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Extreme values of particular nonlinear processes
Dominique Guegan, Sophie A. Ladoucette
To cite this version:
Dominique Guegan, Sophie A. Ladoucette. Extreme values of particular nonlinear processes. C.R.A.S.,
2002, 335, pp.73 - 78. �halshs-00201320�
Extreme values of particular non linear processes DominiqueGU EGAN a , SophieLADOUCETTE a,b a
EcoleNormaleSuperieuredeCachan,GRID,UMRCNRS8534,61avenueduPresidentWilson, 94235Cachancedex, France.
b
BanquedeFrance,41-1391DGEI-DEER-CentredeRecherche,31rueCroixdesPetitsChamps, 75049Pariscedex1,France.
E-mail: guegan@grid.ens-cachan.fr,sophie.ladoucette@banque-france.fr
Abstract. Weareinterestedinthebehaviorofthemaximaofageneralclassofdeterministicchaotic processes-includingthetentmapandthelogisticmap-,ofnoisychaoticprocesses,and ofthelongmemoryGaussiank-factorGegenbauerprocesses.
Valeurs extr^emes pour des processus non lineaires particuliers
Resume. Nousnousinteressonsaucomportementdesmaximad'uneclassegeneraledeprocessus chaotiquesdeterministes -comprenant les applications tentet logistique -,de proces-sus chaotiquesbruites et desprocessus a memoire longuede Gegenbauer a k-facteurs Gaussiens.
Version francaise abregee
Dans cette Note, nous nous interessonsau comportement asymptotique des maxima de certains processus dependants. Nous consideronsd'abord lesprocessus generespar unefamilleparametriqued'applicationschaotiquesde niepar:
X n+1 =' (X n )= 1 (1 2X n ) ; 0X n <1=2 1 (2X n 1) ; 1=2X n 1 (3)
ou 1=2 2(voirHallet Wol [9] et Lawrance et Spencer[10]). Nous considerons aussilesprocessusgeneresparuneversionbruitedusystemechaotique(3)prisen=2, de niepar: Y n =X n +" n ; n2N X n =4X n 1 (1 X n 1 ); n2N (4) avec(" n ) n2N
unesuitedevariablesaleatoiresindependantesetidentiquementdistribuees de densite (a;b) (a;b >0), independantes de (X
n )
n2N
. En n, nous nous interessons auxprocessusdeGegenbauerak-facteursGaussiens de nispar:
k Y i=1 (I 2 i B+B 2 ) di X n =" n (5) avecjd i j<1=2sij i j<1,jd i j<1=4sij i j=1,d i 6=0,pouri=1;;k,et(" n ) n2Z un bruitblancGaussien(voirGrayetal.[6]etGiraitiset Leipus[4]).
Enetablissantprecisementladistributionlimitedeleursmaxima,nousgeneralisonsle Theoreme1deFisher-Tippett [3]pourlesprocessusdependants(3),(4)et(5).
Dans la suite, nous noterons (M n
) n2N
la suite des maxima M n = max(Z 1 ;;Z n ), n1,associeeaunprocessus(Z n ) n2N dedistribution marginaleF.
DanslesTheoremes2et 3,nous donnons lesdomainesd'attraction desdistributions marginalesassocieesauxprocessus (3)et(4), sousl'hypothesedenon nullitedel'index extremalde nien(2).
Theor
eme2.{ Soit(X n
) n2N
unprocessusassocieausysteme(3)avec xe,dontla fonction de densitef est supposee continue etstrictement positive en 1=2. Si (X
n )
n2N possedeunindexextremalveri ant0<1,alorsF appartientaudomained'attraction de Weibull d'index 1=,i.e. :
Pfc 1 n (M n d n )xg! 1= (x) avec c n =(f(1=2)n) etd n =1. Theor eme 3. { Soit (Y n ) n2N
un processus associe au systeme (4) avec (" n ) n2N de densite (a;b). Si (Y n ) n2N
possede un index extremal veri ant 0 < 1, alors F appartientaudomained'attractionde Weibulld'index (2b+1)=2,i.e. :
Pfc 1 n (M n d n )xg! (2b+1)=2 (x) avec c n =( 4 (a+b)n (2b+1) (a) (b) ) 2=(2b+1) etd n
=2,o u estla fonctionGamma.
