Soutenue le ~

devant le Jury compos de 1STI GO L.t4GALAYE DIA, Saint-Lo:i;Fir'nlent rapporteur.

THESE

DEDOCTORAT D'ETA '8. P 1~~b '

s Sciences Mathmatiques

PRESENTEEA L'UNIVERSITE CHEIKH ANTA DIOP DE DAKAR (Sngal)

PAR

MONSIEUR GANE SAMB LPOUR OBTENIR LE GRADE DOCTEUR ES SCIENCES

'"Sujet de la these

CARACTERISATION EMPIRIQUE

DES EXTREMES ET QUESTIONS

STATISTIQUES LIEES

HAMET SEYDIDakar. Examinateur

PAUL DEHEUVELS, PARIS VI, Rapporteur. SAKHIR THIAMDakar. Examinateur

MICHEL BROWNIATOWSKl,strasbourg, Rapporteur.

saint-Louis, le OS Dcembre 1991

~S~~ ~ TI~S ID'~~~ S S~O~~S ~~~~~~~O~~S

1fD~[R ID iL~ ~DiS :Caractrisation empirique des extrmes et problmestatistique lis.

:Gane samb LO.

~v~w :Paul DEHEUVELS (Universit de Paris VI), Michel BRONIATOWSKI

(Universit de Strasbourg), Galaye DIA, (Universit de Saint-Louis)

Hamet SEYDI et Sakhir THIAM (Universit de Dakar).

1DA\.1f, ~~CR 1f tL[] [EV IDIE S

JE DEDIE CE HUMBLE TRAVAIL A MES PARENTS

ET AMIS, ENSUITE A t'-1A TENDRE ET PATIENTE'

EPOUSE, MA COMPLICE EN TOUTE CHOSE.

1

REMERCIEMENTS

Je remercie d'abord le Professeur Paul Deheuvels pour l'intret qu'il a

toujours port nos travaux depuis sept ans malgr ses multiples et

riches occupations. Il a d'emble accept de se dplacer jusqu'

Dakar pour faire partie du jury de cette soutenance.

Mes remerciements sincres vont aussi au Professeur Michel

Broniatowski pour les mmes raisons. Sa courtoisie m'a beaucoup

rassur quand 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

trouvent ici, tous les deux, l'expression de ma trs sincre gratitude.

Enfin, "the last but not the least l1 , le professeur Galaye Dia

m'a toujours encourag depuis que nous tions ensemble au LSTA, lui

prparant une thse 'Etat et moi une thse nouveau rgime. Il n'a

mnag aucun effort pour que cette soutenance soit une russite. Je

lui tmoigne ici ma reconnaissance.

Pour terminer, je remercie notre secrtaire de recherches,

~le Rokhaya sarr, pour la comptence avec laquelle elle a tap une

partie de cette thse.

Gane samb L

Saint-Louis, le 15 Novembre 1991

2

SOMMAIRE

TITRE PAGE

INTRODUCTION GENERALE 4

PREMIERE PARTIE:

CONTRIBUTION A L'INFERENCE DE L~INDEX D'UNE LOI DE PARETO. 10

ASYMPTOTIC BEVAVIOR OF HILL'S ESTIMATE AND APPLICATIONS. Il

ON THE ASYMPTOTIC NORMALITY OF SUMS OF EXTREME VALUES. 26

THE WEAK LIMITING BEHAVIOR OF THE DE HAAN-RESNICK ESTIMATE

FOR THE EXPONENT OF A STABLE DISTRIBUTION. 36

APPENDIX: RESUME DE LA QUESTION DE LA DISTRIBUTION DES SOMMES

DE VALEURS EXTREMES. 50

DEUXIEME PARTIE.

CARACTERISATION EMPIRIQUE DES EXTREt'-1ES. 52EMPIRICAL CHARACTERIZATION OF THE EXTREMES:

1: A FAMILY OF CHARACTERIZING STATISTICS. 53

II: THE ASYMPTOTIC NORMALITY OF THE CHARACTERIZING VECTORS. 70

III: THE LAWS OF THE ITERATED LOGARITHM OF THE CHARACTERIZING 116

STATISTICS.

A GAUSSIAN PROCESS LIMIT OF SUM-PRODUCT STATISTICS BASED ON 150

EXTREME VALUES.

TROISIEME PARTIE.

CONTRIBUTION A L~ETUDE DES ESPACEMENTS. 176

GAUSSIAN APPROXIMATION AND RELATED QUESTIONS FOR THE SPACINGS

PROCESS 177

ON THE INCREMENTS OF THE EMPIRICAL K-SPACINGS PROCESS FOR A FIXED

OR MOVING STEP. 194

QUATRIEME PARTIE.

STATISTIQUES D'ORDRE DANS LES ESPACES LATTICES 212MAJORATION, MINORATION ET CONVERGENCE FORTE DES QUANTILES CENTRALES

ET EXTREMES DE VARIABLES ALEATOIRES DANS DES ESPACES DE TYPE C(S). 213

FIN 220

INTRODUCTION GENERALE

Cette thse regroupe un ensemble de travaux dont le lien vient

est la statistique ctordre. Toutes les recherches dont les rsultats

sont exposs ici nous ont t proposes ou ont germ dans notre esprit

lorsque nous tions dans le Laboratoire de Statistique Thorique et

Appliques (LSTA) de l'Universit de Paris 6 o j'tais tudiant en

thse entre 1983 et 1986. En ce moment, toutes les questions relatives

aux statistiques d'ordre taient systmatiquement tudies soit par

les professeurs Geffroy et 'Deheuvels soit par les membres de leurs

quipes. Je voudrais relater ici les circonstances dans les quelles

j'ai eu m'intresser chacun des travaux expos ici dans ce

document. Ceci nous permettra aussi de classer les articles dans un

ordre chronologique.

PREMIERE PARTIE,La premire partie de ce document est une contribution la thorie

de l'infrence sur l'index d'une loi de Pareto. Au dbut des annes 80,

la jonction entre cette thorie d'une pqrt et le problme de la

determination de la loi limite des sommes de valeurs extremes a t

ralise par l'estimateur de Hill{1975j de l'index d'une loi stable.

cela justifie, entre autres raisons, la plus grande importance donne

cette estimateur parmi la classe des estimateurs d'un tel index.

L'infrence sur l'index d'une loi de Pareto a t trs vite tendue

toute la classe des fonctions de rpartitions ont le maximum des

observations indpendantes (rn.o.i.) est attir par une loi de

Frchet quand la taille devient infinie. Par ailleurs, on sait que

le m.o.i. est attir soit par une loi de Frchet, une loi de Weibull

~u soit par une loi de Gumbel. Le cas de Weibull est immdiatement

trait travers le cas de Frchet par une simple transformation

llgbrique. Ds lors, l'alternative naturelle des rsultats de conver-

}ence et de normalit asymptotique tait l'attraction du m.o.i. par

lne loi de Gumbel. Cette classe contient les trs importants cas

)articuliers des lois normale, log-normale et exponentielle. On

~omprend ds lors l'intret des spcialistes port cette question.

4

Le Professeur Dehauvels m'a demand une aprs-midi d'octobre 1984

de chercher le comportement de l'estimateur de Hill{1975} lorsque la

fonction de rpartition associe tait celle de Gauss. Ce fut mon

premier contact avec les valeurs extrmes. Les rsultats obtenus puis

gnraliss pour toute la classe A des fonctions de rpartitions dont

le m.o.i. est attir par une loi de Gumbel ont t publis dans le

Journal of Applied Probability(1986). J'ai t trs surpris quand

le Professeur Deheuvels m'a demand 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 lorsqu'il s'est agi de prouver la normalit asymptotique de

l'estimateur de Hill; la collision aavec le formidable travail de

csorgo-Haeusler-Mason s'opra. Finalement, je trouvais dans le travail

de Csorgo-Mason(1986) la manire de rsoudre le problme. Il a suffi

d'affiner suffisamment les proprits des lments de A pour recon-

duire les mthodes de Csorgo-Mason. Dans Csorgo-Haeusler-Mason(Ann.

Probab. Vol 19, NZ, 783-811), o les lois limites des sommes de

valeurs extrmes sont entirement dtermines, cette contribution

l'tude de la normalits des sommes de valeurs extrmes a t

reconnue et signale (voir annexe, p. ) .

Dans la foule de ces rsultats, nous nous sommes intresss au

comportement de l'estimateur de DE HAAN-RESNICK(19BO) (voir troisime

article) et des estimateurs de Csorgo-Deheuvels-Mason(1987) {rsultats

non exposs ici). Ces deux papiers furent publis en rapports

techniques au LSTA, Nos 29/1985 et 49/1986).

5

TROI8IEME PARTIE.

Au cours de l'anne 1985. Dr. Van Zuijlen a t invit au sminaire

de 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 une document.at.ion frache et complte non publie sur

le thme.

Je me suis donc intress l'approximation Gaussienne du processus

empirique bas sur les espacements d'un chantillon issu d'une v.a

alatoire uniformment rpartie sur (0,1). Nous avons montr que

l'approximation propose par AlY-Beirlant-H6rvath{1984} tait optimale-

dans l'approche que j'appelerai de 5horack - par la dtermination de

la limite suprieure. Ce rsultat a t obtenu mme avec un pas deve-

nant infini. avec la taJIJ.e de l'chantillon. Ce rsultat indique qu'il

faudrait ncessairement changer d'approche pour atteindre Ulle meilleure

vitesse.

Rcemment, au colloque "arder statistics and Nonparametrics: Theory

and Applications" tenu Alexandrie, 18-20 September 1991, Dr. Aly me

confirma que cette vitesse n'a pas encore t amliore. Je pense

m'intresser ce problme plus tard.

Nous avons tendu certaines proprits du module de continui t. du

processus empirique classique au processus des espacements. Ces travaux

furent publis en rapports techniques LSTA N47 et 48/1986 et exposs

au sminaire Deheuvels-Chevalier(5 Fvrier 1985).

QUATRIEME PARTIE.

Vers la fin de l'an 1986, nous nous sommes intress aux

statistiques d'ordre dans les espaces de Banach lattices; particuli-

rement dans C(S), o 5 est un espace mtrique compact sparable. Cet

article est le premier pas. Il fut d'abord soumis Journal of

Multivariate Analysis. Les deux rapports des r~f~r~~$ taient

concluants. Je n'ai pas pu, malheureusement, l'poque {I988}

envoyer une version rvise. Entre autres, l'un des referee confirma

l'originalit de l'tudue aprs avoir consult les fichiers MATHFILE

6

et . Nous avons l pour l'Universit de Saint-Louis un beau

champ d'investigations pour nos collgues et tudiants intervenant

en Statistique.

DEUXIEME PARTIE.

Cette partie, il est vrai, doit tre classe immdiatement aprs la

premire. Mais, chronologiquement, elle constinue notre dernire srie

de travaux allant de la priode de 1988 1991. Nous voudrions la consi-

drer comme notre conclusion notre contribution l'infrence des

extrmes.

Nous caractrisons l'attraction du rn.o.i. par une classe de huit.statistiques. Deux statistiques jouent les rles principaux. La pre-

mire (notons la comme dans notre premier article T (2,k,L), dj fortnconnue, est celle de Hill(1975). La seconde, note A (l,k,L), est

ndans sa forme, nouvelle. En fait la discrimination entre tous les

cas (Frchet, Weibull, Gumbel) est dj ralise par le couple

(T (2,k,L), A (l,k,L)). rI faut cependant signaler que cette discrimi-n n

aation a dj t obtenue par Dekkers-Einmahl-DE HAAN(Ann. Statist.,

1989, 17, 1833-1855). Des rsultats similaires sont signals par

riago de Oliveira et Arne Fransn (J. Tiago de Oliveira (ed.), statis-

~ical Extrmes and Applications, 373-394).

Nous avions dj dit que An

- 1 /2

(l,k,L) tait nouveau en tant

{u'estimateur de l'index d'une loi deParto. Lors du colloque

l'Alexandrie, le Professeur DE HAAN m'a dit qu'il pensait que l'esti-

lateur M(2) de son papier avec Dekkers et Einmahl avait un lien avecn

.e ntre. De retour saint-Louis, j'ai vrifi que M~2)=2An(l,k,O).

:eci dinimue la porte de notre affirmation. En fait, la forme que

7

nous avons introduite est drive directement d'une intgrale. Elle

a donc l'avantage d'tre plus commode lors de l'tude de la normalitasymptotique.

Une fois la caractrisation obtenue; :es questions naturelles

relatives aux techniques asymptotiques s'imposaient d'elles-mmes.

Elles sont:

@) normalit asymptotique des statistiques.

@) 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 plus loin plus tard en developpant le calcul.

