The Bootstrap Methodology in Statistics of Extremes& ... - Springer Link

Download PDF
0 downloads 0 Views 706KB Size Report
Comment
Key words. bootstrap methodology, optimal sample fraction, semi-parametric .... 4 the performance of a bootstrap selection of k0ЕnЖ of the type of the one ...
Extremes 4:4, 331±358, 2001 # 2002 Kluwer Academic Publishers. Manufactured in The Netherlands.

The Bootstrap Methodology in Statistics of ExtremesÐChoice of the Optimal Sample Fraction M. IVETTE GOMES D.E.I.O. and C.E.A.U.L., University of Lisbon (F.C.U.L.), Department of Statistics, Faculty of Science of Lisbon, EdifõÂcio C2, Piso 2, Cidade UniversitaÂria, Campo Grande, 1749-016 Lisboa, Portugal E-mail: [email protected] ORLANDO OLIVEIRA D.E.I.O. and C.E.A.U.L., University of Lisbon (F.C.U.L.), Department of Statistics, Faculty of Science of Lisbon, EdifõÂcio C2, Piso 2, Cidade UniversitaÂria, Campo Grande, 1749-016 Lisboa, Portugal E-mail: [email protected] [Received February 9, 1999; Revised and Accepted February 7, 2002] Abstract. The main objective of statistics of extremes is the prediction of rare events, and its primary problem has been the estimation of the tail index g, usually performed on the basis of the largest k order statistics in the sample or on the excesses over a high level u. The question that has been often addressed in practical applications of extreme value theory is the choice of either k or u, and an adaptive estimation of g. We shall be here mainly interested in the use of the bootstrap methodology to estimate g adaptively, and although the methods provided may be applied, with adequate modi®cations, to the general domain of attraction of Gg ; g [ R, we shall here illustrate the methods for heavy right tails, i.e. for g > 0. Special relevance will be given to the use of an auxiliary statistic that is merely the difference of two estimators with the same functional form as the estimator under study, computed at two different levels. We shall also compare, through Monte Carlo simulation, these bootstrap methodologies with other data-driven choices of the optimal sample fraction available in the literature. Key words. bootstrap methodology, optimal sample fraction, semi-parametric estimation, statistical theory of extremes AMS 2000 Subject Classi®cation.

1.

PrimaryÐ62G32, 62F40 SecondaryÐ65C05

Introduction and preliminaries

Let X1 ; X2 ; . . . ; Xn be independent, identically distributed (i.i.d.) random variables (r.v.s) from an underlying population with unknown distribution function (d.f.) F, and let X1:n  X2:n      Xn:n denote the associated ascending order statistics (o.s.). We then have the validity of Gnedenko's theorem (Gnedenko, 1943; Galambos, 1987), i.e. if there exist attraction coef®cients fan > 0gn  1 and fbn gn  1 , and a non-degenerate d.f. G such

332

GOMES AND OLIVEIRA

that the maximum Xn:n , linearly normalized, converges to G, then G is of the type of an extreme value (EV) d.f., Gg …x† :ˆ exp

n

…1 ‡ gx†

1=g

o

;

1 ‡ gx > 0;

g [ R:

…1†

The main objective of statistical theory of extremes is the prediction of rare events, and thus the need for an adequate estimation of parameters of rare events, such as high quantiles, return periods, the extremal index, the mean size of clusters of exceedances of high levels, and many other parameters related to natural ``disasters''. The primary question has however been for a long time the estimation of the tail index g. One of the most recent and general approaches has been the semi-parametric one, where it is merely assumed that F belongs to the domain of attraction of Gg , the estimation of g being then based on the largest k o.s. in the sample or on the excesses over a high level u. The question that has been often addressed in the practical applications of extreme value theory is the choice of either k or u. There have appeared some practical applications with a weak theoretical support, and a few nice asymptotic results, which usually require the consideration of new tuning parameters, and thus the need for a sensitivity analysis related to those extra parameters introduced. A great variety of semi-parametric estimators, gn …k†, of the tail index g have the same type of problemsÐconsistency for intermediate ranks, i.e. we need to have k ˆ kn ! ?, and kn =n ! 0, as n ! ?, high variance for small values of k, and high bias for large values of k. A usual approach to optimality problems in statistics is to require that the mean squared error (MSE) is minimal. However for semi-parametric estimators gn …k† the second moment may not exist. So we need to modify this criterion. In this paper we shall concentrate on Hill's estimator (Hill, 1975), g…1† n …k† :ˆ

k 1X ‰ln Xn k iˆ1

i ‡ 1:n

ln Xn

k:n Š:

…2†

Let us put U…t† :ˆ F/ …1 1=t†; t > 1, where F/ denotes the generalized inverse function of F. We shall assume that there exists a function A of constant sign and going to 0 as t ? ?, such that lim

ln U…tx†

t??

ln U…t† A…t†

g ln x

ˆ

xr

1 r

;

…3†

for every x > 0, where r … 0† is a second order parameter. Then we have, asymptotically, the following distributional representation for the Hill estimator in (2), d g…1† n …k† ˆ



g 1 A…n=k† g ‡ p P…1† ‡ 1 r k

 …1 ‡ op …1††;

…4†

333

THE BOOTSTRAP METHODOLOGY IN STATISTICS OF EXTREMES

where P…1† is a standard Normal r.v. For a proof seen de Haan and Peng (1998). The limit in (3) must be of the stated form, and jA…t†j [ RV pr (from theorem 1.9 of Geluk and de Haan, 1987). It thus follows that if k is such that k A…n=k† ? l, ®nite, as n ? ?, then ph …1† k gn …k†

g

i



d

?N

l

1

r

 ;g ; 2

as

n ? ?;

…5†

i.e. we may have a non-null asymptotic bias. This asymptotic normality of Hill's estimator has ®rst been derived by Hall (1982),pHaeusler and Teugels (1985) and Goldie and Smith  (1987), among others. Moreover, if k A…n=k† ? ?, h

…1†

gn …k†

g

A…n=k†

i 1

p

?

1

r

:

…6†

p So we see that whether the limit of k A…n=k† is ®nite or in®nite, we can always specify …1† the rate at which gn …k† converges to g. A ®rst possible substitute for the MSE is (cf. (4))  2 g …1† 1 g2 A2 …n=k† p   A…n=k† AMSE‰g…1† …k†Š :ˆ E ‡ ˆ ‡ ; P n 1 r k …1 r†2 k

…7† …1†

depending on n and k. It is then possible to see that if r < 0 and AMSE‰gn …k†Š is minimal at k0 ˆ k0 …n†, then p g…1 r† k0 A…n=k0 † ? p ; as n ? ? 2r …1†

(Dekkers and de Haan, 1993). This means that the rate at which gn …k† converges to g is minimized under the regime (5) and not (6). This gives rise to the following de®nition of optimality: k ˆ k…n† is called optimal if it minimizes  E g…1† n …k†

2

g Ijg…1† …k† n

gj 1, we have 

‰k0 …n1 †Š  k0 …n2 †

a



na1 n na n2

2r=…1

2r†

ˆ fk0 …n†ga

1

…1 ‡ op …1††:

It is then enough to choose n2 ˆ n…n1 =n†a , in order to have independence of r. If we put n2 ˆ n21 =n, i.e. a ˆ 2, we have 

‰k0 …n1 †Š2 ˆ k0 …n†…1 ‡ op …1††;  k0 …n2 † …1†

as

n ? ?:

