SOUTENUE

REPUBLIQUE DU SENEGAL.

UNIVERSITE CHEIKH ANTA DIOP DE DAKAR

THESE DE DOCTORAT D'ETAT DE MATHEMATIQUES

'-----' ""'-----'.~-=',-_~,.-",.,.-------l:ONSE!l AF:R!CAIN ET MALGACHE \'OUR L'ENSEIGNEMENT SUPERiEUR 1:, i\. tri:.E. S. - OUAG,b,DOUGOU !,{,':"/08 O5FfV;199g .. JarMonsieur Gane Samb L:i1regi.str sous n #02 '3~Q'4 JI....-- ----

en vue d'obtenir le grade de DOCTEUR ES SCIENCES MATHEMATIQUES

SUJET DE LA THESE

CARACTERISATION EMPIRIQUE DES EXTREMESET QUESTIONS STATISTIQUES LIEES

Le 6 Decembre 1991

devant la commission compose de:

Professeur Galaye Dia, Prsident RapporteurProfesseur Paul Deheuvels, Membre Correspondant de l'Acadmie

Franaise, RapporteurProfesseur Michel Broniatowski, Universit de Lille VI, rapporteurProfesseur Hamet Seydi, Universit de Dakar

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JE DEDIE CE HUMBLE TRAvAIL A MES PARENTSET AMIS, ENSUITE A MA TENDRE ET PATIENTE

EPOUSE, MA COMPLICE EN TOUTE CHOSE.

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REMERCIEMENTS

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Je remercie d'abord le Professeur Paul Deheuvels pour l'intret qu'il a toujoursport nos travaux depuis sept ans malgr ses multiples et riches occupations.Il a.rd'emble accept de se dplacer jusqu' Dakar pour faire partie du jury decette soutenance.

Mes remerciements sincres vont aussi au Professeur MichelBroniatowski pour les ~mes raisons. Sa courtoisie m'a beaucoup rassurquand je fus tudiant en thse au LSTA de Paris 6. \

Que dire de mes anciens professeurs depuis la premire anne l'Universit de Dakar, les professeurs Hamet Seydi et Sakhir Thiam.

Le premier mathmaticien mrite, a toujours t affable mon gard.Il vient encore de le prouver en facilitant l'organisation matrielle de cette

soutenance.Le second a personnellement organis mon voyage vers la France aprs

ma matrise. Ce travail n'aurait donc jamais eu lieu sans lui. Qu'ils trouventici, tous les deux, l'expression de ma ts sincre gratitude.

Enfin, "the last but not the least", le professeur Galaye Dia m'a toujoursencourag, depuis que nous tions ensemble au LSTA, lui prparant une thsed'Etat et moi une thse nouveau rgime. Il n'a mnag aucun effort pour quecette soutenance soit une russite. Je lui tmoigne ici ma reconnaissance.

Pour terminer, je remercie notre secrtaire de recherches, Mlle Rokhaya SaIT,pour la comptence avec laquelle elle a tapp une partie de cette thse.

Gane Samb LSaint-Louis, le 15 Novembre 1991

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SOMMAIRETITRE PAGE

INTRODUCTION GENERALE 1PREM'IERE PARTIE: 6

CONTRIBUTION A I:lINFERENCE DE L'INDEX D'UNE LOI DE PARETO.ASYMPTOTIC BEVAVIOR OF HILL'S ESTIMA TE AND APPLICATIONS 6

ON THE ASYMPTOTIC NORMALITY OF SUMS OF EXTREME VALUES 21

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THE WEAK LIMITING BEHA VIOR OF THE DE HAAN-RESNICK 30ESTIMATE FOR THE EXPONENT OF A STABLE DISTRIBUTION.

APPENDIX: RESUME DE LA QUESTION DE LA DISTRIBUTION DESSOMMES DE VALEURS EXTREMES. (Par D. Masan et al.) 44

DEUXIEME PARTIE. 46CARACTERISATION EMPIRIQUE DES EXTREMES. 46

EMPIRICAL CHARACTERIZATION OF THE EXTREMES: 461: A FAMILY OF CHARACTERIZING STATISTICS. 46Il: THE ASYMPTOTIC NORMALITY OF THE CHARACTERIZING 64VECTORSIII: THE LAWS OF THE ITERATED LOGARITHMOF THECHARACTERIZING STATISTICS. 113

A GAUSSIAN PROCESS LIMIT OF SUM-PRODUCT STATISTICS BASEDON EXTREME VALUES 147

TROISIEME PARTIE. 168CONTRIBUTION A L'ETUDE DES ESPACEMENTS.

1- GAUSSIAN APPROXIMATION AND RELATED QUESTIONS FOR THESPACINGS PROCESS 168

II - ON THE INCREMENTS OF THE EMPIRICAL K-SPACINGS PROCESS fORAFIXED OR MOVING STEP. 186

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QUATRIEME PARTIE.STATISTIQUES D'ORDRE DANS LES ESPACES LATTICES

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MAJORATION. MINORATION ET CONVERGENCE FORTE DES QUANTILESCENTRALES ET EXTREMES DE VARIABLES ALEATOIRES DANS DES ESPACESDE TYPE C(S). 203FIN

INTRODUCTION GENERALE

Cette thse regroupe un ensemble de travaux dontle lien vident est la statistiqued'ordre. Toutes les recherches dont les rsultats sont exposs ici nous ont tproposes ou, ont germ dans notre esprit lorsque nous tions dans leLa.boratoire de Statistique Thorique et Appliques, (LSTA) de l'Universit deParis 6 o j'tais tudiant \en. thse entre 1983 et 1986. En ce moment,\toutesles questions relatives aux statistiques d'ordre, taient systmatiquementtudies soit par les professeurs Geffroy et Deheuvels soit par les membres deleurs quipes. Je voudrais relater ici les circonstanc~s dans les quelles j'ai eu m'intresser chacun des travaux exposs ici dans ce document. Ceci nouspermettra aussi de classer les travaux dans un ordre chronologique.

PREMIERE PARTIE.

La premire partie de ce document est une contribution la thorie del'infrence sur l'index d'une loi de Pareto. Au dbut des annes 80, la jonctionentre cette thorie d'une part et le problme de la determination de la loi limitedes sommes de valeurs extremes a t ralise par l'estimateur de Hill(1975)de l'index d'une loi stable. Cela justifie, entre autres raisons, la plus grandeimportance donne cette estimateur parmi la classe des estimateurs d'un telindex.

L'infrence sur l'index d'une loi de Pareto a t trs vite tendue toute laclasse des fonctions de rpartitions dont le maximum des observationsindpendantes (m.o.i.) est attir par une loi de Frchet quand la taille devientinfinie. Par ailleurs, on sait que le m.o.i. est attir soit par une loi de Frchet,une loi de Weibull ou soit par une loi de Gumbel. Le cas de Weibull estimmdiatement trait travers le cas de Frchet par une simple transformationalgbrique. Ds lors, l'alternative naturelle des rsultats de convergence et denormalit asymptotique tait l'attraction du m.o.i. par une loi de Gumbel. Cetteclasse contient les trs importants cas particuliers des lois normale, log-normale et exponentielle. On comprend dss lors l'intret des spcialistes port cette question.

.. 'j '4,

.. 'j 4, -2-

Le Professeur Deheuvels m'a demand une aprs-midi d'octobre 1984de chercher le comportement de l'estimateur de Hill(1975) lorsque lafonction de rpartition associe tait celle de Gauss. Ce fut mon premiercontact avec les valeurs extrmes. Les rsultats obtenus puis gnralisspour toute la classe L des fonctions de rpartitions dont le m.o.. est attirpar un loi de Gumbel ont t publis dans l~ Journal of AppliedProbability(1986). J'ai t trs surpris quand le Professeur Deheuvels m'ademand, de rdiger l'article en Anglais puis de le soumettre J.A.P.Depuis, j'ai pris l'habitude d'crire mes articles en Anglais.

Mais lorsqutil s'est ,agi de prouver la normalit asymptotique del'estimateur de Hill~ la collision aavec le formidable travail de Csorgo-

,Haeusler-Mason s'opra. Finalement, je trouvais da~s le travail de Csorgo-,Mason(1986) la manire de rsoudre le problme. Il a suffi d'affinersuffisamment les, proprits des lments de L pour reconduire lesmthodes de Csorgo-Mason. Dans Csorgo-Haeusler-Mason(Ann.Probab.Vol 19, No2, 783-811), o les lois limites des sommes de valeurs extrmessont entirement dtermines. Cette contribution l'tude de la normalit dessommes de valeurs extrmes a t reconnue et signale (voir annexe, p. ).

Dans la foule de ces rsultats, nous nous sommes intresssaucomportement de l'estimateur de DE HAAN-RESNICK(1980) (voir troisimearticle) et des estimateurs de Csorgo-Deheuvels-Mason( 1987) (rsultats nonexposs ici). Ces deux papiers furent publis en rapports techniques au LSTA,Nos 29/1985 et 49/1985)

TROISIEME PARTIE.

Au cours de l'anne 1985, Dr. Van Zuijlen a t invit au sminairede Deheuvels-Chevalier. Il fit un expos sur les espacements. Ce domaine

m'enchanta. Le Profeseur Deheuvels qui je fis part de cela me remit toute unedocumentation fraiche et complte non publie sur le thme.

Je me suis donc intress l'approximation Gaussienne du processusempirique bas sur les espacements d'un chantillon issu d'une v.a alatoireuniformment rpartie sur (0,1). Nous avons montr que l'approximation

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propose par Aly-Beirlant-Hprvath(1984) tait optimale- dans l'approche quej'appeleraide Shorack - par la dtermination de la limite suprieure. Ce rsultata t obtenu mme avec un pas devenant infini avec la taille de l'chantillon. Cersultat indique qu'il faudrait ncessairement changer d'approche pour atteindreune meilleure vitesse.

Rcemment, au colloque "Order Statistics and Nonparametrics: Theory andApplications" tenu Alexandrie, 18-20 September 1991, Dr. Aly me confirmaque cette vitesse n'a pas encore t amliore. Je pense m'intresser ceproblme plus tard.

Nous avons tendu certaines proprits du module de continuit duprocessus empirique classique au processus des espacements. Ces travauxfurent publis en rapports techniques LSTA N 47 et 48/1986 et exposs ausminaire Deheuvels-Chevalier(5 Fvrier 1985).

QUATRIEME PARTIE.

Vers la fin de l'an 1986, nous nous sommes intress aux Statistiquesd'ordre dans les espaces de Banach lattices, particulirement dans C(S), o S estun espace mtrique compact sparable. Cet article est le premier pas. Il futd'abord soumis Journal of Multivariate Analysis. Les deux rapports desreferees taient concluants. Je n'ai pas pu, malheureusement, l'poque (1988)envoyer une version rvisee. Entre autres, l'un des referee confirma l'originalitde l'tudue aprs avoir consult les fichiers MATHFILE.

Nous avons l pour l'Universit de Saint-Louis un beau champd'investigations pour nos collgues et tudiants intervenant en Statistique.

DEUXIEME PARTIE.

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Cette partie, il est vrai, doit tre classe immdiatement aprs la premire.Mais, chronologiquement, elle constitue notre dernire srie de travaux allantde la priode de 1988 1991. Nous voudrions la considrer comme notrenclusion notre contribution l'infrence des extrmes.

Nous caractrisons l'attraction du m.o.i. par une classe de huit statistiques.Deux statistiques jouent les rles principaux. La premire (notons la commedans notre premier article Tn (2,k,l), dj fort connue, est celle de Hill(1975).La seconde, note An(1,k,I), est dans sa forme, nouvelle. En fait ladiscrimination entre tous les cas (Frchet, Weibull, Gumbel) est dj ralise

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par le couple (Tn(2,k,I), An(1,k,I)). Il faut cependant signaler que cettediscrimination a dj t obtenue par Dekkers-Einmahl-DE HAAN (Ann.Statist., 1989,17, 1833-1855). Des rsultats similaires sont signals parTiago de Oliveira et Arne Fransn (J. Tiago de Oliveira (ed.), StatisticalExtrmes and Applications, 373-394).

Nous avions'dj dit que An(1,k,l) tait nouveau en tant qu'estimateur del'index d'une lo'i'de Parto. Lors du Colloque d'Alexandrie, le Professeur DEHAAN m'a dit qu'il pensait que l'esti mateur Mn de son papier avec Dekkers etEinmahl avait un lien avet le ntre. De retour Saint-Louis, j'ai vrifi que Mn=2A (1,k,O). Ceci dinimue la porte de notre affirmation. En fait, la forme quenous avons introduite est drive directement d'une intgrale. Elle a doncl'avantage d'tre plus commode lors de l'tude de la normalit asymptotique.

Une fois la caractrisation obtenue, les questions naturelles relatives auxtechniques asymptotiques s'imposaient d'elles-mmes.

Elles sont:1) normalit, asymptotique des statistiques.2) lois du logarithme itr.

Les deux articles qui suivent traitent de ces deux questions. En le faisant,nous avons obtenu des rsultats de caractrisation. Ce qui permet d'aller plusloin plus tard en developpant le calcul. Nous devons signaler le role primordialjou par les approximations Csorgo-Csorgo-Horvth-Mason(1986).

