ESTIMATION OF THE FINITE RIGHT ENDPOINT IN THE GUMBEL DOMAIN

Transcription

1 Saiica Siica 4 4, doi:hp://dx.doi.org/.575/.3.83 ESTIMATION OF THE FINITE RIGHT ENDPOINT IN THE GUMBEL DOMAIN Iabel Fraga Alve ad Cláudia Neve Uiveriy of Libo ad Uiveriy of Aveiro Abrac: A imple eimaor for he fiie righ edpoi of a diribuio fucio i he Gumbel max-domai of aracio i propoed. Large ample properie uch a coiecy ad he aympoic diribuio are derived. A imulaio udy i preeed. Key word ad phrae: iferece. Edpoi eimaio, exreme value heory, aiical. Iroducio Le X, X, X, be he order aiic from he ample X, X,..., X of i.i.d. radom variable wih commo uow diribuio fucio F. Le x F deoe he righ edpoi of F. We aume ha he diribuio fucio F ha a fiie righ edpoi, x F := up{x : F x < } R. The fudameal reul for exreme value heory i due i variou degree of geeraliy o Fiher ad Tippe 98, Gedeo 943, de Haa 97 ad Balema ad de Haa 974. The exreme value heorem or exremal ype heorem reric he cla of all poible limiig diribuio fucio o oly hree ype. Thu, if here exi coa a >, b R uch ha for all x, G o-degeerae, he G mu be oe of lim F a x + b = Gx,. Ψ α x = exp{ x α }, x <, α >, Λx = exp{ exp x}, x R, Φ α x = exp{ x α }, x >, α >. Redefiig he coa a > ad b R, hee ca be eed i a oeparameer family of diribuio, he Geeralized Exreme Value GEV diribuio wih diribuio fucio G γ x := exp{ + γx /γ }, + γx >, γ R.

2 8 ISABEL FRAGA ALVES AND CLÁUDIA NEVES We coider F i he max-domai of aracio of G γ ad ue he oaio F D M G γ. For γ <, γ =, ad γ >, he GEV diribuio fucio reduce o he Weibull, Gumbel, ad Fréche diribuio fucio, repecively. A equivale exreme value codiio allow he limi relaio i. o ru over he real lie cf., Theorem..6 de Haa ad Ferreira 6: F D M G γ if ad oly if lim F a x + b = + γ x /γ,. for all x uch ha + γx >, a := a [] ad b := b [], wih [] deoig he ieger par of. The exreme value idex γ deermie variou degree of ail heavie. If F D M G γ wih γ >, he diribuio fucio F ha a power-law decayig ail wih ifiie righ edpoi, while γ < refer o hor ail wih a fiie righ edpoi. The Gumbel domai of aracio D M G ecloe a grea variey of diribuio, ragig from ligh-ailed diribuio, uch a he Normal, he expoeial, o moderaely heavy diribuio, uch a he Logormal. All he amed diribuio have a ifiie righ edpoi, bu a fiie edpoi i alo poible i he Gumbel domai. We give everal example i Secio. Ligh-ailed diribuio wih fiie edpoi, bu o o ligh ha hey are icluded i he Gumbel domai, have bee i demad a feaible diribuio uderlyig real life pheomea. A example i he exreme value aalyi by Eimahl ad Magu 8 of he be mar i Ahleic, aimed a aeig he ulimae record for everal eve. For iace, Table 3 i Eimahl ad Magu 8 ha everal miig value for he eimae of he edpoi which are due o a eimaed exreme value idex γ ear zero. A aemp o fill hee bla pace wih a appropriae framewor for iferece i he Gumbel domai ha bee provided by Fraga Alve, de Haa, ad Neve 3, alhough from he view poi of applicaio o he Log Jump daa e ued i Eimahl ad Magu 8. The eaive eimaor propoed by Fraga Alve, de Haa, ad Neve 3 i virually he ame a he oe we iroduce. The ovely here i i he developme of a imple cloed-from expreio for he eimaor. The problem of eimaig he righ edpoi x F of a diribuio fucio lyig i he Gumbel exremal domai of aracio i acled by he emiparameric aiic X, + X, log log or, i a more compac form, by i= i= + i + + i X i, ˆx F := X, + a i, X, X i,,.3