…15† 

…1†

Also MSE‰gn1 …k1 † gn …kaux †jXn Š is minimal at a value k0 …n1 † asymptotically  equivalent to k0 …n1 †. Then, for an initial sub-sample of size n1 ˆ O…n1 e †, 0 < e < 1=2, we take another sub-sample of size n2 ˆ n21 =n, and consider, for a suitable intermediate kaux , 2h i2 3  k0 …n1 † 6 7 k^0 …n; kaux ; n1 † :ˆ 4  2 5; k0 …n1 =n†

  ^ gn;n1 …kaux † :ˆ g…1† n k0 …n; kaux ; n1 † ;

…16†



where [x] denotes the integer part of x, and for i ˆ 1 and 2, k0 …ni † is once again the …1† …1† empirical counterpart of arg minki MSE‰gni …ki † gn …kaux †jXn Š. The algorithm (for a single data set) is thus the following: 1. Given the sample size n, consider apsub-sample size n1 and n2 ˆ n21 =n. Fix a suitable  value for kaux , for instance kaux ˆ 2 n, like in Drees and Kaufmann (1998); …1† 2. For the sample …x1 ; x2 ; . . . ; xn †, compute gn …k†; k ˆ 1; 2; . . . ; n 1;

339

THE BOOTSTRAP METHODOLOGY IN STATISTICS OF EXTREMES

3. For l from 1 till B ˆ 250, generate independently B bootstrap samples …x1 ; . . . ; xn2 † and …x1 ; . . . ; xn2 ; xn2 ‡ 1 ; . . . ; xn1 †, of sizes n2 and n1 , respectively, from the empirical d.f. 

Fn …x† ˆ

n 1X I ; n i ˆ 1 ‰Xi  xŠ

obtain …tn1 ;l …k†; tn2 ;l …k††; 1  l  B, the observed values of the statistic 



Tni …k† ˆ g…1† ni …k†

g…1† n …kaux †jXn ; i ˆ 1; 2;

and get B  2 X  d * …ni ; k† ˆ 1 MSE tni ;l …k† ; B lˆ1 

4. Obtain k0 …ni † :ˆ arg min1  k  n 5. Compute

1

k ˆ 1; 2; . . . ; ni

1;

i ˆ 1; 2;

d * …ni ; k†; i ˆ 1; 2; MSE

"  # ‰k0 …n1 †Š2 ^ k0 …n; kaux ; n1 † :ˆ ;  k0 …n2 † if k^0 6 [ ‰1; n† skip to another method; otherwise, continue; …1† 6. Obtain gn;n1 …kaux † :ˆ gn …k^0 …n; kaux ; n1 ††. Remarks: i. In simulations, we generate a sample from a given model, in item 2 of the algorithm, and whenever k^0 6 [ ‰1; n† (in item 5), we identify the problem as a non-convergence of the method, and go to item 2, generating a new sample. ii. If there Pare negative elements in the sample, the value of n should be replaced by n‡ ˆ niˆ 1 I‰Xi > 0Š (the number of positive values in the sample). Analogously for n1 and for n2 . iii. The use of the sample of size n2 ; …x1 ; . . . ; xn2 †, and of the sample of size n1 ; …x1 ; . . . ; xn2 ; xn2 ‡ 1 ; . . . ; xn1 †, with independent xj led us to increase the precision of the result with a smaller B, the number of bootstrap samples generated in item 3. This is quite similar to the use of the simulation technique of common random numbers in comparison problems, when we want to decrease the variance of a ®nal answer to z ˆ y1 y2 , inducing a positive dependence between y1 and y2 . iv. The Monte Carlo procedure in item 3 may be replicated if we want to associate easily a standard error to the optimal sample fraction, i.e. we may repeat items 3 and 4 r … j† … j† times, obtain …k0 …n1 †; k0 …n2 ††; 1  j  r, and take as overall estimates ˆ P … j† r k 0 …ni † ˆ 1r j ˆ 1 k0 …ni †, i ˆ 1; 2. The value of B may also be adequately chosen.

340

GOMES AND OLIVEIRA

v. Although aware of the fact that k must be intermediate, in Sections 2, 3 and 4 we have used the entire region of search of arg mink MSE…k†, the region 1  k  n 1. Indeed, whenever working with small values of n; k ˆ 1 or k ˆ n 1 are possible candidates to the value of an intermediate sequence for such n. This does not apply so much to Hill's estimator, for which MSE…k† has a nice U-shaped behavior, but to several other semi-parametric estimators of parameters of rare events and to the auxiliary statistics used in Sections 3 and 4. Indeed, to search through a region for instance of the type ‰ln n; n= ln nŠ, like it is done in Draisma et al. (1999), may be dangerous for some semiparametric statistics and for not very large values of n, due to unrealistic minimal values attained at the border. Here we have again the old controversy between theoreticians and practionersÐkl ˆ ‰c ln nŠ ‡ 1 is intermediate for every constant c, and if we take for instance c ˆ 1=10; kl ˆ 1 for every n  22,026. Also, Hall's formula for the asymptotic optimal level, 2 3 1=…1 2r† k0 …n†*…a 2r …1 r† n 2r = …2b2 … r† †† , valid for Hill's estimator and for models in (10), may lead, for a ®xed n, and for several choices of a and b, to values of k0 …n† either equal to 1 or to n 1 according as r is close to 0 or quite small, respectively. It may now be argued that, while at the beginning we had only a tuning parameter, the value of k, we now have two (if no more) tuning parameters, the values of kaux and of n1 . Some obvious questions are immediately put forward: ®rst, what is the sensitivity of k^0 …n; kaux ; n1 †, relatively to the choice of the subsample size n1 ; second, how does the …1† method depend on the initial choice of kaux , and consequently on gn …kaux †. The simulation study, carried out for different models with heavy tails, enabled us to conclude that given kaux , the sample path of k^0 …n; kaux ; n1 † is ``almost independent'' of n1 . This means that although aware of the need of considering subsample sizes of smaller order than n, the estimation seems to work pretty well up till n, unless n is very large. But all simulations suggest a strong dependence on kaux . In Figure 1 we consider for samples of size n ˆ 1000 from a Burr model with parameters …g; r† ˆ …1; 0:5† and for three different values of kaux ˆ 10; 50; 100, the simulated mean values, based on 1000 runs, with B ˆ 250 bootstrap samples, of k^0 …n; kaux ; n1 †=n (left) and of MSE‰k^0 …n; kaux ; n1 †=nŠ (right), for sub-sample sizes n1 ˆ 50…10†n. In Figure 2 it is shown the path of the MSE of gn;n1 …kaux †; kaux ˆ 10; 50; 100. It is worth noticing that, among the above mentioned values of kaux , and uniformly on n1 for reasonably large values of n1 , the minimum MSE was attained for kaux ˆ 50, whereas the asymptotic optimal number of o.s. corresponding to n ˆ 1000 is 38. Despite the differences obtained for different values of kaux , the results may be considered reasonable from a practical point of view, but it seems important to have urgently an adaptive choice of kaux , similar to the one in Gomes (1999). We still notice the following interesting by-products of Hall's bootstrap methodology: 

…1†

 The mean bootstrap estimator g…1† n …k† ˆ E‰gn …k†jX n Š ``smooths'' the whole sample …1† path of the original estimator gn …k†, for every k.

341

THE BOOTSTRAP METHODOLOGY IN STATISTICS OF EXTREMES

Figure 1. Sensitivity of k^0 …n; kaux ; n1 † in (16), with respect to the choices of kaux n1 ˆ 50 …10† n …n ˆ 1000†, for a Burr parent with …g; r† ˆ …1; 0:5†.