Nous devons signaler le rle primordial jou par les approximations

Csorgo-csorgo-H6 rvth-Mason(1986).

Dans le quatrime article, les statistiques T (2,k,l) et A (l,k,l)n nsont gnralises un rang quelconque. Le rsultat est un processus

dont l'tude asymptotique est tudie. cette tude nous fait dboucher

sur un processus apparemment nouveau dont la fonction de covariance

est entirement dtermine. Ce processus ainsi mis en vidence

mrite, mon avis, une tude particulire. c'est un lment de plus

pour une future quipe de recherche Saint-Louis ou simplement entre

Saint-Louis et Dakar.

CONCLUSION.

D'une manire gnrale, chacune des parties de notre travail contient

des directions de recherche. Aucun travail de recherche ne peut tre

complet et dfinitif. Il faut toujours s'arrter un instant, profiter

de ce qui a dj t fait, puis repartir. Mon souhait est qu'merge

entre les Universit de Dakar et de Saint-Louis une quipe aussi

clbre que l'cole Hongroise ou l'cole Nelandaise dans le domaine

des probabilits et statistiques. La chose n'est pas impossible.,

gn tout cas, tout au long de cet expos, nous avons signal des problmes

statistiques dignes de recherches de niveau international.

8

~ous promettons d'y travailler.

Enfin, un mot sur la forme de la thse. Nous aurions pu tout r-

~crire en un seul corps avec beaucoup de rappels. Nous avons prfr,

avec l'autorisation de Monsieur Deheuvels; laisser apparatre le cher-

~heur dans ses activits. Les articles et rapports techniques sont

rendus tels quels.

Saint-Louis, le 15 Novembre 1991.

Gane 8amb L

9

PREMIERE PARTIE

CONTRIBUTION A L'INFERENCE DE L'INDEX D'UNE LOI DE PARETO

10

- 11 -

J. AppL Prob. 23, 922-936 (1986)Printcd in Israel

lE Applicd Probabilil)' Trust 1986

ASYMPTOTIC BEHAVIOR OF HILL'S ESTIMATE AND APPLICATiONS

GANE SAMB LO,' Universit Paris VI

Abstract

The problem of estimating the exponent of a stable law is recelvmg anincreasing amount of attention because Pareto's law (or Zipf's law) describesmany biological phenomena very weil (see e.g. Hill (1974)). This problem wasfirst solved by Hill (1

-- 12 -

Asymptotic behavior of Hill' s estimait and applications

whose distribution funetion F(x) =P(X ~ x) is such that F(Iog(' satisfies(1.1), where Xl,. ~ X1,A ~ .. ;:;; X..... denote the order statistics of Xl,' " x..

The asymptotic study of T.. (consistency and asymptotic normality) has beenconsiderably developed by various authors such as Hall (1982), Csrgo et al.(1985), etc. In fact, Mason (1982) proved that for an unbounded random variableX, i.e., supfx, F(x) < 1} = +00, and for any real c, 0 < c, one has

T..~ c, in probability(~ ) for any sequence le,. = k

satisfying

(K)

if and only if

as n~+OO

(Ac) 'lit >0, lim 1-F(log(tx=rlfc t +~ 1 - F(log(x

Vx, -oo

- Il-

GANESAMB L

If (Hl) holds, we have the following (see e.g. Lemma 3):(1.2) tbere exist some constant Co and a positive function s() such that

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

Therefore, we can make the following assumption:(H3) (Hl) and (1.2) hold and s(u) is uItimately non-increasing as u ! O.Our main results are as follows.

Theorem 1. Let

(H4) F(log( . E D(A) and F(x) is ultimately continuous as x t Abe satisfied. Then for any sequence Tc.. = k satisfying (K), we have

i-k

-[ " p 0k L.J (Xn-i+1.n - X n-k.n)~ ,i-l

as n~+oo.

Theorem 2. Let (Hl) be satisfied. Let (1.2) be reduced to

1.1 !ill

(1.2') O(l-u)=co+ .. t dt,

then for any sequence satisfying (K), we have

where

an~ 0, C4~ = s(iln),

and ~ denotes the convergence in distributin.

i = 1,"', k

CorolJary 1. Let the assumptions of Theorem 2 and (H3) be satisfied. If ksatisfies

(KI)

we have

s(kln)/s(l/n)~1, as n~+oo,

i-k

Ck.n k- 1 L (Xn - i + 1n - X ..-k... )~ 1, as n ~ +00.i-l

CorolJary 2. Let (H2) aIid (H4) be satisfied. Let k = (n S ), 0< 5 < 1. If forsorne Il, 0 < Il < 5/2, we have : .

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

- 14 -

Asymptotic behavior of Hill' s estimale and applicalions

then we get

i-t

(1.4) Ck." ~ k-I 2: (X.- i ....,. - X'-k.')--+ 1, almost surely. (a.s.);-1

Corollary 3. Let A > 0 and (H2) be satisfied. Then if (Hl) and (1.2') (or(H4 are satisfied, the statement (1.3) (or (1.4 remains true if we replace

X"-i+I,,, by log X.- i .....,,' i = 1,"', k + ls(u) by r(u)-{R(Q(1-u/Q(1-u)} asu!O,

C,. by r(i/n),

and if k satisfies (K) (or (K) and (K2) with s( . ) replaced by r( . Moreover iflog A > 0, we may derive similar results for the second logarithm, etc.

Remark. If A > 0, we have for large values of n

0< X"-HI. < A, 1~ i ~ k. + 1, since k./n --+ 0 and kn --+ +00.

Corollary 3 May be inverted as follows.

Corollary 4. Let R(t)--+O as t f A and let (H2) be satisfied. Then (Hl)and (1.2') (or (H4 are satisfied, the statement (1.3) (or (1.4 remains true if wereplace

X"-HI.. by exp(X"-i+1,n)

s(u) by t(u)- {R(Q(1- uexp(Q(l- u} as u ! 0C,. by t(i/n)

and k satisfies (K) (or (K) and (K2) with s( ) replaced by t( . Moreover, ift(u)--+ 0 as u ! 0, we may repeat the operation, etc.

Now, we give sorne examples via the expressions of the norming constants

Ck.'"

Corollary 5 (particular cases). In each case, (i) (or (ii will correspond to thechoice of le. =(log n) (or kn = (n 5), 0 < 8 < 1).

1. Normal case: X - N(O, l)(i) (2 log ftY12 T" ~ 1,(ii) (2(1-8)logn)Tn--+1, a.s.2. Exponential (or gamma case): exp(X) - E(l)(i) (log n )T"~ 1,() 1- 8)log n)Tn --+ 1, a.s.3. Log,,-normal: X = logp sup(b, Z), Z - N(O, 1), logp stands for the pth log

and logpb =0: Let D" =(2Iog(k./nII::~-llogi(210g(k../n)YI2, logo x =1, 'tr/x.

as t ' , t A

-- 15 -

GANE SAMB L

Then(i) DoT. ~ 1,(ii) D.T. -+ 1, a.s.4. Let T.

t(or T.

2) denote T.. for X = logsup(O, Z), Z - N1(0, 1) (or Z - E(I,

then(i) T. 2/T. t -+ 2, in probability,(ii) TnJT. 1-+ 2, a.s.

2. Proofs of theorems and corollaries

Before we proceed any further, we give two useful lemmas.

Lemma 1. Let FE D(A), then

(t' t )

(k(t')- k(t)-+ a) R (t) -+ a

for any - CXJ ~ a ~ + 00, where k(t) = -log(l- F(t.

Proof of Lemma 1. It is well known that FE D (A) (see e.g. De Haan (1970

iff

(B)NX) I-F(t+xR(t () astjA\ y 1 _ F(t) -+ exp - x ,

which may be written

(B) ('Ix) k(t+xR(t-k(t)-+x, astjA.

(i) Suppose that lirn .._(t~-t,.)/R(t,.)~a. where a is finite and (t~,t.. ) is asubsequence extracted from (t', t) and t', t t A. Then fr any E > 0, we have forlarge values of n: t~~ 1,. + (a - e)R (1,.). The fact that k ( . ) is non-decreasingand (B) imply that lirn._infk(t~)-k(t,.)~a-e."ence

Hm inf k(t~)- k(t,.)~ a..-(ii) In the same manner, we prove that

{~(t~- t,.)/R (t,.)~ al :::} hi~ sup k(t~)- k(t,.)~alNow, suppose that k(t')'- k(t)-a. -oo~ a ~ +00.

(a) Let a be finite: by (i) and (ii) it follows that for any sequence (t~, 1,.) suchthat t~, t. tA, we have necessarily that (t~ - t,.)/R (1,.) is bounded. Furthermoreif (t~ - 1,.)/R (1,.)-+ d, where d is finite, the same points (i) and (ii) imply that

d=a.

- r6 -

Asymptotic behavior of Hill's estimate and applications

(b) Let a = +00. Suppose that there exists a sequence (t~, 1,,), t~, 1" tA suchthat (t~- t..)/R(t..) is bounded. () would imply that k(t~)- k(l,,) is alsobounded, which is not possible at the same time as a = + 00.

(c) Let a = - 00. We use (i) at the place of (ii) in the above case and get that(t' - t)/R (t)- - 00.So, we have proved that

(k(t')- k(t)- a) ~ t' - t)/R(t)- a) at t', t t A.Converse1y, suppose that (1'- t)/R(t)- a.

(1) Let a be finite. Then for any E > 0, we have for t', t near A,t + (a - E)R (t) ~ t' ~ t + (a + E)R (t). Therefore, (B) implies

a - E~ Iim inf k(t')- k(t)~ lim sup k(t')- k (t) ~ a + E.,','tA ,','tA

(~) Let a = + 00, for any d > 0, we have for t', t near A : t' ~ t + dR (t).Therefore (B) implies

liminf k(t')- k(t)~ d.",'tA

('Y) Let a = - 00; similarly to the preceding case, we get

limsup k(t')- k(t)~ - d.,',' t A

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

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

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

gg~::) 1+Qog(1/v)+ 0(1 Rb~?_-u)But, by Lemma 5, R (t)/t-O as t t A, whenever FE D(A). Hencelim.. t 0 Q (1 - u )/Q (1 - uv) = 1, which is the announced result.

Proof of Theorem 1. Since (H5) holds, Lemma 2 implies that the functionH (1 - u) associated with FQog() is SVZ.

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

- 17 ,-

GANE SAMB L

(2.1)

(2.2)

where Ul n ~ U2n~ ... ~ U",n denote the order statistics associated withUl, U2,"', Un, a sequence independent and unifonnly distributed on (0,1).Finally, recall that H(I- u) admits Karamata's representation since it is SVZ:

H(I- u) = c(u)exp (f (r(s)/s)ds) ,c(u)-+c, O

- 18 -

Asymptotic behaviorofHill's estimate and applications

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

Lemma 4. Let the assertion (ii) of Lemma 3 he satisfied, then FE D(A). fiin addition F is continuous as x t A, we get

10 seul 1!N R(Q(l- u .

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

Lemma 5. Let (Hl) and (H2) be satisfied, then R(t)/t~O as t t A andR(Q(l- u is SVZ.

Proof ofLemma 5. First, we remark that Lemma 4 implies that R (Q (1 - uis SVZ and by Lemma 3 of Lo (1986), R(t)/t~O as t t A.

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

Q(l- u) = Co +f (s(t)/t)dLThus, for each ~ 1~ i ~ k, k satisfying (K), one has

(2.5) X..-i+l... -X..-~.. ~ Q(l-V~.)-Q(l-Vi+l ... )= L:~+I"(S(U)/U)dU.

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

s(u)= z(u)exp (f (W(v)/v)dV) ,(2.6)

z(u)-.z, O

- 19 -

GANESAMBLO

independently of ~ 1~ i ~ le. Moreover, sinee a~.. ~ l.l,+l ... ~ Ult+l ...~ 0, we getalso that {z(~.. )/z(i/n)}.!.l, uniformly with respect to ~ .. and with respect to ~1~ i ~ k. Therefore, we cao see that the foUowing equalities are independent ofa,.. and of ~ 1~ i ~ k:(2.9) f... .!!!! dv = 0, (1), {z(a,.)/z(i/n)} = 1 + 0, (1).i/.. vApplying (2.9) to (2.7), and recalling that s(' ) is positive, we get

1 QI - f...!...< ...!.U!L-1 Q2(2.10) +,.,,,;- fi ('/ )=sup ('/ )- +"'.h as nt +00oEJ. sIn oEJ. sIn

where 1. = (V,., Vi+l,.. ) and (Ji,.; = 0,(1), j =1,2, independently of i, 1 ~ i ~ k.Apply (2.10) in an appropriate manner to (2.5) and get

(2.11) iC,. (X.-i+l. - X.-i..) = log (tt::), (1 + y.dwhere Yni = 0, (1) independently of ~ 1~ i ~ le, k satisfying (K). However, it isalso weIl known that

{lOg(VHI.n/V,.. ), 1~ i ~ n} ~ {6/0 ~ i ~ n} with Vn+I,n = 1.

