On the block maxima method in extreme value theory

Transcription

1 On the bloc axiethod in extree value theory Ana Ferreira ISA, Univ Tecn Lisboa, Tapada da Ajuda Lisboa, Portugal CEAUL, FCUL, Bloco C6 - Piso 4 Capo Grande, Lisboa, Portugal anafh@isa.utl.pt and Laurens de Haan Erasus Univ Rotterda, P.O. Box 738, 3 DR Rotterda, The Netherlands CEAUL, FCUL, Bloco C6 - Piso 4 Capo Grande, Lisboa, Portugal ldehaan@ese.eur.nl Abstract: In extree value theory there are two fundaental approaches, both widely used: the bloc axiethod and the peas-over-threshold POT ethod. Whereas uch research has gone into the POT ethod, the bloc axiethod has not been studied thoroughly. The present paper ais at providing conditions under which the bloc axiethod can be justified. In this paper we restrict attention to the independent and identically distributed case and focus on the probability weighted oent PWM estiators of Hosing, Wallis and Wood 985. MSC 2 subject classifications: Priary 62G32, ; secondary 62G2, 62G3. Keywords and phrases: bloc axia, probability weighted oent estiators, extree value index, asyptotic norality, extree quantile estiation.. Introduction The bloc axia approach in extree value theory EVT, consists of dividing the observation period into non-overlapping periods of equal size and restricts attention to the axiu observation in each period. The new observations thus created follow - under extree value conditions - approxiately an extree value distribution G for soe real. Paraetric statistical ethods for the extree value distributions are then applied to those observations. Usually it is taen for granted that the bloc axia follow very well an extree value distribution. In this paper we tae this isspecification into account. Since G is not the exact distribution for those observations, a bias ay appear. The procedure can be justified using a strengthening of the doain of attraction conditions of EVT. In the peas-over-threshold approach in EVT one selects those of the initial observations that exceed a certain high threshold. The probability distribution of those selected observations- under extree value conditions- is approxiately a generalized Pareto distribution GPD. Paraetric statistical ethods for GPD

2 Ferreira and de Haan/On the bloc axiethod 2 are then applied to those observations. Again, a bias ay appear since GPD is not the exact distribution of those selected observations. The bloc axiethod is the older one see e.g. Gubel, 958. The POT ethod has been developed by Picands 975 who provided the theoretical fraewor and devised statistical tools. In the case of the POT ethod, exact conditions under which the ethod is justified are well-nown see e.g. de Haan and Ferreira 26, Chapters 3-4, and Drees 998. The POT ethod pics up all relevant high observations. The bloc axia ethod, on the one hand isses soe of these high observations and, on the other hand, retains soe lower observations. Hence the POT sees to ae better use of the available inforation. However, there ay be reasons for using the bloc axiethod: The only available inforation ay be bloc axiae.g. yearly axia. The bloc axiethod ay be preferable when the observations are not exactly independent and identically distributed i.i.d.. For exaple, there ay be a seasonal periodicity in case of yearly axia or, there ay be short range dependence that plays a role within blocs but not between blocs. The bloc axiethod ay be easier to apply since the bloc periods appear naturally in any situations. Hence the proble of choosing a high threshold in the POT ethod which is a difficult one does not play a role. The present paper ais at forulating exact conditions under which the bloc axiethod can be justified. Since soe of the bloc axiay actually not be very high, one expects soewhat ore strict conditions in this case then in the POT case. However, as it turns out, the conditions are siilar. Throughout the paper we assue that the observations are i.i.d. When woring with bloc axia there are two ajor sets of estiators that are widely used: the axiu lielihood estiators e.g. Prescott and Walden, 98 and the probability weighted oent PWM estiators Hosing, Wallis and Wood, 985. Recently, Dobry 23 has proved consistency of the axiu lielihood estiators. The present paper concentrates on the PWM estiators. The asyptotic norality result for the PWM estiators is stated in Section 2, along with a siilar result for the accopanying high quantile estiator. The proofs in Section 3 are based on a unifor expansion of the relevant quantile process given in Proposition 3.. In future wor we shall extend the results to the non - i.i.d. case and to the axiu lielihood estiator. We shall also develop a theoretical coparison between the peas-over-threshold and the bloc axiethods. A coparative siulation study has been carried out by S. Caires 29.