…1†

and of

…1†

 Since BIAS? ‰gn1 …k†Š ˆ BIAS? ‰gn1 …k†Š ˆ 1 1 r A…n1 =k†, we may also obtain, under the …1† validity of (9), a bootstrap approximation of the BIAS of gn …k†:  d n ‰g…1† BIAS n …k†Š :ˆ 1

d n …k† BIAS 1

2

d n …k† BIAS 2

 d n …k† ˆ Efg…1† where BIAS nj …kj † j obtain the new estimator …1† gU n;n1 …k†jkaux :ˆ gn …k†

;

1  k < n2 ;

…1†

gn …kaux †jXn g;

h i d n g…1† BIAS …k† : n 1

n2 ˆ

n21 ; n

…17†

j ˆ 1; 2. After removing bias, we

…18†

For a sample from a Burr model with g ˆ 1; r ˆ 0:5…n ˆ 1000†, we present in Figure  …1† U 3 (on the left) the sample paths of gn …k†; g…1† n …k† and gn;n1 …k†jkaux , obtained on the basis of 0:975 B ˆ 250 bootstrap samples, for n1 ˆ n , kaux ˆ 50, and the simulated analogs, obtained on the basis of 5000 runs. It is also provided (on the right) information on MSE. The 2 properties of gU n;n1 …k†jkaux ; 1  k < n1 =n for other values of n1 are of the same type. This procedure is easy to implement, and the new estimator has an appealing sample path from a practical point of view. More than that: the MSE of gU n;n1 …k†jkaux is also ``almost'' constant for a large range of k-values, contrarily to what happens to the MSE of the usual semi-parametric estimators, which implies no extra work to choose an optimal sample fraction. Unfortunately, the values of the MSE of gU n;n1 …k†jkaux depend heavily on kaux , as is also enhanced by Figure 3.

342

GOMES AND OLIVEIRA

Figure 2. MSE of the Hill estimator at k^0 …n; k ; n1 †, for a Burr parent with …g; r† ˆ …1; kaux ˆ 10; 50; 100 and for n1 ˆ 50…10†n.

0:5†, for

The method provides consistent estimates of the tail index, and the sample paths of the new estimators for different values of k, enhanced in Figure 3 seem to be appealing from a practical point of view. It is also appealing to have bootstrap estimates of MSE and BIAS, on the basis of the available sample, but we still have a disturbing sensitivity to the initial value of kaux . These factsÐstrong dependence on kaux and almost independence on n1 Ð arose strong con®dence in the suitability of the alternative bootstrap methodology of Draisma et al. (1999) and Danielsson et al. (2001), investigated in the next section.

0:975 Figure 3. Sample path (one-sample) and simulated behavior (5000 runs) of gU , n;n1 …k†jkaux in (18), n1 ˆ n …1† …1† kaux ˆ 50, the Hill estimator, gn …k†, and the mean bootstrap Hill estimator, gn …k† (on the left), and MSE patterns (on the right) for a Burr parent …g ˆ 1; r ˆ 0:5†, and for n ˆ 1000.

343

THE BOOTSTRAP METHODOLOGY IN STATISTICS OF EXTREMES

4.

The use of auxiliary statisticsÐan alternative bootstrap methodology

The problems related to the need of an initial consistent estimation of the tail index g may be overpassed by the consideration of an auxiliary statistic, with null mean value, which consequently has a MSE equal to its variance, and whose asymptotic properties are intimately close to the ones of the estimator under study. The method has been ®rst devised in the papers of Draisma et al. (1999) for the general domain of attraction of the EV d.f., and of Danielsson et al. (2001) for heavy tails. We have several possibilities for the choice of the auxiliary statistic. Different statistics provide usually slightly different answers but, whenever working with one sample, we may sometimes get non-admissible estimates of r or non-admissible estimates of k0 …n† associated to a chosen statistic. It is thus sensible to have different auxiliary statistics at hand, and the most obvious ones seem to be differences of our estimator at two different levels. For heavy tails, if we consider the Hill estimator and two different intermediate levels k1 and k2 , with k1 < k2 ; k2 k1 ? ?, as n ? ?, we have the asymptotic representations: g …1† 1 A…n=ki † ‡ op …A…n=ki ††; g…1† n …ki † ˆ g ‡ p Zn;i ‡ 1 r ki

i ˆ 1; 2;

…1† …1†  where Zn;1 ; Zn;2 is asymptotically Bivariate Normal with null mean and covariance p matrix S1;2 ˆ ‰sij Š, where s11 ˆ s22 ˆ 1, and s12 ˆ s21 ˆ k1 =k2 . p Pki p …1† d ki ; i ˆ 1; 2, This follows from the fact that we may write Zn;i ˆ …1= ki † j ˆ 1 Wj where fWj g is a sequence of independent, unit exponential r.v.s. Moreover, under the r validity of (9), we have that A…n=k2 † ˆ …k1 =k2 † A…n=k1 †…1 ‡ o…1††, as n ? ?. See also Drees et al. (2000) and Gomes et al. (2000). Then, if we consider

Tn …k; y† :ˆ g…1† n …yk†

g…1† n …k†;

0 < y < 1;

…19†

we have the distributional representation g d Tn …k; y† ˆ p Zn k

1

r

y 1

r

A…n=k† ‡ op …A…n=k††;

where Zn is asymptotically Normal …0; …1 y†=y† (for more details see Gomes et al., 2000). The AMSE of Tn …k; y† is thus minimal at a level k0T …n† such that k0 …n† ˆ

1 y…1

!

y y

r 2

†

1=…1

2r†

k0T …n†…1 ‡ o…1††;

as

n ? ?:

…20†

The set of auxiliary statistics Tn …k; y†; y ˆ 1=4; 1=2; 3=4 seems then to be an adequate

344

GOMES AND OLIVEIRA

choice. Here, however, we have decided to consider only one of these statistics, i.e. the statistic Tn;1 …k†:Tn …k; 1=2† :ˆ g…1† n …k=2†

g…1† n …k†;

2  k < n;

…21†

together with the statistic suggested in the paper of Danielsson et al. (2001) Tn;2 …k† :ˆ Mn…2† …k†

h i2 2 g…1† n …k† ;

P …2† where Mn …k† ˆ …1=k† kiˆ 1 ‰ln Xn We have also considered

i ‡ 1:n

…22† ln Xn

2 k:n Š .

g…1† n …k†;

Tn;3 …k† :ˆ ~gn …k† …2†

…23†

…1†

where ~gn …k† ˆ Mn …k†=‰2gn …k†Š is de Vries estimator of the tail index g. For more details on such estimator and its generalizations see Gomes and Martins (2001). The consideration of the ®rst statistic Tn;1 …k† as an alternative to Tn;2 …k† was in part due to the fact that Tn;2 …k† led us often to non-admissible estimates of r and non-admissible estimates of k0 …n†. With the consideration of Tn;1 …k† the problems of non-admissible estimates of r were overcome, but we were still confronted with non-admissible estimates of k0 …n†. That led us to the consideration of the third statistic Tn;3 …k†, and to bootstrap multi-sample Monte Carlo searches. As it was said before, it is always sensible to have extra auxiliary statistics at hand, when, in a practical situation, we have only one-sample available. It is obvious that we may always change the bootstrap experiment trying to get convergence, but we may pay a high price for it in computing time, and may be less expensive to change to another auxiliary statistic. We have the following asymptotic representations for these statistics: g 2r 1 d Tn;1 …k† ˆ p Zn…1† ‡ A…n=k† ‡ op …A…n=k††; 1 r k 2 2gr d 2g Tn;2 …k† ˆ p Zn…2† ‡ A…n=k† ‡ op …A…n=k††; 2 k …1 r† p g r d Tn;3 …k† ˆ p Zn…3† ‡ A…n=k† ‡ op …1= k† ‡ op …A…n=k††; 2 k …1 r† …1†

