Iterative Estimation of the Extreme Value Index

source: https://doi.org/10.7892/boris.117853 | downloaded: 1.2.2022

Iterative Estimation of the Extreme Value Index*

SAMUEL MU¨ LLER samuel.mueller@stat.unibe.ch

Department of Mathematical Statistics and Actuarial Science, University of Bern, 3012 Bern, Switzerland

JU¨ RG HU¨SLER juerg.huesler@stat.unibe.ch

Department of Mathematical Statistics and Actuarial Science, University of Bern, 3012 Bern, Switzerland Received May 15, 2004; Revised November 19, 2004; Accepted January 28, 2005

Abstract. Let {Xn, n Q 1} be a sequence of independent random variables with common continuous distribution functionFhaving finite and unknown upper endpoint. A new iterative estimation procedure for the extreme value indexis proposed and one implemented iterative estimator is investigated in detail, which is asymptotically as good as the uniform minimum varianced unbiased estimator in an ideal model. Moreover, the superiority of the iterative estimator over its non iterated counterpart in the non asymptotic case is shown in a simulation study.

Keywords: extreme value theory, tail index estimation, iterative estimator AMS 2000 Subject Classification: 62G32

1. Introduction

Let {Xn, n Q 1} be a sequence of independent random variables with common dis- tribution functionF, such thatFbelongs to the max-domain of attraction of G, denoted byF 2 Dð ÞG i.e., there exist constantsan> 0,bn2R such that forx2R

G xð Þ ¼ lim

n! 1P a ¹_n

, sup

x2R^jF^nðan

From Gnedenko (1943) it is known that F2 Dð ÞG if and only if G2G :2R , where

Gð Þ ¼x expð1þxÞ¹⁼

; 1þx>0;

andG^{ð Þ}

ð Þ¹⁼!expðxÞ; for !0;

interpretG0(x) as exp(je^jx).

*Supported by Swiss National Science foundation.

We consider the case that the distribution function Fhas finite upper endpoint

!(F) := sup{x: F(x) < 1} <1and thatFis continuous in the left neighborhood of !.

Hence e0.

Estimators for the extreme value tail indexbased on maximum-likelihood go back to the work of Hall (1982) and Smith (1985, 1987) and Smith and Weissman (1985) and it is well known that these estimators are not consistent for < j1/2. In the last three decades the estimation ofwas intensively studied and to list all relevant articles would go beyond the scope of this article. Very recent contributions among others are Ferreira et al. (2003), Mu¨ller (2003), and Paulauskas (2003).

The paper is organized as follows: In Section 2 an iterative procedure for the esti- mation of the extreme tail index is derived and the definition of the iterated tail index estimator is given. In Section 3 the main theorem is given, which shows that the defined estimator is asymptotically as good as the uniform minimum varianced un- biased estimator in an ideal model. Moreover, the superiority of the iterated estimator over its non iterated counterpart in the non asymptotic case is shown in a simulation study. The proof of the main theorem is given in Section 4.

2. Motivation and Definition

Falk (1994, 1995) considered the estimation problem in the setting of independent random variables X1, . . . ,Xnhaving distribution function Fwith a finite right endpoint and under the assumption that Fpossesses a density f in a left neighborhood of!(F), which satisfies for some> 0 the expansion

f xð Þ ¼expðb=Þ

ð!xÞ¹¹1þOð!xÞ⁼

ð1Þ asxtends to!from below for some< 0,> 0,b2R. LetX(1,n)eX(2,n)e. . .eX(n,n)

denote the order statistics ofX1, . . . ,Xnand let (kn,nQ1) be an intermediate sequence of integers, i.e., knY1andkn/n Y0 asnY1. Falk (1994) showed that

b !:¼ 1

kn

X^kn

j¼1

log !Xðnjþ1;nÞ

!Xð_nkn;nÞ

ð2Þ is in an ideal model an uniform minimum varianced unbiased estimator ( UMVUE) if the endpoint!is known. Falk (1995) replaced the endpoint!ofFby the sample maximum X(n,n)in the case of unknown!and it turned out that in the case<j1/2

b

_Falk:¼ 1 k_n 1

X^kn

j¼2

log Xð_n;nÞXð_njþ1;nÞ

Xð_n;nÞXð_nk_n;nÞ

ð3Þ is asymptotically as good as his nonrandom counterpart with known endpoint ! (Theorem 1.2. and 1.3. in Falk (1995)).