Applying this and again applying (2.9), we get

d

ic;'. (X..-HI... - Xn-,.. ) = ~, (1 + Yni)'It follows that

i-Ic i-Ic

Vn = k-l L iC~.. (X.-HI,n - X n-,n) ~ k-l L ~i + an'i-I i-l

It is obvious that a .. ~O, because of (2.11). Therefore, the weak law ofp

large numbers implies that V.. -+ 1. Moreover the central limit theorem givesk '12(V.. -1- a.. ) = k l12(k- l ~::~ ~ -1)~N(O, 1).

Proof of Corollary 1. This corollary is immediate sinee V..~ 1 and

I-Ic i-Ic

k-l L i(X..-i+l... - X .._,..) = k-l L (X..- H1... - X n-"-n)'.-1 i-l

Proof of Corollary 2. If (H4) is satisfied, Theorem 1 implies that T" ~O. Onthe other band, Mason (1982) bas proved that if Tn = T.. (6), where k = le,. =(n li), is bounded in probability, then

T.. (6)- R(Q(l- U,,-..= O(n-) a.s., 0 < v < 8/2,d

where {T;(8), n ~ 1} = {T.. (6), n ~ 1} (as proeesses).By Lemma 7, (H4) and (H2):::? (Hl). Renee, by Lemma 4,

l

- 20 -

Asymptotic behaviorof Hill' s estimate and applications

R (0(1- u1s(u)- 1, as u ~ O. Now, since s(u) is SVZ and (nI k )U"R - 1, a.s.,we get by Lemma 5

R(Q(l- Urc.,,ls(kln)-l, a.s.

which completes the proof.

Proof of Corollaries 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 bas a distribution function FEL,log sup(O, X) has a distribution function GEL and

5(0-1(1- u - {R (Q(l- u/0(1- u)}, as u t 0,where

_J. 108 A (1- G(v5(t)-, (1- G(t dv, - co < t < log A.

Conversely, we have the following.

Lemma 8. Let R (t)- 0 as t t A. Then If X has a distribution fonctionFEL, then exp(X) has a distribution function Z ELand

R(Q(l- u -{T(Z-I(l- u/Z-I(l- u)} as ut 0,

where

-J."l-Z(V)T(t) -, 1- Z(t) dv,

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 Corollaries 3 and 4. By Lemma 7, we see that if for instance (Hl) issatisfied, the same property is also true for 10gsup(0, X). So, we can write (1.3)with s ( . ) replaced by r( . ), where r(u) is derived from De Haan's representationfor G-1(1- u) as in (1.2). But, we see also that 0(1- u)= co+ f~(s(t)lt)dt ~s(u) = uO'(l - u). Since

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

r(u) =: ug~11_-uU/= u(G-I(l- u)')is SVZ by Lemma 2 and (1.2). Renee,

G-I(1- u) = - G-1(1 ~ YI)+ f r~t) dt, for u ~ YI ~ 1. , 'f\g1

1

- 21 :-

GANESAMB LO

The preceding means that the De Haan representation of 0-1(1-.) is reduced.We May apply Theorem 2 for G(). Note that sup(X..- i +1... , 0) = X..- Hl... , a.s., ask - 00, n - 00, kln _ O. Corollary 4 is proved in a similar way.

Proof of Corollary 5. It May he easily verified that in ail our particular cases,we have that l(u) =uO'(1 - u) is slowly varying at 0, where Q'(u) is thederivative function of 0 for u near 1. So, forsome y near 0,< y < 1, we have(2.12) 0(1-u)= - 0(1- y)+ f (l(t)lt)dt, 0< u < y.That means that (1.2) is reduced. As remarked in the proof of Theorem 2, (2.12)May entirely replace (1.2). Moreover, if (2.12) holds with a function SVZ, itfollows that FE D(A) and l(u)- R(0(1- u (see Lo (1986), Lemma 4). Notealso that 0 (1- u) is continuous as u ! whenever (2.12) holds.

Hy the above expianations, the proof of Corollary 5 consists in determiningthe function I(u). The verification of (K1) or (K2) is immediate.

1. Normal case: X - N(O, 1). We use the well-known expansion of the tail ofthe standard Gaussian random variable:

Using that, one verifies that W(x) = (1- F(x 1F'(x) has a negative derivativefor x ~ xo. Then uO'(1- u) = (1 - F(x /(F'(x is strictly decreasing as u =1- F(x) ! O. This shows that (H3) is satisfied. Moreover, it is also known that

(2.13) 0(1-U)=(210 !)112{1+IOgIOg(1IS)-41T+O(1)} as Si O.g s 4 . log(l/s) , ~

Routine calculations show that

0'(1- s) = s(IOg(~/sl12 (l + 0(1 as s ! O.

So l(u)=(210g(1/s)f l12(1+0(1.At this step, an appropriate application of Corollaries 1 and 2 gives the stated

results.

2. Exponential case: exp(X) - E(1). Here Q (1 - s) = log log(l/s); there-fore, we apply Corollaries 1 and 2. Remark that for a general gamma law,0(1- s) -loglog(1/s).

3. This part follows from a typicalapplication of Corollary 3 p times.4. This part follows from Part 2 and Part 3 in the case where p =1.

- 22 -

Asymptotic behavior ofHiU's estimate and applications

3. Applications and simnlations

As remarked above, Hill (1974) described some basic models which follow(1.1) or (Ac). We note that ail these models are closely related to problems basedon extreme values. As already noticed, the works of several authors (Csrgo andMason (1985), Csorgo et al. (1985), Hall (1982), Hill (1970), (1974), (1975), Mason(1982 have entirely settled the properties of T.. under the assumption (Ac). Itfollows from their results that if Xl, X 2, " X.. are the observations of ~ we canverity if (Ac) holds. In that case, we proceed as follows:

3.1. Identification of the upper tail of a distribution.(i) Choose B, 0 < B< 1,(ii) Choose le" = (n 3),(iii) Calculate T.. = k-l L::~(X~-i+1.~ - X~-t... ) for large values of n.(iv) If T~ is very near c, 0 < c < + 00, then by Theorem 1 of Mason (1982),

(Ac) holds.(v) But (Ac) ~ FED(A) ~ c-l(X..... - Q(l-l/n~A,

and therefore, we can use (v) for predictions about a critical value of X......However, it is not always certain that T.. converges to a finite strictly positivenumber. For example, if we want to know whether F satisfies (Ac) or if F(log( . is the distribution function of the standard Gaussian random variable, how couldwe proceed?

3.2. Comparison between a regular tail and a gaussian tail. We want to knowif

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

(L) Xl =log sup(O, Z), Z - N(O, 1)where F(' ) is the unknown distribution function associated with the observa-tions Xl, 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 n 1l4(T.. - c)~ N(O, 1).

(ili) If (L) holds, we have(a) (Corollary 5, Part 3, p = 1), log nT.. -+ 1, a.s.(b) (Lo (1986 n 1l4(D..T.. -1)~ N(O, 1), D .. = log n(l + 0(1.

Thus, we see that we are now able to test (LI) against (L). If we chooseD = {(c - E) ~ T.. ~ (c + e)} as the accepted region of our test, (i) and () givethe characteristics of that test. To test (.) against (LI), one can choosejj = {1 - e ~ T.. . log n.~ 1+ e} as the accepted region with a small value of e.

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

- 23 -

GANESAMBLO

Xl, X 2 , " X"' YI, Y2 ,' " Y" be two independent samples respectively as-sociated with the distribution functions F and G. We want to know if

(L2) F{log(x = 1- exp( - x) = G(log(xor

(L) XI =10gsup(O, Z), Z - N(O, 1), G(log(x = l-exp(- x).In Lo (1986), we have also given the limiting law of T" under the assumptionF{log(x = 1- exp( - x). Denote by Td (T"2) the Hill's estimate associated withX I,X2," ',X" (YI, Y2,"', Y,,). Then from Corollary 5, we get T"dT,,2 .... 1/2,a.s. under (L) and T"dT,,2 .... 1, a.s. under (L2). The independence of the twosamples and the limiting laws obtained in Lo (1986) thus enable us to constructstatistical tests. After the simulations, numerical applications of that test will begiven.

Finally, we give numerical applications using simulations.

3.4. Simulations. Here we have used an ordered sample generated from auniform random variable Uh U2,"', U4OQO. We have constructed the followingorder statistics:

Exponential case: Yi = -log(l- Ui)'

Normal case: Xj = (2 log (1 ! Uj))1/2 , for large values of ~ 1~ i ~ 4000.Pareto case: Zi = (1 - u;ri.Define for k = k,. = (n 1/2),

i-k

Td = k- I ~ (log X"-i+l -log x,,_,,), 3990 ~ n ~ 4000.f=1We write T"I = B" (Xi). Therefore, we define T"2 = B" (YI) and T"3 = B" (Zi)'

Our simulations are given in Table 1.

TABLE 1

1 2 3 4 5

N lognT., ~lognT.2 T UaJo-.+1

3991 0.5222 0.5222 0.6780 0.0024353992 0.6090 0.6090 0.6988 0.0016313993 0.6225 0.6225 0.7215 0.0016203994 0.6443 0.6443 0.7035 0.0013373995 0.6204 0.6204 0.7162 0.0009883996 0.6464 0.6464 0.7460 0.000437

~ 3997 0.6683 0.6683 0.7182 0.0004183998 0.6898 0.6898 0.8102 0.0003083999 0.7093 0.7093 0.8412 0.0002974000 0.7554 0.7554 0.9086 0.000095

- 24 -

Asymptotic behaviorofHill's estimate and applications .. ~..

Remaries.1. One might he surprised to find that our simulations are not sufficiently

good, considering the large size of the sample space (n = 4000). However, ornythe k extreme observations (k ~ 62 or 63) are used for the caleulation of T...Taking that into account, the theoretical part of this paper is relatively weIlillustrated by the simulations. Specifically:

2. Column 4 illustrates the aImost sure convergence of Hill's estimate for thePareto law: T.., converges almost surely to 1.

3. The identity of columns 2 and 3 is a consequence of the choice of Xi. It isclear that if Xi = (- 210g(l- U;W12, we get T..2 = ~T.I' However, this choice is notarbitrary. Indeed, if T..2 is the true value obtained from the use of the truequantile function, we have T..i = B.. (1;), where (see e.g. statement (2.13.

log 1; =log {( - 210g(l- 11,112 (1 + log( -I01\~;(~~:i~1T + 0(1)}

= ~log2+!log Yi + 0 ((lOg log 1~ u,) /410g 1~ 11,) .With our data, 3937~ i ~ 4000, k = 62 or 63, we have

log~ =Hog2+!logYi 0.07,

therefore

r..; = !Tnl 0.07.4. We now test (L3): (Xi) are the order statistics of a N(O, 1) random variable,

against (L4): (Xi) are the order statistics of the standard exponentiallaw. If wechoose R .. = {1- a ~ log nT.t ~ 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 {3 - 0.74,and R.. will he R.. = {0.75~ T.. t log n ~ 1.2533}. Here, we accept (13) since thetable gives the value 0.7554 for (log n Till) for n = 4000.

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

{1- u;, 1~ i ~ 4000} :. {"4000-Hl, 1~ i ~ 4OOO}.

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

5. Remarks and forther genenlizatloDS

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

l

1

,

1

25 -

GANE SAMB LO

(This remark is derived from a discussion with Professor D. M. Mason and Dr.Haeusler. Many thanks to them.)

Rema,k 2. We have used the continuity of F(') just to obtain thatF(rl(x =x for large x. But this i's true for any distribution fonction.

Remark 3. We have proved in Lo (1986) (see Lemma 12') that the assump-tion K2 in Corollary 2 is always satisfied.

These three simple remarks yield the following generalization.

Generalization. Throughout the paper, the hypothesis (H3) may be replacedby FE D(A), (H4) by F(log( E D (A), and both (KI) and (K2) may bedropped.

Acknowledgments

1 should like to thank Professor Deheuvels for having suggested to me theproblem which led to this investigation, and for his moral support. 1 am alsoindebted to M. Der Mergreditchian for giving me permission to use the EERM'scomputers, to N. Nader, M. Davis and R. Moradkhan for their commentsconceming the English text, and finally to the referee for his helpful suggestions.

References

BOUNGER, G. A. (1885) Catalogue of the Lizards in the British Museum. British Museum,London.

CsORGO, S. AND MASON, D. (1984) Centrallimit theorems for sums of extreme values. Math.Pme. Camb. PhiL Soc. 98, 547-558.

