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Asymptotic Normality of Hill’s Estimator under Weak Dependence

By Boualam Karima and Berkoun Youcef

Submitted: November 24th 2018Reviewed: January 18th 2019Published: February 26th 2020

DOI: 10.5772/intechopen.84555

Downloaded: 43

Abstract

This note is devoted to the asymptotic normality of Hill’s estimator when data are weakly dependent in the sense of Doukhan. The primary results on this setting rely on the observations being strong mixing. This assumption is often the key tool for establishing the asymptotic behavior of this estimator. A number of attempts have been made to relax the assumption of stationarity and mixing. Relaxing this condition, and assuming the weak dependence, we extend the results obtained by Rootzen and Starica. This approach requires less restrictive conditions than the previous results.

Keywords

  • tail index
  • Hill’s estimator
  • regularly varying function
  • linear process
  • weak dependence

1. Introduction

Extreme value theory (EVT) is a branch of statistics which focus on modeling and measuring extremes events occurring with small probability. Rare events can have severe consequences for human and economic society. The protection against these events is therefore of particular interest. EVT have been extensively applied in various many fields including hydrology, finance, insurance and telecommunications. Unlike most traditional statistical analysis that deal with the center of the underlying distribution, EVT enables us to restrict attention to the behavior of the tails of the distribution which is strongly connected to limiting distribution of extremes values, i.e., maximum or minimum of a sample.

Let X1,X2,,Xnbe i.i.d random variables with a common distribution Fand let XnX1the order statistics pertaining to X1,X2,,Xn, where Xn=MinXiand X1=MaxXi. Suppose that there exist two normalizing constants an, bn,bn>0and a nondegenerate distribution function Hsuch that Fnbnx+anHx, for every continuity point x of H, then Hbelongs to one of the three types

  • Type I (β>0): Φβx=expxβifx>00ifx0

This class is often called the Frechet class of distributions (fat tailed distribution).

  • Type II (β<0): Ψβx=expxβifx<01ifx0

This class is called Weibull class of distributions (short tailed distributions).

  • Type III (β=0): Λ0x=expex,xR,

called the Gumbel type (moderate tail).

This result is known as Fisher-Tippett theorem (see [14]) or the extreme value theorem.

These three family of distributions can be nested into a single representation called the generalized extreme value distribution (GEV) and is given by

Hγx=exp1+γx1γ,1+γx>0

This representation is useful in practice since it nets three types of limiting distributions behavior in one framework.

For a positive γ=1β, we recover the Frechet distribution with, negative γ=1βcorresponds to the Weibull type, and the limit case γ0describes the Gumbel family. The shape parameter γgoverns the tail behavior of the distribution. The extreme value theorem remains true if condition of independence of the rv’s is replaced by the requirement that the form a stationary sequences satisfying a weak dependence condition called distributional mixing condition (e.g., Leadbetter et al. [20]).

The problem of estimating the tail index has received much attention and a variety of estimators have been proposed in the literature in the context of i.i.d observations, see Hill [16], Pickands [23], Dekkers and De Haan [12]. We focus on the popular Hill estimator (available only for β>0) based on the k- upper order statistics and defined as follows.

Hk,n=1ki=1klogXiXk+1E1

where k=knand knnis an intermediate sequence that is, kn,knn0asnThe asymptotic behavior of this estimator has been extensively investigated in the i.i.d setup. Mason [22] proved weak consistency of Hk,nfor any intermediate sequence knand Deheuvels et al. [11] derived strong consistency under the condition that kloglogn,as n. Under varying conditions on the sequence knand the second-order behavior of F, asymptotic normality of Hk,nwas discussed among others in Hall [15], Davis and Resnick [7], Csörgo and Mason [5], De Haan and Peng [10], De Haan and Resnick [9].

Hill estimator can still be used for dependent data. In this context, we give below the asymptotic behavior of this estimator.

Hsing [19] and Rootzen et al. [26] established the consistency and the asymptotic normality of Hk,nunder some general conditions for strictly stationary strong mixing sequences. Brito and Freitas [3] also gave a simplified sufficient condition for consistency, appropriate for applications.

Resnick and Starica [24, 25] proves the weak consistency of Hill’s estimator for certain class of stationary sequences with heavy tailed observations which can be approximated by m-dependent sequences. Using this result, they also proved consistency and asymptotic normality of this estimator for an infinite order moving average and autoregressive sequences with regularly varying marginal distribution. However, Ling and Peng [21] extend their results to an ARMA model with i.i.d residuals, based on the estimated residuals, this method can achieve a smaller asymptotic variance than applying hill’s estimator to the original data.