3 ENDPOINT ESTIMATION IN GUMBEL DOMAIN 83 where a i, := log log + i log + i + >, uch ha i= a i, =. Here ad hroughou i i fac a equece of poiive ieger goig o ifiiy a bu a a much lower rae ha. Thu, we are defiig ˆx F a a fucioal of he op obervaio of he origial ample, relyig o a iermediae equece =, wih, = o, a. From he o-egaivee of he weighed pacig i.3, we ee ha ˆx F i greaer ha he maximum X, wih probabiliy oe. Thi i a advaage i compario wih he uual emiparameric eimaor for he righ edpoi of a diribuio fucio i he Weibull domai of aracio. We refer o Hall 98, Fal 995, Hall ad Wag 999, ad o de Haa ad Ferreira 6 ad referece herei. To he be of our owledge, oe of hee eimaor have eured he exrapolaio beyod he ample rage. There have bee, however, ome developme of edpoi eimaor coeced wih γ < i he ee of bia reducio ad/or correcio. Li ad Peg 9, Li, Peg, ad Xu ad Cai, de Haa, ad Zhou 3 are a few of he wor. Recely, Girard, Guillou, ad Supfler devied a edpoi eimaor from he high-order mome peraiig o a diribuio wih γ < ; Li ad Peg propoed a boorap eimaor for he edpoi evolvig from he oe of Hall 98 i cae γ /,. The pree paper addree he cla of diribuio fucio belogig o he Gumbel domai of aracio, for which o correpodig ha ye bee provided. The appropriae framewor i developed i Secio. The remaider of he paper i a follow. The raioale behid he propoal of he ew eimaor for he righ edpoi i expouded i Secio 3. Coiecy ad he aympoic diribuio of he eimaor, are wored ou i Secio 4, aig advaage of a form of eparabiliy bewee he maximum ad he um of higher order aiic. I order o perform aympoic, we require ome baic codiio i he coex of he heory of regular variaio. Thee are laid ou i Secio. I Secio 5 we gaher ome imulaio reul. Secio 6 i devoed o ome applicaio ad cocluio.. Framewor Le F be a diribuio fucio d.f. wih righ edpoi x F := up{x : F x < }. For ow we aume x F. Suppoe F aifie he exreme value codiio F + x f lim = + γ x /γ,. x x F F for all x R uch ha + γ x >, wih a uiable poiive fucio f hi i equivale o., ee Theorem..6 of de Haa ad Ferreira 6. For γ = he limi i. read a e x.

4 84 ISABEL FRAGA ALVES AND CLÁUDIA NEVES Le U be he geeralized ivere fucio of / F. If F aifie. wih γ = he we ca aume here exi a poiive fucio a uch ha, for all x >, Ux U lim = log x.. a Hece U belog o he cla Π ee Defiiio B..4 of de Haa ad Ferreira 6 ad a i a meaurable fucio uch ha lim a x/a = for all x >. The a i a lowly varyig fucio ad we wrie a RV ee Theorem B..7 of de Haa ad Ferreira 6. The fucio a ad f of. ad. are relaed o each oher by a = f U ee Theorem B.. of de Haa ad Ferreira 6. We ue he oaio U Πa o emphaize he auxiliary fucio a. We aume he followig: A U Πa. A U = U + a d + o a, for ome, wih a poiive fucio a RV aifyig a a a. B x F := U = lim U exi fiie. Uder A, Propoiio B..53 of de Haa ad Ferreira 6 guaraee he exiece of a wice differeiable fucio f, wih f RV, uch ha U = f + o a. Le f = f + f d be hi fucio. Hece, U = U + f f + o a, wih a a, ad where we e f = a/. Thi i A. Coverely, A implie A by Propoiio B..55 of de Haa ad Ferreira 6 wih g = a/ RV herei. Uder codiio A ad B, we have a U U = d + o a,,.3 x lim U/d U/d = log x,.4 a/d for all x >. Hece U d/ i alo Π-varyig wih he auxiliary fucio q := a d = a d = a d..5 I he uual oaio, he above i U d/ Πq. The q i lowly varyig while.3 eail ha q a cf., Lemma C. from Appedix C. Some example of diribuio belogig o he Gumbel domai of aracio wih fiie righ edpoi, where.3 hold, are lied below.