CsORGO, S., DEHEUVELS, P. AND MASON, D. (1985) Kemel estimates of the tail index of adistribution. Ann. Statist. 13, 1467-1487.

DE HAAN, L. (1970) On Regular Variation and Applications ta the Weak Convergence ofSampleExtnmes. Mathematical Centre Tracts, 32, Amsterdam.

DE HAAN, 1. AND REsNICK, S. 1. (1980) A simple asymptotic estimate for the index of a stablelaw. J. R. Statist. Soc. B 44, ~.

DBHEUVELS, P., HAEUSLER, E. AND MASON, D. M. (1986) Laws of the iterated logarithm whenthe maximum is attraeted to a Gumbellaw. Unpublished.

GALAMBOS, J. (1978) T1ae Asymptatic Theory of Extnme Order Stalistics. Wiley, New York.RAu, P. (1982) On some simple estimates of an e~nentof regular variation. J. R. Statist. Soc.

B 44, 37-42. 'HIu., B. M. (1970) Zipf's law and prior distributions for the composition of a population. J.

Amer. Statist. Auoc. 65, 1~1232.Hn.L, B. M. (1974) The rank-frequency form of Zipf's law. LAmer. Statist. AUDe. 69,

1017-1026.HaL, B. M. (1975) A simple general approach ta inference about the tail of a distribution. Ann.

Statist. 3, 1163-1174.Lo, G. S. (1986) Ph.D. Dissertation, University of Paris VI.MASON, D. (1982) Law of large numbers for sums of extreme values. Ann. Prob. 10,754-764.

- 26 -

C.S. Lo 1 Asymplolic normalily of extremes

The case (1) is contained in the theorem of Csorgo and Mason (1986) and the case(2) is Theorem 1.5 of Csiirgo and Mason (1985).

An application of Theorem 204.1 of de Haan (1970) (Lemma 1 below) combinedwith Fact lA of Csorgo and Mason (1985) shows that (2) implies the existence ofsequences of normalizing constants an and centering constants bn such that

da;; \ (X",n - b,,) -----+ G as n --+ 00,

where G is a Gumbel random variable with distribution function

P(G:$X) = exp(-e--X ) for -OO

- 27 :- 1

Journal of Statistical Planning and Inference 22 (1989) 127-136NorthHolland

A NOTE ON THE ASYMPTOTIC NORMALITY OF SUMS OFEXTREME VALUES

Gane Sarnb L

Universit Paris 6, L.S. T.A. TAS-55. E.3.. 4 Place Jussieu, 75230 Paris OS. France

Received 15 January 1988; revised manuscript received 16 March 1988Recommended by P. Deheuvels

Abstract: Let X" X2 be a sequence of independent random variables wth common distribu-tion function F n the domain of attraction of a Gumbel extreme value distribution and for eachimeger n 2: 1. let X ,.n :$ .. , :$X..n denote the order statistcs based on the first n of these randomvariables. Along with related results it is shown that for any sequence of positive integers k n ~ 00and kn/n~O as n-oo, the sum of the upper knextreme values Xn_k.+I.n++Xn.n. whenproperly centered and normalized. converges in distribution to a standard normal random variableN(O. 1). These resuhs constitute an extension of resuhs by S. Csrg and D.M. Mason (1985).

AMS Subject Classification: 62E20. 62G30. 6OF05.

Key words: Drder statstics; extreme values; Gumbel law; asymptotic normality.

1. Introduction

Let XI' Xz, ... , be a sequence of independent randorn variables with cornmondistribution function F and for each integer n?:. 1. let XI,IlS'" sXIl 1l denote theorder statistics based on the first n of these random variables. Csrgo and Mason(1985, 1986) have recently shown among other results that if

-0''''.,'"~ ,o. :~:

'~)',;.'

1- F(x) =L *(x)x-Q as X-' 00, (1 )

where L * is a slowly varying function at infinity and Q?:. 2, or if F has exponential-Iike upper tails, rneaning

lhere exist sequences A" > 0 of normalizing constants and en of centering constantssuch that

t:

r(1- F(y dy/(I- F(x) -. c as X-' 00,where O

- 28 -

G.S. Lo / Asymp/o/ic nornlQlily of ex/reines

For any sequence of positive integers kIl such that (K) holds and FE D(/I). set forn= 1,2...

'1

Pn(k,,)=n \ Q(s)ds., 1 - k.ln

The following theorem contains our main results.

Theorem. On a rich enough probability space there exist a sequence of independentrandom variables XI' X 2, ... , with common distribution function F and a sequenceofBrownian bridges BI' 8 20 such that for any sequence kIl satisfying (K), when-ever FED(/I),

k14 1k n-

1I1c(knIn)-I L~I X n- i+ I,n - Pn(kn)j

= -(nlkn)ll2c(knln)-1 ri Bn(s) dQ(s) +op(1) :=Zn+op(l), (5)JI-k.ln

and whenever FE D(A).

k~/2c(knIn)-I {Xn-k.,n - Q(I - k nln)}

=-(nlkn)II2Bn(l-knln)+op(l):= Yn+op(l).and

(6)

= Zn - Yn+ op(l). (7)

Furlhermore. the random variables on the /eft side of (5). (6) and (7). respective/y.converge in distribution to N(O,2). N(O. 1) and N(O. 1), respective/y. as nT 00.

Remark. With the choice An =(2kn)-1/2c(knln)-1 and en =p,,(kn). we see that ourIheorem implies (3) ~henever FED(/I). Our theorem also eXlends Theorem 1.5. 1.7and 2.1 and Corollary 2.5 of Csrgo and Mason (1985). The random variable onthe left side of (7) is related to the Hill (1975) estimator of the tail index of a distribu-for this random variable'was motivated by the work of Mason (1982) (see alsoDeheuvels. Haeusler and Mason (1988.

3. Proof of the theorem

We use the following integral convention:When Osasbs 1. g is left continuous and fis right continuous.

rfdg= \' fdg and,b ~(Q,b)

l'b gdf= \' g df,Q ,(a,b) 1

-29 -

G.s. Lo / Asymptot;c norma/ity of extremes

whenever these integrals make sense as Lebesgue-Stieltjes integrals. ln this case, theusual integration by parts fonnula

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

The proof of our theorem will follow cIosely the proofs of the results of Csorgand Mason (1985), substituting their tcchnicallemmas concerning properties of thequantile functions of distribution functions satisfying (2), by those describingproperties of the quantile functions of FED(/I). We therefore begin with thesetechnical lemmas.

Lcmma 1. FE D(A) if and only if for each cho;ce of 0 s x, y, w, z< 00 flxed, y:;t: w,

Q(I-sx)-Q(I-sZ) __ logx-Iogz ass!O.Q(I -sy) - Q(I-sw) logy-Iog w

(8)

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

Lemma 2. Whenever FED(A), c(s, P) ;s slowly vary;ng at zero for each cho;ce of

O

r1

- 30 -

O.S. Lo / Asymptotic normalily uf ex/reines

Thus for ail s>O sufficiently small,

(1 + c)c(s, p):s -----or c(s, /3).

Observing that for ail s> 0 small enough,

(12)

\

'1 00

1 -As (1 - u)p dQ(u) ~ i~O PS P8(i+ I)P {Q( 1 - s8 i + 1) - Q(I - }.s8 i )},

we see that by an argument very much like the one just given, we have for aIl s>Osufficiently small,

(13)

Assertion (9) now follows from inequalities (12) and (13) by the fact that 8 can bechosen arbitrarily close to one and c arbitrarily close to zero. This completes theproof of Lemma 2.

The following lemma is related to Theorem lA.3.d of de Haan (1970) and ilsproof is based on a modification of the techniques used to prove this theorem. Fordetails see Deheuvels et al. (1986).

Lemma 3. Whenever FE D(/I), there exists a constant -00 < b< 00 such that for 011O

C.S. La 1 Asymplolic normalily of ulrt!mes

- 31 -

(I7)

(18)

Lemma 6. Whenever FeD(A).

u 2(s)l(2sc 2(s -+ 1 as s10.

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

Proof. The proof is based on Lemma 5 and follows almost exaetly as the proof ofLemma 3.3 of Csorgo and Mason (1985). Therefore. the details are omitted.

Lemma 7. Let an be any sequence of positive constants such that an -- 0 andnan-> 00. Also let L be any slowly varyingfunction at zero. Thenfor any O

r-I

- 32 -

G.S. Lo / Asymp/o/ic normality of ex/rernes

and the quantile process

,Bn(s)=n l12 {s- Vn(S)} , Os;ss; l,where

and, with V I ns; .,. 5: Vn.ndenoting the order statistics corresponding to VI' ... , V n.

Vn(S) = Vkn ~f (~-I)/n

,i!

- 33 -

G.S. Lo / Asymptotic normality of extremes

From (21), wc obtain

c(k ln .1_ v}R =0 (n-V}n" n ,l k- v

I,n p c(knln) ".

which by Lcmma 5 cquals op(l}.Also

ERL == a2(1ln}/(:n c 2( ~ )).

which by Lemma 6 is

-2a!(lln}/a 2(k"ln} as n -> 00.

From Lemmas 2 and 6 we infcr that a 2(s} is rcgularly varying of exponent one arzero. Hence, by Lemma 7,

a 2(1ln}/a 2(k"ln)-O as n->oo,

which yiclds R2n =op(l).Thus we have proved R,,==op(I).

Ne"t we show that L1 2,,,=op(l). Choose any I

- 34 -

G.S. Lo / Asymplolic: normafity of exlremes

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

lim Hm sup ET" () = O.;11 "-00

(24)

(26)

The fact that ,d2." =op(l) now folIows by an elementary argument based on (23)and (24). This completes the proof of (5).

Next consider (6). Notice that since Fe D*(A),

C(S)=S-I fi r(l-u)du ..11-5

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

r(s)1c(s) - l as s10. (25)The lefl side of (6) equals

k~/2 r'-u r(U) J/2r(k"ln) }---- --du=-k" ---{log(l-U"_k.I")-log(k,,ln) ,

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

k~/2 JI-U...... du * *- --- (r(u) - r(k" ln - :=,d, +,d2 .

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

The same argument based on (20) as given in Csrgo and Mason (1985) shows that

-k~I2{log(l_U"_k. ")-log(k,,ln)} = Y,,+op(1).

Therefore by (25) and the fact that Y,,=oil) we have

,d~" = Y" + op(I).

Since r is slowly varying at zero we get for each 1

- 35 -

O.S. Lo / Asymprolic normalily of eXlrell/es

Acknowledgement

The author is extreme[y gratefuI to David Mason, Erich Haeusler and PaulDeheuvels, for helping him to complete this work.

References

Balkema, A. and L. de Haan (1975). Limil Iaws for arder statistics. In: P. Rvsz, Ed., Limit Theoremsof Probabilily Theory, Coll. Math. Soc. Bolyai. Vol. Il. 17-22.

Csorg, M., S. Csorg, L. Horvth and D.M. Masan (1986). Weighted empirical and quanlile processes,Ann. Prob. Il, 31-85.

Csorg, S. and D.M. Mason (1985). Central limit theorems for sums of extreme values, Malh. Proc.Cambridge Philos. Soc. 98, 547-558.

Csrgo, S. and D.M. Mason (1986). The asymptolic distribution of sums of extreme values from aregularly varying distribution, Ann. Probab. 14, 974-983.

Deheuvels, P., E. Haeusler and D.M. Mason (1988). Almost sure convergence of the HiII estimator.Malh. Proc. Cambridge Philos. Soc. 104,371-381.

Deheuvels, P., E. Haeusler and D.M. Mason (1989). Laws of the iterated logrithm for sums of extremevalues in the domain of auraclion of a Oumbel law. Bull. Sei. Math., to appear.

de Haan, L. (1970). On Regular Varialion and ils Applicalion to Ihe Weak Convergence of SampleExlremes. Malhematical Centre Tract No. 32 (Amsterdam).

Hill, B.M. (1975). A simple general approach to inferenee about the tail of a distribution. Ann. Sialisl.3, 1163-1174.

Lo, O.S. (1986). Asymptotic behavior of HiII's estimate and applications. J. Appl. Probab. 23,922-936.Mason, D.M. (1982). Laws of large numbers for sums of eXlreme values. Ann. Probab. 10, 756-764.

- 36 -

THE WEAK LIMITING B~~VIOR OF THE DE HAfu~/RESNICK'S

ESTIMATE OF THE EXYO~ENT OF A STABLE DISTRIBUTION.

by

La Gane Samb, Universit Paris 6 .

.L.S.T.A.

Abstract.:

The problem of estimating the exponent of a stable law receives a conside-

rable attention in the recent literature. Here, we deal wi th an estima te 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 be applied ta many biological phenomena. Exarnples

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

4, Place Jussieu. 75230, Paris Cedex 05.