Hill [17] proved that Hk,nstill asymptotically normal for dependent, heterogeneous processes with extremes that form mixingale sequences and for near-Epoch-dependent process.

Hill [18] extends the results of Resnick and Starica [24] and Ling and Peng [21] to a wide range of filtered time series satisfying β- mixing condition. Without using the strong mixing condition, Zhang and McCormick [28], established the asymptotic normality of Hill’s estimator, for shot noise sequence provided some mild conditions on the impulse response function.

As mentioned above, the asymptotic normality of Hill’s estimator has so far been proved for dependent data under various mixing conditions, but not for weak dependent which is the aim of this note. This notion of weak dependence is more general than the classical frameworks of mixing, associated sequences and Markovian models. This type of dependence covers a broad range of time series models.

In order to establish the asymptotic behavior of Hill’s estimator in this setting, we first extend the result of Rootzen et al. [26] to random variables, which fulfill the weak dependence condition. Secondly, we derive the asymptotic normality of the Hill estimator when the observations are generated by a linear process satisfying the η- weak dependence condition (see Boualam and Berkoun [2]). This result extends the work of Resnick and Starica [24].

The novelty of using weak dependence instead of mixing dependence lies in the fact that conditions ensuring the normality of the Hill estimator are weaker than the existing conditions.

To make the chapter self-contained, we present definitions and some important results that we need in the sequel.

2. Definitions and auxiliary results

2.1 Regularly varying functions

We start with some background theory on regular variation.

2.1.1 Regularly varying

A positive measurable function 1Fis called regularly varying function at infinity with index β,β>0(written 1FRVβ) if

limt+1Ftx1Ft=xβ,x>0

Recall that Fbelongs to the domain of attraction of H1β,β>0if and only if 1FxRVβ.

To establish the asymptotic normality of Hk,n, a second order regular variation is imposed on the survival function distribution 1F.

2.1.2 Second-order regular condition

A function 1Fis said to be of second-order regular variation with parameter ρ0, if there exists a function gthaving constant sign with limt+gt=0and a constant c0such that

limt+1Ftx1Ftxβgt=cxβ1xμρ1,x>0E2

Then it is written as 1F2RVβρand gtis referred as the auxiliary function of 1F. The convergence in (2) is uniform in xon compact intervals of 0+.

Under this assumption, de Haan and Peng derived the asymptotic expansion

Hk,n=γ+γkZk+g(n/k1ρ(1+op1

where Zk=ki=1kEi1kand Eiis a sequence of i.i.d standard exponential random variables. Hence, choosing ksuch that kgn/k=λ0leads to asymptotic normality of kHk,nγwith mean λ1ρand variance γ2.

2.2 Strong mixing condition and weak dependence

Several ways of modeling dependence have already been proposed. One of the most popular is the notion of strong mixing introduced by Rosenblatt [27].

2.2.1 Strong mixing

The sequence Xnnis called strongly mixing with mixing coefficient

αn,l=supPABPAPB:AF1,pBFp+l,n1pnl
iflimnαn,l=0

Fi,jis the σfield generated by Xp:ipj

It turns out certain classes of processes are not mixing. Inspired by such problems, and in order to generalize mixing and other dependence, Doukhan and Louhichi introduced a new weak dependence condition.

Recall that random variables U, Vwith values in a measurable space χare independent if for some rich enough class Fof numerical functions on χ

CovfUgV=0,f,gF

Weakening this assumption leads to definition of weak dependence condition. More precisely, assume that, for convenient functions fand g, covf"past"g"future"converge to zero as the distance between the “past” and the “future” converge to infinity. Here “past” and “future” refer to the values of the process of interest. This makes explicit the asymptotic dependence between past and future.

Now we describe the notion of weak dependence (in the sense of Doukhan and Louhichi) considered here (see [13]).

2.2.2 Weak dependence

A process Xnnis called ε£nΨ-weakly dependent if there exists a function Ψ:R+2R+and a sequence ε=εllNdecreasing to zero at infinity, such that for any hk£u×£vand uvN2

CovhXi1XiukXj1XjvΨLiphLipkuvεl

For any i1iuand j1jvwith i1<.<iuiu+lj1<.<jv.

£ndenotes the class of real Lipschitz functions, bounded by 1 and defined on RnnN. Lipfdenotes the Lipschitz modulus of continuity of function f, that is

Lipf=supxyfxfyxy1

with xy1=i=1nxiyi.