3 Ferreira and de Haan/On the bloc axiethod 3 2. The estiators and their properties Let X, X 2,... be i.i.d. rando variables with distribution function F. Define for =,2,... and i =,2,..., the bloc axia X i = ax i <j i X i. Hence, the observations are divided into blocs of size. Write n =, the total nuber of observations. We study the odel for large and, hence we shall assue that n ; in order to obtain eaningful liit results, we have to require that both = n and = n, as n. TheainassuptionisthatF isinthedoainofattraction ofsoeextree value distribution G x = exp +x /, R, +x >. Then X i, when noralized, will follow approxiately this extree value distribution i.e., for appropriately chosen > and b and all x li P Xi b x = li F x+b = G x, i =,2,...,. This can be written as li logf x+b = +x/, which is equivalent to the convergence of the inverse functions: Vx b li = x, x >, with V = /logf. Hence b can be chosen to be V. This is the first order condition. For our analysis we also need a second order expansion as follows. Condition 2. Second order condition. Suppose that for soe positive function a and soe positive or negative function A with li t At =, li t Vtx Vt at At x = x s s u ρ duds = H,ρ x, for all x > see e.g. de Haan and Ferreira, Corollary Note that the function A is regularly varying with index ρ. The PWM estiators to estiate, as well as the location b and scale, are defined as follows. Let X,,...,X, be the order statistics of X,...,X. Let M r = i...i r... r X i,, for r =,,2; > r. 3 i= 2

4 Ferreira and de Haan/On the bloc axiethod 4 The PWM estiators are siple functionals of M, M and M 2. The estiator ˆ, for is defined as the solution of the equation The estiator â, of is and the estiator ˆb, of b is 3ˆ, 2ˆ, = 3M 2 M 2M M. 4 â, = ˆ, 2ˆ, 2M M Γ ˆ,, 5 ˆb, = M +â, Γ ˆ, ˆ,, 6 where Γx = t x e t dt, x > Hosing, Wallis and Wood, 985. For ˆ, = the estiators follow by continuity: ˆ, = if â, = 2M M log2 3M 2 M = log3 2M M log2, and ˆb, = M +â, Γ. Note that Γ is Euler s constant. Clearly, the given estiators are quite different fro the ones in the POT case. Rear 2.. There are other variants for M r, e.g. rxi, i i= +, but they give theoretical probles. 2.. Asyptotic norality Next we state conditions for the asyptotic norality of the entioned estiators. Theore 2.. Assue that F is in the doain of attraction of an extree value distribution G with < /2 and that Condition 2. holds. Let = n and = n as n, in such a way that A λ R. Then r +Mr b d r + D r s r Bsds+λI r,ρ =: Q r, 7 as n, jointly for r =,,2, where d eans convergence in distribution, B is Brownian bridge, D r ξ = r +ξ Γ ξ, ξ < ξ