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Dans le quatrime article, les statistiques Tn (2,k,l) et An (1,k,l) sontgnralises un rang quelconque. Le rsultat est un processus dont l'tudeasymptotique est tudie. Cette tude nous fait dboucher sur un processusapparemment nouveau dont la fonction de covariance est entirementdtermine. Ce processus ainsi mis en vidence mrite mon avis, une tudeparticulire. C'est un lment de plus pour une future quipe de recherche Saint-Louis ou simplement entre Saint-Louis et Dakar.

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CONCLUSION.

D'une manire gnrale, chacune des parties de notre travail contient desdirections de recherche. Aucun travail de recherche ne peut tre complet etdfinitif. Il faut toujours s'arrter un instant, profiter de ce qui a dj t fait,puis rpartir; Mon souhait est qu'merge entre les Universit de Dakar et deSaint-Louis une quipe aussi clbr que l'cole Hongroise ou l'coleNerelandaise dans le domaine des probabilits et statistiques. La chose n'estpas impossible. En tout cas, tout au long de cet expos, nous avons signal desproblmes statistiques dignes de recherhes de niveau international.

Nous promettons d'y travailler.

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Enfin, un mot sur la forme de la thse. Nous aurions pu tout recrire en unseul corps avec beaucoup de rappels. Nous avons prfr avec l'autorisation deMonsieur Deheuvels, laisser apparatre le chercheur dans ses activits. Lesarticles et rapports techniques sont rendus tels qu'ils ont t prsents au public.

Saint-Louis, le 15 Novembre 1991.Gane Samb L

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PREMIERE PARTIE~

CONTRIBUTION A L'INFERENCE DE L'INDEX D'UNELOI DE PARETO

J /\PPi. jJroh ~3, q=~__~c\ ( !qt~(~.lPr!!l!ed in /srnc!

.-\SYMPl'Ol'IC nEHA VIOR OF HILL'S ESTJ\L\ TE AND APPLIC-\ l'IONS

.\ nstrocl

The :,:\,hlern of eSllnl{lll~g. the t\pi.Jnent (lf 3. sta~ie la\\' is rec)\-ing ;::1inc:::~~5-!;:~ JiilOuni: (If Jilntinn hl.';:;~~:': Pi1reto's 12.... \~)r Zipf's 13\\") d(,;"C,ib:..'~Ill:!::~, t-:Cl)~;f!Cjl rhL'rH-lr:H~r::i \"Cl:' W;: (5~C c_~ Hill (J:+)). This r;(~h!::'.;~; 1,;:',:-Grs; .'(lh~J n~' Hill (11.)'7)), \\"ho proposed an estJn12[e. 2!id the (()n .... er,::cnc l,fth2: c,:;li;-',L";(C 10 ~Oin~ rt)Sill .... e e:tnd hnl1c nU.mber was snl)\';n ta DC fi ch3ra:k.ris-li,~ \..',; t;:~,~~ihl~::on ;'lJrlct!nrls bel(ln~in~ {\..) th~ Frchtt dom~llll 0; at:,3;"';:()'

(\L~~(l~l (1(;0~) ,-\s a conlribution io 3 cump!cte thCOi)' of inkrtncc for !!l('u!)~:::r I~)I ,Ji a gcncral distribution functrn, ',.I.e gi\'c the asymptotic bcha\':e"(\\e:1k 2:ld S1ron~) of Hil\"s estimate when the associatcd distribution func

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GANE SAMB LO

If (Hl) holds, we have (he following (see e.g. Lemma 3):(1.2) there exisl sorne constant Co and a positive function s( ) su ch lhat

0(1 - u)= co+ s(u)+ f (s(t)/t)dt,where 5 ( . ) is a function slowly varying at (SYZ).

Therefore. we can rnake the following assumption:(H3) (Hl) and (1.2) hold and s(u) is ultirnately non-increasing as u t 0Our main results are as follows.

Theorem 1. LeI

(H4) F(log(' )) E:: D(A) and F(x) is ultirnately continuous as x 1A

be salisfied Then for anv sequence "n = k satisfying (K), we havc

k ! 'f PC .. '0 - X,- + 00,

1 = Il

-1 " Pen' k ~ (Xn-i.,o - Xn-k.n)- l,1 =1

as n ~ + 00.

Corol/ary 2. Let (H2) and (H4) be satisfied. Let k = (n'), 0< 0 < 1. If forsorne Il, 0 < Il < 0/2, we have

(K2) n"s(k/n)->+oo, as n->oo

AsymplOtic behavior of Hil/'s estimare alld dpplIC

.-tn mploric behavior of Hi/l's esrimare and app/icariolls

(b) Let a = +00. Suppose that there exists a sequence (1;, 1,),1;, ln A suchthat (I~- tn)/R(tn) is bounded. (ii) would impil that k(t~)~ k(tn) is alsobounded, which is not possible at the same time as a = + x.

(c) Let a = - 00. We use (i) at the place of (ii) in the above case and get that(l' - 1)/R (t ) _ - 00.

50, we have proved that

(k(t')-k(I)-a)=> ((l'-I)/R(I)-a) al 1'.1 i A.

Converse/y, suppose Ihal (l' - 1)/ R (1)_ a.(0) Let a be finite. Then for any E > 0, we have for t', 1 near A,

(+ (a - E)R(I)~ l' ~ 1 + (a + E}R(I). Therefore, (13) implies

a - E ~ lim inf k (l') - k (1) ~ lim sup k (r') - k(t) ~ a + E.t',r t A (./ i A

(13) Let a = + 00, for any ri> 0, we have for ('. 1 near A : 1';::; 1 + dR (1).Therefore (B) implies

lim inf k (l') - k (1);::; ri.,',1 i A

(-y) Let a = - 00; similarly to the preceding case. we gel

limsup k(t')- k(I)~ - d_,',/ T A

By letting E - 0, and d - + 00, we get the other implication of the equivalencewe had to prave.

Lemma 2. Let FE D(A). If in addition F(x) is continuous for x near A,then F-'(I - u) = 0(1- u) is slowly varying at (SYZ).

Proof of Lemma 2. Let l' = 0(1- u) and 1 = 0(1- uv), with v fixed andv> O. Beeause of the eontinuity of F(), we have u = 1- F(t') = exp( - k(t'))and uv = 1 - F(t) = exp( - k(I. Hence k(I') - k(l) = 10g(1/v). Lemma 1 im-plies then (t'-t)/R(t)-log(l/v), which in tum implies that

O(l- u) _ R(O(l- uO(l_uv)-I+(Iog(l/v)+o(l O(1-u)

But, by Lemma 5, R(t)/I-O as ItA, whenever FED(A). HeneelimuloO(l- u)/O(l- uv)= l, which is the announced result.

Proof of Theorem 1. Since (H5) holds, Lemma 2 implies that the functionH (l - u) assoeiated with F(log( is SYZ.

Note that O(l - u) = log H(1 - u) as u 10. Recall also the well-knownrepresentations

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Asymproric behavior of Hill' s esrimare and applicarions

Proof of Lemma 3. The proof fllows from Theorem 1.4.1 and Theorem2.4.1 of De Haan (1970).

Lemma 4. Let the assertion (ii) of Lcmma 3 be satisfied. then FE D(A). Ifin addition F is continuous as x i A, we get

. s(u)_~IP R (00 - u - 1.

Praof of Lemma 4. The proof is given in Lo (1986), Lemma 4.

Lemma 5. Let (HI) and (H2) be satisfied, then R(I)/1-1-0 as 1 i A andR(O(I- u is SVZ.

Praef of Lemma 5. First, we remark that Lemma 4 implies that R (00 - uis SVZ and by Lemma 3 of L (1986), R (1)/1 -1- 0 as 1 i A

Proof of Theorem 2 (continued). Since (HI) is satisfied, suppose that (1.2) isreduced to

0(1- u)= c + f (s(I)/t)dt.Thus, for each i, 1~ i ~ k, k satisfying (K), one has

(2.5) X"-i+I."-Xdn ~ OO-Vi.n)-O(I-V'+ln)=fu'"(S(U)/U)dU.U'"

Now, since s(u) is SVZ, it admits the Karamata representation:

s (u ) = z ( u )exp (f (w (v )/ v )dv) ,

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(2.6)Z(u)-1-Z, O

Asymploric behavior Dj Hill" s eslimare and applicarions

R (0(1- u))/s(u)~ 1, as u! O. Now, since s(u) is SVZ and (n/k)Uk." ~ 1, a.s.,we get by Lemma 5

R(O(l- Ukn ))/s(k/n)-'.>1, a.s.

which completes the proof.

Proof of Corol/aries 3 and 4. These two corollaries will follow from thelemmas below. Define by L the set of distribution functions F satisfying (Hl)and (H2).

Lemma 7. Let A > O. Then if X has a distribution function FEL,log sup(O, X) has a distribution function 0 ELand

S(O-'(I- u))~ {R(O(l- u))/O(1- u)}, as 1< L0,where

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_J'''F' (1 - 0 (v))S(I)-, (l-O(I))dv,

Conversely, we have the following.

- 00 < 1 < log A.

Lemma 8. Let R (I)~ 0 as 1 i A. Then if X has a distribution functionFEL, then exp(X) has a distribution function Z ELand

R(O(I-u))-{T(Z'(I-u))/Z-'(1-u)} as u LO.where

_J'AI-Z(V)T(t) -, 1- Z(I) dv,

for u ~ YI ~ 1.

Proof of Lemmas 7 and 8. These lemmas are proved in Lo (1986), viaLemmas 9 and 10. Note that (H2) implies that log sup(O, X) exists almost surely ifA >0.

Proof of Corol/aries 3 and 4. By Lemma 7, we see that if for instance (Hl) issatisfied, the same property is also true for log sup(O, X). So, we can write (1.3)with s (. ) replaced by r('), where r(u) is derived from De Haan's representationfor O-I(1-U) as in (1.2). But, we see also that 0(1- u)= co+J~(s(t)/I)dt =>s(u)= uQ'(l- u). Since

0'(1- u) = log 0(1- au), a = P(X > 0),

r(u) = ug~~l_-u~) = u (0-'(1 - u)')

is SVZ by Lemma 2 and (1.2). Hence,

0-'(1 - u) = - 0-'(1- YI) + f !ljJ dl,

AsympiOcic behavior of Hill' s estimate and applications

3. Applications and simulations

As remarked above, Hill (1974) desribed sorne basic models which follow

(1.1) or (Ac). We note that ail these models are c10sely related to problems based

on extreme values. As already noticed, the works of several authors (Csorg6 and

Mason (1985), Csorg6 et al. (1985), Hall (1982), Hill (1970), (1974), (1975), Mason

(1982 have entirely settled the properties of T" under the assumption (Ac). Itfallows from their results that if X io X 2 , " Xo are the observations of X, we canverify if (Ac) holds. In that case, we proceed as follows:

3.1. Identification of the upper tail of a distribution.(i) Chaose 0,0< 0 < 1,(ii) Choose k. = (n 6 ),(iii) Calculate T. = k ~1 L::~ (X.-'+l . - X"-".) for large values of Il.(iv) If T. is very near c, 0 < c < + 00, Ihen by Theorem 1 of Mason (1982),

(Ac) holds.1 d

(v) But (Ac)? FE D(A)? c- (X,," - Q(l-l/n~A,and therefore, we can use (v) for predictions about a critical value of X,,".However, il is not always certain that T. converges 10 a finile slrictly positivenumber. For example, if we want to know whether F salisfies (Ac) or if F(log( is the distribulion function of the standard Gaussian random variable, how could

we proceed?

3.2 Comparison between a regular rail and a gaussian tai/. We want ta knowif

(Ll) F(log(x = x -Ile (1 + DO (x -b, c > 0, and b = 1/2eor

() XI = log sup(O, Z), Z ~ N(O, 1)where F() is Ihe unknown distribution function associated wilh Ihe observa-tions XI, X 2 , " X. of X.

(i) Choose k = (n 112), then(ii) If (LI) holds, we have

(a) (Mason (1982 T" ~ c, a.s.(b) (Hall (1982 nI/4(T"-c)~N(0,1).

(iii) If () holds, we have(a) (Corollary 5, Part 3, p = 1), log nT. ~ 1, a.s.

(b) (Lo (1986 n 1I4(D"T" - 1)~ N(O, 1), D" = log n(l + 0(1.Thus, we see that we are now able to test (Ll) against (LI). If we chooseD = {( C - t:) ~ T" ~ (c + t:)} as the accepted region of our test, (i) and (ii) givethe characteristics of that test. To test (LI) against (LI), One can choose15 = {1 - t: ~ T" . log n ~ 1 + d as the accepted region wilh a small value of t:.

3.3. Comparison between an exponential and a gaussian tail. Let

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Asymptotic behavior of Hill' s esrimate and applications

Remarks.1. One might be surprised to find that our simulations are not sufficiently

good, considering the large size of the sam pie space (n = 4(00). However, onlythe k extreme observations (k = 62 or 63) are used for the calculation of Tn

Taking that into account, the theoretical part of this paper is relatively weilillustrated by the simulations. SpecificaIly:

2. Column 4 illustra tes the almost sure convergence of HiIl's estima te for thePareto law: TnJ converges almost surely to 1.