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

tion, norming constants, arder Statistics, Limiting law.

!1

iNTRODUCTION AND RESULTS:

- 37 -

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

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

~st city ( see e. g. Hill (l974) ). This motivated considerable works on the pro-1

lem of estimating c. More generally, if X1,X2 , .... , Xnare independent and

)

distribution ( i )

as

, in

k/n + 0

order statistics of X1 'X2 ' ... , Xn and k is

c, in probability-

( T -c) + An

T = (X . - X k )/Lg kn n,n n- ,n

(i) Tn

(ii) Log kc

De Hah and Resnick(1980) have proved that (Ac) implies under sorne conditionsp+

~entical copies of a random variable ( rv ) X such that F(x)= P( X< x) satisfiesi~c) ~ t>O, Lim 1 - F(Log(tx)) = t-1/ c1 xt+oo 1 - F(Log(x))

rveral estimates ( Hill(1975), S.Csrg8-Deheuvels-Mason(1983) ) have been pro-1

josed. De Haan and Resnick (1980) introduced11

1

and

that:

1Jhere Xl ' X2 , ,X are thei ,n ,n n,nr sequence of integers satisfying

~K) O

- 38

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

Define

A = inf {x, F(x) = l } B = Sup {x, F(x) = 0 }

R(t) = (l-F(t-l J~ (l-F(v dv, B~t

- 39 -

rue if we replace

Xn,n ,X kn- ,nby Log Xn,n , Log X k. n- ,n

Q(l-s)

r(s)

by

by

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

s(u) = R(Q(l-u)/Q(l-u)

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

the operation.

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

and (1.4) hold ifwe replace

x , X kn,n n-,nQ(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 ( particular cases)

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

integer part, and ~ is any positive number.

1) Normal case: X'\>N(O,l)

(i) ~(2Log n)!(LogLog n ). T .(1+0(1) - ~(LogLog n)(l+o(l)) ~ An

'(ii) (2LOgn)!Tn

-- 1

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

(i) (1+0(1. ~ ( "Log n) (LogLog n) Tn - ~ (LogLog n) (1+0(1 , !l fi(ii) (Log n) T ~ 1

n

3) Suppose that X a=s-Log Sp{e _1(1), Z), Z'\>N(O,l),p>l, Lo"g (resp. e )-- p p - p p

denotes the p-th 1 10garithm -( resp: exponential") with by c?nvention ~ogo x =1,~h-p l .!

Let Cn '" (21og n) TI h:1- 10gh(2Log nrl

(i) ~(Log1og n) C T - ~(LogLog n)(l+o(l ~ fin n

(ii) C T ~ 1.n n

- 40 -!1

1"'markfi

( important): Mason(1982) has proved, that the Hill' s estimate

H = k~l Li~k ( X - X )n i=l n-i+1.n n-k.n

i1s characteristic of distributions satisfying (Ac) in the following sense: sup-

c. for any sequence k satisfying (K)Tn

In order to compare H and T we remark that this property is not obtained forn n

Indeed. the Mason's distribution defined by

F-I (1_2-m) = m. m=0.1.2 ....

F-1(I-u) = m +(2-m_u) 2m+1 if 2-m- 1

- 41 -

-1 R(Q(l-u)) Log AS(G (l-u)) "'(f(i-u) , as u+O, where S(t)= f

t

l~G(v)l-G(t) dv, -oo

- 42 -

implies

sequen-

SUPO

- 43 -

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

A 4/an nU

A 4 = r(k/n)-r(Uk 1 ) + l k+1,n ( r(s)/s ) ds,n + ,n k/nthat since r(u) is slowly varying at 0, we have on account of (2.10) that

1

ite an and bn are defined in the statement of the theorem. First, we provethat!l2)it1.2), we haveJ

~rk

1

1

:thermore,i

Let r(u) be a positive function S.V.Z: Let (u ) te a sequence of rv~sn

as n ++00= 1+0 (1),P

r(s)r(1/d )

nSupsEJ

n

(d . li ) = 0 Cl), and (d .u ) -1 = 0 (1) an n ,++00n n p n n p

r(s)= lnf r(l/d) l,

AS(3.19) Zn-Xn~G-l(1-Sn)-G-1(1-ASn)==S(~Sn)-s(En)-fn t-ls(t)dt,

sn

for large n, where & =U . Now, the properties of 5VZ functions easilyn \.+1, n

yield for any fixed s, 0

- 66 -

1.28) Tn(6)~Zn(2){R(Z )/Z (l)(z -x - T (8,~)O (l)}.n n n n n p

lUS, the convergence of Tn (8J3/2) to zero, Fact 4 and (2.28) imply

'.29) R(Zn)/(Zn-Xn)~pO, as n~+oo,

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

R(Zn)/(zn-X )S(y -z )/(z -x )=l/(-l+l/T (9.non n n n

follows that R(X ,2 )/R(x )~ 1 as n~+w. By the same methods (replacen n n p

[(6) par Tn (7 and remark that lim sup W(x)~(lim sup R(X2) , we alsox~Yo x~Yo

Lve W(x ,z )/W(x )~ 1 as n~+oo and thusn n n p~.30) Iim W(x )/R(x )2=K, in probability_n-i>+CO n n

view of Le~~s i and 2, Thaorem 2 wi2 2 be proved if we show that

\.31) Iim W(x)/W(x _ ,)(resp R(x)/R(x ( )=1 in probability.Y~Yo nlm) n ID

le proof of this is quite direct and technical. We state it in the

'PENDIX.

'PENDIX

calI basic facts

n-P~n-P + n-l~(k+l)Sn-P + 2n-1 for k=[nal, p=l-a.

fact 2,

-p -6 ~ ~ -p -1n -sn ~Uk 1 ~n +2sn , a.s., as n~+oo, 0... makes a

[O,l~. For X~Yo' u=l-G(X)~O. There exists at each

~p of this limit an integer n such that a 2~~~a lSa. Letn+ n+ n

2- Proof of Theorem 2. - 67 -

first use Fact 2 in (3.3a) ang get for any O

- bU

=(n)=n-(n7], T=2-a-6, p(n)=n+(n

T], T=2-a-6.

:emark that O.55,c'

-0::1.--=-;---------------------,lrnportant rerrva.rh.. dg H.aa.n and l

- 70 -applications., J.Appl. probab., 23, 322-936.

15] - L, G.S. (1989). A note on the asymptotic normality of surns of

extrerne values. J. statist. Plann. Inference, 22, 12-136.

16] - L, G.S.(1991). Empirical characterization of the extremes:

parts l & II. Technical Reports LSTA-CNRS,

Universit paris VI.

17] - Mason, D.M.(1982). Law of large nlmbers for extreme values.

Mln. probab, 10, 754-764.

18] - Pickands, J.(1975). Statistical inference using extreme value

theory. Ann. Statist. 119-131.

19] - Smith, R.L.(1987}. Estimating tails of probability distribution.

Ann. Statist. 15, 1174-1207.

20] - Resnick, S.I. (1987). Extrema VaL-t.les, ReB"ul.ar Variat.ion and

Point Processes. Springer verlag, New-York.

as

symbol-

of

- 71 -

EMPIRICAL CHARACTERIZATION OF THE EXTREMES II :

THE ASYMPTOTIC NORMALITY OF THE CHARACTERIZING VECTORS.Gane Samb LO.

Universit saint-Louis & Universit Paris VI-LSTA.

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

ith cornmon distribution function F such that F(l)=O and for each

~, x~ r- S ..s ...SX denote the order statistics based on the n first of.i,1 n,n

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

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

l'U i=k 2j =i . (1' /2) (l l ) (l l }1 \. J -o~.. ogX ... - ogX. ogX. 1 - ogX . )i=L+l j=L+l i] n-l+~,n n-l,n n-J+ ,n n-J,n

l1ere (k,L) is a couple of integers such that k-HOO, k/n~O, t 2 /k-+O,

~+oo, log stands of the Natural logarithm and 6 .. is Kronecker's8 lJ

rom which he set ~ -vectors that characterize the whole domain

ttraction of the sample extreme and each particular domain of attraction

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

re completely determined in this paper. Tt is shown that the single

lements of this family of Characterizing statistics of the Extremes

fACSEXT) and their ratios are asymtotically normal as L-++oo. But sorne

atios become asymtotically extremal whenever L is bounded. The use of a

rrified approach enabled to obtain the multivariate asymptotic normality.

~s 1990 subject classifications prirnary 62E20, 62310 ; secondary 60FOS.

~y words and phrases. Order statistics; extreme value theorYi dornain of

:traction of the sample extremesi extremal and Gaussian lawsi

!presentation of distribution functions; invariance principes;

Itivariate normalitYi characterization.

iling address.UER Math & Informatique, BP 234, USL, Saint-Louis, sngal.

search addres. LSTA, Universit Paris VI, T.45-55, 8.3, 8.3, Place Jussieu.

75230 paris cdex 05. France.

-72-

- INTRODUCTION

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

ttracted to sorne nondegenerate d.f. M (written FE D(M by four statistics

hile no condition was required on F. This empirical and unified appoach1

ncludes detection procedures of the extremal law of a sample and statistica

ests. In bath cases, one has to determine the limiting laws of this Familyi

f characterizing statistics of the Extremes (FACSEXT) which is our aim.

he reader is referred to L (1990) as a general introduction ta this

aper and ta Leadbetter and R6tzen (1982) and Resnick (1987) for detailed

eferences on extreme value theory. However, we recall that if F lies on D(M

here M is not degenerate, then M is necessarily the Gumbel type of d.f.,

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

r the Frchet type of d.f. of parameter y>O,

-yr(x}=exp(-x )X[0,+400 [(X), for xeR,

r the Weibull type of d.f. of parameter y>O,

y(x')=exp(-(-X)r) X l - oo , O](X)+(I-X]_oo, O](x, XER,

here XA denotes the indicator function of the set A.

Several analytic characterizations of D(.p)::: Ur>o D(.p...J, D(lp)=UY>OD(lpy)'

(A) and r = D(A) U D(.p) U D(lp) exist. We quote here only those of them

lvolved in our present work.

leorem A.

Karamata's representation (KARARE).

FED(.p ), y>O, iffy 1

.1) F- 1 (1-U} ::: c(l+f(u u- l / Y exp(J b(t)t-1 ),O

- 73 -

) FED(~y)' y>O, iff XO(F) = SUp{x, F(x)

- 74 -

1( 4} = y .n,n' = (Y y-l)n-L,n n-k,n

-2(7}=(Y l -y k ) A (l,l,l) jl n- ,n n- ,n n T (8,v}=n-v

(y l -y k }-ljn n- ,n n- ,n

y

for

"Csorgo and Hason (1985), L (1989) imposed f=O. In

- 75 -

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 constants

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

F=(f,b) D(A) iffi -1 1_1(1.4) G (1-u)=d-s(u)+f s(t)t dt, 0

- 76 -

r a combination of (1. 7) and (1.8kn (l), ... , .kn(p),

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

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

.p 0, i = l, ... ,p,

here p is a positive integer, kn(i) is a sequence of integers such that for

ach fixed i, ISisp, kn(i)/n~O and Uk,n is the k th minimum among n

ndependent r.v. 's uniformly distributed on (0,1) (see (2.2) below).

et r(o), r(kn(l), ... ,kn(p and r(O,kn(l), ,.kn(P be the class of all

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

respectively. Each of these three conditions is quoted as a

egularity condition.

related problem consists in replacing the centring sequence by l/r for

rD(~r)' It has been studied by Haeusler and Teuqels (1985).They obtained arneral analytic condition for the asymptotic normality. This problem willi~t be considered here for space reasons.

Each single statistic is systematically treated apart in sections III,

J and V while the ratios are studied in Section VI. section VII is devoted

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

~to unified invariance principles based on the same Brownian bridge. We

terefore begin to describe the structure of the limiting laws.1

Description of the limitinq laws.

i " "1 Csorgo, Csorgo, Horvth and Mason (1986) have constructed a probability

lace carrying a sequence u1 ' u2 ' ... of independent and uniform r.v. 's oni1,1) and a sequence of Brownian bridges {Bn(S), OSsSl}, n = 1,2, ... 8uch\~t for aIl v, O

- 77 -

11/2 1 1/2-u -u

(2. 2) sUP(1/n)~s~1-1/n n (s-Vn(S)-Bn(S) /(s(1-s)) = 0p(n ),

where U (s) = j/n for u. ss

- 78 -z

= {(n/k)1/2.J n B (1-G(t dt } dt}/R1(X) ;n nxn

1/2'n(2,.) =-(n/.) Bn(./n), n2:1;

z z'n(3,k,l) = {(n/k)1/2J n f n Bn(l-G(t)dt dy}/R2 (Xn ) ;x n y=n(t)=nu1+1,n/l , n2::l.

le prove in the next sections that each Tn(i) is asymptotically one of

:hese r.v.ls or a linear combination of them. Their asymptotic laws are

Lescribed in

~heorem C. 1) For each n, the vector (W (k), Wn(l, with Wn(k)=

Nn(O,k,l), Nn (3,k,l), Nn (2,k and Wn(l) = Nn (3,l,1), Nn (2,l is Gaussian.:f further l~+OOf k~+oo, k/n~Of l/k~O n~+oo, and F e: r, then Wn(k), Wn(l)

fonverges in istribution to an R6 -Gaussian r.v. (W(l), W(2 where W(l) and

r(2) are independent vectors with the same covariance matrix :!,

[

2(r+l)/(r+2 )

3(r+l)/(r+3)

-1

6(r+l) (1"+2)

(r+3)(r+4)

-1

~d!

