instance Weissman , Dekkers and de Haan , Matthys and Beirlant  and Gomes et al. . Next, we introduce one of the most popular quantile estimators. Let and be sequences of integers (called trimming sequences) satisfying , , , and , as . Weissman's estimators of extreme quantiles and are defined, respectively, by

where

are two forms of Hill's estimator  for the stability index which could also be estimated, using the order statistics associated to a sample from , as follows:

with and being an intermediate sequence fulfilling the same conditions as and Hill's estimator has been thoroughly studied, improved, and even generalized to any real-valued tail index. Its weak consistency was established by Mason  assuming only that the underlying distribution is regularly varying at infinity. The almost sure convergence was proved by Deheuvels et al.  and more recently by Necir . The asymptotic normality has been investigated, under various conditions on the distribution tail, by numerous workers like, for instance, Csörgő and Mason , Beirlant and Teugels , and Dekkers et al. .

##### 3.2. A Discussion on the Sample Fractions and

Extreme value-based estimators rely essentially on the numbers and of lower- and upper-order statistics used in estimate computation. Estimators and have, in general, substantial variances for small values of and and considerable biases for large values of and Therefore, one has to look for optimal values for and 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 see, for example, Dekkers and de Haan , Drees and Kaufmann , Danielsson et al. , Cheng and Peng  and Neves and Alves . Graphically, the behaviors of , , and as functions of , and , respectively, are illustrated by Figures 1, 2, and 3 drawn by means of the statistical software R . According to Figure 1, is much more suitable than when estimating the stability index of a distribution which is skewed to the right whereas Figure 2 shows that is much more reliable than when the distribution is skewed to the left In the case where the distribution is symmetric both estimators seem to be equally good as seen in Figure 3. Finally, it is worth noting that, regardless of the distribution skewness, estimator based on the top statistics pertaining to the absolute value of works well and gives good estimates for the characteristic exponent

It is clear that, in general, there is no reason for the trimming sequences and to be equal. We assume that there exists a positive real constant such that as If the distribution is symmetric, the value of is equal to otherwise, it is less or greater than depending on the sign of the distribution skewness. For an illustration, see Figure 4 where we plot the ratio for several increasing sample sizes.

##### 3.3. Some Regularity Assumptions on

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

(H1) is differentiable on (H2)(H3)both and are regularly varying at zero with common index (H4)there exists a function not changing sign near zero such that where 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  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 in (2.6).

Remark 3.2. Assumptions (H1)–(H4) are satisfied by all weight functions with (see Section 4.1) and by function in (1.9) with These two examples show that the constants and may be positive or negative depending on application needs.

Remark 3.3. -functionals exist for any and such that However, Lemma 5.4 below shows that for we have . Then, recall (1.3); whenever the trimmed L-moments exist however Likewise, recall (1.9); whenever the two-sided deviation exists while

##### 3.4. Defining the Estimator and Main Results

We now have all the necessary tools to introduce our estimator of given in (1.1), when with . Let and be sequences of integers satisfying , , , , , , and the additional condition as First, we must note that since (see Remark 3.3) and since both and are consistent estimators of (see, Mason ), then we have for all large

Observe now that defined in (1.1) may be split in three integrals as follows:

Substituting and for and in and , respectively and making use of assumption (H3) and (3.5) yield that for all large

Hence we may estimate and by

respectively. As an estimator of we take the sample one that is

with the same constants as those in (1.11). Thus, the final form of our estimator is

A universal estimator of may be summarized by

where is as in (1.11). More precisely

where and is its complementary in

Note that for the particular case and the asymptotic normality of the trimmed mean has been established in Theorem of Csörgő et al. . The following theorem gives the asymptotic normality of for more general trimming sequences and and weighting function . For convenience, we set, for any and

and let

Theorem 3.4. Assume that with . For any measurable function satisfying assumption (H3) with index such that and for any sequences of integers and such that, , , , , , as , there exists a probability space carrying the sequence and a sequence of Brownian bridges such that one has for all large and therefore

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

Theorem 3.5. Assume that with . For any measurable function satisfying assumptions (H1)–(H4) with index such that and for any sequences of integers and such that , , , , , , and as , one has where

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

Corollary 3.6. Under the assumptions of Theorem 3.5 one has where with as in statement (ii) of Section 2 and as in assumptions (H2)-(H3) of Section 3.