Specific functions Ψyield variants of weak dependence appropriate to describe various examples of models:

  • η-weakly dependence for which ΨLiphLipkuv=uLiph+vLipk

  • λ-weakly dependence for which ΨLiphLipkuv=uLiph+vLipk+uvLiphLipk

  • κ-weakly dependence for which ΨLiphLipkuv=uvLiphLipk

  • ζ-weakly dependence for which ΨLiphLipkuv=minuvLiphLipk

Several class of processes satisfy the weak dependence assumption, as the Bernoulli shift, a Gaussian or an associated process, linear process, GARCHpqand ARCHprocesses (more examples and details can be found in the Dedecker et al. [8]).

The coefficients of weak dependence have some hereditary properties. If the sequence Xttis κ,λor θweakly dependent, then for a Lipschitz function h, the sequence hXttis also weakly dependent.

Mixing conditions refer to σalgebras rather than to random variables. The main inconvenience of mixing coefficients is the difficulty of checking them. The weak dependence in the sense of Doukhan is measured in terms of covariance which is much easier to compute than mixing coefficients.

3. Asymptotic normality of Hill’s estimator under strong mixing condition

In order to proof the asymptotic normality of Hill estimator, we use the approach of Rootzen described in the following.

Let Ynnbe a sequence of stationary strong mixing random variables with mixing coefficients αn,lntending to zero at infinity and ln=on.Suppose that the common distribution function Fof Ynis such that

limt+1Ft+x1Ft=eβx,x0E3

i.e., 1Fxdecays approximately in an exponential manner eβxas xor (by log transformations) as an approximate Inverse power law in the sense of regular variation.

Rootzen et al. [26] considered the estimator

βn=1ki=1kYiYkE4

Where Y1,Y2,,Ynare the order statistics pertaining to a sample Y1,Y2,,Yn.

Under certain conditions, they proved that

knλnβnβndN01E5

where βn=nknEY1un+and λn=nknrnvari=1rn{Yjun1Yjun01β1Yjun0}.

The sequences un,rnare chosen such that limnpnαn,ln+lnn=0, limnn1Funkn=1and

rn=npn,withpn,rnn0asnE6

Note that if we replace Y=logXin (13), we find the expression of the Hill estimator.

3.1 Hill’s estimator in case of infinite order moving average process

Resnick and Starica [24] generalize the Hill estimator for more general settings with possibly dependent data especially for infinite moving average model and AR(p) process.

For a sequence Xnnof random variables generated by a strong mixing linear process, with common distribution Fsatisfying the following von Mises condition

limt+tft1Ft=βE7

Resnick and Starica [24] have adopted the approach of Rootzen applied to Ynn=logXnnfor proving the normality of Hill?s estimator. It is well known that if (15) holds then 1FRVβ.

Let Xttbe a strictly stationary linear process defined by

Xt=i0ciεtiE8

εttis an i.i.d sequence of random variables with marginal distribution satisfying

G¯x=1Gx=xβlx,x>0,β>0E9

lis a slowly varying function at infinity and ciiis a sequence of real numbers satisfying certain mild summability conditions.

Throughout this paper, assume that:

j=0cjδ<,forsome0<δ<1βE10

then (Cline [4]) j=0cjεj<which implies that j=0cjεj<(Datta and McCormick [6]). Next, assume that

j=1k=0cj+kβckβlogcj+kckcj+kck<E11

then Xt=i0ciεtiis regularly varying.

As a direct consequence of the lemma 2.1 for Resnick and Starica [24] we have

λnλ=1β21+2j=1k=0ckβck+jβj=0ckβ,asnE12

Note that λis finite and depends only on the coefficients cj.

Theorem 3.1 (Resnick and Starica [24]) Let Xttbe a strongly mixing linear process and assume that conditions (7), (9), (10) and (11) hold. If the intermediate sequence kis such that

liminfnnk32>0orlimsupnnk32<E13
and 1F2RVβρwith the auxiliary function gsatisfying:
ngbnk0,asn  wherebis the quantile functionE14

Then

kHk,n1βN0λ

Note that the second order condition imposed on F implies condition (11) required by Rootzen (see Rootzen et al. [26], Appendix. p44). Condition (13) on the intermediate sequence kallows us to prove the existence of sequence rnnpreviously defined.

3.2 Hill’s estimator in case of AR(p)process

Similar result to 3.1 where obtained by Rootzen et al. [26] for ARpprocess.