3. The identity of columns 2 and 3 is a consequence of the choice of Xi. It isc1ear that if Xi = ( - 21og(1 - u; If2, we get Tn 2 = ~ Tn 1. However, this choice is notarbitrary. Indeed, if Tnz is the true value obtained from the use of the truequantile function, we have Tn*z = Bn(li)' where (see e.g. statement (2.\3)).

1 =1 {(-'21 (1- .1/2(1 [Og(-IOg(I- Ui)-47T+O(l)}og l, og og u, + 4 log(l - ui

)

= ~ log 2 + ~ log Yi + 0 ((lOg log 1~ uJ / 410g 1~ uJ .

With our data, 3937 ~ i ~ 4000, k = 62 or 63, we have

log li = 410g 2 + ~ log y, :!: 0.07,

therefore

4. We now test (U): (Xi) are the order statistics of a N(O, 1) random variable,against (L4): (Xi) are the order statistics of the standard exponential law. If wechoose R. = {l- a ~ log nTnl ~ 1 + a} as our accepted region and choose a =0.29 as the significance level of our test, the power of the test will be f3 ~ 0.74,and Rn will be Rn = {0.75 ~ Tnl 'Iog n ~ 1.2533}. Here, we accept (U) since thetable gives the value 0.7554 for (log n . LI) for n = 4000.

5. Column 5 gives the ten first values of the order statistics of the uniformrandom variable. One may work with the highest or the lowest values since

{l- u;, 1~ i ~ 4000} ~ {U4X)-'+I' 1~ i ~ 4000}.

Conclusion. De Haan and Resnick (1980) and Csorg6 et al. (1985) have alsogiven estimates of c under the assumption (Ac). In future papers, we shaHdescribe their asymptotic behavior using the assumptions of this paper.

5. Remarks and furtber generalizations

Remark 1. Deheuvels et al. (1986) have recently shown that the De Haanrepresentation (1.2) holds whenever F(' ) belongs to D (A).

-19-

- 2~ -

Journal of Sialislicai Planning and Inference 22 (1989) 127-136NonhHolland

A NOTE ON THE ASYMPTOTIC NORMALITY OF SUMS OfEXTREME VALUES

Gane Samb L

Univers Paris 6, L.S. T.A. T.4555, E.3., 4 Place Jussieu, 75130 Paris 05, Fronce

Recei\'td 15 January 1988; revised manuscripl received 16 March 1988Recommended by P. Deheuvels

Absrracr: Lei X I,X2 , be a sequence of independenl random variables wilh common distribulion funnion Fin Ihe domain of allraction of a Gumbel exneme value distribulion and for eachimeger n 2: 1. let X I ft " "X",n denole the order statislics based on lhe first n of these randomvariables. Along wilh related results il is shown that for any sequence of positive imegers k. - 00

and k./n-O as n-oo. lhe sum of lhe upper kn extreme values Xn-k.'l.n+'+X. n' whenproperly cenlered and normalized, converges in disl ribulion 10 a slandard normal random variableN(O.I). These results conslilule an extension of resuhs by S. Csorgo and D.M. Mason (1985).

AMS Subjecr Classification: 62E20, 62G30. 6OF05.

Key words: Order stalistics; extreme values; Gumbel law; asymplotic normality.

1. Introduction

Let X"X2 be a sequence of independenr random variables with commondistribution function F and for each integer n ~ 1. Jet Xl,n ~ '" ~Xn.n denote theorder statislics based on the first n of these random variables. Csorg and Mason(1985. 1986) have recently shown among other results that if

l-

I-F(x)=L*(x)x-a as x-oo, (1)

where L is a slowly varying function at infinily and a~ 2, or if F has exponentiallike upper tails, meaning

l'~ (1 - FlY dyl(l -F(x --+ C as x --+ 00.. :r

where 0 < c< 00, then for any sequence of integers satisfying

(2)

(K)

there exist sequences An> 0 of normalizing constants and Cn of centering constantsi such thatf!

as n-cx>, (3)

03783758/89/$3.50 1989. Elsevier Science Publi,hers n. V. (Norrh.Holland)

C.s. Lo / Asymplolic norma/ily of ulremes

-22-

"'1'..

The case (1) is contained in the lheorem of Cs6rgo and Mason (1986) and the case(2) is Theorem 1.5 of Cs6rgo and Mason (1985).

An application of Theorem 2.4.1 of de Haan (1970) (Lemma 1 below) combinedwith Fact 1.4 of Cs6rgo and Mason (1985) shows that (2) implies the existence ofsequences of normalizing constants On and centering constants bn such that

(4)

where G is a Gumbel random variable with distribution function

P(G::5x)=e.\p(-e- X ) for -oo=s-p\ (I-u)PdQ(u), forO::5s::51.,,1 - 5

For convenience, when P= l, we set c(s) =c(s, 1). (Refer 10 the nexl section for ourinlegral convention.)

Let D*(A) denote the subclass of D(A) consisting of ail distribution funclions Fwhose quantile function Q satisfies

'1

Q(I -s)=a+ \ u-1r(u)du,, s

for ail s~ 0 su fficielllly small, ois a fixed constant and ris a striclly positive functionslowly varying at zero. The fact thal D*(A) is a subclass of D(A) follows fromTheorem 2.4.1 of de Haan (1970).

-23-

- 29 -

G.S. Lo / Asymp/o/ir no,malilY olex/,emes

whenever these integrals make sense as Lebesgue-Stieltjes integrals. In this case, the

usual integration by parts formula

rf dg +rg df=g(b)f(b) - g(a)f(a)is valid.

The proof of our theorem will follow c\osely the proofs of the results of Csorgo

and Mason (1985), substituting their technical lemmas concerning properties of the.quantile functions of distribution functions salisfying (2), by those describing

properties of lhe quantile functions of FED(II). We therefore begin with thesetechnical lemmas.

Lemma I. FE D(II) if and only if for each choice of 0 s x, y, w, z < 00 fixed, y * w,Q(I -sx)- Q(I -sz)

Q(I -sy) - Q(I -sw)

log x-log zas sLo.

logy-Iog w(8)

(9)

(II)

(This is Theorem 2.4.1 of de Haan (1970).)

Lemma 2. Whenever FED(il), c(s,{J) is slowly varying ar zero for each choice of0

-24-

- 30 -

ri C.S. La / AsymplOIic narmaliry uf extreme.0 sufficienlly small,

(1 + e)c(As, p):5 -----rJi c(s, Pl.

Observing that for ail 5>0 small enough,

1 00

\ (1 - u)1l d Q(II)~I ).t'sIlO(i+ \lB {Q(I - ).sOi + 1) - Q(I - AsB i )},,1 - ..h 1=0

(12)

we see that by an argument very much like the one jusl given, we have for ail 5 >0suflciently small,

(13)

Assertion (9) now follows from inequalities (12) and (13) by the fact that B can bechosen arbitrarily close to one and e arbitrarily close 10 zero. This completes theproof of Lemma 2.

The following lemma is related 10 Theorem IA.3.d of de Haan (1970) and itspmof is based on a modification of the techniques used 10 prove this theorem. Forderails sec Deheuvcls et al. (J 986).

Lemma 3. Whenever FE D(1), there exists a constant - 00 < b < 00 sllch that Jor 011O

-25-

- 31 -

c.s. Lo 1 Asymptotic normolity of extremes

Proof. Let Q(I - s) =0 Q(l - Slip). Since by Lemma 1 for any choice of 0 (sf!)-----

c(s) c(s)

Q(l - 2s) - Q(I - s) Q(l- (2s)P) - Q(I - sP)=0 X --=---------

c(s) Q(l - 2s) - Q(l -s)

which by Lemmas 1and 4 converges to 1/pas s 10, completing the proof of Lemma 5.

Lemma 6. Whenever FE D(/I),

a2(s)/(2sc2(s))~1 as s10. (17)

Proof. The proof is based on Lemma 5 and fol1ows almost exactly as the proof ofLemma 3.3 of Csorg and Mason (1985). Therefore, the details are omitted.

The proof of the following lemma is an easy consequence of the Karamatarepresentation for a slowly varying function.

Lemma 7. Let an be any sequence of positive constants such that an ~ 0 andnon ..... 00. Also let L be any slowly varying funetion at zero. Then for any O

r')

- 32 -

C.S. Lo / Asymp/o/c normali/y of ex/remes

-26-

and the quantile process

where

and, with UI.n:s. ... :s. Un.n denoting the order slatistics corresponding to UI' ... , Un'

_[Uk.n if (k-I)ln

- 33 -

C.S. Lo 1 Asympfofic normalif." of exfremes

From (21), we oblain

c(k /n 1_ v)R =0 (n-" n" " ,) k' V

l.n P ) c(kn/n) n'

which by Lemma 5 equals op(I).

Aiso

which by Lemma 6 is

-2a~(I/Il)/a~(k,,/Il) as n~oo.

-27-

"11!i1

From Lemmas 2 and 6 we infer lhal a 2(s) is regularly varying of exponenl one al

zero. Hence, by Lemma 7,

a2(I/Il)/a2(kn/n)~O as n~oo,

which yields R2n = ope 1).Thus we have proved R" =op(l).Nexl we show lhal Ll ~.n = ope 1). Choose any 1< A< 00 and set

T,,(A) = IIk,; 1I2C(k" ln) -III - Gn(I - knln) - kIl /n\ { Q(r,; (A - Q(r,;(A},

whereAk

r;(A) = 1 - _n andn

NOlice lhal since for ail 5 in the closed interval formed by U n - k n and I-k,,/n,

Il - G,,(s) - k,,/nl ~ Il - Gn(I -kn/n)-kn/nl,

we have for any ) < A< 00,

Since (K) implies (cf. Balkema and de Haan (I975 thal

pn(l- Un-k. ,,)/kn--+ 1 as n-+oo,

lhe lower bound in lhe above inequalily equals one.

Hence for each 1< A< 00,

Observe for each 1

- 34 -

C.S. Lo / Asyrnp/o/ic normoli/y 01 ex/rernes

Applying Lemma 4 we see that this last expression converges to 2 log A, which yields

-28-

lim lim sup ETn(A)=O. 11 n ..... 00

(24)

(26)

The facl lhal .1 2.'1 = op(l) now follows by an elementary argument based on (23)and (24). This compleles lhe proof of (5).

Nexl consider (6). NOlice lhal since FED*(A),

\

')

C(S)=S-1 r(l-u)du .. 1-5

Thus since r> 0 and slowly varying al zero Theorem 1.2.1 of de Haan (1970) gives

r(s)/c(s) --+ 1 as s10. (25)Th~ !eft sidc of (6) equals

k~f2 \'1- u... r(u) 1/2 r(knln)- -- - du = -kn -- {Iog(l- Un_k ln) -Iog(knln)}.

c(knln) .. k.ln U c(knln)

k~f2 J.I-U...... du * *---- (r(u)-r(knln-:=.1ln+.12n'

c(k" ln) k.ln u' .

The same argumenl based on (20) as given in Csrgo and Mason (1985) shows lhal

_k~i2{ log(l - Un _k. n) -Iog(kn ln)} = Yn+ op(l).

Therefore by (25) and lhe facl thal Yn=op(l) we have

.1tn= Yn+op(l).

Sincc r is slowly varying al zero we get for each 1< A< 00 as n --+ 00,

[ ~ A~]sup Ir(s) - r(knln)llc(knln): - :ss:s - --+ O.An nThe facl lhal .1rn = op(l) now follows easily from Yn = Op (1 ), (22) and (26) com-pleling lhe proof of (6).

Since FE D*(/J) we have

\

1

JJn(kn)-knQ(I-knln)=n r(l-s)ds.... I-k,,/n

Asserlion (7) is now a direcl consequence of (5) and (6).finally we prove lhe convergence in dislribulion of Zn' Yn and Zn - Yn 10 N(O, 2),

N(O, 1) and N(O,I), respeclively, as n --+ 00. NOlice lhal lhe Zn random variable in(5) is normal Wilh mean zero and second moment (J2(knln)/(knc

2(kn)/n), which byLcmma 6 converges 10 1 as n --+ 00. The Yn random variable in (6) is normal Wilhmean zero and second momenl 1- k nln --+ 1 as n --+ 00.

The Zn - Yn random variable in (7) is normal wilh mean zero. Applying Lemmas5 and 6 il is easy 10 verify lhal E(Zn - Yni --+ 1 and n:-' 00.

This compleles lhe proof of the lheorem.

-29-

- 35 -

G.S. Lo / Asymptotic normality of extremes

Acknowledgemenl

The aUlhor is eXlremely gralefui 10 David Mason, Erich Haeusler and PaulDeheuvels, for helping him 10 complele Ihis work.

References

Balkema. A. and L. de Haan (1975). Limit laws for order slalislics. In: P. Rvsz, Ed., Limit Theorems

of Probabilit)' Theory, Coll. Malh. Soc. Bolyai, Vol. Il, 17-22.Csorgo, M., S. Csorgo, L. Horvlh and D.M. Mason (1986). Weighled empirical and quanlile processes,

Ann. Prob. Il,31-85.

Csorgo, S. and D.M. Mason (1985). Central limil Iheorems for sums of extreme values, Math. Proc.

Cambridge Philos. Soc. 98, 547-558.

Csorgo, S. and D.M. Mason (1986). The asymplotic distribution of su ms of extreme values from a

rcgularly varying distribution, Ann. Probab. 14,974-983.Deheuvels, P., E. Haeusler and D.M. Mason (1988). Almosl sure convergence of Ihe Hill eslimator.