5 Ferreira and de Haan/On the bloc axiethod 5 D r = logr + Γ as defined by continuity, and I r,ρ ρ D r +ρ D r, ρ, = D r = r+ Γ +logr +Γ Dr r+,,ρ =, D r = log 3 r ++Γ 3logr +Γ, =,ρ =. 2 Note that Γ = u e u logudu. Rear 2.2. Explicit coplicated expressions for the liiting covariance atrix can be found in Hosing, Wallis and Wood 985, cf. v r,r, v r,r+ and v r,r+s C.9 C. in their Appendix C. Fro there, varq r = r+ 2 v rr and covq r,q s = r +s+v r,s. Rear 2.3. The condition A λ R eans that the growth of n, the nuber of blocs, is restricted with respect to the growth n, the size of a bloc, as n. In particular this condition iplies that log/, as n. The asyptotic norality of ˆ,, â, and ˆb, follows fro Theore 2.: Theore 2.2. Under the conditions of Theore 2., as n, ˆ, d { log3 log2 Γ Q 2 Q } 2 Q Q =:, â, d { log2 2 Γ Q Q } + Γ =: Λ log2 Γ ˆb, b d Q + Γ +Γ 2 + Γ Λ =: B; where for = the forulas should read as defined by continuity: ˆ, d log3 2 log2 2 log3 Q 2 Q log2 Q Q

6 Ferreira and de Haan/On the bloc axiethod 6 â, log2 d log2 Q Q + +Γ 2 ˆb, b d Q Γ +Γ Λ Rear 2.4. The second order condition for the asyptotic norality in the POT case see e.g. de Haan and Ferreira, Theore 3.6., is forulated in ters of the function U = / F, not the function V. For a coparison between the two second order conditions see Drees, de Haan and Li High quantile estiation High quantile estiation is discussed in the next theore. Let the total nuber of observations be n, that is, n = as before. Let = n, = n as n. We want to estiate x n := F p n with p n sall and write x n = V / log p n. Our estiator for x n is cˆ, n ˆx, :=ˆb, +â, ˆ, with c n = log p n and we obtain the following result: Theore 2.3. Assue the conditions of Theore 2. with the second order paraeter ρ negative, or zero with negative. Moreover assue li c n = and n logc li n =. n Then, as n, ˆx, x n d + 2 B Λ λ q c n +ρ where := in, and q t := t s logsds. Rear 2.5. This is the result for a high quantile of the original distribution F. One ay also want to estiate a high quantile of the distribution of the bloc axiu. In that case we need to estiate x n := V / log p n. The result is as above with c n replaced by c n.

7 Ferreira and de Haan/On the bloc axiethod 7 3. Proofs Throughout this section Z represents a unit Fréchet rando variable, i.e. one with distribution function Fx = e /x, x >, and {Z i, } i= are the order statistics fro the associated i.i.d. saple of size, Z,...,Z. Siilarly, {X i, } i= represents the order statistics of the bloc axia X,...,X fro and, X u, := X r, for r < u r, r =,...,. Recall the function V fro Section 2. The following representation will be useful, X = d VZ. 8 We start by forulating a nuber of auxiliary results. Lea 3... As, sup /+ s /+ logz, P. 2. Csörgő and Horváth 993, p. 38 Let < ν < /2. With {B } an appropriate sequence of Brownian bridges, s s s ν Z s, B s s = o P, as u represents the sallest integer larger or equal to u. 3. Siilarly, with < ν < /2 for an appropriate sequence {B } of Brownian bridges and ξ R, sup s s ν /+ s /+ Z ξ s +ξ s, ξ ξ ξ as. B s = o P, Lea 3.2. Under Condition 2., there are functions At A t and at = a t+oa t, as t, such that for all ε,δ > there exists t = t ε,δ such that for t,tx > t, Vtx Vt at At x H,ρ x εax x +ρ+δ,x +ρ δ. 9 Moreover, and atx at x At x xρ ρ εax x +ρ+δ,x +ρ δ Atx At xρ εaxxρ+δ,x ρ δ.