Consider a stationary, pth-order autoregression Xttsatisfying

Xt=i=1pϕiXti+εt,tNE15

We assume that the common distribution of i.i.d sequence εisatisfy condition (9). Under mild conditions the process (15) has a causal representation of the form (8); if these conditions are not verified then the procedure of applying the Hill estimator directly to an autoregressive process is first to estimate the autoregressive coefficients and then estimating βusing estimated residuals.

We assume that we have a sequence ϕ̂n=ϕ̂1nϕ̂pn,n1,of consistent estimators for the coefficients of the autoregression such that dnϕ̂nϕSwhere Sis nondegenerate random vector and dn. So that εtε̂tn=i=1pϕ̂inϕiXti.

Applying the Hill estimator to the estimated residuals ε̂1n,ε̂2n,,ε̂nn, Resnick and Starica [24] obtained that, if the distribution G¯ε2RVβρand the sequence kis chosen to satisfy the condition (13) and kbnkbnk=odn,as nthen,

k1ki=1klogε̂inε̂k+1n1βdN01β2

For AR(p) process, the approach used is quite different than the previous one. Instead, of working with the original observations, the authors used the estimated residuals in order to get the asymptotic normality of Hill’s estimator. This method achieves a smaller variance of the Hill estimator than the first one.

4. Asymptotic normality of Hill’s estimator under weak dependence

Following the approach of Rootzen et al. [26], we investigate the asymptotic normality of the Hill estimator when the observations are drawn from a causal weakly dependent process in Doukhan sense. In order to check the asymptotic normality of the Hill estimator, we first extend the normality asymptotic of βndefined by (13) for η-weakly dependent random variables. Therefore, applying this to the process Yttwhere Yt=logXt, we obtain the desired result.

Let Ynnbe a stationary sequence of random variables η-weakly dependent. We suppose that for each sequences pnnand rnnthe condition (6) is satisfied and lnnis such that

limnlnn=0,limnlnrn=0andlimnnln12μrn=0,μ>12E16

To establish the asymptotic normality of Hill’s estimator, we need to show that under suitable conditions and even if the function logarithmdoes not satisfy the conditions of proposition 2.1 of [8], the sequence Ytt=logXttis η-weakly dependent and possess the hereditary property.

Lemma 4.1 (Boualam and Berkoun [2]) Let Xttbe a stationary sequence of positive random variables η-weakly dependent. Suppose that there exists a constant C>0, such that X1pC, with p>1then Yttwhere Yt=logXtis also η-weakly dependent with ηYr=Oηpp1r.

Let Xttbe a causal linear process given by (8) where

ck=Okμ,withμ>1/2E17

then Xttis η-weak dependent with ηln=O1lnμ1/2(see Bardet et al. [1]).

Now, we extend theorem 4.3 of Rootzen et al. [26] obtained for strong mixing sequences to η-weakly dependent random variables.

Theorem 4.1 (Boualam and Berkoun [2]) Let Ynnbe a stationary sequence of η-weakly dependent random variables. If condition limnpnαn,ln+lnn=0of theorem 4.3 of Rootzen et al. [26] is replaced by (16), then

knλnβnβndN01E18

The above results allows us to state our main result which extend the result obtained by Resnick and Starica [24] for strong mixing to weak dependent sequences.

Theorem 4.2 (Boualam and Berkoun [2]) Let Xttbe a linear process given by (16) with common distribution F, satisfying assumptions (7), (10), (11), (13),(14), (16) and (17) then

kHk,n1βN0λ

5. Conclusion

In a primary work, Hsing showed the asymptotic normality of Hill’s estimator in a weak dependent setting under suitable mixing and stationary conditions. Similar results have derived for data with several types of dependence or some specific structures. These conditions have been considerably weakened in Hill. We extend the results obtained by Rootzen and Resnick and Starica. The contribution of this note is threefold. First, the weak dependence in the sense of Doukhan is more general than the framework of mixing and several class of processes possesses this type of dependence. It is important to stress that this dependence allows us to prove the asymptotic normality of the Hill estimator without requiring the assumption that the linear process enjoys the strong mixing property. Consequently, the conditions ensuring the asymptotic normality are weakened with our approach. Second, mixing is hard to verify and requires some regularity conditions. However, using weak dependence which focus on covariances is much easier to compute and this assumption is more often checked by several process. Third, our work can be extended to linear process with dependent innovations (under mild conditions, linear process with dependent innovations is η-weak dependent).

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Boualam Karima and Berkoun Youcef (February 26th 2020). Asymptotic Normality of Hill’s Estimator under Weak Dependence, Statistical Methodologies, Jan Peter Hessling, IntechOpen, DOI: 10.5772/intechopen.84555. Available from:

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