Math. Proc. Cambridge Philos. Soc. \04,371-381.Deheuvels, P., E. Haeusler and D.M. Mason (1989). Laws of the iteraled logrithm for sums of extreme

values in rhe domain of allraclion of a Gumbcl law. /Juil. Sci. Math., to appear.de Haan, L. (1970). On Regular Variation and its Application to the Weak Convergence of Sample

E~tremes. Malhemalical CenirC Traci No. )2 (Amsterdam).Hill, B.M. (1975). A simple general approach 10 inference about che tail of a dislribution. Ann. Starist.

J, 1163-1174.Lo, G.S. (1986). AsymplOlic behavior of Hill's eSlimale and applications. J. Appl. Probab. 23,922-936.

Mason, D.M. (1982). Laws of large numbers for su ms of e."reme values. Ann. Probab. \0,756-764.

-30-

THE l,vEAK Ll'IITH:G B':::AVIOR OF THE DE HAAN /RESNICK ' S

ESTIMATS Of THS EXPO~ENT OF A STABLE DISTRIBUTION.

by

LO Gane Samb, Universit Paris 6.

-L.S.T ..~_.

Abstract~

The problem of estimating the exponent of a stable la\" receives a conside-

rable attention in the recent literature. Here, we deal with an estimate of

such an exponent introduced by De Haan and Resnick when the corresponding dis-

tribution function belongs to the Gumbel's domain of attraction. This study

permits to construct new statistical tests for the inference about the upper

tail of a distribution and may beapplied to many biological phenomena. ~xamples

and simulations are given. The limiting law are shown to be the Gumbel's law

and particular cases are ,given with norming constants expressed with itera-

ted logarithms and exponentials.

Address: LO Gane Samb, Universit Paris 6. L.S.T.A. Tour 45-55, E.3.,

A, Place Jussieu. 75230, Paris Cedex 05.

Keys words and phrases: Regulary and slowly varying functions, Domain of attrac-

tion, norming constants, Order Statistics, Limiting law.

::

. .. '

11 -31-

11iTRODUCTION AND RESULTS:

Many biological phenomena seem to fit the Zipf's form:

l - G(x) = C x-l/c, c>Q and C>O.r .1)r instance, we can cite the plot against r of the population of the r-th lar-

st city ( see e.g. Hill(l974) ). This motivated considerable works on the pro-

em of estimating c. More generally, if XI ,X2, .... , Xn are independent and

ientical copies of a random variable ( rv ) X such that F(x)::: P( X~ x) satisfies

osed. De Haan anG ~esnick (1980) introduced

In order to contribute to a complete asymptotic theory for the inference

Domain of attraction of the Gumbel's law D(fI). The results we have obtained

as n -r + cok ::: ken) -r +

-32-

true if we replace

X ,X k by Log X , Log X kn,n n- ,n n,n n- ,n

Q(l-s) by Log Q(l-as), a = P(X>O)

r(s) by s(u) = R(Q(l-ufQ(l-u)

and if k satisfies Ks() at the place of Kr( ). Moreover, if Log A>O, we may

repeat the operation.

Conversely

Corollary 2: Let r(u)+O as ufO. then if (Hl) and (H2) are satisfied, (1.3)

and (1.4) hold if we replace

Xn,n , Xn_k n,Q(l-u)

r(u)

by

by

by

exp( X ), exp(X k )n,n n- ,n

exp( Q(l-u) )

t(u) = exp(Q(l-u R(Q(l-u

and if k satisfies Kt() at the place of Kr().

Corollary 3 ( particu1ar cases)

Here, we restrict ourselves to the case where k=(.Log n).Q.). ( )denotes the

integer part, and .Q. is any positive number.

1) Normal case: X'VN(O,l)

(i)

(H)

.!..Q.(2Log n)2(LogLog n ). T

n

(2Log n)~ T -- ~ 1n

.(1+0(1 - .Q.(LogLog n)(l+o(l ~ A

2) exponential case : exp(X) 'V E(l). ( or general gamma case, see the proof )

(i) (1+0(1 . .Q.( "Log n) (1ogLog n) Tn - t(LogLog n)(1+o(l - j A

A

1

C Tn n

(i) .Q.(LogLog n) C T - .Q.(LogLog n)(l+o(l jn n

~ 1.(H)

(ii) (Log n) Tn

3) Sup'pose that X a,;.s-L08p Sp(ep_151), 2), Z'VN(O,l),p~l, Lo-g p (resp. ep )

denotes the p-th~logarithm -( resp: exponential-) with by ~onvention LogO x =1,~

L C ( TI h=p-1 !et n 'V 2Log n) h=l Logh(2Log n)

1

\

11

1

111

-33-

Before the statements of the results, we need sorne notations.

Define

A = inf h, F(x) 1 } B = Sup h, F(x) o }

R(t) (l-F(t))-l fA (l-F(v)) dv, B~tO. Then, if (Hl) and (H2) hold, (1.3) and (1.4) remain

1

-34-

pose that A= +00, then for any real number c, O

S(G-l(l-u)) '\,R(Q(l-u , as u+O, whereQ(l-u)

Log AS(t)= S

t

-35-

l~G(v) dv, -oo

ce of independent rv's uniformly distributed on (0,1). Therefore, (2.3) implies

o=uo .s. U1 -S U2 .s .... oS U .s U 1 =1 are the order statistics of a sequen-,n- n- ,n- - n,n - n+ ,n

We recall tha t

( 2. 3) {X. , l.~iSn} ~ {Q (U . .), 1S,.sn }, U.l, n - - l , n - - l,n

-36-

SUPO

-37-

w~ere a and b are defined in the statement of the theorem. First, we prove thatn n

oA 4/an nU

By (1.2), we have An4 = r(k/n)-r(Uk+l ,n) + Jk~~l,n ( r(s)/s ) ds,

(2.12)

Remark that since r(u) is slowly varying at 0, we have on account of (2.10) that

r(k/n)!r(Uk 1 ' ) + l, in probability+ ,n

F'.lr thermore ,

(2.14)

where

U

a~l Ifk~~l,n ( r(s)/s ) dsl ~ ~Og(nk-luk+l,n)ls~ilr(~/~)1n

l =(Min(~, Uk 1 ), Max(~ ,Uk 1 )) is a random intervaln n + ,n n + ,n

At thiS step, we need a lemma

Lemma 5: Let r(u) be a positive function S.V.Z: Let (u ) te a sequence of rv'sn 1 -,\

~p-d (d ) be a sequence of real numbers such thatn

u J 0,n

-1(d .u ) = 0 (1), and (d .u )n n p n n o (1) an n+t=pthen

1+0 (1)P

r(s)InJf r(1/ d )SE nn

< SupsEJ

n

r(s)r(l/d )

n1+0 (1),

pas n + +00

.,.. ,.(

where un)' Max (~ ,un))n

:~!; 'j: ';,:. ;~;~

.( .;

Proof of lemma 5: The proof is the same'as that of lemma 3.5 in LO(1985b)

Proof of the part (ii) ofthe.theorem ( continued )

By (2. lOb) , ~ U ~ 1. Sa, we may apply lemma 5 ta (2.14) and getn k+l,n

(2.15)

Combining

U-1 If k+l n (/ 1a k/' (r s) s) ds.'S-o(1).n n - p

(2.13) and (2.15), we get (2.12)

(1+0 (1))P

Now; we concentrate on An3 and show that

(2.16) A 3/an n .A ,if k satisfies (Kr())

We have A 3 = Q(l-Ul )- Q(l-l/n)n ,n

-38-

(2.17) r(Ul,n)-r(l/n) - I0/ n (r(s)/s) ds,l,n

Recal1 that

(2.18) P( n Ul n~ x) +,

This means that nUl = (1) and,n p

-xe as n ++ co

-1(nUl,n) = 0p(l) . Thus, we may apply lemma

5 and get

(2.19) r(l/n)!r(U l ) X lasn++co,nThen, if k satisfies (K(r()), ~e have

(2.20)P+ 0, as n t + .xl

Now, let

We have

B = I l / nn Ul,n

( r(s)/s ) ds

B /an n= {r(l/n)/r(k/n)}. I l / n { r(s)

U r(l/n)l,n

dss

Then, it fol1ows from lemma 5 and the fact that r(u) is positive that if k satis-

fies (Kr()), we have

(2.21) {B / a} + Log nUln n ,n Log nUl . 0 (1),n p

= fi (x)-x

-ee) =< xLim P ( - Log nUl ,nn t+ ex>

We get finally that if k satisfies (Kr()), one has

By (2.18), we see that

(2.30) dB /a + ./\n n

(2.16) and (2.12) together imply the theorem.

Proof of the corollary 3:

Previously in the occasion of our study of the same particular cases for the

Hill's estimate ( see LO(1985a), lemma 5 and corollary 5 ) we have proved that

(3.1) Lim R(Q(l-u))/p(u) + l, where peu) = u Q'(1-u), Q'(u) = dl(u) foru-}O

values of u near 1.

With (3.1), we may handle the different points of corollary 3. Here, we

concentrate on the case where k =(Log n)~) , ~>O

-39-

J,- 2/2Jx -2-tI-Normal case: F(x) = (2n) e dt_00

Remark that F(Log x) is the distribution function of the log-normal law.

It follows that F(Log(.))~D(A). (H2) is obviously true. On the other hand, it

is well known that

(3.2) Q(1-s)l

(2Log(1/s)) 2 + L08L08(1/s) + 4n+o(1)l2 (2Lo~(1/s))2

1 .S si-O

(3.3) p(s) == s Q~ (1-s)l

(2Log(1/s))-2 (1+0(1)), as s~O.

Then

Notice that we might have used ( see Galambos(lS7G), p.66 )

-1 -3R(t) == t (1+o(t )) and P(s)""R(Q(l-s))

1 1 la- == P- (k/n) (1+0(1)) == (2Log n)2 (1+0(1)) and

n

an

bn

Q(l-l/n)-Q(l-k/n) t~08L08 ~1 ( 1 + O((2Log n)2

L08L08 n )Log n

Therefore, b =={tLogLog n } (1+0(1)n

-ax2-Exponential case: F(Log x) = 1-e , a > 0

More generally, since the tail of the quantile function associated with a ge-

neral gamma law y(r,a), r>O, a> 0, admits the expansion

(3.4) -1 1H (l-u) =(Log u) (1+0(1))the behavior of T is the same for all gamma laws because (3.4) doesn't depend

n

on r or o.. Tha t' s why we only consider

F(Log x) = 1 - e-x

Therefore

Q(l-u) = LogLog(l/s) , -1peu) == (Log(l/s))

then a == (Log n) (1+0(1)), b =(tLogLog n ) (1+0(1))n n

At this step, we apply the theorem.

3-1n this case, we have,

for large values of n,

T = {Log 2 - Log 2 k }/ Log kn p n,n p n- ,n

where 21 ,22 ,n, ... ,2 are the order statistics of a,n n,n

sequence of independent and standard Gaussian rv's

-40-

butiori functionassociated to Sup( e 10), Z), Z'VN(O,l) and m=P(Z > e 10 ))p- p-

1

11

r

1

We have also

(3.6)

Since X= Log Sup(e 1(1), Z), one hasp-

Q(l-u) = Log G-1Gl-mu)p

_t2/2e dt ), where G is the distri-

Log {(2Log(1/ms)1+ LogLog(l/s~)+Cil)}p 2 (2Log(1/ms)2

It follows from (3.3) and (3.6) that

and from (3.6) we deduce after sorne calcultions

b = ~LogLog n (1+0(1))n

Then

p -l(s) = (2Log(1/s)) TI;:r- l Logj(2Log(1/s))1

'- 1a 'V 2Log n n~-Pl- Log, n

n J= J

Remark that the part 3 of the corollary might have been derived from th~ part 1

of the same corollary after p applications of corollary 1. We have given the

normal case as example but such an operation is possible whenever Z, has a dis-1

tribution function F such that F(Log(. ))E. D(A) and Log 1 A > O. Even F(Log('))p-

E: D(~), where ~(x)=e-l/x is the Frchet's law, we can have the part 3 since

F(Log())E.D(~) impliesthat F()E.D(A).

III- Simulations

Her~, we will illustrate the behavior of T in the three casesn

Ci) exp(X)'VE(l), (ii) exp(X) = Sup(O,Z), Z'VN(O,l) , (iii) F(Log x) = l-l/x

For making our simulations, we have generated an ordered sample ui

' 1~i~4000

from a uniform rv. Therefore, we have constructed

Ci) an order sample of the standard exponential law

y. = - Log(l-u.)1 1

.1and defined Tn1 = ( Log Yn - Log Yn-k )/Log k, k=(n

2)

Tni = Cn ( Yi )

111

J

,

-41-

(ii) an ordered samp1e of the standard Normal law for U.t 1l

1-x. =(-2Log(1-u.) )2 ( see e.g. 3.2)

l l

and defined T 2 = C (x.)n n l

(iii) an ordered sample of the Pareto'S law

-1z. = (l-u.) and defined T 3 = C (z.)

l l n n l

Before we proceed any farther, we remark that with x_= (-2log(1-u.)), we getl l

1*1Tn2= "2 Tn1 In fact, we have Tn2 2 Tn1LogLog(l/(l-u k )

+ 0 ( n- n )4 Log k LogO/O-un_k,n)

where T~ is the exact value of Tn2 if we use the true quantile function. With

* * lour data, we get Tn2 = ! T;l 0.0179 and (Log n) Tn2 =(2 Log n)Tnl 0,14

The simulations are given as follows:

n (~Log n) Tn1(Log n)Tn2 U4000-i+1

3991 0.3294 0.3294 0.3302 0.002435 ".