Nousetablissonsmaintenantunresultatsimilairepourleprocessus(5),enremarquant que la fonction d'autocovariance de ce processus veri e la condition de Berman [1] (n)logn!0quandn!+1. Theor eme 4. { Soit(X n ) n2Z
unprocessusde Gegenbauer ak-facteursGaussien (5) d'ecart-type,telquemax(d
1 ;;d
k
)>0. Alors,F appartientaudomained'attraction de Gumbel,i.e. : Pfc 1 n (M n d n )xg!(x) avec c n =(2logn) 1=2 etd n =((2logn) 1=2
loglogn+log4 2(2logn)
1=2 ).
1. Introduction and background
Extremevaluetheoryisanareaofstatisticsdevotedtothedevelopmentofmodelsand techniquesforestimatingthebehaviorofunusualorrareeventswhich areimportantin severaldomainsas nance,internettraÆc,meteorology,physicandbiology. InthisNote, weareinterestedtoinvestigatethistheoryforafamilyofchaoticmaps(thegeneralized tentfamily) representingdeterministicdynamicalsystemswithorwithoutmeasurement noise,andforaparticularclassoflongmemoryprocesses.
ThroughoutthisNote, wedenote(M n
) n2N
thesequenceofmaximade ned byM n = max(Z 1 ;;Z n
), n1, associated with aprocess (Z n
) n2N
with marginaldistribution function(d.f.) F,x
F
therightendpointofad.f. F,andg2R +1 Æ
apositiveLebesgue measurablefunctionon]0;+1[whichisregularvaryingat+1withindexÆ2R.
Let(X n
) n2N
beasequenceofindependentandidenticallydistributed (i.i.d.) random variables. Theclassicalextreme valuetheorydealswiththedistribution ofM
n
asn! +1, andits central result is thefollowingtheorem, derived by Fisherand Tippett [3] and rstprovedrigorouslybyGnedenko[5],whichgivesthepossiblelimitingdistribution formaxima.
Theorem 1. { Let(X n
) n2N
be a sequence of i.i.d. random variables. If there exists someconstants c n >0andd n 2R suchthat: Pfc 1 n (M n d n )xg!H(x) (1)
for somenondegenerate d.f. H,thenH belongstothe typeof oneof the following three d.f.s: Frechet: (x)= 0; x0 exp( x ); x>0; >0 Weibull: (x)= exp( ( x) ); x0; >0 1; x>0 Gumbel: (x)=exp( e x ); x2R:
We say that the d.f. F of a sequence of random variables (Z n
) n2N
belongs to the domain of attraction of H, and we write F 2 D(H), if (1) holds for some sequences c
n
>0andd n
2R. NecessaryandsuÆcientconditionsareknownforeachtypeoflimit, involvingthebehaviorof thetail1 F(x)(denoted byF(x))asx increases.
Beforede ningthemodelsonwhichwework,werecallthede nitionofextremalindex whichturnsouttobeusefulinthesequel. FollowingLeadbetteretal.[11],astationary process(Z
n )
n2N
hasextremalindex(01)ifforevery >0thereexitsasequence (u n ) n2N such that: lim n!+1 n(1 F(u n ))= and lim n!+1 P(M n u n )=e : (2)
First, we consider the processes generated by a parametric family of chaotic maps de nedby: X n+1 =' (X n )= 1 (1 2X n ) ; 0X n <1=2 1 (2X n 1) ; 1=2X n 1 (3)
where1=22. Thisfamilycontainsthetentmapandthelogisticmap,respectively for = 1 and = 2, and it is referred to the generalized tent family (see Hall and Wol [9]andLawranceandSpencer[10]). Wealsoconsideranoisyversionofthesystem (3)for=2,de nedby:
Y n =X n +" n ; n2N X n =4X n 1 (1 X n 1 ); n2N (4) where (" n ) n2N
is a sequence of i.i.d. random variables independent of (X n
) n2N
with (a;b)density(a;b>0). ThelastmodelwhichweconsideristhelongmemoryGaussian k-factorGegenbauerprocess(X
n ) n2Z de ned by: k Y i=1 (I 2 i B+B 2 ) di X n =" n (5) withjd i j<1=2ifj i j<1,jd i j<1=4ifj i j=1,d i 6=0,for i=1;;k, and(" n ) n2Z a Gaussianwhitenoise(seeGrayet al.[6]andGiraitisandLeipus[4]).