6

-1 J , if Fe:D(A) Uo(~),i

i1~ere the symmetric matrices are given only in one side.1

ir> Let l be fixed, then for aIl x~, ~(En(l>SX) converges to

- 79 --

o elsewhereWe do remark that Var (W(l is obtained for FED(A)Uo(~) by putting

r = + 00 in Var(W(l for FED(~r). This fact occurs almost always in thispaper.Then, throughtout, for any expression depending on rwith O

- 80 -

(with n = 0 if FeD(~) U D(~), n>O if FeD(A, then

1l(k,l)k1 / 2 {T (2,k,L) - }l(k,L)} = N (O,k,L)+o (1) 9 N(O,02(r),n n p 0

where 02(r) = 2(Y+l)/(Y+2), O

- 81 -

(3.0a) 1-y (o)=f(U +1 )-f(o/n);n o,n

T*(2,k,l.) = R1

(X )-lk1/2(T (2,k,l}-1l(kn n n

(3.0c) Nn (1,k,L)=Nn (1,k,L)+e1 (Y)Nn (2,k), e 1 (Y)=(Y+1)/y,e2(Y)=Y+1,e

2(+00)=1.

Theorem 3.2. Let F~(f,b) E r and let (k,L) satisfy (2.4). We have

1/2N (1,k,l.)+e 2 (y)k y {k) (1+0 (1) )+0 (1),n n p p

1 with Nn (l,k,L)-N(O,Oi (y) = (y3 + y2 + 2)/Y(Y + 2), O

40 by what is above.p

- 82 -

one proves this corollary from Theorem 3.2 by using (2.2). Let (1.7)

Je satisfied, thus

{1/2y (k) = nk- 1/ 2 (Uk

1 -k/n) (u fl(U (1+0 =N (2,k)+0 (l(u fI (un)n ,+ ,n n n p n p n

1+0p (1 4 0, where nu /k 4 1. Now, if (1.8k) holds, thusp n p

[f (1.8k) holds, thus

Ik1

/2

Yn(k) 1::S0p(l)+un f i (un) X Nn(2,k)+k1/2If1 (k/n) Isup( If2 (k/n) 1 ~1

1

)f2 (Uk +1 n) \); ,ile should remark that ~~~ uf'(u)=O is a fairly general condition since it

fo1ds whenever the limit exists. Also limiting laws of Tn (5)=Tn (2,l,1)

1

}re particular cases of results stated above.J

11ROOE'S. 'rhey largely use technical results in L (1990). Use (2.1)

r\2.3) to get R

1(X

n) -1k 1/2 (Tn

(2 ,k, l) -ll(:k,l) ) =Nn

(0, k, L) +Zn1 (k) +(l/k) 1/2

with

z= R1(Xn)-1f-n{an(1-G(t-Bn(1-G(t}dt,

xn

r~re an(s) (resp. rn(s = n 1 / 2 (un (S)-S) (resp. nl

/2

(s-Vn (S ;

i G-1 (1-U 1 )1 (0) = {(n/o)1/2 f o+,n Bn(l-G(t dt}/R1(X

n),r1 G- 1 (1-0/n)

hdj1

and

z ( c. ) = {n2 c.G- 1 (1-U )

J ~+l,n B (l-G(t dt1

}/R (x ).G- 1 (1-c. In) n n:

(3.3) "ri ->l, lim IP(L/-n)SU S(-1)/n) = l.n4+00 L+1,n

For convenience, if (3.3) holds, we say "for aIl >l, one has

We shall treat each error term into statements denoted (Sl.3),(2.3),etc ...

(Sl.3) Zn3 ~ 0, for L~+oo.

First, we have by (2.2)

-1/2 d(3.2) nL (UL+1 -L/n)=N (2,1)+0 (1) ~ N(O,l).,n n p

so that nUL+1 /L -l> 1 and hence,n p

L/ (-n)SU 1L+ ,

~(-L)/n with Probability as Near One as Whished (PNOW) for large values

of n". Hence,

(3.4) G(t)Sl-l/n, uniformly for x ~tSz ,n n

with PNOW as n is large. Secondly,

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

-1G ( 1-01.)

If_oo

-1G (l-a )

h1-G(t)h(t)dtl~suPassS1Ih(s)f-00

Ip(t) Idt.

combining (2.1), (3.4) and Lemma 1 yieds for some v, O

either GED(~r}' O

b) Let GED(A). By (3.7), GrED(A) with xo(G) = Xx(Gr ) = Yo and thus(cf. Lemma A in L (1990), for any tER,

-t(3.10)(1-Gr (X+tR 1 (X,Gr /(1-Gr (x ~e , as x~Yo'

(3.11) (l-G(X+tR(X,G/(l-G(x ~ e- t , as x~y .o

Combining (3.10) and (3.11) imp1ies

(3.12) (1-G(X+tR1(X,Gr/(1-G(X~e-t/r,as X~Yo.

If for a sequence x ~y , one has R1 (X ,G)/R1 (X ,G )~v, Oo, which is absurd because of (3.10).

Whence, 1im inf Rl(X,G)/Rl(X,G )~r. simi1ar1y, one gets 1im SUpx~y~~ r ~o

R1(x,G)/Rl(x,Gr)~r. Fina11y R1(x,G)/Rl(x,Gr)~r, as x~Yo' which combined

with (3.9) proves Part ii).

Lemma 3.4. i) If eD(A) U D(~), then Rl(G-l(l-U is Slow1y Varying

at Zero (SVZ).

ii) If FED(~y)' then R(G-l(l-U) is Regu1ar1y Varying at

Zero -1with exponent y -RVZ).

We return back to'our proofs of Theorems. By (3.5) and Lemma 3.2, 3.3

ii Proof of Lemma 3.4. a) Let FED(A) U d(1'), thus GED(A) by Lemma 1 in LU~ (1990). Next, Lemma 3.2 above and Lemma in L (1986a) yie1d for A>O, ~~1,

1 (3.13) {G-l(l-AU)G-l(l-U)}/R(Gl(l-U~-log A, as u~O,

and Lemma 4 in L (1989) imp1ies

\ 3.14) {G-l(1-AU)-G-1(1-U)}/S(U)~-log A, as u~O,

1 ~:~::)S:;:_~;1:::)::;u~e::n::o:s in (1.3). Hence'j so that R(G- 1 (l-u is' SVZ and Part i is proved.l,

III', f d (b) Part ii) is easi1y derived by Formula 2.5.4 0 e Haan 1970) (cf.

~ Lemma 4.1 below) and Formula (1.6) above.li\i'1li

1Illi!

- 86 -

and 3.4, ( ~; - 16) i Z;_ c.- i :c, Op ( kdJ

where we have taken (3.8) into account. ~his ploves (51.3).

(52.3) Z_~ -, 0, when t i8 fixed.Li~' P

we have

(3.17) Z ( 2 )'-1 c.. ,L J

with

Z ( 1) =n,::

also has G(t)~1-1/n)(one

-1 (1 ' ..1/2 ~G -Ljnj

(n/k) {! lo_(l-G(t))-B_(l-G(t))!dt}/R~(X)., 1 -- l.J Il . 1. il

X n

~) by same arguments used in0 __ ( k1J

I( 3. 18)i1,!\'lhich i51

1

:and

IZ ~(2)Tl..:'

7

(' /' \ 1 / 2 r~ni' ~ . ~ .. ' . ..,.= \(11 hj t... IC~ l-L-(;{LJj-E~~li-'C::'{tj;!" -, . 11 1 i '

G - (1-L/n "-

k,v Tleor-ern ('1 ~ - i

l/'j 2!~1) ;-:.~ LJ--"(t//"_ll) :~:--u. ::{/.. t)/.i-~)_-=~_~

~:~:~':,:",:~':i"3 ,,'" ,. '."JI' '.", llll'-',+.i j,' (" Jin hol::; wi.r.h PNOW [or laIl]E'. ,1 ... ~

hus, with PNOW for large values of il and,

\j''- Co .

;3.21)

1

'1 '?Z,.:,(2}::: {(n/k)~/~

.Ll_.'

*= '] , 2 'J: L,rd l

t (l /.\. )r n ,'o. (l-G(t) j-B (l-G(tj) idt}/R., (x )

,, Il '-, '1 r-, "') L _

G _1 J~. { t j dt: /J. / 1, ir (Z". (k) )40hl.

in aIl cases and fic011y (53.3) holds by Markov's inequality and (3.23).

(54.3)' i,"t/k)1/2 Z (l') ~ O.nI ,'. P

This ls proved axactly as (53.3) when L~+oo. When L ls fixed, one uses

(3.20) instead of (3.2j and the proof of (53.3) ls valid again.

(55.3)

a) Let

Lk- 1 / 2 z il) ~ 0n2 \ ' P .FeD(~)' thus GED(w ).y , . y By Lemma 4.1 below, Lemmas 3.2, 3.3, 3.4

above and Theorem Ci

3.25) Lk- 1 / 2

- 88 -

Z .2(l):::lk- 1/ 2 {(1+0 (l)(y -z )jy -x )-0 il'}'n p 0 n 0 Tl p\

(y -z ) /y -x )1~ no il' 0 il j p-'

rhenever lim In this specia.l case, one can choose

r ,~,

SUD( I y -z 1 ly- - z 1 i/iv -1{ 'j~ 4 Ion' Ion . \~o n'

1/-,.-

proof of Lemma 3.5 By Theorem

- 89 -

C, N. (,k,,\) is!l

a normal random varjable.

Its Val iance i:='" Z._2 1 !. t - i~ il

i.'3.32',i S !U\J=! {nJ-K)!...... \ , . 1

~, '. Xil

f\ (l-G( t) .1-G'\' S J\ 'l' - (1-'-;( L 'Jo j' ('l-Gi ~ i -, J"dc:dt.l, ,. .. \ - - '\ " -- 1 1 - 1

.!

{~

FED(~ }. Use R1 (X }~(y -x }/(Y+l) and gety non

1/2 - - 1/2(k}-k {(y -x }-(y -x }}/{(y -x )/(y+l}}~(y+l}knon 0 non

- 90 -

is the closed interval formed by k/n and Uk +L . Thus,n

U1/2 -1 k+l,n -1 -1/2

(3.36) k s(k/n) f s(t}t dt=(l+o (l})+o (nk (Uk 1 -k/n).k/n P P + ,nAIso,

(3.37) k 1 / 2{S(Uk

+1 }/s(k/n}-1}=k1/ 2y (k}(l+o (l}}+o (nk-1/ 2 (Uk 1 -k/n)}.,n n p p + ,n

(See (3.38) - (3.40) or more details).1

b} let FeD(~}. This case is exactly the preceding since GeD(A} and (1.4)i

rOldS.

p} Let1

\(3.38)i

X({y -x }/(y -x )}-1}.o non1

!~ow, by (1. 6 ) ,

1

1j(3.39}

11

y - xo n(1 ( ' l'en ,-& +1/1' ~- 1-(1 (k,,(n+1"' K l -U 1 n -.l::: -::: +1' -ll-n k k+1 11 n k

1 Y - xo n

& +l/YU, ~ ) fi -l,

K+.l,n

11

Nhere s = sup{b(t}, tSmax(Uk 1 ,k/n)} ~ o. Since nUk 1 /k ~ l,: n + ,n p + ,n p1

1

t3.40}1

1/2 -1 -1/2e (k)=(Y+1)K y (k}(l+o (l)}+y (Y+l)nk (Uk 1 -k/n)(l+op (l}}.n n p + ,nJ

1

~ow (2.2),(3.36),(3.40) and Point b) just below complete the proof of Lemma1 .

13 . 6 .