…2†

…3†

where Zn , Zn and Zn are asymptotically standard Normal r.v.'s. If for j ˆ 1; 2; 3, we denote by k0j …n† the value of k where the AMSE of Tn; j …k† is

345

THE BOOTSTRAP METHODOLOGY IN STATISTICS OF EXTREMES

 minimal, and by k0j …n1 † the value of k1 where the MSE of its bootstrap analog T n1 ; j …k1 †jXn is minimal, we have

k0 …n† ˆ k01 …n†‰1  ˆ k0j …n† 1

2=…1

2r Š  1 r

2r†

2=…1

…1 ‡ o…1††;

2r†

…1 ‡ o…1††;

…24† j ˆ 2; 3;

…25†

and  k0j …n1 † n1 2r=…2r ˆ k0j …n† n

1†

…1 ‡ op …1††;

as

n ? ?;

j ˆ 1; 2; 3:

…26†

Then, for n2 ˆ n21 =n, h

 k0j …n1 †

i2 ˆ k0j …n†…1 ‡ op …1††;

 k0j …n2 †

as

n ? ?;

j ˆ 1; 2; 3;

…27†

enables us to estimate k0j …n†, j ˆ 1; 2; 3. In order to estimate k0 …n† we need ®rst to estimate r. Several estimators of r have been proposed in the literature. We use here the bootstrap estimator of Danielsson et al. (2001): 

rj :ˆ



ln k0j …n1 † 

2 ln…k0j …n1 †=n1 †

;

j ˆ 1; 2; 3;

…28†



 , j ˆ 1; 2; 3. where k0j denotes, as before, the sample counterpart of k0j We then have the estimates

" #  2  ‰k01 …n1 †Š …1† ^ k 0 …n; n1 † :ˆ Cr1  2 ; Cr1 ˆ 1 k01 …n1 =n† " #  ‰k0j …n1 †Š2 … j† ^ k 0 …n; n1 † :ˆ Crj  2 ; Crj ˆ 1 k0j …n1 =n†



2 r1 1 rj

2=…1

!

2=…1

2r1 †

;

…29†

2rj †

;

…30†

for j ˆ 2; 3, and   … j† … j† gn;n :ˆ g…1† k^0 …n; n1 † ; n 1

j ˆ 1; 2; 3:

…31†

A few practical questions may be raised under the set-up developed: How does the asymptotic method work for moderate sample sizes? What is the type of the sample path of the new estimator for different values of n1 ? What is the dependence of the method on the

346

GOMES AND OLIVEIRA

choice of n1 ? What is the sensitivity of the method with respect to the choice of r's estimator? Although aware of the need of n1 ˆ o…n†, what happens if we choose n1 ˆ n? The algorithm here is similar to the algorithm in Section 3, with …1† …1† Tni …k† ˆ gni …k† gn …kaux † replaced by any of the statistics Tni;j …k†, the bootstrap version of Tni;j …k† in (21), (22) and (23) respectively, j ˆ 1; 2; 3; i ˆ 1; 2, and steps 5 and 6 replaced by: 







50 . Compute rj ˆ ln k0j …n1 †=…2 ln…k0j …n1 †=n1 ††; if rj  0 skip to another method; otherwise go to 60 ;  0 6 . Obtain the scale constant Crj associated to the auxiliary statistic;    … j† … j† 70 . Compute k^0 …n; n1 † ˆ ‰Crj ‰k0j …n1 †Š2 =k0j …n2 †Š; if k^0 6 [ ‰1; n† skip also to another 0 method; otherwise, follow to 8 ; … j† …1† … j† 80 . Obtain gn;n1 :ˆ gn …k^0 …n; n1 ††. Remarks similar to (i), (ii), (iii) (iv) and (v) also apply here. We make now illustrations for a sample size n ˆ 1000. It is, however, worth mentioning that for smaller sample sizes the method also works pretty well. In Figure 4 we present … j† … j† …1† … j† the mean values of k^0 …n; n1 †=n and of gn;n1 :ˆ gn …k^0 …n; n1 ††, j ˆ 1; 2; 3, for n1 ˆ 50…1†1000, associated to a Monte Carlo simulation of 5000 runs. The bootstrap Monte Carlo procedure underlying this ®gure was based on a multi-sample simulation with 10 replicas of samples of size B ˆ 100. This choice of a multi-sample Monte Carlo bootstrap was due to the few nonconvergence problems mentioned before. In Figure 5, we present the percentage of ``nonconvergence'' obtained for two different bootstrap Monte Carlo simulations: a one-sample procedure, with samples of size B ˆ 1000 (left), and a multi-sample procedure with 10 replicates of samples of size B ˆ 100 (right). Notice thus the importance of the experimental design associated to the Monte Carlo search of the minimum bootstrap MSE.



… j† … j† …1† … j† Figure 4. Simulated mean values of k^0 …n; n1 † and of gn;n1 :ˆ gn …k^0 …n; n1 ††, j ˆ 1; 2; 3, based on 5000 runs, for a sample of size n ˆ 1000 from a Burr parent, with g ˆ 1 and r ˆ 0:5, and for n1 ˆ 50…10†n.

THE BOOTSTRAP METHODOLOGY IN STATISTICS OF EXTREMES

347

Figure 5. Percentage of ``non-convergence'' in the bootstrap selection of k0 …n†Ð5000 runs.

A few additional remarks: vi. All simulated characteristics were calculated on the basis of those samples for which all procedures yield some admissible result. vii. We would like to notice again the ``almost independence'' on the choice of the subsample size, n1 , which enhances the practical value of the method. Consequently, although aware of the need of n1 ˆ o…n†, it seems that, once again, we get good results up until n, for a wide range of r-values. There appear some unstability close to n only for values of r close to zero. A general choice seems then to be for instance n1 ˆ n0:95 , the one used later in Section 5. … j†

In Figure 6 we picture the MSE of gn;n1 :ˆ gn …k^0 …n; n1 ††, j ˆ 1; 2; 3, and for n1 ˆ 50…10†n. …1†

… j†

Figure 6. Simulated MSE of the adaptive Hill estimator at the optimal estimated levels.

348

Figure 7. Simulated mean values of rj , j ˆ 1; 2; 3, ‰r ˆ

GOMES AND OLIVEIRA

0:5Š.

Notice that, for this model, the best results seem to be achieved for the statistic Tn;1 …k†, although not a long way from Tn;3 …k†. Notice also that, may be it is worth a slightly more intrincate procedure: use an auxiliary statistic to estimate the optimal level, then use that optimal level as kaux , and use Hall's bootstrap methodology to estimate MSE and BIAS. This has been suggested by a small scale simulation, performed merely as a curiosity. Finally, it is worth mentioning that, despite the unexpected surprisingly ``good'' practical results obtained regarding the ``almost independence'' on the tuning parameter n1 , the estimator of r, the second order parameter has a very bad performance (with a high bias), as may be noticed in Figure 7. The simulation was again based on 5000 runs. To illustrate the dependence of the method on the choice of the estimator of r, we picture Figures 8 and 9, which are equivalent to Figures 4 and 6, respectively, but where r is assumed to be known, and equal to its true value. The better performance of the adaptive estimator of the tail index g, enhanced by these



… j† …1† … j† … j† Figure 8. Simulated mean values of k^0 …n; n1 † and of gn;n1 :ˆ gn …k^0 …n; n1 ††, j ˆ 1; 2; 3, based on 5000 runs, for a Burr parent, g ˆ 1, r ˆ 0:5, for a sample of size n ˆ 1000, and assuming that r is known.

THE BOOTSTRAP METHODOLOGY IN STATISTICS OF EXTREMES

349

Figure 9. Simulated MSE of Hill estimator at the optimal estimated level, given the exact value of r.