5 ENDPOINT ESTIMATION IN GUMBEL DOMAIN 85 Example. A radom variable X i Negaive Fréche wih parameer β > if i ha diribuio fucio F x = exp{ x F x β }, x x F, β >. The aociaed ail quaile fucio i U = F / = x F log /β, he arrow ad for he geeralized ivere. The U Πa wih a = /βlog /β, a. The auxiliary fucio i.4 i q = log /β, β >. Example. Le F x = exp{ ax/β}, x < βπ/, β >. The U = β arcalog,. Hece U aifie.3 wih a = / log +β ad U Πa where U = βπ/ = x F. Example 3. Le F x = exp{π/ β arci x/β β }, x < β, β >. The U = β { i [ /π β + log ] /β },, ad.3 hold wih a = log /β+ colog /β, U Πa ad U = β = x F. 3. Saiic Le X, X,..., X be a radom ample of ize from he uderlyig diribuio fucio F wih fiie righ edpoi x F. Le X, X,... X, be he correpodig order aiic. We iroduce he eimaor ˆq/ for he auxiliary fucio q a i.5, evaluaed a = /. Thi eimaor ha he propery ha, a, =, ad / provided ome uiable, mild rericio o he ecod order refieme of U/ d hold, q / ˆq / a / q / d N, where N i a o-degeerae radom variable. Several eimaor for he righ edpoi x F = U < ca be devied from.3, i he ee ha hee migh evolve from a uiable eimaor ˆq/ for q/, a ˆx F = Û + ˆq = X, + ˆq. 3. Here ˆx F carrie aalogou large ample properie o ˆq/. I paricular, he coiecy of ˆx F i eeially eured by he coiecy of ˆq/. Theorem i Secio 4 accou for hi. We evaluae relaio.4 a x = /, ogeher wih q a = / ee la equaliy i.5, ad wrie, for large eough, U d U q log. Eimaio of q/ arie aurally from he empirical couerpar Û /θ = X [θ],,, ], θ =,, o

6 86 ISABEL FRAGA ALVES AND CLÁUDIA NEVES ˆq := log X [], X [], d. 3. Simple calculaio yield he aleraive expreio ˆq = X, + + i log X i,. 3.3 log + i + i= Combiig 3. wih 3.3 lead o he righ edpoi eimaor ˆx F := X, + X, + + i log X i,. 3.4 log + i + Afer rearragig compoe, i i poible o expre ˆx F a he maximum X, added o ome weighed mea of o-egaive ummad a ˆx F = X, + a i, X, X i,, i= wih a i, := log log + i log + i + >, i =,,..., N, uch ha i= a i, =. Remar. The propoed eimaor for he righ edpoi reur value alway larger ha x,. Thi coiue a major advaage i compario o he available emi-parameric eimaor for he edpoi i he cae of Weibull domai of aracio, for which he exrapolaio beyod he ample rage i o guaraeed. Thi iadequacy of he exiig eimaor ofe lead o ome diappoiig reul i pracical applicaio, wih eimae-yield ha may be lower ha he oberved maximum from he daa. i= 4. Aympoic Reul Our reaoig here i ha he large ample properie of he eimaor ˆx F are eeially govered by he aympoic properie of he eimaor ˆq/. Thi how up i Theorem wih repec o coiecy. The coiecy of ˆq/ i acled i Appedix A, wih ˆq/ defied i 3. ee alo 3.3, for a iermediae equece =. Similarly, he limiig diribuio of ˆq/ i Theorem reder he aympoic diribuio of ˆx F via Propoiio. Thee proof regardig ˆq/ are popoed o Appedix A. Theorem. Le X, X,... be i.i.d. radom variable wih ail quaile fucio U aifyig A ad B. Suppoe = i a equece of poiive ieger uch ha, /, a, ad ˆq//q/. p The ˆx F := X, + ˆq/ i a coie eimaor for x F <, ˆx F p xf.