8 Ferreira and de Haan/On the bloc axiethod 8 Rear 3.. This is an easily obtained variant of Theore B.3. of de Haan and Ferreira 26. Note that, H,ρ x = x +ρ ρ +ρ x, ρ x logx x, ρ = x ρ ρ ρ logx, ρ = 2 logx2, ρ = =. Proposition 3.. Assue the conditions of Theore 2.. Let < ε < /2 and {X i, } i= be the order statistics of the bloc axia X,X 2,...,X. Then, X s, b + = B s s ++ AH,ρ s /2 ε s /2 ρ ε +Bss ρ ε o P, as n, where the o P ter is unifor for / + s / +. Rear 3.2. This proposition should also be useful when analysing other estiators for the bloc axia approach, lie the axiu lielihood estiators. Proof of Proposition 3.. By representation 8, X s, b V Z s, b V = d + V b b = I rando part+ii bias part. We start with part I, Z s, I = V V a a a = I. I.2.

9 Ferreira and de Haan/On the bloc axiethod 9 According to of Lea 3.2, for each ε,δ > there exists t such that the factor I.2 is bounded above and below by { ρ +A ±εax ρ+δ, ρ δ} ρ provided t and s e /t. According to 9 of Lea 3.2, for factor I. we have the bounds Z s, +A { H,ρ Z s, Z s, +ρ+δ, +ρ δ } ±εax Z s, = I.a+I.b provided s e /t and /log t the latter inequality eventually holds true since A is bounded. Note that /log t iplies Z, 2t which iplies Lea 3. Z s, 2t for all s. For ter I.a we use Lea 3..3: Z s, is bounded above and below by B s s + ± ε s s ν s +, for soe ε >, < ν < /2 and all s [/ +,/ +]. Next we turn to ter I.b. By Lea 3.2, A above and below by A { ρ ±εax ρ+δ, ρ δ} is bounded provided s > e /t and /log > t. Furtherore for ρ and s [/ +,/ +], by Lea 3..3, H,ρ Z s, { = +ρ } Z s, Z s, ρ +ρ [ = { Z +ρ +ρ s, ] ρ ρ +ρ +ρ [ Z s, ]}

10 is bounded by ρ Ferreira and de Haan/On the bloc axiethod { +ρ [ [ = ± 2ε ρ s s ν s, B s s ++ρ ± ε s s ν s ++ρ B s s + ε s s ν s + and siilarly for cases other than ρ. The reaining part of I.b, naely Z s, +ρ+δ, +ρ δ ±εax Z s,, is siilar. Part II, by the inequalities of Lea 3.2, is bounded by { A H,ρ ±εax ρ+δ, ρ δ} hence it contributes AH,ρ to the result. Collecting all the ters, one finds the result. Proof of Theore 2.. Let, for r =,,2, J r s... s r s =, s [,].... r Note that J r s sr, as, uniforly in s [,], and, i= i...i r... r = J r sds = r + = ] ]} s r ds.

11 Ferreira and de Haan/On the bloc axiethod Then, r +Mr b r + Γ = r + X s,j r sds b r + = X s, b r + r + s r J r s ds = r + + r + + r + /+ /+ /+ /+ r + = I+I2+I3+I4. For I4: since For I, note that J r sds X s, b X s, b X s, b s r J r s ds J r sds J r sds J r sds s r J r s = O/ uniforly in s, I4= O/. /+ s r ds X s, b s r ds = o P. This follows since, the left-hand side of equals, in distribution, + r+ V Z, V which, by Lea 3.., Lea 3.2 and the fact that /log, is bounded below and above by { Z, } + r+ +AH,ρ Z, ±εaax Z +ρ+δ,,z +ρ δ,. This is easily seen to converge to zero in probability, since Z ξ, / = {logz, } ξ log ξ / P for all real ξ and A λ. Hence, I= o P. Next we show that /+ X s, b J r sds = o P. 2