~e return to the proof of Theorem 3.2 By (3.1), (Sl.3), ... (S6.3},(3.41)!

i xnrn(2){Tn(2,k,L)-Cn(2)}=Nn(O,k,L)+kl/2{iJ; l-G(t); n

~Sing Lemmas 3.2 and 3.6 gives

1i

dt}/R1 (X }+o (1).n P

- 91 -

from whicl1 the characterizat i.l :LS (,1) v ions. 'l"O CC>lpU'c",'

bridge, and an easy calculatian yields

(3.43) !.t(N_(o,k,i)N,)) -, -l.Il 11

This and i.emma 3.5 Sllfflc-

complete,

lV -- Limit laws for A. (l,k,').~-.-----~.- ---_.~~_. ---~-------------- :, ,-_.-

j';-.

or ta L(1990).

"

j J whenever F~D ( .

_L -~r.~j\)LJ

-~ ./ .L

; )

d ..ijY

'

1.17 . '}. ')"D' o_f Q~P_ Haal-.' ('19'-1_-'(j ') ,'-, WC -- ~ ".- _.' _ _ _ _~ '-- b!.l() ...L.U~ Ll.lc.C-

1 Y0jc~ ~ = ':: .;=- D ( i\ ) ~-.;. ? ( ,. ) = 1 - r j _. c.; _. ( ;. J : l ,~- ',_ .. j \ '\)\ .l ..~

1

IAPplying this p tiIliS 'Ji V'=-':-;

iLe"@" ';, 2, Let t'-,i

Cl,.l c:. 1

- 92 -'roof of Lemma 4.2.

i} is easily derived from Part . . illJ of Lemma 4 -j above and Lemma 8 of L

1990). To prove (ii), put

dt dL::..a,b

yo

r-I

Jr

c~~.J l-G(t)

x a - 1 YI Yb - 1

zr

.... 1.,z

r./

z4.1) m(a,b,x) = J

x

here df.J.a,b= dx1 .. dx, :xdy_, ... dYb ~ = dLiaXd~ and.-l. 1. -.1 JJ

i\ -mC1

yo

m(C,b,x) =Jx

Zr

m(a,O,x) =.Jx

v~o

r.J y ...1

zr

. . '.Jx ci-l

1 - G(t) dt dD. a

traightforward manipula.l_ions yi.,,,,ld fUl any p2:2, for z

- 93 -

where -, (k, l }~ z_ z= "-.Lf li r 11 ~ '-,-,

llX .1 -'.J 1. -G ( l- Jx y

n

dt '- R (x2 111 in

probability and u~(r) = 6(.r+l)(;

- 94 -

*"ii) A (4,k,t) ~ + 00 (rep. -00) lff N (-00) 4 + 00 (resp.-oo).n p n }.)lemark 4.3. 'l'he cl1aracterizatioDE: are 5.dentical in 'rheorems 3.2 and 4.2.

le pointed out in section III that we have the asymptotic normality

rhenever (1.7) holds. The following examples concentrate on the case where

lf ' (u) has not a limit as u~O.

:xamples 4.1. Let f(u)=u sin(l/u).

:' exists and uf' (1..1)=1..1 sin(l/u) - cos(l/u) does noL converge as 1..1~O.

;ut if there exists a sequence of integers (Pn)n~l such that (l/Un )-

:np ~b, -n

- 95 -

~ /~ r- Z - Z Z zQ ~(L)=(n/L}l L {Jf 1IJf n B (l-G(t})dt.dy}-{1 n1' n B ,(l-G(t)n .1. 1 , il ,) ., 1.1

'- X n y x n ydtdy} ~j /R~ (x),

_ L Il

l/"r' Z Z-~ '; ) ._', " ~ r f n r n B ' 1 - r ~ ,',.!, ~ l' ~IL/" l\J l '- -lC'\L-111 c!.. '" ,,~ ../ rI .. ..- x y

n

and

Zr li

,j J.

Y- G(t} 't - }! / ' .d ,ey J' R~ { X" j ,

_ L !..!

( , l ,li2 [' f Zn rZ.n r ( ~ ( t ' ) ( 1 ..... ) ) \ ' d -) l ' )=- ni ''-1 .J - 't':::< .L-G 1 -B \ -G( L. J Cli y . R~) ~ X ,,x y n n _

- 96 -

:54L 1/2( t. ( \. p, ) - t ( \.) ) 2

1. n 1 12 -1() x(k / R 2 (X1 )/R,,(Z ..... ) .R, Z J ~ u_ n

y Lemma J jn L (1990) and Lernmas 4.1.2 and Formulas (3.13) and (3.15)

(t (l/y}-t. (Ln n

R (~ )~(loo \~ Ra that ~ Q 3(3) ~ O. Bv the same arguments ~n* l'3')~O2 -

- 97 -

"*concludes that ~ ~3 (2)~O and hence (51.4) halds.

( 5 2 . 4) Qn1 (k j

We have

~O.

P

dt dY}/R.,(X )=:Q l(k,l)+Q ~(k,2).G n n D..l

It follows that for any ~>1, one has with PNOW as n is large,

* *(4.14) 1Qn1(k,l) Is Qnl(k,l); !Qnl(k,2) ISQnl(k,2)

!with

, (4 1 ~) lE * 'k ~. - 3 ' ,-1. 1 / 2 " "/') t (' k) ) 2 / ( ) 2i .:J ~1 l ,i. ) ::: fi: (:;V ) .. 1 \1 (K /', - n /', RI Xn '1

1

land1

1 t (kjl,-) -t (.\k) R~ (X , t (L p, 1 G~ f~) RA (X _ ,G)IL Il i. TI n 1., L ' li

-l{l-(i~~(k }-}---->~ RI (Xn

, Gi

/;;:) .._-->~~ l (xn

,_~ 1 z!

O'IC1

(L,l)S3f1:(y)-1(.\l/k)1/2(t (k/A1)-t (k

Il n n

(t (L fi, )-t (L /R (X ) 2n 2 il 1 il

llin (3.24) show that that il Q_~ (k,l)-"O and a combination of these samei Li."':

!arguments and Lenuncts 7 .3)](': .-, in ~_~, (1.390) and Lmrna.~) ) ,~. ~n(/ --=:"J,c;j~()ve!

-k * "Ji;lensure that ~ ~11(k,2)~O. We conclude tbat Qnl(k,I)+Qnl(k,2)tO and

1

Iithus, by (4.8), (52.4) halds_

1 \ ( S 3 . 4) ( LIk) 1/2 ~1l (L) pO.

1poe has with PNOW as n is large1

\(4.17) 1(L/k)'!./2%1(L) ISQn1(L,l) + Q:1(L,2)

Ikith\(4.18) [[

\1

1

land

li * -1 1/2 '" 2\~4.19) [[ ~1(L,2)S 3;>e(y) (L/k) (tn(L/;"2)-tn(LRl(Zn,G1/2)/Rl(Xn)

l~here ;'1>1 and either ;"2 == 1'1 ( for L~ + (0) or 2

is taken large (for L

i'il1

*Q 2(L,1) -+ 0,n P

i

i:ed). By the arguments many time~l ;~.sed above, one has lFQ~ l C! ,1 J'O and

;~nl{L,2)-+0 and eons~qUentlY, (l/k)-/LQn1(l)=60 whenever

\ . 20) PI1

( 1, '.S ) = ( li k ) "".) R ~ ( z. }! R ~ ( x. ) -+ 0 ,1 J. E ~ U

~r aIl 6>0 and for aIl d.t. FEl. But, G-"'D{A) U D('f') and by Lemma 3.4,1

l 21) ,-; (l .

- 99 -

:\11 these conditions on (k, 1) a:u~ i'i1.:pJied by (2. 4c). Thi ~ complf'tes the1

1

>roof of (54.4: j

~t remains ta praye this important !esult.

! ~~~~~_~!~i?_~_~~... Id:~t. F\;::::r-': y.~--++::j k/ll--~lJj I.j/]{-o;dJ J tl1e11 l'lJn (3,k,L) is a (;aUssia.Il

:.V. with mean zero snell that li rlj l'3 1; i)2~6R {x z )/R l'X }2,~, 1,,1"n ,"', 4\ n' n 211 2\'11

ot} give;=;

(4.30)1

Il~,'he second term in brackets is

",dDi

~ 1

,)

,

rith~=q-Pi~

z Zu=f ne n I-G(t} dtdy. Finally,

q y

k1/21{T(k,l)-T(k)}/R~J(X )l lk- 1 / 2 (t (l)-t P,k))R1,(t (l)/R~.(t (k))L. n n n n ~ n

! ~ 2(4.3) s (3) "-'6R

4(x , z ) /R

2(x.) .

1 n n n li

~emmas 4.1 and 4.2 thus complete the proof of Lemma 4.3.

~heorem 4.1 is proved by (51.4), ... , (54.4) and Lemma 4.3.

the first part of Remark 4.2 follows from the lines just below (4.26).'1

1rhe second part of Remark 4.2 follows by remarking thatI-I'[4.32)

1)

- 100 -

lor A>l, with PNOW as il is large. Both terms at right tend ta zero1

IxactlY as in (4.24) and (4.25). Remark 4.2 is now completeJy justified.

l? prove THEORE_M 4.2 -=- remark tllat t by 'l'heorem 4:.1 t!

:,,4.33) An*=Nn(3tktL)+kl/2{glr~nrZnI-G(t)dt dy - ~ J,Zn j"Zn 1-G(t) dtdyjh '> ,/ k ' ,

x v X y.. ~ n

N (3, k ; l ) +Q. ~ (k) +c:..J (1) .n ll~ p/R" (x.. ) +0 (1) =;G It P(3.41);

il',; -:-- . r l' '11i'-""; d.0

14.34) Qn2(k) = k1/2{(~J~n J:{n 1-G(t)dt + kl/2(~ J~nJZn I-G(t.) dtdyj

,1

x y x xil il n

/R2

(Xn

)=(2k l / 2 ",(;) )-1" (k)2(1) )+H(y)-l( (k) (1+0p t :..));

~ere E (y) is defined in Lemma 3.6 by which'n

1. 3-) * .. 1/2, )\-1, .' .~,' ') ( )(1 ( .. ),2 (' , ,l,li: ':J A =~ K 'H. \ y 1 ( e 1 U iN.. \ L , K ) + e~, t y N - co + 0 .i. J +e 4 ) ) 1'111

\ -, l) .i n l.! L n p

lit remains to compute ~N (4,k,!j2. Remark that h(~ ~\=(1-~(t\1{!-k/D\! n -'n' ~ 'CO ",.- ./-,

!or x St::;;z (see (4.28)) aJld tllus! n n! n. ( ) rZn rZ'n 'l"l(t t) ~.\(4.36) u:.Nn (3,k;l)N1._... (2,k)=- n/k .J,,' , dt dyjR,,(X,)4 - ,''-n 1 y n L d1

riS easily implies that

114.37) lLN (4,k,L)2=u~() )-2e-:-(l)2, v~('(')=5.. n .J ..' .:'!

~ have proved the first part of Theorem 4.1. The characterizatiuII i8i

fViOUS now when we remember that k-1

/2

Nn (-OO) = f(Uk+I,n)-f(k/n) Po.1

~ - Limit laws for C = y -y =z -x1. n n-l,n Il-K, n n

i!

1 We assume from now on that the regularity conditions (1.7) or

i.8k,l) or (1.9k,L) hold for for sake of simplicity. But il will

ppear in the proofs how optimality results may be obtained. Notice

hat (1.8l) or (1.9l) is require only when L~+oo. Put C =z -x .! n n n

- 101 -

bheorem 5.1. Let F~(f,b,Er' satisfying the regularity conditions. Lett-------!(2.4a) holds.11

~) If FED(A) U D(9), theni1 - ~ dle -C ')/R (z )=z -z )/R (z )+0 1 l'J' =-logE1".\'l)'+Op(1) ~ -logE l,' lJ'r n n 1 non 1 n p' _-,

~hen is fixed and

Il/2 (c -c )/ R1 (z ) =l 1 / 2 (z__ -z )!R l ( Z ) +0 _ ( 1 ) =e 1 (y ) N (2, L) + a (1),1 n n n Il]) li unD

i wwhen L-1o+C and L/k-100.1

2) Let FED (1'.,) .