7 ENDPOINT ESTIMATION IN GUMBEL DOMAIN 87 Proof. Le U +, be he + -h acedig order aiic from he radom ample U, U,..., U of uiformly diribued radom variable o he ui ierval. The X, d = U /U+,, where U i he uderlyig ail quaile fucio ad = d ad for equaliy i diribuio. Wrie x F ˆx F d = U U q U U U +, q ˆq/ q/ = I II III, where I := U U/ q/ = o a/, from.3. We have II := U U U +, = o p a becaue U Πa ad Smirov Lemma eure / U +,..3 i de Haa ad Ferreira 6. Moreover, ˆq / III := q/ q / = o p by Propoiio ad he fac ha.3 implie q/ = o. P ee Lemma The limiig diribuio of ˆq/ ad, laer o, he aympoic diribuio of ˆx F i aaied uder a uiable ecod order refieme of.: uppoe here exi fucio a, poiive, ad A edig o zero a, uch ha for all x >, Ux U/a log x lim = A log x. 4. Remar. 4. follow direcly from Theorem B.3.6, Remar B.3.7, ad Corollary.3.5 of de Haa ad Ferreira 6 becaue he former ae ha, i our eup of γ = ad x F <, he oly cae allowed i he cae of he ecod order parameer ρ equal o zero. The ecod order auxiliary fucio A coverge o zero, o chagig ig for ear ifiiy, ad for every x >, Ax/A, oaio: A RV. Example 4. Coider he Negaive Fréche diribuio fucio F x = exp{ x F x β }, x x F, β >. The auxiliary fucio here i q = log /β, β >, while raighforward calculaio yield A = + /β log, which implie ha a /q = A / + β, for ear ifiiy. Theorem. Aume A, B ad 4. hold. Le = be uch ha, a,, /, a/a/, ad A/ = O. If U / U / d lim A/ q/ log = λ R, 4.

8 88 ISABEL FRAGA ALVES AND CLÁUDIA NEVES he q/ ˆq/ a/ q/ d Λ log where Λ ha he Gumbel diribuio exp{ e x } for all x R. λ log, 4.3 Here 4. cocer a ecod order refieme of.4, x U/d U/d /q log x lim = Q log x, 4.4 ae i he poi x = for large eough = /. Thu 4. em from he heory of exeded regular variaio. We refer o Appedix B of de Haa ad Ferreira 6 for a good caalog of reul i he heory of exeded regular variaio. The aumpio ha a//a a i more rericive i erm of creeig for a adequae value o deermie he umber of op order aiic o which o bae our iferece. For example, for he Negaive Fréche wih = p, p,, a a/ = log /β+ = p /β+, log which i approximaely if ad oly if p approache zero. A more appropriae choice i = log r, r, ], for which a a/ = log log /β+ = r log log log /β+ +. log The upper boud r i impoed i order o comply wih he aumpio A/ = O. We believe ha uch a choice for =, wih log = olog, i a feaible oe for mo model aifyig.3. We brig forward he fac here ha a mi-pecificaio of, i he ee ha a/ /a coverge o a coa differe ha, ha a direc impac o he aympoic variace of he ormalized relaive error preeed i Theorem raher ha upo he aympoic bia. Remar 3. The aumpio log = olog i a commo oe i he heoreical aalyi of eimaor for Weibull-ype ail, which form a rich ubcla of he Gumbel max-domai of aracio, albei wih x F =, ee Goegebeur, Beirla, ad De We ad Garde, Girard, ad Guillou. Example 5. The Negaive Fréche diribuio ha a ail quaile fucio give by U = x F log /β,, < β <, aifyig 4.4, wih Q = β log.

9 ENDPOINT ESTIMATION IN GUMBEL DOMAIN 89 Propoiio. Uder he codiio of Theorem, ˆx F x F q/ ˆq/ a/ a/ q/ P. Proof. We wrie ˆx F x F a/ q/ ˆq/ a/ q/ = ˆxF ˆq/ xf q/ a/ a/ = X, U/ U U/ q/. a/ a/ Uder 4., Theorem.4. of de Haa ad Ferreira 6 eure ha X, U/ /a/ = O p / = op, ad he re follow from.3. The ex reul give a aleraive formulaio of he reul of Theorem ad Propoiio aimed a providig cofidece bad for ˆx F. Theorem 3. Le X, X,... be i.i.d. radom variable wih ail quaile fucio U aifyig 4.. Le â/ be a coie eimaor for a/. Suppoe = i a equece of poiive ieger uch ha, a,, /, a/a/, ad A/ = O. If U / U / he lim A/ â/ ˆx F x F q/ d Λ log d log = λ R, λ log. Proof. The reul follow from Theorem ad Propoiio, applyig Sluy heorem. There are everal poibiliie for eimaig he auxiliary fucio a/. A obviou choice i he Maximum Lielihood Eimaor MLE, preedig ha he exceedace over a cerai high radom hrehold follow a Geeralized Pareo diribuio cf., Secio 3.4 of de Haa ad Ferreira 6: â/ = ˆσ MLE := X i, X,. i=