2. Maxima'slimitingdistributionof processes(3),(4) and (5)
IntheTheorem2and3,weprovidethemaximaldomainofattractionofthemarginal d.f.softheprocesses(3)and(4),undertheassumptionthatextremalindexde nedin (2)isdi erentfromzero.
Theorem2. { Let(X n
) n2N
beaprocess associated withthe system(3)for a xed, withinvariantdensityf assumedtobecontinuousandstrictlypositiveat1=2. If(X
n )
n2N hasextremalindex0<1,thenF belongstothemaximaldomainofattractionofthe Weibulldistributionwithindex 1=,i.e.:
Pfc 1 n (M n d n )xg! 1= (x) withc n =(f(1=2)n) andd n =1. Forinstance, wegetF 2D( 1 ),c n =(n) 1 and d n
=1forthetentprocess( =1 in (3)), and weget F 2 D( 1=2 ), c n =(=(2n)) 2 and d n
=1 forthe logistic process (=2in (3)).
Theorem 3. { Let (Y n
) n2N
be a process associated with the system (4), assuming ("
n )
n2N
has (a;b) density. If (Y n
) n2N
has extremalindex 0<1, thenF belongsto themaximal domainof attractionof theWeibulldistributionwithindex (2b+1)=2, i.e.:
Pfc 1 n (M n d n )xg! (2b+1)=2 (x) withc n =( 4 (a+b)n (2b+1) (a) (b) ) 2=(2b+1) andd n
=2, where denotethe Gamma function. Now,weprovideasimilarresultfortheprocess(5),remarkingthatitsautocovariance function veri estheBerman'scondition (n)logn!0asn!+1(seeBerman[1]).
Theorem4.{ Let(X n
) n2Z
beaGaussiank-factorGegenbauerprocess(5)with stan-darddeviation,suchthatmax(d
1 ;;d
k
)>0. Then,F belongstothemaximaldomain ofattraction ofthe Gumbeldistribution, i.e.:
Pfc 1 n (M n d n )xg!(x) withc n =(2logn) 1=2 andd n =((2logn) 1=2
loglogn+log4 2(2logn)
1=2 ).
Inestablishingpreciselythelimitingdistributionoftheirmaxima,wehavegeneralized theTheorem1ofFisher-Tippett[3]forthedependentprocesses(3),(4)and(5). Proofs aregiveninSection 4,anddetails canbefoundin GueganandLadoucette[8].
3. Estimationof extremal index
InTheorems 2and 3 above,the extremal index de ned in (2) appears in the nor-malizingconstantc
n
. Sincewedon't provide thevalueof analytically,weproposeto estimateit. Wechoosetheblocksmethod which involvesdividingN givendatainto m blocksoflengthnandsettingahighthresholdu. Anaturalestimatorofisthen:
^ =n 1 (log (1 K u =m))=(log(1 N u =(mn))) whereK u andN u
arerespectivelythenumberofblocksandthenumberofobservations thatexceedthethreshold. Wedon'thavetheasymptoticnormalityofthisestimator,but statisticalpropertiesaregiveninSmithandWeissman[12]. Forthesystem(3)or(4),we simulatesrealizationsoftheassociatedprocessinconsideringsdi erentinitialconditions X
0
inordertocompute ^
anditsstandarddeviation. Then,weobtainsestimatedvalues for , and we compute
^ = ( P s i=1 ^ i )=s, where the ^ i
's are the estimated valuesfor a singlerealization,and ^
= q ( P s i=1 ( ^ i ^ ) 2
)=s. Thus, wecandeduce ^c n
, anestimate forc
n .