Figures, claims thus for the need of a more sophisticated estimation of the second order parameter, but that study is beyond the scope of this paper. 5.

Comparison of alternative choices of the optimal sample fraction

In this section, apart from the bootstrap adaptive Hill estimators investigated before, p …1† …2† …3† D D D  gn;n1 …2 n†, gn;n1 , gn;n1 , gn;n1 obtained for n1 ˆ n0:95 , and denoted here gnH , gn 1 , gn 2 , gn 3 , respectively, we shall consider the adaptive estimators suggested by Hall and Welsh B (1985), by Beirlant et al. (1996a,b) and by Drees and Kaufmann (1998), denoted gHW n , gn DK and gn , respectively, whose algorithms are synthetized in the sequel. The comparison will be done in terms of

EFF‰gn Š ˆ

h ˆ i …1† RMSEs gn k 0 …n† RMSEs ‰gn Š

;

…32†

where with the index s we denote the simulated RMSE. The size of the simulation here is smaller than in Section 2, due to the fact that particularly Beirlant et al.'s (1996a,b) choice (but also the second bootstrap methodology) is quite time-consuming. We have however decided to use a 100065 simulation for all values of n, skipping Beirlant's method for values of n  2000 whenever r 6ˆ 1, and for n  1000 whenever r ˆ 1. The general algorithm was implemented for each model, with all the methodologies running sequentially, usually in an order related to a decreasing percentage of nonconvergence (in a FreÂchet case): Beirlant et al. (1996a,b) method, Danielsson et al. (2001) bootstrap approach, Drees and Kaufmann (1998) approach, Hall's (1990) bootstrap

350

GOMES AND OLIVEIRA

approach, and ®nally Hall and Welsh (1985) approach, denoted B, D, DK, H and HW, respectively. Indeed, to carry out a fair comparison of the methods, we have taken, as mentioned before in remark (vi), the same 100065 samples for all procedures. For samples from the model outside Hall's class, we have placed in ®rst place Drees and Kaufmann's algorithm, due to the same reasons presented before. Hall and Welsh's (1985) choice may be sinthesized in the following algorithm: 1. For s ˆ 0:5, t1 ˆ 0:9, t2 ˆ 0:95 obtain s ˆ ‰ns Š, t1 ˆ ‰nt1 Š, t2 ˆ ‰nt2 Š; 2. Obtain 1=g…1† …t † 1=g…1† …s† t1 n 1 n ^n ˆ log …1† r = log ; 1=gn …t † 1=g…1† t2 n …s† 2 3. Put p n  r^n g…1† …s† g…1† …t † n n 1 ^ lˆ 2^ rn …1† t1 gn …t †

2=…1

2^ rn †

;

1

2^ rn =…1 HW 4. Compute k^0 …n† ˆ ‰^ ln continue; …1† 5. Obtain gHW ˆ gn …k^HW n 0 …n††.

2^ rn †

HW Š; if k^0 6 [ ‰1; n† choose another method; otherwise,

Consistency of the method holds only for   s s ; r[ ; 2…1 t1 † 2…1 s† and the method is quite sensitive to changes in the parameters s, t1 and t2 , but the choice suggested by Hall and Welsh (1985) seems to work well. The algorithm related to Beirlant et al.'s (1996a,b) regression diagnostics procedure may be written as: 1. Compute ajk ˆ

kX j ‡ 1 lˆ1

and cik

ˆ

k X jˆ1

wijk

k

 1 ; l‡1

 bjk ‡ ajk

bjk ˆ

kX j ‡ 1 lˆ1

k

1 l‡1

2 ;

 2 ! k‡1 log ; j

i ˆ 1; 2; for w1jk ˆ 1 and w2jk ˆ j=…k ‡ 1†; 1  j  k  n

1;

351

THE BOOTSTRAP METHODOLOGY IN STATISTICS OF EXTREMES

…1†

2. Given the sample …x1 ; x2 ; . . . ; xn †, compute gn …k†, k ˆ 1; 2; . . . ; n 3. With  Xn j ‡ 1:n aj …g† ˆ log Xn k:n

 2 k‡1 g log ; j

obtain k0 :ˆ arg min‰n=100Š  k  n 4. Compute

1

 1 Pk

jˆ1

k

…1† …1† aj …gn …k†† , and put g0 ˆ gn …k0 †;

   …1†  …1† m ‡ k0 m ‡ k0 g   g n n 4 2 2m = log rn …m† :ˆ log ; …1† …1† m ‡ k0 gn …m=2† gn …m† 5. With Cj …m† ˆ

…1† gn … j† …1† gn …m†

g…1† … j=2† n log …1† gn …m=2†

 !2 m rn …m† log ; j

obtain ( m0 :ˆ arg mink0 ‡ 1mn

1

1 m

m X1

k0 j ˆ k

) Cj …m† ;

0

and put r0 :ˆ rn …m0 †; 6. Obtain 0 B xjk …r0 † ˆ @

j k‡1



r0

r0

1

12 C A :

Compute dk1 : dk1 …r0 † ˆ

…1

2 k r0 † X xjk …r0 † k jˆ1

…1

k r0 †2 X j x …r †: k ‡ 1 jk 0 k jˆ1

and dk2 : dk2 …r0 † ˆ

1;

352

GOMES AND OLIVEIRA

Obtain 1 wopt jk …r0 † ˆ wjk

dk2 1 ck dk2

c2k c1 ‡ w2jk 1 k2 2 1 ck dk c k dk

dk1 ; c2k dk1

7. For i ˆ 1; 2; . . . ; nite ˆ 10, Pk opt 1 d  Compute MSE…kjr i 1 ; gi 1 † ˆ k j ˆ 1 wjk …ri 1 †aj …gi 1 † d  obtain ki :ˆ arg min‰n=100Škn 1 fMSE…kjr i 1 ; gi 1 †g; …1†

 put gi :ˆ gn …ki †;

 Pm 1  obtain mi :ˆ arg minki 1 ‡ 1  m  n 1 m 1k j ˆ ki 1 Cj …m† ; i 1  put ri :ˆ rn …mi †, k^0B …n† ˆ ki whenever ki ˆ ki 1 , and go to 10; 8. If the previous stopping rule is not attained or if the ®nal ri is non-admissible, choose …1† another method; otherwise, obtain gBn ˆ gn …k^0B …n††. The method provides interesting by-products: estimates of the MSE and of squared bias. If kp is the ®nal solution provided by the algorithm for an available sample, we have the estimates: k X d n …k† :ˆ 1 MSE wopt …r k j ˆ 1 jk p

 Xn j ‡ 1:n † log 1 Xn k:n

gp

 2 k‡1 log ; 1 j

…33†

and k X d 2n …k† :ˆ 1 BIAS wbias …rp k j ˆ 1 jk

 Xn j ‡ 1:n 1 † log Xn k:n

gp

 2 k‡1 ; 1 log j

…34†

where wbias jk …r† ˆ

c2k w1jk

c2k dk1

c1k w2jk c1k dk2

;

…35†

but we have been confronted with a few computational problems related to this method. 1. The procedure is quite time-consuming, particularly if we intend to obtain properties of the possible by-products in (33) and (34). 2. For some generated samples, the ®nal estimate of the MSE, which is a nice by-product of this method, turns out to be negative for some values of k, due to negative weights wopt jk . Those samples were not discarded from the simulation, because the procedure seems to yield reasonable estimates for k, but its percentage is registered in Table 3. 3. A high percentage of estimates of r are non-admissibleÐthose samples were discarded in the simulation, and information on this percentage is given in Table 4.