1 a) If'r>2, then

is fixed andl/l(r+l)(l-E(l) ), when L

'[e -C )/R1

(Z ) =(2 -z )/R~(Z )+0 (1)=(r+1)(1-E (L)l/Y)+a (1) ~,n 0 0 0 Il .i. 0 pop

i

i

!.1/2(C -C )/R~(Z )=ll/2(i -z )/R1

(Z }+o (1)=e1

(r)N (2,l)+a (1),i n 0 l 0 n Il Il P n P

J wheo L-1o+w, L/k~o.i1

1 ~ _._. b) If 0

- 102 -

( 8\(nv

T (8)-1_C,(8 = -N (2 L\+O fl)' . N (0,1) when L-++(O, L/k-+O.l J n Il n ' J p\ il

FED('!') Y>2, then (nuT.(8)--J_ c (3) l'Y+l) (l-E(L)l/;')' when L isy , :1 11

.xed and

Y*(8)(nv

T (8)-1- c (8=e1

(J ")N (2,'}+0 (1) when L-++C,n 11 n n p

, FED(~) r>2, then V*l'8\ = k 1 / 2 ,- (x )r ' 11' 110 as n~'+''-', k~+:.:', kjn-'>O, L being fixed and1 nI{3,1/2)-+O as n-++ oo , L-+ +

- 103 -

5.2b) v*,g)((: -C ':=e_f?)N (2 lJ'TO {'l'n' ,. 11 nI J' n' p J

~or F-='D (',' ) 1 O

- 104 -

lready studied includiny the majo~ elemen~ of the FACSEXT which 18

(l,k"j.

l - Lin; i t i aw for ratios from the !,'Acsr;XT.

Case of T (1, K, l) .n

1

~ .._--- - .-.- -------------------

4: (; + 1 ) U' +3) U + 4 )

2J3

+101,2 +32;--+24

( 2 . 4: ), then

t.}} d. t

-~ (2N. (0, k, l) -K, ( 3 ,k, L) ) +0 (1).

11, li P

0

- 105 -

::: T(k) + T\'k)k- 1 / 2N (3 k l) + 0 (-1/2Tfk )\n " P \,!i6 . 4) An ( 1 , k , L )1

la that r

16 . 4) A (1, k , L) ::: T (J

- 106 -

:;.11) 2N (l,k,L)-N (4,k,l)=2Nr

(O,k,l)-Nl'(3,k,l)+! n il _i .i1

Nn

(2, k) ,

i~ich, together with (3.43) and (6.10). permits ta compute 0s(r).1

Case of Tn (3,k,L).

~ already noticed in Theorem 5.1 that when zn-xn intervenes, the1 r"

tntribution of in(normal or extremal) domindtes that of Xu (normal).'-'>-'

tre lgain, this is the case except when F"=D ('I-"y) where each of zn and

1 may get the better of the other with possibilities of a mixture oft

bth. We begin with.11

~rnma 6. 1. Let FEr' . Putit'Ch k;.+., L/k---O, k/n~O

n i 4 J', 11 \

; 1. ls

__ -1/2( )'. .-1- K Z.- x. R, lz.,G} ,

il li .i. Il

fixed in 0 (4) and 1.~+oo inn

'5) 1/2 '4',on ( =1. .on l J..1

Let F~"D('J ), r>O.J

-if"

2 )

1'.

v (7'; ('~ (7) - .t ,1 ,'.

-' -1v (8) (v -rr (8 'u)n n n"

C '1 ( 7 ) )

c __ ( 8) )il

v (9)(T (9)-c (9.n 11 11

li(F2) by replacig v;1(3l, V. (6) and v (7) byn . D n

[(3L v:1' (6) and v:(7) respectively.

1 1 .1.1

1 obtain1

leorem 7.1. Let =(f,b}:r and (k,L) satisfies (2.4). We assume that thej------,~ularity conditions hold.

i If FED (

- III -1,

,..,,** ~-....p (0) (al' li"" -Ii. V.) aJe 'i.sY]lltoticallyindeI,endent witb cespect~lvern1

limiting covaliance matrices.1

i!"

1!

, ' lL 'Z

* ( ') **12 ( ,'-' ) 1 1 and 2: ( ct) j u -J 1= I-! '-' :J ! = l,

L1 1

Ci 'J .1) " .- :.: 2 ','11

U 0 0 )

fith tH denoting the transpose of the matrix M.

1

=(2:(Y>21,

respective

:1

lf'~-r.v) and(an

with

2

=\ (Y>2+o (1) 1L.n p*then T (r>2)n) >2 1

zf' l, Cl

J +:r ~2}

l' 1 '. F + 1 ;, .: }- -+ 3 :. ":} +:"1 .* 1 j' .,. 2 , . 7 Y ,:r - ; )

(Y>2)= ,1 2 (,-,-----1~ - ----,---)'.,.J l ':.0""3'(]-"'1)

!

,) j If E""'D (J', J 11 t

!h** 8 ,*~ (r>2)) 1s an ~ -Gaussian vector such tha~ 1 (1">2)~~ ~n

1** .i~ (1' >2) (an [pC> -[.'. v.) are !;ymptotical1y ir!.dependentLXI.1

Limiting covariance matricesi

".~

21"

. r -+- 1 . -: ,y+ 2 ;.

2 jO

z

/....- -1}" + 1 H Y + :1 H

2+:r + Z

J

- 112 -

.: Y+1;'.' y

--.: 01 ~ 1 :', or

OK

... :r >2 ) = 'l' . 0,'0 ::, ~ 1 ,,i

1

1 21"1 Hl1 2l''-(I +lj/Y

3 2Y +j' +2

--- :2: - -.

:r 31-;.~ +:.:t~ +~'" +2

zr (y+ z;

3 Z2:1" +,ty +lSY+IH

2

-2Y

:.J Z}- -t-) + ::

d Z2.1 ~!} +lH}+18

-2

.'1 ~ 2~} +11Y-+1.]"+7}'+12

- 2 -z

If FED{ 'l,l

Y>2, then T*{Y

- 113 -*-J:

2: (r

(r

- 115 -

13,743-756.

Hall, P. (1982). One SOffie simple estimates of an exponent of regular

variation J. Roy. statist. Soc Ser.B, 44,37-42.

distributions J. Appl. Probab., 15; 639-644.

Hall, P.1i

1

(1978). Representation l imi t th(~orl2ms foc extremC' value

G.S. (1986a). Asymptotic behavior of Hill's estimate.J.Appl. Probab.

, 23, 922-936.

G.S. (11986b). The weak limiting behavior the Haan-Resnick estimate

of the exponent of a stable distribution. Technical report,

49,LSTA, Universit Paris VI.

~, G.S. (1989). A note on the asymptotic normality of surns of extreme

11 values. J. statist. Pann. Inf., 22,127-136.

l~. G.S. (1990). EmpL"icd..l Ch.C.c'l.>:=rization of the extremes 1: A" .

ramily of characteriziny statistics. Technical report, LSTA; Paris

VI.

Proce55es. Spr in,"'. 1 New York.

snicK; S.I. (1987). ..->crTP,c,c,Ee "O"."""S.

Extremal theory st.OChdstic(1988).

processes. Ann. ?robab . 16, 431-478.

adbetter, M.R. and 80c}~Zell; H.

statist.; 15,1174-1027.

- 116 -

MODIFICATIONS

LA PREMIERE VERSION DU PREMIER ARTICLE DE CETTE PARTIE,

'ESTIMATEUR T (3) A ETE MODIFIE PAR LA SUITE. LE LECTEUR SEn

RENDRA COMPTE DE CETTE DIFFERENCE. CEPENDANT, LA LOI ASYMPTOTIQUE

'OBTIENT AXACTEMENT QUE CELLE DU VECTEUR TN(G}.

- 117 -

EMPIRICAL CHARACTERIZATION OF THE EXTREMES III :

LAWS OF THE ITERATED LOGARITHM FOR THE CHARACTERIZING STATISTICS.

Gane Samb LO

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

)stract. Let Xl' X2 , ... be a sequence of identical and independent random

3riables with P(Xl~x)'=F(X), l~x

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. INTRODUCTION.

~t X , X , be a sequence of i ndependent random vari ab les:1. z (r.v. 's)

th common distribution function (d.f.) F such that F(1)=0 and for

lch n~ 1, X S .. x denote the order statistics based on the n first1~n n,n

: these random variables. It is shown in L (1990) that the following

:atistics

-1/2(1,k,L)=T (2,k,L)A (l,k,L) ;l n n

-:1. i=k j=i .( 1 , k , L) =k Ei =L+1Ej =L+ 1] ( 1- 6 i j / 2 ) ( Vn _i + 1 , n - Vn - i , n ) ( Vn - j +1 , n - Vn - j , n )

( 2 k L)=k- 1"i=k -(V -V )., , ""'i=L+l 1 n-i+1,n n-i,n '-"l.J

(3,k,L,"l.J) = n (V L -V k )/T (2,k,L)1 n- , n n-, n n

-:1.(4) = V ; T (5) = T (2,L,1) ; T (6) = (V L -V k ) T (2,L,1);

~ n,n n n n n-,n n-,n n

\

1 (7) = (V L - V k ) -2A (1, L, 1 ) T (8) =n-"l.J ( V L - V k ) - 1;n- ,n n- ,n n n n- ,n n- ,n

\(9) =(Yo-Vn-L,n)/(Yo-Vn-k,n)

Inlv defined when V o ' log(sup[x,F(x)

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n them. Generally, asymptotic normality implies laws of the iteratd

~garithm which, from the statistical point of view, determine the

trong accuracy of confidence intervals derived from the asymptotic

~nsi stency.

This motivated us to determine the strong laws of the iterated

~garithm related to the central limit theorems given in the cited

3per above. Our best achievement is the characterization of the d.f.is

ithin the domain of attraction, for which such laws hold.

Our results are closely related to results by Haeusler and Mason

1987a-b), Deheuvels-Haeusler-Mason (1989), Kuelbs-Le douxC19S7) etc

The reader is referred to Leadbetter and R60tzen (1988) and

rsnick (1987)

P (1990,1991)j

for detailed references on extreme value theory and to

as a general introduction to this paper. We only recall

n..,)oo or equivalently (X -b )/an,n n n

the

is

M(x) as

ddistribution (..,) to M. It is well-known that M

i~at F is attracted to sorne non degenerate d.f.M (written FED(M iff1

jere exists two non-random sequences (an>O)n~1and (bn)n~1 such that

>r any real x, F(a x+b ) ..,)n n

;i _Inverges ln1

!jCeSSaril Y the Gumbel type of distribution (A(x)=exp(-e-x), XE~,

.chet type of d.f of parameter y>O (~y(x)=exp(-x-r), x~O) or the1

'ibull type of d.f. of parameter y>O (~ (x) = exp (-(-x)Y), x$O).y

We now classify the elements of the whole domain of attraction r

First define the quantile function of Four convenience.i

l(u)=inf{x, F(x)~u},

~ the endpoint of F

-1 -10oD(~y) and

ke Theorem A and Formulas (1.4-6) in L (19911

1

1

i1

-:11.1a) FED(A) ... GED(A) and G (1-u)

- 120 -:1= d-s(u)+fu

-:1s(t)t dt, O

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K2) 1im inf 10gkn/10g

2n > 2.

0 .... + 00

K3) k(n)~kl(n)~+oo. t(n)~tl(n)~ , t(n)/n~~ ~O. k(n)/n~~ ~.O as n +00.n n

K4) 3 p >0. ~ O

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1 the resu1ts.

heorem 1A. Let F=(f,b)~r. Let (K1-2-3-4) and (L1-2) be sBtisfied.

"'en,

3nd

k) 1/2 (T (2,k,I.)-11(k,l.

n

B.S=

klim inf ( ) 1/2

n~+oo 2 n2R 1( xn ) 1 og 2

lere o2(Y)=2(Y+1)/(Y+2), O

Therefore,

A (1,k,L)n

*T (2,k,L).n

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t ;5 very easy to derive useful corollaries from Theorem 2A

pplicable for nstance to all the models studed in Haeusler and

eugels (1985), Hall(1982), etc ...

orrollary lA. Suppose that the assumptions of Theorem 2 A hold.

) If, in addition, f admits a derivative fi such that ufl(u) has a

* .imit as u~O, then Tn (2,k,L) satisfies the LIL(-0'1(r)'0'1(r, o

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and nu /k~l, a.s. This proves then

irst part of Corollary lA.

) The second part is obvious snce Uk

~(k/n), a.5., as n~+oo and, n

Nn

(-oo)/(21o92 n ) 1/21~k 1/2 If (k/n)/(2109 2 n) 1/2+ k 1/2f (U k

,n)/(2109 2n) 1/2 1

- n y0 rY0;ut T(k,l )::k- f J l-G(t) dt dy. The results obtained for~ A (1 ,k,1.)

Y Y nn-k,n

te of the type of those related to Tn (2,k. t ).~eorem lB. Let F~(f,b)Er. Let (Kl-2-3) and (Ll-2) be satisfied. Then

1 -1 1/2 -k(x n ) k (A n (l,k,t)-T(k,t satisfies the LIL(-02(r),o2(r)),

~th O~(r)::6(r+l)(r+2)/(r+3)(r+4),O

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~e note that Coro11ary lB is derived from Theorem 28 exact1y as

Co