##### 3.5. Computing Confidence Bounds for

The form of the asymptotic variance in (3.20) suggests that, in order to construct confidence intervals for an estimate of is needed as well. Using the intermediate order statistic , de Haan and Pereira  proposed the following consistent estimator for

where is a sequence of integers satisfying , and , as (the same as that used in (3.3)

Let be a given weight function satisfying (H1)–(H4) with fixed constants and Suppose that, for large enough, we have a realization of a sample from rv with df fulfilling all assumptions of Theorem 3.5. The confidence intervals for will be obtained via the following steps.

Step 1. Select the optimal numbers , , and of lower- and upper-order statistics used in (3.2) and (3.3).

Step 2. Determine , and .

Step 3. Compute, using (3.2), and Then deduce, by (3.10), the estimate

Step 4. Use (3.3) and (3.20) to compute and Then deduce, by (3.19), the asymptotic standard deviation Finally, the lower and upper -confidence bounds for respectively, will be where is the quantile of the standard normal distribution with

#### 4. Applications

##### 4.1. TL-Skewness and TL-Kurtosis When

When the distribution mean exists, the skewness and kurtosis coefficients are, respectively, defined by and with , , and 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  and Hosking . Recently, Elamir and Seheult  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

where are the trimmed L-moments defined in Section 1. The corresponding weight functions of (1.3) are defined as follows:

If we suppose that with then, in view of the results above, asymptotically normal estimators for and will be, respectively,

where

with and

Theorem 4.1. Assume that with For any sequencesof integers and such that ,, , , , , and as , one has, respectively, as where with

##### 4.2. Risk Two-Sided Deviation When

Recall that the risk two-sided deviation is defined by

where

An asymptotically normal estimator for when , is

where

Theorem 4.2. Assume that with such that , for any . Then, for any sequences of integers and such that, , , , and as , one has, as where

#### 5. Proofs

First we begin by the following three technical lemmas.

Lemma 5.1. Let and be two continuous functions defined on and regularly varying at zero with respective indices and such that . Suppose that is differentiable at zero, then

Lemma 5.2. Under the assumptions of Theorem 3.5, one has for any

Lemma 5.3. For any and one has where ,

Lemma 5.4. Under the assumptions of Theorem 3.5, one has where

Proof of Lemma 5.1. Let denote the derivative of Applying integration by parts, we get, for any Since the product is regularly varying at zero with index then as Therefore By using Karamata's representation (see, e.g., Seneta ), it is easy to show that Hence It is clear that (5.8) implies that is regularly varying at zero with index therefore is regularly varying with index . Then, Theorem by de Haan [60, page 15] yields This completes the proof of Lemma 5.1.

Proof of Lemma 5.2. We have By taking, in Lemma 5.1, and with and we get Likewise if we take and with and we have Note that statement of Section 2 implies that The last two relations, together with assumption (H2) and the regular variation of imply that 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 ]. For any we set Then may be rewritten into and the result of Lemma 5.3 follows immediately.

Proof of Lemma 5.4. From Lemma 5.3 we may write where By the same arguments as in the proof of Lemma 5.2, we infer that Therefore Next, we consider the third term which may be rewritten into Observe that It is easy to verify that function is regularly varying at zero with index Thus, by Theorem by de Haan  we have Hence By similar arguments we show that Therefore By analogous techniques we show that as we omit details. Summing up the three limits of , 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.  have constructed a probability space carrying an infinite sequence of independent rv's uniformly distributed on and a sequence of Brownian bridges such that, for the empirical process,

where is the uniform empirical df pertaining to the sample we have for any and for all large

For each , let denote the order statistics corresponding to Note that for each the random vector has the same distribution as Therefore, for we shall use the rv's to represent the rv's 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 defined in (3.9) may be rewritten into

where Integrating by parts yields

where

Next, we show that

(see, e.g., Balkema and de Haan [61, page 18]). Next, we use similar arguments to those used in the proof of Theorem by Csörgő et al. . For any write