353

THE BOOTSTRAP METHODOLOGY IN STATISTICS OF EXTREMES

The main tuning parameters introduced are the initial weights w1jk and w2jk . The method is quite sensitive to these weights, but they may be used to overcome the non-convergence problems referred before, and the method, although time-consuming, provides nice results, particularly for small samples. Finally, we present the algorithm related to the Drees and Kaufmann (1998) procedure, with the choices of the different tuning parameters proposed by these authors: …1† p 1. Obtain ~gn ˆ gn …2 n†; 2. For rn ˆ 2:5~gn n1=4 , compute n p kn …rn † :ˆ min max i‰g…1† n …i† 1kn

1

1ik

o g…1† …k†Š > r : n n

If this previous condition is not satis®ed, replace repeatedly rn by 0.9rn until kn …rn † is well-de®ned; 3. For x ˆ 0:7 obtain kn …rnx †, and for l ˆ 0:6 obtain ^ˆ r

^n;l …rnx † r

ˆ

p …1† ijgn …i† log p …1† max1  i  ‰kn …rnx †Š ijgn …i† max1  i  ‰lkn …rnx †Š

…1†

gn …‰lkn …rnx †Š†j …1† gn …‰kn …rnx †Š†j

^  0 choose another method; otherwise, continue. if r ^Š1=…1 2^r† …k…rnx †=‰kn …rn †Šx †1=…1 4. Put k^0DK …n† :ˆ ‰…1 2^ r†1=^r ‰ 2~g2 r choose another method; otherwise, go to 5. …1† ^DK 5. Obtain gDK n :gn …k0 …n††.

x†

,

1 log l ‡ ; 2

Š; if k^0DK 6 [ ‰1; n†

Apart from the need to estimate r, which relies on the choice of a tuning parameter l, and of an initial estimate of the tail index g, there are two other tuning parameters, rn and x. The method is quite sensitive to the choice of rn and x, but the suggestions given by Drees and Kaufmann (1998) seem appropriate, and the method has an overall positive performance. We still add a ®nal remark, related to the bootstrap algorithms: viii. Due to the high computational time of the general comparison algorithm, we have here restricted the search region of k in the bootstrap approaches to the region ‰max…1; ‰ln n=2Š†, min…n 1; ‰2n= ln nŠ†Š, paying attention to a possible minimum value attained at the border, which would then lead the algorithm to widen the search region of k-values in the adequate direction. Doing so, we gain in ef®ciency, and we no longer obtain non-admissible bootstrap estimators of r, in the second bootstrap methodology. In Table 2 we present the ef®ciencies in (32), and 95% con®dence intervals for those ef®ciencies, for the estimators described before and for the models simulated in Section 2. In the bootstrap procedures we have used n1 ˆ n0:95 . Finally in Tables 3, 4 and 5 we report the percentage of failures of each procedure.

354

GOMES AND OLIVEIRA

Table 2. Ef®ciencies of the adaptive Hill estimators. n

100

200

500

1000

Student (8) parentÐr ˆ  gnH  gnD1  gnD2  gnD3 gHW n gBn gDK n

0.560 + 0.011 0.836 + 0.015 0.861 + 0.024 0.847 + 0.016 0.906 + 0.030 1.040 + 0.024 0.929 + 0.014

0.552 + 0.011 0.848 + 0.006 0.857 + 0.017 0.870 + 0.013 0.936 + 0.008 0.995 + 0.008 0.908 + 0.016



0.520 + 0.012 0.814 + 0.018 0.801 + 0.021 0.808 + 0.024 0.882 + 0.021 1.023 + 0.017 0.905 + 0.031

0.527 + 0.012 0.822 + 0.011 0.802 + 0.020 0.847 + 0.012 0.914 + 0.010 0.979 + 0.020 0.900 + 0.016

0.567 + 0.013 0.834 + 0.012 0.863 + 0.012 0.883 + 0.016 0.948 + 0.011 0.933 + 0.011 0.882 + 0.008

0.560 + 0.006 0.801 + 0.015 0.854 + 0.016 0.873 + 0.018 0.924 + 0.011 0.894 + 0.010 0.844 + 0.017

0.536 + 0.014 0.978 + 0.032 0.911 + 0.027 0.934 + 0.024 0.831 + 0.019 0.917 + 0.025 0.862 + 0.029

0.565 + 0.022 0.907 + 0.013 0.859 + 0.027 0.889 + 0.012 0.835 + 0.008 0.862 + 0.011 0.834 + 0.027

0.558 + 0.013 0.786 + 0.013 0.814 + 0.014 0.847 + 0.010 0.922 + 0.015 0.890 + 0.016 0.823 + 0.031

0.816 + 0.025 0.917 + 0.020 0.800 + 0.024 0.880 + 0.032 0.784 + 0.022 0.600 + 0.008 0.776 + 0.024

0.813 + 0.018 0.901 + 0.025 0.752 + 0.027 0.824 + 0.018 0.771 + 0.020 0.590 + 0.027 0.758 + 0.044

0.598 + 0.018 0.831 + 0.011 0.832 + 0.031 0.853 + 0.020 0.857 + 0.025 0.865 + 0.010 0.830 + 0.028

0.816 + 0.019 0.895 + 0.027 0.689 + 0.040 0.795 + 0.041 0.741 + 0.017 0.571 + 0.017 0.680 + 0.078 Cauchy parentÐr ˆ



gnH  gnD1  gnD2  gnD3 gHW n gBn gDK n

0.451 + 0.041 0.968 + 0.029 0.746 + 0.021 0.843 + 0.024 0.731 + 0.073 0.715 + 0.028 0.789 + 0.054

0.495 + 0.028 0.928 + 0.047 0.728 + 0.031 0.825 + 0.036 0.744 + 0.049 0.679 + 0.028 0.739 + 0.060

0.510 + 0.031 0.859 + 0.030 0.638 + 0.035 0.747 + 0.031 0.722 + 0.016 0.615 + 0.021 0.689 + 0.071 Out-Hall parentÐr ˆ

 gnH  gnD1  gnD2  gnD3 gHW n gBn gDK n

0.960 + 0.004 0.863 + 0.004 0.909 + 0.008 0.948 + 0.006 0.915 + 0.004 0.814 + 0.005 0.853 + 0.009

0.981 + 0.008 0.831 + 0.013 0.940 + 0.008 0.982 + 0.005 0.908 + 0.007 0.757 + 0.014 0.758 + 0.014

0.973 + 0.006 0.796 + 0.014 0.953 + 0.012 0.989 + 0.010 0.844 + 0.021 0.696 + 0.013 0.600 + 0.010