Replacing the endpoint!by the sample maximumX(n,n)in Equation (2) is not the only way to get an estimate for the tail index, actually any endpoint estimator could be used

instead. From an algorithmic point of view the following iterative procedure has the potential to produce better estimates for finite sample size than the tail index estimator b

_Falkdefined in Equation (3).

Step 1:Estimatebyb₁, where b₁is any tail estimator for negative. Step 2:Estimate!by an endpoint estimator!!b(b₁).

Step 3:Re-estimatereplacing!in Equation (2) by!!b(b₁) .

There exists numerous ways of implementing this iterative procedure. In this paper we will consider only one iterated estimator in detail. For the first step, let b₁be any con- sistent estimator for. For the second step we take the endpoint estimator of Hall (1982), which is based on a linear combination of the mlargest order statistics which has the property that for knownits asymptotic distribution has mean!and a variance which is a minimum among all such linear combinations if<j1/2. Thus our endpoint estimator used in the second step is

b

!

!_Hall;mð Þb₁ :¼ X^m

j¼1

a_jð ÞXb₁ ðnjþ1;nÞ; ð4Þ

where the weightsa= (a1, . . . ,am)^Tare given by a¼ ¹ ðv^T1vÞ1m1^T_m¹v

v v^T1v

ð Þ1^T_m¹1_m

1^T_m¹v2; ð5Þ

withL=_ijthe symmetricmmmatrix given by _ij¼ð2b₁þiÞðb₁þjÞ

ðb₁þiÞð Þj ;ji; and with 1_k:¼ð1; : : : ;1Þ^T

|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}

ktimes

;

v:¼ ðb₁þ1Þ

ð Þ1 ; : : : ; ðb₁þmÞ ð Þm

T

;

ð5Þ

whereG(

I

DEFINITION1 The iterated tail index estimatorb_m;n is defined by b

_m;n:¼ 1 kn1

X^kn

j¼2

log !!b_Hall;mXðnjþ1;nÞ

b

!

!_Hall;mXð_nkn;nÞ

: ð6Þ

REMARK 1 From Definition 1 it follows directly thatb_m;n is location and scale invariant.

3. Results

In the following Theorem it turns out that given < ¹₂ this iterated tail index estima- tor is asymptotically as good as the best estimator b_Falk.

THEOREM 1 Suppose that F satisfies(1)for some> 0, b; !2R and < ¹₂. If knis an intermediate sequence of integers, which satisfies

logn ffiffiffiffiffi kn

p !0; as n! 1; then

ffiffiffiffiffi kn

p b_m;nb_Falk¼oPð Þ:1

However, the efficiency for finite samples shows another behavior. To investigate the finite sample size performance of the iterated tail index estimator we perform simulation studies each based on 5,000 replications. We generatekupper order statisticsk2{1,000, 2,000, 4,000} from a power-function distribution with tail index ranging from j1.2 to j0.4 in steps of 0.05 andb_Falkas well asb_m;nform= 5 are calculated. Asymptotically the estimator is suitable for the cases < j1/2. But in the finite sample we can still consider this estimator for Q j1/2. Figure 1 shows the simulated relative efficiencies (re), which are calculated by the ratio of the simulated mean squared errors (mse) of the iterated tail index estimator form= 5 and of Falk’s estimator that is

reb_5;n;b_Falk

¼mseðb_FalkÞ mseb_m;n:

Figure 1. Relative efficiencies ofbm;n¼bb!!Hall;5ðbFalkÞ

and Falk’s estimator.

The efficiency curves suggest, that the iterated tail index is superior in the left neighborhood of =j0.5 and slightly less efficient for in a neighborhood ofj0.9.

Moreover, a closer look reveals that for increasing n the efficiency curve is approximating the horizontal line of equal efficiency. From Tables 1 and 2 we conclude that this approximation process is very slow. The simulated values fork= 16,000 andk= 64,000 are both based on 5,000 replications.

The influence of the number of upper order statistics used for estimating! is rather important as can be seen in Figure 2, which shows the simulated efficiencies for k = 2,000 in dependence of m 2{2, 3, 4, 5}. It indicates that the efficiency is increasing withm.

Overall the simulation results suggest that the proposed iterative procedure is superior than its non iterated counterpart.

REMARK 2 Theorem 1 is still holding, if b₁is replaced by any other consistent estimator for .