10 8 ISABEL FRAGA ALVES AND CLÁUDIA NEVES Figure. row: Righ ail for probabiliy deiy fucio of Negaive Fréche Model wih righ edpoi x F = ad β = olid lie, β =.7 dahed lie, β =.5 doed lie, β =.3 dodah lie. d row: Mea eimae lef ad empirical Mea Squared Error righ of ˆx F defied i 3.4, for he referred model wih ample ize =,. All plo are depiced agai he umber = of op obervaio ued i he eimaor. The aive maximum eimaor, x F := X,, i alo depiced horizoal lie. 5. Simulaio The Negaive Fréche diribuio fucio F x = exp{ x F x β }, x x F, β >, ha he ail quaile fucio U = x F log /β,, wih a = β log /β, β >. The rage of β offer variou ail hape, a how i he graphic draw i Figure row. We imulaed, ample of ize =,,,,, from he Negaive Fréche wih righ edpoi x F = for β =.3,.5,.7,. The reul for =, are depiced i Figure d row. The commo approach o elecig he umber of op order aiic ued

11 ENDPOINT ESTIMATION IN GUMBEL DOMAIN 8 i he eimaio or = i he pree cae i o loo for a regio where he plo are relaively able. Give he coiecy propery of he adoped eimaor, oe hould i priciple be away from mall value of ha are uually aociaed wih a large variace, ad o o far off i he ail a o iduce bia due o large. A appropriae choice for a iermediae = i = log r, wih r, ]. If we are uig =, ad e r =, he maximum allowed for r, we obai 48 ad hu 96. Wih aroud, he plo i Figure loo quie able i a viciiy of he arge value x F = repreeed by he olid horizoal blac lie. Hece, he low covergece impoed by = log eem o have lile effec o he fiie ample behavior of he eimaor ˆx F. Thi i paricular rue i cae < β <. The graph i Figure d row ugge beer eimaio uder he Negaive Fréche model if he parameer β i le ha, which correpod o he cae where he ihere ecod order codiio are aified. If β, he Negaive Fréche diribuio aifie he fir order codiio bu o he ecod order. Moreover, he geeral paer for he mea eimae of ˆx F ivolve a moderaed bia wih i he upper par of he ample, ad a quicly icreaig bia wih aroud 4% of he ample ize. Similar imulaio have bee carried ou for he oher wo model i Example ad 3, leadig alo o favorable reul. For ay model wih fiie righ edpoi, he ample pah of ˆx F depar from he ample maximum, alway reurig value beyod he ample. We hould highligh ha our eimaor ˆx F yield beer reul ha he maximum eimaor x F := X,, which alway udereimae he rue value x F. The relaive performace of boh x F ad ˆx F ca alo be eaily oberved if we compare he MSE graphic i Figure d row, righ: for he op par he ample, depedig o he β value, he eimaor ˆx F alway ouperform he maximum x F, preeig he ew eimaor a lower mea quared error ha he aive maximum eimaor. 6. Cae Sudy ad Cocluio Thi ecio i dedicaed o he eimaio of he fiie righ edpoi i he Gumbel maximum domai of aracio, which embrace ligh-ailed diribuio wih fiie edpoi. Here exreme value aalyi demad he eimaio of he righ edpoi, alhough he uderlyig diribuio ail i o o ligh a o be icluded i Weibull domai of aracio. Fraga Alve, de Haa, ad Neve 3, worig wih ahleic record daa, filled he gap, highlighed i Eimahl ad Magu 8, o aeig he ulimae record for everal ahleic eve. We here coider a applicaio o aiical exreme value aalyi of Achorage Ieraioal Airpor ANC Taxiway Ceerlie Deviaio for Boeig 747