;ifi~~:3992 0.3391 0.3391 0.4042 0.001631f;~~~,i;~r{

3993 0.3625 0.3625 0.4334 0.001620

3994 0.3550 0.3550 0.4324 0.001337

3995 0.4598 0.4598 0.5954 0.000988

3996 0.4693 0.4593 0.6124 0.000437.'~""~

3997 0.4689 0.4689 0.6130 0.000418

3998 0.4977 0.4977 0.6625 0.000308

3999 0.5116 0.5116 0.6963 0.000297

4000 0.6942 0.9942 1.0332 0.000095

1 2 3 4 5

-42-

Comments :

1) The right co1umn gives the first values of (u i ). With the symmetry of the

uniform law, we have {l-ui , 1~i~4000} ~ {u4000-i+1 ' 1~i~4000 l,1

2) if k~(n2), simi1ar calculations as in the proof of the part 2 of corollary

3 show thata'" Log n and a . b '" Log 2. Therefore, if 1 0.p.notes the De Haan/n n n nI

Resnick estic8te for exp(X) '" E( 1) ,~ we get

(4.1 ) (~Log n) Tn1 Log 2

the same considerations from the part 3 of coro11ary 3 ( p=l) yield

(4.2) (Log n) 1n2 Log 2

where Tn2

denotes the De Haan/Resnick estimate for X = Log Sup(O,Z), Z"'N(O,l).

Notice that (4.1) and (4.2) are weIl illustrated by our simulations since

Log 2 '" 0.69314

3) The column 4 illustrates the results of De Haan/resnick (1980)

T-- 1i.3

IV- Conclusion

R,We have proved that for a nice choice of k ( for instance k"'{Log n} ),

we can find the norming constants d such thatn

. ~'.

T d .n n 1In addition, we havegiven the limit law as the Gumbel distribution. The same

work has been already done by De Haan and Resnick(1980) under the hypothesis

(Ac). So, for a wide range of distributions be10nging in D(A), we can der ive

from that statistical tests. For instance, we may obtain a recognition code

between a Gaussian sample and an Exponential sample based on (4.1) and (4.2).

Similar tests are specified in LO(1985a)

-43-

REFERENCES:

- Balkema., A.A and De Haan, L.(1974): Limitlaws for order statistics. In:

Colloquia. Math. Soc. Boylai. Limit theorems of Probabibility, Keszthely,

17-20.

_ CsorgH, S., Deheuvels, P and Masan, D.M.(1983): Kernel estimates for the tail

index of a distribution. Technical report, L.S.T.A., University Paris 6.

Ta appear in Annals of Statistics.

- CsorgH, S. and Masan, D.M.(1984): Central limit theorems for sums of extreme

values. Proc. Car;;br. P;:il. ~~ath. Soc.

- Davis, R. and Resnick, S.I.(1984): Tail estimates motivated by extreme value

theory. Ann. Satist., Vol 12, N4, 1467-1487":1

De Haan, L.(1970): On regular variation and its application ta the weak con-

vergence of sarnple extreme. Mathematical Centre, Amsterdam.

l, .,~.. -~.: :.'~ ' .c.

lilil~~k- De Haan, L.and Resnick, S.1. (1980): A simple asymptotic estimate for the tail index .

of a stable distribution. J. ~oy. Statist. Soc. D 42, 83-37

- Galambos, J. (1978): The asymptotic theory of extreme arder Statistics. Hiley,

tfew-York.

Hill, B.M.(l974): The ran~,:-frequency form of Zipf's law. Journ. Amer. Assac.

Dec., Vol 69, nO 348. Theory and Methods Section.

- Hill, B.M.(1975): A simrle general approach to the inference about the tail

index of a distribution. Ann. Statist. 3. 1163-1174

LO, G.S.(1985a): Asymptotic behavior of Hill's estimate. Tech. Report, n028,

L.S.T.A., University Paris 6. Submitted

- LO, G.S.(1985b): On the CLT for sums of extreme values. Tech. Report. n030,

L.S.T.A., University Paris 6. Submitted.

- Mason, D.M.(19~2) Laws of large numbers for sums of extreme values. Ann.

?';/~".

. :t.(

Probabb. Vol 10, n03, pp.754-764.

- Teugels, J.L.(1981); Limit theorems on order Statistics. Ann. Probab.9, 868-880. ~i

,]

,~

~I

) ''',: 1 1

) ",

DEUXIEME PARTIE.

CARACTERISATION EMPIRIQUE DES EXTREMES.

" ,) ''',

-44-

The Annals of Probahil ity1991. Vol. 19. No. 2. 783-811

THE ASYMPTOTIC DISTRmUTION OF EXTREME SUMS

By SANDOR CSORG6,1 ERICH fIAEUSLER AND DAVID M. MAsON 2

University of Szeged, University ofMunich andUniversity of Delaware

Let X I n -'> .. -'> Xn n be the order statistics of n independent ran-dom variables with a common distribution function F and let k n bepositive integers such that kn ..... co and kn/n ..... a as n ..... co, whereo -'> a < 1. We find necessary and sufficient conditions for the existence ofnormalizing and centering constants An > 0 and Cn such that the se-quence

converges in distribution along subsequences of the integers ln} to nonde-generate limits and completely describe the possible subsequential limitingdistributions. We also give a necessary and sufficient condition for theexistence of An and Cn such that En he asymptotically normal along agiven subsequence, and with suitable An and en determine the limitingdistributions of En along the whole sequence ln} when Fis in the domainof attraction of an extreme value distribution.

1. Introduction and statements of results. Let X, Xl' X 2 .. be asequence of independent nondegenerate randorn variables with a cornmondistribution function F(x) = P{X s x}, X ER, and for each integer n z 1, letXl n S ... S X n n denote the order statistics based on the sample Xl"'" X n Th'roughout the paper k n will be a sequence of integers such that

as n -> oc;

with 0 < a < 1,

andk -> 00Tl

k n = [na]or(1.1 )

\1

where [.] denotes integer part. (We shaH refer to the first case as the casea = 0.) The study of the asymptotic distribution of the (properly normalizedand centered) surns of extrerne values ~ ,

was initiated in [6) for the case when a = 0 in (1.1) and under the restrictive1( 1.2)

k n

L X Tl + 1 - i . n =i~l

n

L X i Tli=n-k n + 1

111

Reeeived April 1988; revised December 1989. .Ipartially supported by the Hungarian National Foundation for Scienti~c Research, Grants

1808/86 and 457/88.2Partially supported by the Alexander von Humboldt Foundation, NSF Grant DMS-88-03209

and il Fulbright grant. ,.AMS 1980 subject classification. Primary 60F05.Key words and phrases. Sums of extreme values, asyrnptotic distribution.

783

1

\1,,,'

,'1 1

J ''\

.. 1

J ''\

-46-

EMPIRICAL CHARACTERIZATION OF THE EXTREMES:

A FAMILY OF CHARACTERIZING STATISTICS.

Gane Samb L

Universit Saint-Louis - Universit Paris VI (LSTA).

Abstract. Given 'only a sequence of i.i.d. random variables with common

unknown distribution function F, we provide a class of four statisticsi

which character~zes the limit law of the largest or ~he smallest1

observation. No icondition is required on F.i

AMS 1990 subject classifiation. Primary 62G, Secondry 62F.

Key words and phrases. Ext~eme value theory, order statistics,

empirical distribution function, domain of attraction of the maximum,

characterizatio~.

Mailing and permanent address. UER de Mathmatiques Appliques et

d'informatique, Universit de Saint-Louis, bp 234. Sngal.

Research affiliation. LSTA, Universit Paris VI, T44/45, 3E,

4 Place Jussieu. Paris Cdex 05. France.

1

,'1 1

) '4,

I- Introduction and statement of results.

a) Statement of the problem.

,", , -:-47-) '4,

Let Xl' X2 , be aisequence of independent copies (s.i.e) of a real

rahdom variable (r.v.) X with p(x~x)=F(x), xem. A major step of the

extreme value theory consisted in determining necessary and sufficient11

conditions on F for/the convergence in distribution to ai non-degenerate: 1l '

distribution M of the maximun X =max(x1 , ... ,X ), when lt is; n,n n1 ;

appropriately centered and normalized by two sequences of real nurnbers

(a >0) 1 and (b) l' that isn n~ n n~

(1.1) IyIxem, lim p (X ,', ~a x+b ) =M (x) .n~+oo n,n n n

,.

~If (1.1) holds, it is said that F is attracted to M or F belongs'to the

'domain of attractior of M, written FeD(M) . \

Several authors such as Frchet (1927) , Fisher and Tippet(1928),

Gnedenko(1936) led to a complete solution of the probabilistic problem

of finding conditions under which F belongs to D(M) along with the

determination of the sequences (a ) and (b ). Further developments andn n

other references may be found in de Haan (1970) and Galambos (1978) .

Pickands(1975) also gave necessary and sufficient conditions on F in

order that FeD(M). We summerize here the classical characterization of

the extreme in

Theorem A. If FeD(M), where M is not degenerate, then M is

necessarily one of these three types of distribution

A(x)=exp(-e- x ), xe~, (Gumbel's type);

-0~ (x)=exp(-x ), xe~ , 0>0, (Frchet's type of parameter 0);o +~ (x)=exp-x)O), xe~_,o>O, (Weibull's type of parameter 0).oSpecifically,

(i) FeD(A) iff, IyIte~, lim (l-F(x+tR(x,x ,F))/(l-F(x))=e- t ,x~x 0

o

2

" 1 1

J ''\

.,,,;J ''\ -48-

.,";

x~+lXl

-1 JXwhere R(X'XO,F)=(l-F(X)) 0 l-F(t) dt, -oo

,'1 1

} ''\,'J t

}'\ - 49-

On an other hand, many theorem limits for statistics under hypotheses

like FeD'(L\), FeD (q,) or FeD (t/J) ,have been continuing ta be established;1/

particularly when FeD(q,), (see Mason(1982), Csorgo, S,-Mason(1984),

1/

L(1986-89), Csorgo, S.-Deheuvels-Mason(198S), Smith(1987), Davis and

Resnick(1984) etc ... ). It is clear that a correct

(ql), (q2) and ~q3) will yield a general rule for1

such results. This was our second motivation.1

Mason(1982) partly answered the problem. He proved

an?wer of questions,i

the application of1

i11

1

1

Theorem B. Let k=[na ], O

} .,,": 1

} .,," -50-" 1 } .,,"

T(9) =(y-Y )/(y-Y k )n 0 n-l,n 0 n- ,n

(only defined when y = log(sup{X,F(x)

2v=min(l-cx, 0+[3-1).

II- Technical lemmas.

In the next section, we shall recall or introduce sorne technical

lemmas. Sorne of them are known or may be easily derived from results

" ," 1-51-

then FeD (A) ii

as n4+oo,i 2c

thenlFeD(~ ), where ~=-2+ --2--'~ c -1

,'1 1

J ''',

~p lA as n~ with YO

" ,J ''\

-52- ,', )J ''\

(iii) FeD(W ) iff FeD(W ); and then R(x,G)~O as x~Yo;a a(iv) FED(W ) iff x (F)

1 , -53-J ''\

due to de Haan(1970). It is now generalized by Deheuvels-Hauesler-Mason

(1988) so that il will be qoted as the DDHM representation.

i

The following two lemmaswill be used to overcome discontinuity

(i) for any u, O

, ' ,', 1J '4,

Proof. Direct computations give

(2.4) R(x,z)=R(x) (1-E1) and 0~E1~(1+(Z-X)/R(z))-1

and

-54-

2 -1(2.5) W(x,z)=W(x) (1-E2-E3)j 0~E2~(1+(z-x) /W(z)) j 0~E3~R(z)/(z-x).

These two remarks combined with Points (i) and (ii) prove Lemma 7.

z~Yo

\ i

Lemma 8. Let !Fer, then for u=l-G(x) , v=l-G(z), v/J~o as x~y ,! i 0

(i) (z-x)/R(z)G)~+ooj (ii) (z-x)2/W(x,G)~+00, as x~Yo'

z~Yo'

Proof. Lemma 1 in L(1986) implies Part (i) for GeD(A). If FeD(w ),~

KARARE combined:"withFormula (2.4.6) of de Haan(1970) yields Part (i),.

Part (ii) follows from Part (i) and Lemma 2 above.\ \

To finish with .this section we recall properties of empirical distribu-

tion functions (EDF). The EDF associated with Y1 ,. .'Yn is defined by

(2.6) G (x)= #{i, l~i~n, y.~x}/n, xeR.n l

Let U (s), O~s~l, be the EDF associated with the n first observationsn

of a s.i.c. of a uniform r.v. on (0,1), U1 ' U2 ' .....

These basic facts will be constantly used.