0.573 + 0.015 0.764 + 0.019 0.852 + 0.010 0.856 + 0.021 0.891 + 0.034 Ð 0.818 + 0.028

0.566 + 0.007 0.757 + 0.017 0.796 + 0.017 0.832 + 0.019 0.926 + 0.016 Ð 0.816 + 0.012

0.595 + 0.011 0.737 + 0.019 0.802 + 0.016 0.827 + 0.013 0.942 + 0.019 Ð 0.823 + 0.015

0.658 + 0.011 0.765 + 0.014 0.781 + 0.022 0.805 + 0.021 0.886 + 0.008 Ð 0.825 + 0.025

0.684 + 0.014 0.732 + 0.006 0.730 + 0.024 0.767 + 0.017 0.905 + 0.010 Ð 0.814 + 0.020

0.774 + 0.025 0.890 + 0.018 0.633 + 0.017 0.779 + 0.044 0.674 + 0.031 Ð 0.703 + 0.186

0.736 + 0.036 0.877 + 0.013 0.689 + 0.100 0.819 + 0.018 0.664 + 0.019 Ð 0.783 + 0.027

0.551 + 0.056 0.843 + 0.013 0.579 + 0.049 0.738 + 0.028 0.692 + 0.017 Ð 0.665 + 0.153

0.545 + 0.026 0.838 + 0.023 0.591 + 0.041 0.730 + 0.034 0.615 + 0.167 Ð 0.674 + 0.262

0.952 + 0.013 0.828 + 0.014 0.945 + 0.005 0.970 + 0.004 0.690 + 0.030 Ð 0.505 + 0.022

0.916 + 0.012 0.797 + 0.010 0.878 + 0.019 0.925 + 0.014 0.670 + 0.017 Ð 0.430 + 0.031

0:5, g ˆ 1

FreÂchet parentÐr ˆ  gnH  gnD1  gnD2  gnD3 gHW n gBn gDK n

0.569 + 0.018 0.787 + 0.020 0.847 + 0.020 0.871 + 0.018 0.906 + 0.015 Ð 0.833 + 0.018

0:40, g ˆ 0:20

0.548 + 0.010 0.814 + 0.012 0.807 + 0.014 0.857 + 0.008 0.931 + 0.012 0.920 + 0.010 0.856 + 0.029 Burr parentÐr ˆ

 gnH  gnD1  gnD2  gnD3 gHW n gBn gDK n

5000

0:25, g ˆ 0:125

Student (5) parentÐr ˆ gnH  gnD1  gnD2  gnD3 gHW n gBn gDK n

2000

0.627 + 0.013 0.790 + 0.030 0.794 + 0.035 0.816 + 0.028 0.876 + 0.015 0.842 + 0.020 0.816 + 0.053 1, g ˆ 1 0.836 + 0.026 0.901 + 0.015 0.687 + 0.029 0.799 + 0.054 0.722 + 0.018 Ð 0.770 + 0.022 2, g ˆ 1 0.539 + 0.015 0.805 + 0.022 0.562 + 0.034 0.696 + 0.022 0.661 + 0.060 0.571 + 0.012 0.656 + 0.078 1, g ˆ 1 0.973 + 0.008 0.822 + 0.014 0.969 + 0.009 0.993 + 0.008 0.783 + 0.026 Ð 0.554 + 0.028

355

THE BOOTSTRAP METHODOLOGY IN STATISTICS OF EXTREMES

Table 3. Values of p1 ==p2 in Beirlant et al.'s (1996a,b) method, p1 ˆ percentage of non-convergent iterative procedures (10 iterates), p2 ˆ percentage of samples where arg mink MSE…k† < 0. n

100

200

500

1000

Student (8) …r ˆ 0:25; g ˆ 0:125† Student (5) …r ˆ 0:4; g ˆ 0:2† Burr …r ˆ 0:5; g ˆ 1† FreÂchet …r ˆ 1; g ˆ 1† Student (1) …r ˆ 2; g ˆ 1† Out-Hall …r ˆ 1; g ˆ 1†

16.13//9.85 17.68//9.52 18.75//10.90 18.92//8.30 18.02//7.91 22.28//8.51

13.05//6.99 14.46//6.48 16.55//7.29 19.03//4.86 18.12//4.94 20.80//8.25

10.79//3.54 11.50//3.50 15.01//3.37 18.78//3.07 17.90//3.23 24.48//4.88

11.25//1.83 11.43//2.46 16.16//2.17 Ð//Ð 19.28//2.18 Ð//Ð

In Hall's class of models, the adaptive estimator gHW behaves pretty well for large n values of n and small values of jrj, although, given the choice of the parameters t1 , t2 and s, consistency of the optimal sample fraction holds only for 2:5  r  0:5. The estimator gBn , although terribly time-consuming, compares favorably with the others for DK small values of n and for small values of jrj. The bootstrap adaptive estimators gD n and gn have an overall nice performance, and in general they have a much better performance than the bootstrap estimator gH n , which shows the worst performance of all data-driven choices of k. In the model outside Hall's class, the worst performance is shown by gBn and gDK n . Among all the methodologies herewith investigated for the selection of the optimal sample fraction of the Hill estimator, we would elect the bootstrap technique of Danielsson et al. (2001) as the most appealing in the sense that there is no need for an initial estimation of g (due to the consideration of the auxiliary statistic which converges to 0), and apart from the estimation of r, which needs to be further investigated, there is, for a large class of models useful in applications, a high insensitivity with respect to the subsample size n1 , the unique extra tuning parameter introduced in the construction of the adaptive Hill estimator.

Table 4. Values of q1 ==q2 , q1 ˆ percentage of non-admissible estimates of r in Beirlant et al. (1996a,b) method, q2 ˆ percentage of non-admissible estimates of r in Drees and Kaufmann's (1998) method. n

100

200

500

1000

2000

5000

Student (8) Student (5) Burr FreÂchet Student (1) Out-Hall

8.17//0.06 7.26//0.24 10.51//1.54 10.63//3.99 9.72//3.36 14.27//36.28

8.35//0.16 8.24//0.30 8.59//1.56 9.18//2.03 7.58//2.27 12.75//38.79

8.68//0.14 7.50//0.20 8.19//1.12 7.36//1.18 7.06//1.53 14.37//38.48

8.61//0.16 8.06//0.06 8.00//0.87 Ð//1.01 6.85//1.38 Ð//36.15

Ð//0.08 Ð//0.06 Ð//0.71 Ð//0.54 Ð//0.93 Ð//33.17

Ð//0.02 Ð//0.00 Ð//0.24 Ð//0.32 Ð//0.62 Ð//22.21

356

GOMES AND OLIVEIRA

Table 5. Percentage of samples where k^0 < 1 // k^0  n. n

100

200

500

1000

Student (8) parentÐr ˆ H D1 D2 D3 HW DK

0.00//0.02 48.96//0.00 44.04//0.00 4.32//0.00 0.00//0.30 0.40//0.00

0.00//0.02 30.07//0.00 34.93//0.00 0.88//0.00 0.00//0.00 0.60//0.00

0.00//0.06 41.06//0.00 38.81//0.00 2.86//0.00 0.00//0.34 0.32//0.00

0.00//0.00 23.34//0.00 30.19//0.00 0.82//0.00 0.00//0.00 0.71//0.00

0.00//0.00 13.57//0.00 32.63//0.00 0.80//0.00 0.00//0.00 0.79//0.00

0.00//0.00 5.69//0.00 26.40//0.00 0.57//0.00 0.00//0.00 0.50//0.00

H D1 D2 D3 HW DK

0.00//0.02 9.70//0.00 21.14//0.00 0.67//0.00 0.00//0.04 0.71//0.00

0.00//0.04 4.18//0.00 16.92//0.00 0.31//0.00 0.00//0.00 0.71//0.00

0.00//0.00 2.80//0.00 19.97//0.00 0.89//0.00 0.00//0.00 0.34//0.00

0.00//0.18 2.48//0.00 10.58//0.00 0.25//0.00 0.00//2.87 2.33//0.00

0.00//0.17 0.59//0.00 7.36//0.00 0.34//0.00 0.00//1.50 0.55//0.00

0.00//0.00 0.87//0.00 13.55//0.00 0.43//0.00 0.00//0.00 0.48//0.00

0.00//0.00 0.07//0.00 5.02//0.00 0.14//0.00 0.00//0.54 0.18//0.00 Cauchy parentÐr ˆ

H D1 D2 D3 HW DK

0.00//0.93 9.65//0.00 16.20//0.00 0.34//0.00 0.00//1.88 1.51//0.00

0.00//0.40 2.60//0.00 11.47//0.00 0.32//0.00 0.00//0.79 0.96//0.00

0.00//0.08 0.25//0.00 7.01//0.00 0.15//0.00 0.00//0.32 0.38//0.00 Out-Hall parentÐr ˆ