12 8 ISABEL FRAGA ALVES AND CLÁUDIA NEVES Figure. ANCr daa: EVI eimaio wih Mome eimaor ad 95% cofidece badwidh, ploed agai. Aircraf, ee Scholz 3. The goal wa o provide a bai for uderadig he exreme behavior of ceerlie deviaio of he Boeig-747. Tha repor addreed he ri of a aircraf deviaig a a fixed locaio alog he axiway beyod a cerai hrehold diace from he axiway ceerlie. The B-747 axiway deviaio daa were colleced from 9/4/ o 9/7/ a ANC; durig hi period, 9,767 deviaio were recorded a ANC wih a rage of [-8.5, 8.863] fee, i boh direcio of he axiway. Baed o he exreme value limiig aumpio, poiive deviaio ANCr daa wih ample ize = 4, 9 were exrapolaed uig he = 385 mo exreme deviaio a ANC, he choe value of op obervaio o EVI eimaio, amely ˆγ =.395, Scholz 3. Figure depic he ample pah of ˆγ, he EVI eimae, uig he Mome eimaor of Deer, Eimahl, ad de Haa 989, alog wih he 95% cofidece badwidh, for he ANCr daa. I i eaily checed ha he raigh lie correpodig γ = i iide he cofidece badwidh for a very large upper par of he ample, i he graphic,. Coequely, he Gumbel domai of aracio cao be dicarded. Moreover, he eig procedure for deecig a fiie righ edpoi cf., Neve ad Pereira ugge he preece of a diribuio wih fiie righ edpoi uderlyig he ANC deviaio daa. Figure 3 depic he ample pah of our edpoi eimaor ˆx F agai. I he rage of 65, 3 he graph i quie able. If we rely o ha

13 ENDPOINT ESTIMATION IN GUMBEL DOMAIN 83 Figure 3. edpoi eimaio. regio we ugge for a edpoi eimae a value of approximaely 9. f, hi for he period 9/4/ o 9/7/. We coclude ha he propoed eimaor ˆx F perform reaoably well for pare diribuio i he Gumbel domai wih fiie righ edpoi x F. The robue of he edpoi eimaor.3, uder Weibull domai of aracio, i a opic of furher reearch, bu beyod he pree cope. Acowledgeme We are graeful o Profeor Laure de Haa for iroducig he appropriae characerizaio of diribuio wih fiie righ edpoi i he Gumbel domai, a he origi of he propoed eimaor. We ha Friz Scholz Uiveriy of Wahigo, George Legarrea FAA, ad Jerry Robio Boeig for harig he axiway deviaio daa a he Achorage Ieraioal Airpor. We ha he aoymou referee for heir comme ad uggeio. Reearch parially uppored by FCT: PE-OE/MAT/UI6/ ad 4, EXTREMA-PTDC/MAT/736/8, DEXTE - EXPL/MAT-STA/6/3. Appedix A: Coiecy of ˆq/ Le U,..., U be idepede ad ideically diribued uiform radom variable o he ui ierval ad le U, U,... U, be heir order aiic. Sice = i a iermediae equece = o a, we ca

14 84 ISABEL FRAGA ALVES AND CLÁUDIA NEVES defie a equece of Browia moio { W } uch ha, for each ε >, 3 up +ε θ θ θ U [θ]+, W = o p, A. for all θ cf., Lemma.4. of de Haa ad Ferreira 6, wih γ =. Le X, X,... be i.i.d radom variable wih diribuio fucio F DG, wih fiie righ edpoi x F, uch ha.3 hold. Noe ha U/U i d =X i, i =,,.... I view of relaio.3, he followig hold: Ux U a = /x a a d + ax o + o, a for all x >. Give ha a RV, we obai for ufficiely large ha X [θ], U θ a U θ d θ U U [θ]+, θ = θ a a θ x θ a dx x. θ U [θ]+, θ The uiform iequaliie i Lemma C..ii ell u ha, for ay ε >, a /θ/ a /θ = ± ε ε, <. Sice U [θ]+, [, ] ad for every, ], we ge he upper boud X [θ], U /θ a /θ log log + = log U [θ]+, θ U[θ]+, θ U[θ]+, θ U [θ]+, θ + P, U[θ]+, θ ε + op + ε + o p, wih he o p -erm edig o zero uiformly for [θ, ]. A imilar lower boud i alo poible. We ca apply Cramér δ-mehod o relaio A. o obai X [θ], U a θ θ = log + θ W +o p / ε ± ε +o p, A.