Fact 1. We may WLOG and do assume that

{l-G (x), xeR, n~l}={U (l-G(x)), xeR, n~l}.n n

Fact 2. For any~, 0

) '4," 1 , )4,

Un(s) Un(s)------

,,', 1

j '~

It is easy to check that

, -lJZ n(3.3a) Tn (2,k,l)=nk (l"':Gn (t)); x n

: '

dt, n~l.

j .~

-56-

is SVZ when FeD(~ ). It may~

SVZ. Now using (3.0) and

By Fact 2, for any~, 0

(3.8) and (3.9) together prove (Sl).

-57-)4," 1

a.s., as n~.

) '4," ,

(3.9) n~/2R(X )_n~/2R(G-1(1-k/n))~n

(3.3b), (3.5), (3.6),

2 2(S2) A (l,k,l)=(T (2,k,n/T (l,k,l)) =W (x ) (1+0(1) )=KR(x) (1+0(1)),n n n n n n

a.s., as n~, wh~re K=l if FeD(A)U D(~) and K=1-1/(7+2) if FeD(~ ).1 7

1-G (t) dt dy.n

By Fact 2,

(3.11) A (l,k,l)=W(x ,Z ) (1+0(n-/l(z -x )2/W(x ,Z )), a.s. asn n n n n nn

By Lemmas ~ and 8 and Statements (3.0) and (3.1), we have

n-7+00

as n-7+oo.(3.12) W(x ,Z )/W\ (x ) -7 l, a.s.,n n n

Hence, Lemma 2 yields

(3.13) W(X )/R(x )2 -7 l, a.s. as n-7+oo.n n

It follows from (3 .11), (3.12) and (3.13) that

(3.14) A (l,k,l)=W(x)(1+0(n-/l(z -x )2/R (x)2)), a.s. asn-7+oo.n n n n n

But the calculations that led to (3.6) and (3.8) showed that for all

p>O, O~, for any Fel,

(3.15) (z -x)( n- P R(x )- -7 0,n n n a . s ., as n-7+oo.

Thus (3.13) and (3.14) ensure (S2).

-1/2(S3) T (l,k,l) -7 K ,a.s. as n-7+oo.n

(Sl) and (S2) prove (S3).

(S4) T (3,k,l) -7 0, a.s. as n-7+oo.n

The proof is direct using KARARE and DDHM's representation and SVZ

functions properties. We therefore omit it.

(S5) T (4)~ Y , a.s. as n-7+oo.n 0

12

,'\ 1

j '4,

This is obvious.

.. 1

-58-

: ,

{

1/0 if FED(cP )(S6) T (2,t,l) ~ 0

n p 0 if FED(W)U D(A) .

See Mason(1982) for FED(cP), L(1986a) for FED(A) and Dekkers and al.

(1989) for the remainder case.

(S7) T (6) ~ 0, as n~+oo.,n p

We use the device of Fact 5 in (3.3a) by consedering the integral

as an improper one with respect to the upper bound. Remarking that1

(t/n)-(l-G(Y )), and putting Zn(l)=sup{IUn (s)/sl, U ~s~l}, we getn-l,n l,n

(3.17) T (6)~2 Z (1) R(z )/(z -x).n n n n n

This together with Fact 4 and Lemma 8 prove (87).

(S8) T (7)~ 0, as n~+oo.\ n p

As for T (6), we haven

(3.18) T (7)~2 Z (1) wez )/(z -x )2 ~ 0 as n~+ro.n n n n n

(S9) If FED(A)U D(cP), then T (8)~O, a.s. as n~+oo.n

Recall that T (8)=n- v /(z -x ). Now by DDHM's representation (cf. Lemma 4)n n n

and by (3.0), we have for aIl >l,

C

(3.19) z -x ~G-1(1-c )-G- 1 (1-E )=S(E ) -S(E )-J n t- 1s(t)dt,n n n n n n

En

for large n, where c =U 1 . Now, the properties of SVZ functions easilyn 1+, n

yield for any fixed c, 0

,,,;J ''\

8tatements (81)-(810) together prove Theorem 1.

2- Proof of Theorem 2.

" 1 1

J ''\

-59-

First us~ Fact 2 ih (3.3a) ang get for any 0

,'. 1

J ''\

,'1 1

J ''\

-60- : 1

YonJ 1-G (x)

Yn,n

Yodx=nJ 1-G(x) x

G- 1 (l-U1

),n

-1(nUl ) (U

1,n ,n1 -1J (l-s) dG (s) =0 (1).1-U P

l,n

Thus "

(3 . 28) T (6 ) 2:Z (2) {R (z ) / Z (1) (z - x - T (8, #3) 0 (1) } .il n n n n n n p

Thus, the convergence of T (8,~/2) to zero, Fact 4 and (2.28) implyn

(3.29) R(z )/(z -x )~ 0, as n~+oo,n n n p

whicn is also immediately implied by T (9)~ 0 as n~+oo sincen i p

1

R(z )/(z -x ):s(y -z )/(z -x )=l/(-l+l/T (9)).n nn 0 n n n n

, we alsoW(x):s(lim sup R(x))2)x~yo

It follows that R(x ,z )/R(x )~ 1 as n~+oo. By the same methods (replace, n n n p

T (6) par T (7)): and remark that lim supn n'

x~yo

have W(x ,z )/W(x )~ 1 as n~+oo and thusn n n p

(3.3D) lim W(x )/R(x )2=K, in probabi1ity.1 n~+oo n n 1

In view of Lemma,s 1 and 2, Theorem 2 will be proved if we show that

for Y~Yo' thereexists a subsequence xn(m) with Y~Yo iff n(m)~ and

(3.31) lim W(x)/W(x ( )) (resp R(x)/R(x ( )))=1 in probability.y~yo n m n m

The proof of this is quite direct and technical aand requires the

convergence of Tn (3,k,f) to zero. We state in the

15

.. 'J 'C\,

APPENDIX

Recall basic facts

n-Psn- P + n-1s(k+1)sn-P + 2n- 1 for k=[na ] , p=l-a.

By fact 2,

-P -0 -p-1n -cn SUk' sn +2cn ,a.s., as n~+oo, 0" .makes an J J+ 'partition of [0,1]. For x~y , U=l-G(x)~O, there existsat eacho

-61-

step of this limit an integer n such that a susa ::s:a.n+2 n+1 n Let

~ ~m=(n)=n-[n ], ~=2-a-o, p(n)=n+[n ], ~=2-a-o.

Remark that O. 5 sp

,'1 1

) ''\ -62-

and

x -x0::$1+0 (1) - (1+0 (1)) R(x) 1+0 (1)) m p 1+0 (l))T (3,k,t).

P p R (x ) P R (x ) p nm m

() ( ) 1[ T (3,k,l)0::$1+0 (1) -(1+0 (1)) W x < 1+0p

(1) K- n __p p W(x ) R (x )

m , m

These twoi last formulas yield

W(x ) W(x)Lim

n $=> lim R (x ) 2

=R(x)2n--7 + i x1'y olXli n

2

]

Important remark. De Haan and Resnick(1980) proposed

C =(Y -y k )/log k as an estimator of the index of a stable law. 'n n~n n- ~n

It is shown ini16(1986) that this estimator does not characterize D(I)\

as Hill~s estimator does.

REFERENCES.Il

[1] Csorgo, M. and Rvsz, P. (1981). Strong Approximations in

Probability and Statistics. Academic Press, New-York.Il

[2] - Csorgo, S~ Deheuvels, P. and Mason, D.M. (1985); Kernel estimates

of the tail index of a distribution. Ann. Statist. 13, 1050-1077.Il

[3] - Csorgo, Sand Mason, D.M. (1985). Central limit theorem for sums

of extreme values. Math. Proc. Cambridge Philos. 98, 547-558.

[4] - Dekkers, A.L.M., Einmahl, J.H.J. and ne Haan, L. (1989). A Moment

estimator for the index of an extreme-value distribution. Ann.

statist., 17, ',1833-1855.

[5] - ne Haan, L. (1970) .On Regular Variation and its Applications to

the Weak Convergence of Sample Extreme. Mathematical Centre Tracts,

32, Amsterdam.

[6] ne Haan, L. and Resnick, S.I. (1980). A simple asymptotic estimate

for the index of a stable law. J. Roy. Statist. Soc. Ser.B, 83-87.

17

-63-J '~

'~

Deheuvels, P., Hauesler, E and Mason, D.M. (1990). Laws of

iterated logarithm for sums of extreme values in the domain of

attraction of a Gumbel law. Bull. Sc. Math., 2 srie, 114, 61-95.

Fisher, R.A. and Tippett, L.H.C, (1928). Limiting forms of the

frequency distribution of the largest or smallest member of a

sample. Proc; Cambridge Philos; Soc. XXIV, Part II, 180-190.

Frchet, M. (1927). Sur la loi de probabilit de l'cart maximum.! iAnn. Soc. Polonaise de Math., 6, 93-116.l '

- palambos, J. (1978). The Asymptotic Th~ory of extreme Order1 :

Statistics. Wiley, New-York. 11 1

bnedenko, B.V. (1943). Sur la distribution ::',~,mite du terme maximum

.. ":

[10 ]

[11 ]

[7J

[9]

[8]

d'une srie alatoire. Annals Math. 44.

[12] - Leadbetter, M.R. and R6otzen, H. (1988). Extremal Theory for

stochastic processi~s. Ann. Probab., 16, 431 .. 478.

[13]L, G.S.(1986a). T:hse de doctorat, Univers:i.t Paris t;

[14] - L, G.S. (1986b) .Asyr:1ptotic behavior of Hill,' s estimau' and\ 0 \applications., J. Appl. Probab.'~ 23, 322 - 93 ':. "

[15] - L, ,G.S. (1989). A note on the asymptotic nc".~mality of sums of

extreme values. J. Statist. Plann. Inference, 22, 127-136.

[16] - Mason, D.M. (1982). Law of large numbers for extreme values.

Ann. Probab, 10, 754-764.

[17] Pickands, J. (1975). Statistical inference using extreme value

theory. Ann. Statist. 119-131.

[18J - Smith, R.L. (1987). Estimating tails of probability distribution.

Ann. Statist. 15, 1174-1207.

[19] - Resnick, S.I. (1987). Extreme Values, Regular Variation and

Point Processes. Springer Verlag, New-York.

; 1 \

) .",: 1

) ",

EMPIRICAL CHARACTERIZATION OF THE EXTREMES IlTHE ASYMPTOTIC NORMALITY OF THE CHARACTERIZING VECTORS.

Gane Samb La.Universit Saint-Louis & Universit Paris VI-LSTA.

-64-

Abstract. Let Xl' X2 , .. , be a sequence of independent random variables

with cornrnon distribution function F such that F(l)=O and for each

n~l, let Xl ~ .. ~ ... ~X denote the order statistics based on the n first,n n,n

of these random variabes. L (1990) introduced a class of four statistics

-including this new estimator of the square of the index of a stable law,

1/{-k1~ i=k ~j=i j (1-0 . . /2) (logX . 1 -logX . ) (logX . 1 -logX , )},L . 0 IL. 0 1 lJ n - l + ,n n - l , n n - ] + ,n n - J , nl={.+ J={.+

where (k, e) . 2is a couple of integers such that k-Hoo, k/n~O ,e /k~O, as

n~+oo, log stands of ,the Natural logarithm and o.. is Kronecker's symbol-lJ

from which he set 1R 8 -vectors that characteribe the wholedomain of

attraction of the sample extreme and each particular domain of attraction

(the Gumbel, Frchet and Weibull one) .The limiting laws of these vectors

are completely determined in this paper. It is shown that the single

elements of this family of Characterizing Statistics of the Extremes

(FACSEXT) and their ratios are asymtotically normal as e~+oo. But sorne

ratios become asymtotically extremal whenever e is bounded. The use of a

unified approach enabled to obtain the multivariate asymptotic normality.

AMS 1990 subject classifications:Primary 62E20, 62GI0 i secondary 60F05.

Key words and phrases. arder statisticsi extreme value theorYi domain of

attraction of the sample extremesi extremal and Gaussian lawsi

representation of distribution functionsi invariance principesi

multivariate normalitYi characterization.

Mailing address.UER Math & Informatique, BP 234, USL, Saint-Louis, Sngal

Research address. LSTA, Universit Paris VI, T.45-55, E.3, Place Jussieu.

75230 Paris Cdex 05. France.

1

L (1990) characterized the class of distribution functions (d.f,) F

statistical tests. In both cases, one has to determine the limiting laws

includes detection procedures of the extremal law of a sample and

attracted to some nondegenerate d.f. M (written Fe D(M)) by four statistic

-65-

) '\

'", ') '\

" ,

while no condition was required on F. This empirical and unified approach

l - INTRODUCTION

1

11

r

1

of this Familiy of characterizing Statistics of the Extremes (FACSEXT)

which is our aim.

The reader isreferred to L (1990) as a general introduction to this

paper and to Leadbetter and Rootzn (1982) and Resnick (1~87) for detailed

references on extreme value theory. However, we recall that if F E D(M) 1

A(x)=exp(-e- x ), for xER,

where M islnot degenerate, then M is necessarily the Gumbel type of d.f.,1

or the Frchet type of d.f. of parameter 0>0,

(x)=exp(-x-O)X[O [(X) 1 for xER,o ,+~or the Weibull type of d.f. of parameter 0>0,

oI/Jo(x)=exp(-(-x) ) X]_oo, 0] (X)+(l-X]_oo, 0] (x)), xER,

where XA

denotes the indicator function of the set A.