H D1 D2 D3 HW DK

0.00//0.02 0.47//0.00 0.75//0.00 0.04//0.00 0.00//2.78 49.97//0.27

0.00//0.00 0.17//0.00 0.23//0.00 0.00//0.00 0.00//1.48 52.11//0.01

0.00//0.00 0.06//0.00 0.18//0.00 0.00//0.00 0.00//0.46 44.72//0.00

0.00//0.00 0.88//0.00 15.40//0.00 0.91//0.00 0.00//0.00 0.06//0.00

0.00//0.00 0.95//0.00 16.03//0.00 0.87//0.00 0.00//0.00 0.16//0.00

0.00//0.00 0.20//0.00 9.40//0.00 0.70//0.00 0.00//0.00 0.10//0.00

0.00//0.00 0.00//0.00 6.14//0.00 0.31//0.00 0.00//0.00 0.24//0.00

0.00//0.00 0.02//0.00 2.85//0.00 0.30//0.00 0.00//0.00 0.08//0.00

0.00//0.00 0.00//0.00 0.80//0.00 0.08//0.00 0.00//0.16 0.00//0.00

0.00//0.00 0.00//0.00 0.33//0.00 0.02//0.00 0.00//0.06 0.02//0.00

0.00//0.04 0.00//0.00 1.63//0.00 0.06//0.00 0.00//0.12 0.12//0.00

0.00//0.00 0.00//0.00 0.49//0.00 0.02//0.00 0.00//0.08 0.02//0.00

0.00//0.00 0.00//0.00 0.02//0.00 0.00//0.00 0.00//0.08 21.20//0.00

0.00//0.00 0.00//0.00 0.00//0.00 0.00//0.00 0.00//0.00 8.49//0.00

0:5, g ˆ 1

FreÂchet parentÐr ˆ H D1 D2 D3 HW DK

0.00//0.00 2.25//0.00 22.31//0.00 0.93//0.00 0.00//0.00 0.24//0.00

0:40, g ˆ 0:20

0.00//0.00 8.80//0.00 27.70//0.00 0.70//0.00 0.0//0.00 0.50//0.00 Burr parentÐr ˆ

5000

0:25, g ˆ 0:125

Student (5) parentÐr ˆ H D1 D2 D3 HW DK

2000

0.00//0.00 0.13//0.00 8.29//0.00 0.53//0.00 0.00//0.00 0.12//0.00 1, g ˆ 1 0.00//0.02 0.00//0.00 1.87//0.00 0.00//0.00 0.00//0.28 0.10//0.00 2, g ˆ 1 0.00//0.12 0.02//0.00 3.24//0.00 0.08//0.00 0.00//0.24 0.10//0.00 1, g ˆ 1 0.00//0.00 0.02//0.00 0.14//0.00 0.00//0.00 0.00//0.60 33.17//0.00

THE BOOTSTRAP METHODOLOGY IN STATISTICS OF EXTREMES

357

Acknowledgments The authors are grateful to the editor and to the referees, who helped to improve an earlier version of this manuscript. They also thank M. Isabel Fraga Alves and Laurens de Haan, who encouraged them to submit a revision of the paper. Research partially supported by FCT/PRAXIS XXI/FEDER and POCTI.

References Beirlant, J., Vynckier, P., and Teugels, J.L., ``Excess function and estimation of the extreme-value index,'' Bernoulli 2, 293±318, (1996a). Beirlant, J., Vynckier, P., and Teugels, J.L., ``Tail index estimation, Pareto quantile plots, and regression diagnostics,'' J. Amer. Statist. Assoc. 91, 1659±1667, (1996b). Danielsson, J., Haan, L. de, Peng, L., and de Vries, C.G., ``Using a bootstrap method to choose the sample fraction in the tail index estimation,'' J. Multivariate Analysis 76, 226±248, (2001). Dekkers, A.L.M. and Haan, L. de, ``Optimal choice of sample fraction in extreme-value estimation,'' J. Multivariate Analysis 47, 173±195, (1993). Draisma, G., Haan, L. de, Peng, L., and Pereira, T.T., ``A bootstrap-based method to achieve optimality in estimating the extreme-value index,'' Extremes 2, 367±404, (1999). Drees, H. and Kaufmann, E., ``Selecting the optimal sample fraction in univariate extreme value estimation,'' Stoch. Proc. and Appl. 75, 149±172, (1998). Drees, H., Haan, L. de, and Resnick, S., ``How to make a Hill plot,'' Annals of Statistics 28, 254±274, (2000). Fishman, G.S., Concepts and Methods in Discrete Event Digital Simulation, Wiley, New York, 1972. Galambos, J., The Asymptotic Theory of Extreme Order Statistics (second edition), Krieger, Malabar, Florida, 1987. Gnedenko, B.V., ``Sur la distribution limite du terme maximum d'une seÂrie aleÂatoire,'' Ann. Math. 44, 423±453, (1943). Geluk, J. and Haan, L. de, Regular Variation, Extensions and Tauberian Theorems, CWI Tract 40, Center for Mathematics and Computer Science, Amsterdam, Netherlands, 1987. Goldie, C.M. and Smith, R.L., ``Slow variation with remainder: theory and applications,'' Quart. J. Math. Oxford 38, 45±71, (1987). Gomes, M.I., ``Metodologias Jackknife e bootstrap em estatõÂstica de extremos.'' In: Actas do II Congresso Anual da Sociedade Portuguesa de EstatõÂstica (Mendes-Lopes et al., eds), 31±46, (1994). Gomes, M.I., The bootstrap methodology in statistical extremesÐthe choice of the optimal sample fraction, Notas e ComunicacËoÄes C.E.A.U.L. 15/1998. Gomes, M.I., ``The Jackknife and bootstrap methodologies in the estimation of parameters of rare events,'' Revista de EstatõÂstica 99-I, 7±23, (1999). Gomes, M.I. and Haan, L. de, ``Approximations in extreme value theory,'' Extremes 2, 71±85, (1999). Gomes, M.I. and Martins, M.J., ``Generalizations of the Hill estimatorÐasymptotic versus ®nite sample behavior,'' J. Statistical Planning and Inference 93, 161±180, (2001). Gomes, M.I., Martins, M.J., and Neves, M., ``Alternatives to a semiparametric estimator of parameters of rare eventsÐthe Jackknife methodology,'' Extremes 3, 207±229, (2000). Haan, L. de, On Regular Variation and its Application to the Weak Convergence of Sample Extremes, Mathematical Centre Tract 32, Amsterdam, 1970. Haan, L. de and Peng, L., ``Comparison of tail index estimators,'' Statistica Neerlandica 52, 60±70, (1998). Hall, P., ``On some simple estimates of an exponent of regular variation,'' J. Roy. Statist. Soc. B44, 37±42, (1982). Hall, P., ``Using bootstrap to estimate mean squared error and selecting parameter in nonparametric problems,'' J. Multivariate Analysis 32, 177±203, (1990).

358

GOMES AND OLIVEIRA

Hall, P. and Welsh, A.H., ``Adaptive estimates of parameters of regular variation,'' Ann. Statist. 13, 331±341, (1985). Haeusler, E. and Teugels, J., ``On asymptotic normality of Hill's estimator for the exponent of regular variation,'' Ann. Statist. 13, 743±756, (1985). Hill, B.M., ``A simple general approach to inference about the tail of a distribution,'' Ann. Statist. 3, 1163±1174, (1975). Martins, M.J., Gomes, M.I., and Neves, M., ``Some results on the behavior of Hill's estimator,'' J. Statist. Comput. and Simulation 63, 283±297, (1999).