Forinstance, weconsiderthesystem(3)and xm=n=200ands=100. For=1 and=2,werespectivelychooseu=0:9950andu=0:9999.Weget
^
=0:98,^
for=1,and ^
=0:99,^
=0:01for =2. Similarestimationsforthesystem(3),with di erentvaluesof,andforthesystem(4),areprovidedinGueganandLadoucette[8]. Now,forspeci cdensitiesf inTheorem 2,andspeci cvaluesofaandbin Theorem3, itiseasytocomputec^
n .
4. Proofs
Proof of Theorem 2. { Wesuppose that (X n
) n2N
hasextremal index0< 1. It is clearthat x
F
=1. FollowingHall andWol [9],weconsider thepre-imageofageneral point1 x 1 ; x>1,andweobtain: f(1 x 1 )=(2) 1 x 1 1= (f(1=2(1+x 1= ))+f(1=2(1 x 1= ))):
Then, wehave the asymptoticf(1 x 1 ) 1 f(1=2)x 1 1= asx !+1 i.e. f(1 x 1 )2R +1 1 1=
. UsingaKaramata'stheorem(seeTheorem1page281inFeller[2]),we haveF(1 x 1 )f(1=2)x 1= asx!+1i.e. F(1 x 1 )2R +1 1= .
Thus, byTheorem 1.6.2. and Corollary3.7.3. in Leadbetteret al. [11],weconclude thatF 2D( 1= ),c n =(f(1=2)n) andd n
=1. Thetheoremisthenproved.
Proof of Theorem 3. { Suppose that (Y n
) n2N
has extremalindex 0< 1. Letf, g andhdenoterespectivelytheinvariantdensitiesof(Y
n ) n2N ,(X n ) n2N and(" n ) n2N . Itis obviousthatx F
=2. For x1,wehave(seeGueganandLadoucette[7]):
f(2 x 1 )= Z 1 1 x 1 g(y)h(2 x 1 y)dy = (a+b) (a) (b) Z +1 x y 2 g(1 y 1 )(x 1 y 1 ) b 1 (1+y 1 x 1 ) a 1 dy:
Then,usingaKaramata'stheorem,weobtainthatf(2 x 1 ) 2 (a+b) (a) (b) x b g(1 x 1 )as x!+1:Sincef(2 x 1 )2R +1 (1 2b)=2
,weapplyasecondtimeaKaramata'stheorem. WeobtainthatF(2 x 1 )(2=(2b+1))x 1 f(2 x 1 ) 4 (a+b) (2b+1) (a) (b) x (b+1) g(1 x 1 ) 4 (a+b) (2b+1) (a) (b) x (2b+1)=2 asx!+1,i.e. F(2 x 1 )2R +1 (2b+1)=2 .
Thus, by Theorem 1.6.2. and Corollary3.7.3. in Leadbetteret al. [11],weconclude that F 2D( (2b+1)=2 ), c n =( 4 (a+b)n (2b+1) (a) (b) ) 2=(2b+1) and d n
=2, andthe theorem is proved.
ProofofTheorem4.{ AstationaryGaussianprocesswhichveri estheBerman's condi-tion (n)logn!0asn!+1,belongstotheGumbeldomainwiththesame normal-izingconstantsasin thei.i.d. case(seeTheorem4.3.3. in Leadbetteretal.[11]). Since ithas acausalrepresentation, the process(5) is Gaussian. Fromthe expressionof the asymptoticofits autocovariancefunction , givenin Giraitis andLeipus[4], weobtain that: (n) X i=1;;k :di>0 An 2d i 1 (B+o(1)) as n!+1 with d i = d i if j i j <1 and d i = 2d i ifj i j= 1(i =1;;k : d i > 0), and where A andB aretwo nitepositiveconstants. Since2d
i 1<0,wehaven 2d i 1 logn!0as n!+1(i=1;;k:d i
>0), andthen,wehave (n)logn=o(1)asn!+1. Thus, theBerman'sconditionholds,and theproofiscomplete.
References
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