Several analytic characterizations of D(.)= U 0 D(. ), D(I/J)=U OD(0> 0 0>

D(A) and = D(A) U D(.) U D(I/J) exist. We quote here only those of them

involved in our present work.

Theorem A.

1) Karamata's representation (KARARE).

a) FeD(. ), 0>0, iffo(1.1) F-l(l-U) = c(l+f(u))

1u- 1 / o exp (J b(t)t- 1 ) ,O

-66-J '4,-1

F (0+).

" 1J '4,

,., 1

J '4,

b) FED(~ ), r>O, iff x (F) = sup{x, F(x)

: 1 1

J .", J ", -67-

T (2,f,1)n

T (6,) = (y _y-I )n n-f, n n-k, n T (2,f,1)n

-2T (7)=(y n -y k ) A (l,f,l)n n-l:-,n n- ,n n-u -1

T (8, u) :::n (y 0 -y k ) in n-l:-, n n- ,n

T (9)n

(only defined when x (G)o

y < +(0)o

where k and f are integers such that l~t

: 'J '\

-:68-,- -'\

Let us now classify the elements of r in our convenience. From (1.1),

(1.2) (1.3), it is clear that for each of the three domains, the couple

(f,b) represents a subset having elements only distinguishable by constant

We then write for this class (f,b) (cf. Lemma 1 in L (1990)),

F=(f,b) E D(A) iff

-1 Il -li(1.4) G (l-u)=d-s(u)+ s(t)tu

where s(.) verifies (1.3b)

dt, O

) ''\ " , ) '\-69-

). ''\

(1.8k (1),n

... , k (p)) k (i)1/2 sup(f(k (i)/n), f(Uk (') 1 ))~O, i=l, ... ,n n n n l + ,n p

or a combinat ion of (1.7) and (1.8kn (1), ... ,.kn(p)),

(1.9kn (1), .... ,kn(p)) f(u) f 1 (u) (1+f 2 (u)) where f 1satisfies (1.7),

f 2 (u) ~ 0 as u ~ 0 and kn (i)1/2 f1 (kn (i)/n)sup(f2 (kn (i)/n) ,f 2 (Uk (i)+l,n))n

pO, i = l, ... , p,

where p is a positive integer, k (i) is a sequence of integers such that fn

each fixed i, l~i~p, k (i)/n~O and Uk is the kth minimum among nn ,n

independent r.v.~s uniformly distributed on (0,1) (see (2.2) below)

Let 1(0), l(k (1), ... ,k (p)) and l(O,k (l), ... ,.k (p)) be the class of aln n n n

.d.f.'s F=(f,b) satisfying (1.7), (1.8kn (1), ... ,kn(p)) or (1.9kn(1), ... k

n(};:

\respectively . Each of these three confitions is quoted as a

regularity condition.

A related problem consists ln replacing the centring sequence by 1/0 for

FED(~ ). It has been studied by Haeusler and Teugels (1985) .They obtainedogeneral analytic condition for the asymptotic normality. This problem will

not be considered here for space reasons.

Each single statistic is systematically treated apart in Sections III

IV and V while the ratios are studied in Section VI. Section VII is devote

to multivariate limit laws as best achievements. AlI the results are given

into unified invariance principles based on the same Brownian bridge. We

therefore begin to describe the structure of the limiting laws.

II - Description of the limiting laws.Il II

Csorgo, Csorgo, Horvth and Mason (1986) have constructed a probabili

space carrying a sequence U1 , U2 , ... of independent and uniform r.v.'s on

(0,1) and a sequence of Brownian bridges {B (s), O~s~l}, n = 1,2, ... suchn

that for aIl v, 0

: 'J '\ J\ -70-

11/2 1/2-v

(2. 1) sUP (l/n):SS:Sl-l/n n (Un(S)-S)-Bn(S) 1/(s(l-s))

and

11/2 1 1/2-v

(2. 2) sUP (l/n):ss:Sl-l/n n (S-Vn(S)-Bn(S) /(s(l-s))

-va (n ),p

a (n- v )p ,

where U (s) = j/n for U. :Ss

: ,J '"

:' 'J '"

-71-

If the d.f. is not specified in Rp (')' it is assumed that R ( )-R ( G)p'-p""

for G (x)

N (O,k,i)n

F(ex ). Finally, put

{(n/k)1/2sZn B (l-G(t)) dt } dt}/R1

(x) ix n n

n

N (2,.)n

1

1(2=- (n1 .) : B (. 1n), n~1 i

i n

~ (i)=nU1 1 li, n~l.n + ,n

We prove in )the next sections that each T (i) is asymptotically one ofn

these r.v.'s or la linear combination of them. Their abymptotic laws are

described in

Theorem C. 1) For each n, the vector (W (k), W (i)), with W (k)=n n

(N (O,k,i), N (3,k,i), N (2,k)) and W (e) =(N (O,i,l), N (3,i,l), N (2,i))n n n n n n n

is Gaussian . If further i~+oo, k~+oo, k/n~O, ilk~O n~+oo, and FEl, then

Wn(k), Wn(i)) converges in distribution to an R6 -Gaussian r.v.

(W(l), W(2)) where W(l) and W(2) are independent vectors with the same

covariance matrix :

[

2(/y+1)/(O+2)

3(/Y+1)/(/Y+3)

-1

and

6(/y+1) (0+2)

(/y+3) (/Y+4)

-1

Jif

8

FED (l/J ),/Y

,'1 1

J .,,"

6

-1

: ,

, if FED(A) UD(~),

-72-

where the symmetric matrices are given only in one side.

2) Let e be fixed, then for all XE~, P(E (e)~x) converges ton

- ex ,j = e ( . 1 ) - 1 (n ) j

{

1 - e L j = 1 J.

.. ,J .,,"

below yield Var(i)), i == 1,2.

.. 1 1

J .,," -73-,'1 1

III -. Limit laws for T (2,k,t) and T (5).---"----------n n--

Theorem 3'.1 : Let Fer let (k,t) be a couple of integers satisfying

(2.4a) l

.. ,) 4,

. , -74-

"Theorem 3.1 was proved by Csorgo and Mason (1985) and L (1989) when1

FED(A) U D'(t!. But their proofs used u- 1 I (1-s)dG- 1 (s)=r(u) instead of1-u

R1

(x,G) .Remark that R1 (x)=r(u) for u=l-G(x) if G is ultimately continuous

and increasing. We do not require at aIl such assumptions and since the

elements of the proof of Theorem 3.1 are great(y used in aIl the remainder

of the paper, we should reprove it rigorously ;ln a simultaneous treatment

of aIl cases FED(A), FED(~) and FED(~) .

We now characterize the asymptotic normality of T (2) when attemptingn

to replace (R) by (k). For this, put

D.Oa) 0 (o)=f(U 1 )-f(o/n)n 0+ , n

* -1 1/2T (2, k, t) = R1

(x) k (T (2 , k, t) -

,'1 1

J ':>," ,

J ':>, -75-

k/n-70. Examples such as ln Hauesler and Teugels (1985) may be treated ln !

very simple ways since in all their models f (u) =:f l (u) (l+f,2 (u)) with

f~ (u)-70 as U-70 [for instance f l (u)=a.up,p>o, f l (u)=a. (log(l/U))-P,

P>O, f l (u)=a exp(-b/u)] and f 2 (u)-70]. For all these models, we have

Corollary 3.3.Let F=(f,b) E rand (k,e) satisfies (2.4) 'If further (1.7)

or (1.8k) or (1.9k~ holds, then1

*, d, 2T (2,k,e)=:v (2) (T (2,k,e)-c (2))=N (l,k,e)+o (1) -7' N(O,(J'l(~))' forn n n n n p

O

-)6-

) .'" ..' ') ~'"PROOF8. They largely use technical results in L (1990). Use (2.1) and

(2.3) ta get

(3.1 ) R1

(x )-lk 1/2(T (2,k,e)-(K,e))=N (O,k,e)+Z 1(k)+(e/k)1/2n n n n

-Z

(~)1/2 Z = R1 (X ) -lJ n{a: (l-G(t)) -B (l-G(t)) }dt,n n3 n - n nxn

where a: (s) (resp. (3 (s)) = n 1 / 2 (U (s) -s)n n n

(resp. n~/2(s-V (s))n

Z (0)n1

and

G- 1 (1-U 1 ){ (n/ 0) 1/2 J 0 + ,n

G- 1 (1- 0 /n)B (l-G(t)) dt}'/R

1(x ),

n n

Zn2(0)={nk1 / 2

o

1\

G- 1 (1-U )

Jo+l,n B (l-G(t))

-1 nG (l-o/n)

We shall treat each error term into statements denoted (81.3), (2.3) ,etc ...

(81.3) Zn3 ~ 0, for e~+oo.

First, we have by (2.2)

(3.2) ne- 1 / 2 (Un 1 -e/n) =N (2,e) +0 (1)l:-+ ,n n p

d~ N(O,l).

sa that nUn 1 /e ~ 1 and hencel:-+ ,n p

(3.3) V >l, lim 1P(e!n):5Un 1 :5:(e)/n) = 1.n~+oo ~+ ,n

For convenience, if (3.3) holds, we say "for all >l, one has e/ (ll.n):5Ue+1,

:5(e)/n with Probability as Near One as Whishe (PNOW) for large values

of n".

(3.4)

Hence,

G(t):5:1-1/n, uniformly for x :5:t:5:Z ,n nwith PNOW as n is large. 8econdly,

13

: 'J .~

: ,).~

-77-

Lemma 3.1. (Cf. Fact 5 in L(1990). Let h(o) be a bounded function on (a,l)

a>O and G any d.f. If the integrals below make sense as improper ones, the

G- 1 (l-a) G- 1 (l-a)1J-00 h ( (l-G (t) p (t) dt 1:ssuPa:ss~:l1 h (s) 1J 1p (t) 1dt.

-00

Combtning (2.1), (3.4) and Lemma 1 yields for sorne v, O

.. 1 1

J ''\

(ii) GED(A) ~ GrED(A) with x (G)=x (G )=y ando 0 r 0

,'1 1

z~y ,o

-78-

1-G(z)1-G(X)~0.

-1~ r as x~y , z~y ,

o 0 (l-G (z) ) / (l-G (x) )~O.

Proof of Lemma 3.3. a) Let GED(I/1"1)' Thus (3.6) holds for Gr by putting

rc (x) c(x), ~ ="1r and p (t) = rp(t). Hence G ED(I/1 ) and x (G) =r !r r r"1r 0

x (G ). Further, by Formula 2.5.4 of de Haan (1970) or Lemma 4.1 below,o r-1 -1

(3.8) RI (x,G)/(yo-x) ~("1+1) , RI (x,Gr)/(Yo-x)~("1r+1) , as x~yo'

By Lemmas 7 and 8 of L (1990),1-G(z)

(3.9) RI (x, z,G) /R1 (x,G)~ l, R2 (x, z,G) /R2 (x,G)~l as x~yo' 1-G(x) ~O

whenever GED (1/1) lor GED(A) (hence whenever FEr). (3.8) \ and (3.9) together

prove Part i).

as x~y .o

b) Let GED(A). By (3.7), GrED(A) with xo(G)

(cf. Lemma A in L (1990), for any tER,

(3.10) (l-G (x+tR1 (x,G ))/(l-G (x)) ~e-t, as x~Yo'r r r

(3.11) (l-G(x+tR(x,G))/(l-G(x)) ~ e- t ,

x (G )x r y and thuso

Combining (3.10) and (3.11) implies

-tir(3.12) (l-G(X+tR1(x,Gr))/(l-G(x))~e , as x~Yo'

If for a sequence x ~y , one has R1 (X ,G)/R1 (X ,G )~v, O

" ,) ''\ " , '-7~-

-1Lemma 3.4. i) If eD(A) U D(cf , then R1 (G (l-u)) lS 810wly Varying

at Zero (8VZ).

ii) If FeD(~ ), then R(G- 1 (l-U)) lS Regularly Varying at'a'

Zero -1with exponent 'a' -RVZ).

Proof of Lemma 3.4. a) Let FeD(A) U d(cf, thus GeD(A) by Lemma 1 in L

(1990). Next, Lemma 3.2 above: and Lemma in L (1986a) yield for i\>0, i\~1,1

{-1 -1 i 1

(3.13) G (l-i\u) -G (l-u) }/R(G (l-u))~ -log i\, as u~o,

and Lemma 4 in L (1989) implies

( (3 d 4 ) {G- 1 (1 - i\ u) - G- 1 (1 - u) }/ s (u) ~ - log i\, as u~,wheie s(.) is 8VZ and defined as in (1.3). Hence

(3.15) R(G- 1 (1-U))_S(u) as u10'

so- that R(G- 1 (l-U)) is 8VZ and Part i is proved.

b) Part ii) is easily derived by Formula 2 _5.4 of de Haan (1970) (cf.

Lemma 4.1 below) and Formula (1.6), above.

We return back to our proofs of Theorems. By (3.5) and Lemmas 3.2, 3.

and 3.4, (3.16) IZn31:S0p(k-U) R1 (xn,G1/2_U)/R1 (xn,G) = 0p(k-U

) = 0p(l),

where we have taken (3.8) into account. This proves (81.3).