Moreover, it is interesting to note that even if a rather inefficient initial tail index estimator is used the superiority of the iterated tail index estimator over Falk’s estimator is still holding. For example letb₁ be Pickands estimator (Pickands, 1975) defined by

b

_Pðn;rÞ:¼ 1

log 2log Xð_nr;nÞXð_n2r;nÞ

Xðn2r;nÞXðn4r;nÞ

;

which is well known to be less efficient than Falk’s estimator if e j1/2. For sample sizek= 1,000 we continue the simulation study from above. Based on 5,000 simulation runs we calculate Falk’s estimator, Pickands estimator forr=)k/42, the iterated tail index estimator using Falk’s estimator (FalkYFalk), as well as Pickands estimator

Table 1. Relative efficiencies for=j0.6 in dependence ofk.

k mseðbFalkÞ mse b5;n

re b5;n;bFalk

1,000 8.1271

I

I

4,000 1.8412

I

I

16,000 0.3968

I

I

64,000 0.0912

I

I

Table 2. Relative efficiencies for=j0.9 in dependence ofk.

k mseðbFalkÞ mseb5;n

reb5;n;bFalk

1,000 8.0910

I

I

4,000 2.0248

I

I

16,000 0.4887

I

I

64,000 0.1247

I

I

(FalkYPickands) as initial tail index estimator. For comparison purpose we also calculate the shift and scale invariant tail-index moment estimator from Ferreira et al. (2003, Section 2.3) defined by

b

_FdHPð Þ ¼k N_n^{ð Þ2} 2N_n^{ð Þ1}2

2 Nn^{ð Þ1}

2

2Nn^ð2Þ

; ð7Þ

withN_n^{ð Þj} ¼¹_kPk1

i¼0 Xð_ni;nÞXð_nk;nÞj

, j= 1, 2. The relative efficiencies are shown in Figure 3 which underscores the superiority of the iterated tail index estimator.

Figure 2. Relative efficiencies ofbm;n¼b!!bHall;mðbFalkÞ

andbFalkfork= 2,000.

Figure 3. Relative efficiencies of Falk’s estimator compared to Pickands, the shift and scale invariant moment estimator, and two versions of the iterated tail index estimator form= 5 andk= 1,000.

REMARK 3 In the simulation study we generated the k upper order statistics directly from the power-function distribution and therefore, we do not have to specify n. However, for real data the choice of k has to be addressed. One possibility is to apply the automatic choice procedure of Reiss and Thomas (2001, p. 149) that is the optimal k* is given by

k*¼argmin

k

1 k

X

ik

ib_i;nmedb_i;n; : : : ;b_k;n

with 0 e b < 1/2, where b_i;n is some tail-index estimator based on the upper i order statistics.

4. Proofs

Let Xi= F^j1(1 jUi),i = 1, 2 , . . . , whereU1,U2, . . . are iid U(0;1) random variables.

From Proposition 1.1 (ii) in Falk (1994) the expansion F¹ð1qÞ ¼const

is obtained as qY0. With the Renyi representation U_ði;nÞ

n

i¼1¼^L S_i Snþ1

n i¼1

;

where Sj = ₁ + . . . +_j is the sum of jindependent and standard exponential random variables (see Corollary 1.6.9 in Reiss (1989)) we get the following results.

PROPOSITION 1 For1ejeknthe following relation holds uniformly for every1ejekn

F¹ 1Uð_1;nÞ

F¹ 1Uðjþ1;nÞ

¼

U_ð1;nÞ1þO U _ð1;nÞ U_ðjþ1;nÞ1þO U _ðjþ1;nÞ

¼^L S₁ Sjþ1

1þOðS₁=S_nþ1Þ 1þO Sjþ1

Snþ1

¼OPðð1=jÞÞ:

Proof of Proposition 1: TheOP((1=j)) part is taken from page 119 in Falk (1995).

The relation is holding uniformly since Sj=jis obviously more and more concentrating around 1 asjis increasing. More precisely, we can simply apply Chebychev inequality to give a uniform bound for the probabilityP(Sj=jew) for jQj0and smallw.

Í

Proof of Theorem 1: From Theorem 1.3 in Falk (1995) it is known under the same conditions as in our theorem ffiffiffiffiffi

kn

p jb_Falkb!j ¼oP(1):Hence ffiffiffiffiffi kn

p b_m;nb_Falk¼oPð Þ1 holds if and only if

ffiffiffiffiffi kn

p b_m;nb!

¼oPð Þ:1 ð8Þ

Moreover, by the definition of Hall’s endpoint estimator it follows that!!b_Hall;mis greater thanX(n,n). By Definition 1 the iterated tail index estimatorb_m;n is bounded from below byb_Falk

b

_m;n 1 k1

X^k

j¼2

log Xð_n;nÞXð_njþ1;nÞ

X_ðn;nÞX_ðnk;nÞ

¼b_Falk; ð9Þ

which follows from the inequality z₁

z2

z₁þ"

z2þ"; if " >0 and 0<z1 z2: ð10Þ Hence, for!!b_Hall;m2[Xð_n;nÞ; !] the results follows immediately from Theorem 1.3 in Falk (1995) and from the inequalities (9) and (10). Thus it suffices to prove the Theorem for the case that

b

!