12 Ferreira and de Haan/On the bloc axiethod 2 The left-hand side equals, in distribution since J r s for s +, V Z, V. + Lea 3. yields V Z, V = Z, { +A H,ρ Z, ±εz +ρ+δ, which is since Z, / converges to a positive rando variable of the order O P. Hence the integral is of order + which tends to zero since < /2. Finally, I2 has the sae asyptotic behaviour as r + /+ /+ which, by Proposition 3. tends to r + X s, b s r Bsds+λr + s r ds For the evaluation of the latter integral note that for ξ <, r + Moreover, note that r + } H,ρ s r ds. s r ξ ds = r + ξ v ξ e v dv = r + ξ Γ ξ. s r ξ ξ ds = r +ξ Γ ξ, ξ < ξ D r = logr + Γ as defined by continuity, and r + H,ρ s r ds = ρ D r +ρ D r, ρ, = D r = r+ Γ +logr +Γ Dr r+,,ρ =, D r = log 3 r ++Γ 3logr +Γ, =,ρ =. 2

13 Ferreira and de Haan/On the bloc axiethod 3 Proof of Theore 2.2. Fro Theore 2. we obtain, 2M M 2 Γ d Q Q 3M2 M hence, by Craér s deltethod, 3ˆ, 2ˆ 3, 2 = d 3 Γ 2 3 Γ It follows that ˆ, P and hence rˆ, r has the sae liit distribution as It follows that ˆ, 3ˆ, 2ˆ 3, 2 = 3 2 d Q 2 Q 3M2 M 3 2M M 2 3 Q 2 Q 2 Q Q [ 3ˆ, 3 = rˆ, r logr r, r = 2,3. 2ˆ, ] 2 has the sae liit distribution as 3 log3 log2 ˆ, and, consequently, ˆ, d log3 log2 Γ Q 2 Q 2 Q Q. For the asyptotic distribution of â, we write, â, ˆ, = a 2ˆ, Γ ˆ, { 2M M 2 Γ + 2 2ˆ }, Γ Γ ˆ,, ˆ,.

14 Ferreira and de Haan/On the bloc axiethod 4 and the stateent follows e.g. by Craér s deltethod. For the asyptotic distribution of ˆb, we write, ˆb, b Γ ˆ, â, ˆ, = M b Γ Γ + Γ and the stateent follows e.g. by Craér s deltethod. â, Proof of Theore 2.3. The proof follows the line of the coparable result for the peas-over-threshold ethod see e.g. de Haan and Ferreira 26, Chapter 4.3 ˆx, x n = q c n = ˆb, b q c n q c n cˆ, n ˆb, +â, q c n ˆ, V + â, log p n q c n V cˆ, n c n ˆ c n V log p n + c n q c n â,. Siilarly as on pages of de Haan and Ferreira 26 this converges in distribution to + 2 B Λ λ +ρ. Acnowledgeents. Research partially supported by FCT Project PTDC /MAT /277 /29 and FCT- PEst-OE/MAT/UI6/2. We than Holger Drees for a useful suggestion. References [] Caires, S.29 A coparative siulation study of the annual axia and the peas-over-threshold ethods. SBW-Belastingen: subproject Statistics. Deltares Report [2] Csörgő, M. and Horváth, L. 993 Weighted Approxiations in Probability and Statistics. John Wiley & Sons, Chichester, England [3] Dobry, C. 23 Maxiu lielihood estiators for the extree value index based on the bloc axiethod: arxiv:3.56 [4] Drees, H. 998 On sooth statistical tail functionals. Scand. J. Statist. 25, [5] Gubel, E. 958 Statistics of Extrees, Colubia University Press.

15 Ferreira and de Haan/On the bloc axiethod 5 [6] de Haan, L. and Ferreira, A. 26 Extree Value Theory: An Introduction. Springer, Boston. [7] Hosing, J.R.M., Wallis, J.R. and Wood, E.F. 985 Estiation of the Generalized Extree-Value Distribution by the Method of Probability Weighted Moents. Technoetrics 27, [8] Picands, J. III 975 Statistical inference using extree order statistics. Ann. Statist. 3, 9-3. [9] Prescott, P. and Walden, A.T. 98 Maxiu lielihood estiation of the paraeters of the generalized extree-value distribution. Bioetria 67,