(82.3) Zn3 p 0, when t is fixed.

We have

(3.17) 1 Zn3 1 =Zn3 (1) + Zn3 (2) ,

with

G- 1 (1-t/n)(3.18) Zn3(1) = (n/k)1/2{J_ ICX n (l-G(t))-Bn (l-G(t)) \dt}/R1 (Xn ) ,

xn

which is 0 (k- u ) by same arguments used in (3.5) (one also has G(t):Sl-l/n)p

and

16

, ,} ''\

(3.19) Zn3(2)

1

By'Theorem C,

: 1

-80-

{(n/k)1/2JZ~1 la (l-G(t))-B (l-G(t))1 dt}/R1

(Xn

).G (l-t/n n n

: 1

(3.20) lim lim lP(t/n) ::sUn 1 ::S(t)/n)=l,. ~+oo n~+oo

: ,J '\

with PNOW as n is large. Furhermore,

.. 'J '\

-81-

'< 1/2J t n(k/)(3.24) IZn1 (k)I-3{(n/k) t (k) 1-G1 / 2 (t) dt }/R1 (Xn )

i n

R1

(xn ,G1 / 2 )

:53x X{R1 (tn (k) ,G1 / 2 ) (1+0(1 -R1 (tn (k/) ,G1/

2) (1+0(1}

R1

(xn

,G)

, /R1 (tn (k), G1 / 2 ).

The term in:brackets tends to zero for FED(A) U D:(~) and to 1 /_- 1 / for

FED(~o)' alr by Lemmas 3.3 and 3.4. 8ince is arbitrary >1, [(Z:l (k~O

in all cases and finally (83~3) holds by Markov's inequality and (3.~3).

(84 .3) (t/k) 1/2 Z 1 (t) ~ O.n p

\ \This is proved exactly as (83.3) when t~+oo. When t is fixed, one uses

(3.20)

(85.3)

a) Let

instead of (3.2) and the proof of (83.3) is valid again.

tk- 1/ 2Z 2(t) ~ O.n p

FED(~ ), thus GED(~ ). By Lemma 4.1 below, Lemmas 3.2, 3.3, 3.4o 0above and Theorem C,

(3.25) -1/2{Z 2 (t) =tk (1 +0 (1) (y - z ) / y - x ) - 0 (1 )n p 0 non p

(y -z )/y -x )}~ 0,o non p

whenever lim sup t/k1 / 2

1,

J ''\

1r

,

",

:' 'J,\ -82-:'"

(3.27) Itk-1

/2

Z 2 (t) l=cx(l+o(l)) IR (z ) (1+0 (1)) -0 (l)R (z ) 1 (tk- 1 / 2 )n 1 n p pl n '

where the 0 (1) is 1+0 (1) as t~+oo. Whence,p p

(3 . 2 8) tk - 1/2 Z 2 (t) ~ 0 l' f f l' nk - 1 / 2n p lm sup n~+oo ~

and 2.6.1 of de Haan (1970) (see Lemma 4.1 below) and Lemma 3.2, above,

2, 2(3.34) s (O)-2R2 (x )/Rl(x )-2K(Q), where K(Q)=(Q+l)/(Q+2), O

1

holds.

(See (3.38) - (3.40) or more details).

b) let FED(~}. This case is exactly the preceding since GEID(A} and (1.4)

c) Let FED(~ }. Use R:1 (X }-(y -x )/(1'+1) and getl' 'n 0 n

-84-

.. ,.;J.",

: 1

Also,

(3 . 3 7 ) k1

/ 2 {s (Uk 1 ) / s (k / n) - 1 }== k1 / 2 l' (k) (1 +0 ( 1) ) +0 (nk - 1/2 (Uk - k / n) ) .

+ ,n n p p +l,n

1111r

x({y -x )/(y -x ) }-1}.o nonNow, by (1. 6) ,

(3.39)

where E:n sup{b(t), tsmax(Uk 1 ' k/n)} ~ O. Since nUk 1 /k ~ l,+ ,n p +,n p

(3.40) (k)=(1'+1)k1 / 21' (k) (1+0 (1)}+o-1(1'+I)nk- 1 / 2 (Uk

1 -k/n) (1+0 (1)).n n p + ,n p

Now (2.2), (3.36), (3.40) and Point b) just below complete the proof of Lem!

3.6.

We return to the proof of Theorem 3.2.By (3.1), (S1.3), ... (86.3), (3.41)

x

v (2) {T (2,k,i)-c (2) }=N (O,k,i)+k1 / 2 {kQI-n l-G(t) dt}/R1 (X }+o (1).n n n n x n pnUsing Lemmas 3.2 and 3.6 gives

=N (O,k,i)+e 1 (1'}N (2,k)+e 2 (1')k1 / 21' (k)(1+0 (1)}+0 (1),n n n p p

from which the characterization is obvious. To compute [(N (O,k,i)N (2,~n n

21

" ,

Also,

) .'" " ,-85-

(3.37) k1

/ 2 {S(Uk 1 )/s(k/n)-1}=k1 / 2 0 (k) (1+0 (1))+0 (nk- 1 / 2 (Uk -k/n)).+ ,n n p p +1, n

-(See (3.38) - (3.40) or more details).

b) let FED(~). This case is exactly the preceding since GED(A) and (1.4)

holds.1

c) Let FED(I/J ). Use R1 (X ) .. (y -x )/(0+1) and get'on 0 n .

(3 . 38) (k) - k 1 / 1;{ (y -x )- (y - x )} /{(y - x ) / (0 +1) } _ (0 +1) k1/2n on 0 non

Now, by (1. 6) ,

(3.39)- x

n

l~(l+l'n(k)) (~k U )C n +l/l'_l,k+l,n

where cn

sup{b(t), t~max(Uk 1 ,k/n)} ~ O. Since nUk 1 /k ~ l,+ ,n p + ,n p

(3.40) (k)=(l'+1)k1 / 2l' (k) (1+0 (1) ) +0'-1 (l'+1)nk- 1 / 2 (Uk 1 -k/n) (1+0 (1)).n n p + ,n p

Now (2.2), (3.36), (3.40) and Point b) just below complete the proof of

3.6.

We return to the proof of Theorem 3.2.By (3.1), (S1.3), .. , (S6.3), (3.41)

xn

v (2){T (2,k,t)-c (2)}=N (O,k,t)+k1 / 2 {nk-J- I-G(t) dt}/R1 (x )+0 (1).n n n n x n pn

Using Lemmas 3.2 and 3.6 gives

(3.42) *T (2,k,t)n

from which the characterization is obvious. To compute ~(Nn(O,k,e)Nn(2,k)

21

: 'J .",

: ,J", -86-

we recall that (B(s)B(t)=min(s,t) -st if {B(t), O~t~l} is a Brownian

bridge, and an easy calculation yields

(3.43) (N (O,k,l)N )) ~-l.n n

This and Lemma 3.5 suffi~e to calculate ~l (0). All the proofs are now

complete.

1

1IV - Limit laws for A (l,k,l).------------n,-'-'-'---'--

We, need sorne generalized forms of Lemmas due to de Haan (1970)

or to L( 1990) .

Lemma 4.1. Let Fel, then for any integer p~l,. - . -1

(i) R (x,F)-(x -x)P{n~-81(o+j) }, as x~x whenever FeD(w ),p 0 J= 0 0

and

Proof of Lemma 4.1. (i) is obtained by routine calculations from (1.2).

ii) is easily proved from Lemma 2.5.1 and Theorem

2.5.2b of de Haan (1990) showing that

yG1=G e D(~)~G2(o)=1-J 0 1-G1 (t)dt eD(~) and R1 (X,G1 )-R1 (x,G2 ) as x~xo(G).

o

Applying this p times gives (ii).

Lemma 4.2. Let Fel. Then for any p~l,

( i ) ( z - x) p / R (z , G ) ~ +00,p

(ii) R (x,z,G)/R (x,G)~l,P p

as x~x (G), z~x (G), (l-G(z))/(l-G(x))~O.o 0

22

Proof of Lemma 4.2.

: ,) '~ " ' ...}p-

(i) is easily derived from Part ii) of Lemma 4.1 above and Lemma 8 of L

(1990). To prove (ii), put

z z z Yo Yo(4.1) m(a,b,x) J J . ..J J ...J 1-G(t) dt d~a,b'

x xl xa - 1 YI Yb-1

where d~a,b= dx1 dXa_1xd~1" . dYb_1= d~axd~b and

Yom(O,b,x) =Jx

and

m(a,O,x) - G(t) dt d~ .a

Straightforward manipulations yield for any p~2, for z

: 1

) '\.' ,

) '\ -88-

z zwhere T(k,t) = nk-

1J n J n 1-G(t) dt - R (x z )-R (x ) inX y 2 n' n 2 nn

2probability and 0'2 (r) = 6 (0+1) (0+2) / {(o+3) (o+4)}, 0

.. ";J ''\

, , -89-

x (1 +0 (1) +0 (1) ,P P

where

i)

ii)

2Nn (4,k,l)-N(0,3(0)). Then,

* d 2A (l,k,l) ~ N(m,3 (0)) iff N (-00) ~ mie (0)n n p 4* d 2 . d 2A (4,k,l) ~ N(m, lff N (-00) ~ N(m/e (0), (-oo) )n c n 4 '

limn~+oo~(0)+(-00)2~2e4(0)COVNn(4,k,l)Nn(-00) )=~

with

* 1iii) A (4,k,l) ~ + 00 (rep. -00) iff N (-00) ~ + 00 (resp.-oo).n p n p

Remark 4.3. The characterizations are identical in Theorems 3.2 and 4.2.

We pointed out in Section III that we have the asymptotic normality

whenever (1.7) holds. The following examples concentrate on the case where

uf' (u) has not a limit Js u~O.

Examples 4.1. Let f(u)=u sin(l/u).

f' exists and uf' (u)=u sin(l/u) - COS(l/u) does not converge as u~O.

But if there exists a sequence of integers (p) 1 such that (l/u )-n n~ n

2rrp ~b, -rr

.,,'1 1

J ''\

.. 1 1

J ''\

-90-

PRF8. We proceed as for T (2,k,t) by general statements (81.4), (82.4),n

etc ... First, use (2.1) and (2.3) to obtain

(4 .5)

where

R2

(X )-lk 1/2(A (l,k,t)--c(K,t)=N (3,k,t)+Q 1(k)+(t/k)1/2n n n n-1/2

+ Qn1 (t) +tk Qn2 (t) +Qn3 '

26

1

J '~ -91--' 1J '~

- -Qn2(i)=(n/i)1/2({J

Z

nJZ

n Bn (1-G(t) dtdy}-{J:n JZn 1 - G(t) dtdY})/R2 (Xn ),xn y xn yand

- -(n/k) 1/2 (JX:n JyZn { ( () )' ( ()} )/Qn3 =_ an 1-G t '-Bn 1-G t) dtdy R2 (Xn ) ,

n

We show that each of these error terms tends ta zero in probability.

(81.4) Qn3 -7 O.P

If i~, we get, as in (3.5), for sorne v, 0

.' , --92-J ."

By Lemma 1 in L (1990) and Lemmas 4.1.2 and Formulas (3.13) and (3.15)

above, FED(q>~) implies that R2(xn)-7~-2, R2(xn)-7~-2, (tn(t/~)-tn(t))

2 */R2 (zn)-7(log i\) so that IE.Qn3 (3) -7 O. By the same arguments IEQn3 (3)-701when FED(A) since

p-1(0)= k 1 / 2R (x )/R :(Z ) ~ k 1 / 2 - Ct C/2n 2 n 4 n '

for any C, 0

) ''\

*concludes that ~ Qn3 (2)~O and hence (81.4) holds." , " ,

) ''\ -93-

(82 .4) Qn1 (k)

We have

I~'P

B (l-G(t))n

/ x x xQn1 (k) {(n/k)l 2J_n J n Bn (l-G(t)) dtdy+(n/k)1/2J_nxn y xn

dt dy}/R2 (xn )=:Qn1 (k,l)+Qn1 (k,2).

It follows that fbr any >l, one has with PNOW as n is large,

(4 . 14) * *1Qn1 (k,l) I~ Qn1 (k,l); 1Qn1 (k,2) I~Qn1 (k,2)

with

(4.15) ~ Q*1(k,l)~3K(r)-l1/2(t (k/)-t (k))2/R1 (X )2,n n n n

and

where '= fort~+oo and 'is taken large for t fixed. Arguments given

*in (3.24) show that that ~ Qn1 (k,l)~O and a combination of these same

arguments and Lemmas 7 and 8 in L (1990) and Lemmas 3.2 and 3.3 above

* * *ensure that ~ Qn1(k,2)~O. We conclude that Qn1(k,l)+Qn1(k,2)pO and

thus, by (4.8), (82.4) holds.

(83.4) (t/k) 1/2 Q 1 (t) ~ O.n p

One has with PNOW as n is large

(4.17) 1 (t/k)1/2Qn1 (t) I~Qn1 (t, 1) + Q~l (t,2)

with

(4 . 18)

and

~ Q*1(t,l)~3K(r)-1(t/k)1/2(t (k/1)-t (k))

n n n

(t (t/2

) -t (t)) /R1

(x ) 2n n