!_Hall;m> !: ð11Þ

Because of the location invariance of the iterated tail index estimator we assume without loss of generality that

!¼0: ð12Þ

Then we consider b

_m;nb!¼b_m;n 1 kn

X^kn

j¼2

log X_ðnjþ1;nÞ Xð_nkn;nÞ

1 kn

log X_ðn;nÞ Xð_nkn;nÞ

|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

¼:An

: ð13Þ

Hence using Equation (8) the statement of the theorem holds if ffiffiffiffiffi kn

p

ffiffiffiffiffi k_n p kn1

k_n b_m;n 1 k_n

X^kn

j¼2

log Xð_njþ1;nÞ

Xð_nkn;nÞ

¼o_pð Þ:1 ð14Þ For (14) we have

0 1 k_n

X^kn

j¼2

log !!b_Hall;mXð_njþ1;nÞ

b

!

!_Hall;mXð_nkn;nÞ

1 k_n

X^kn

j¼2

log Xð_njþ1;nÞ

Xð_nk_n;nÞ

¼ 1 kn

X^kn

j¼2

log !!b_Hall;mXðnjþ1;nÞ

b

!

!_Hall;mXð_nkn;nÞ

!

1 kn

X^kn

j¼2

log X_ðnjþ1;nÞ!!b_Hall;m Xð_njþ1;nÞ

¼ 1 kn

X^kn

j¼2

log 1 !!b_Hall;m Xð_njþ1;nÞ

:

Moreover from the assumption ofb₁we have thata1, . . . , amareOP(1). Regarding (11) and (12) Hall’s endpoint estimator is bounded from above by

b

!

!_Hall;m dXðnmþ1;nÞ; ð15Þ

where d = OP(1) is chosen properly, for example any d Q 1 + (jX(n j m + 1,n))² is suitable. For anyd⁰ Q1 andx> 0 we have

d⁰log 1ð þxÞ ¼logð1þxÞ^d0

log 1ð þd⁰xÞ log 1ð þxÞ; ð15Þ thus for any fixedd⁰> 0 we have

1ffiffiffiffiffi k_n p X^kn

j¼2

log 1þXð_nmþ1;nÞ

X_ðnjþ1;nÞ

¼o_Pð Þ1

, 1 ffiffiffiffiffi kn

p X^kn

j¼2

log 1þd⁰_ðnmþ1;nÞ X_ðnjþ1;nÞ

!

¼oPð Þ:1 ð15Þ

Hence withd⁰ =dand inequality (15) we get 1ffiffiffiffiffi

k_n p X^kn

j¼2

log 1þdXð_nmþ1;nÞ

Xðnjþ1;nÞ

¼o_Pð Þ1

) 1 ffiffiffiffiffi k_n p X^kn

j¼2

log 1þ !!b_Hall;m Xðnjþ1;nÞ

¼o_Pð Þ:1 ð15Þ

Hence we have to show, that 1ffiffiffiffiffi

k_n p X^kn

j¼2

log 1þXð_nmþ1;nÞ

X_ðnjþ1;nÞ

¼ 1 ffiffiffiffiffi k_n p X^kn

j¼2

log 1þF¹1Uð_m;nÞ F¹1Uð_j;nÞ

!

¼:Bn¼oPð Þ:1

ð15Þ

Proposition 1 implies that

F¹1U_ðm;nÞ F¹1U_ðj;nÞ¼

U_ðm;nÞ1þO U _ðm;nÞ U_ðj;nÞ1þO U _ðj;nÞ

¼^L Sm

S_j

1þOðSm=Snþ1Þ 1þO Sj

Snþ1

¼O_Pððm=jÞÞ

ð16Þ

holds uniformly. Using the approximation log(1 +z)õzas zY0

Bn¼OP k¹⁼²_n X^kn

j¼2

j

!

¼OPk_n^1=2þ

¼oPð Þ;1 ð16Þ

and also ffiffiffiffiffi kn

p

n

p logkn

¼oPð Þ:1

Í

Acknowledgments

The authors thank a referee for his constructive suggestions which lead to improvement in the readability and clarity of the paper. This research was supported by the Schweizer Nationalfonds.

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