15 ENDPOINT ESTIMATION IN GUMBEL DOMAIN 85 a, uiformly for θ, θ. We coider he ormalized differece bewee a ample iermediae quaile ad correpodig heoreical quaile R θ := X [θ], U θ a A.3 θ = X [θ], U θ a X [θ], U θ + θ + U θ U θ. a θ a θ a θ a θ Bearig o A. combied wih he uiform iequaliie i Lemma C., we hu ge for ay ε >, R θ = log + W + / ε o p θ ± ε + o p ± ε ε log + log ± ε ε = W ± ε + o p ε ε log, A.4 θ for [θ, ], all θ. Thu he diribuio of deviaio bewee high large ample quaile ad heir heoreical couerpar i aaiable, wih a vaihig bia, by mea of a differe ormalizaio ha o he lef had-ide of A.. The wea covergece of ˆq/ i uppored o he laer. Propoiio. Le X, X,... be i.i.d. radom variable wih ail quaile fucio U aifyig.3. Suppoe = i a equece of poiive ieger uch ha, /, a. The ˆq / q / Proof. We begi by oig ha ˆq / q / = Û / Û / log q / = log p. d { X [], U q d X, U q We wrie ee Eq. A.3 wih θ = X [], U q d d + X [], U q d U U q X [], U q d d A.5 }.A.6

16 86 ISABEL FRAGA ALVES AND CLÁUDIA NEVES where = X [], U q I, = a/ { q R d + d =: I,, a a/ R d A.7 }. A.8 From Lemma C.., X [], U I, = q d a/ q R d a/ + R log d q/ a/ q + a/ q/ log R d, A.9 wih high probabiliy, for ufficiely large. We ca provide a imilar lower boud. Owig o A.4, for ay poiive ε, R d Sice ε > i arbirary, meaig ha W d + < ε ε log d. ε d = ε ε ε d ε d log ε, ca be dicarded. A imilar lie of reaoig applie o Le ε ε log d = ε log ε. ε ε Y := W d, be a equece of ormal radom variable wih zero mea, ad V ary = log. +op

17 ENDPOINT ESTIMATION IN GUMBEL DOMAIN 87 Thu {Y } i a equece of degeerae radom variable, eveually, ad he wo iegral i A.5 vaih wih probabiliy edig o oe a. A a//q/ = o, oe ha a/ a/ I, = o p = o p = O p. q/ q/ For he fir iegral i A.6, X, U I, := q X, U = / d = a q/ d d q { U U, U a = a { log U, log + q/ log U U a log d } + op = a q/ log log U, log + op. The he probabiliy iegral raformaio yield } d log U, d = E, log, A. where E, i he maximum of i.i.d. adard expoeial radom variable. Hece, he radom variable A. coverge i diribuio o a Gumbel radom variable wih diribuio fucio exp{ e x }, x R. Moreover, a/q/, a, becaue a//q/ = o ee Lemma C. i Appedix C, where he auxiliary poiive fucio a aifie a, a, by aumpio. Therefore a I, = o p = O p. A. q/ We how ha he la iegral i A.6 i bouded. boud, U U d q/ U ad he lower boud, U / U / q/ d U q/ We eablih he upper d, A.

18 88 ISABEL FRAGA ALVES AND CLÁUDIA NEVES = U / + / U / + q / U +/ U + q d + d + / U +/ U q + d +. Maig = / ru o he real lie oward ifiiy, he Π variaio.4 follow lim U x d U d = log x, x >, A.3 q ad clearly eail he limi for he upper boud i A.: U d U d U d U q/ = Regardig he lower boud, we wrie U U d U q/ q q + U q/ U + q U + + d log. d + d + ad oe ha, for every ε >, here exi N uch ha for, + / < ε. I ur, U + U q + d + > U + U q + A.4, A.5 A.6 ε d. For he fir par of he righ-had ide here we ue A.3, while he ecod par i deal by Theorem B..9 of de Haa ad Ferreira 6 ivolvig he fac ha U Πa: U U + + q = log + o ε a / d + εa q U q / log + o log. + U a + d

19 ENDPOINT ESTIMATION IN GUMBEL DOMAIN 89 For he laer, we recall ha a/ = o q/. Now we wrie δ = / > everywhere i A.5. Furhermore, we aume ha here exi N uch ha, for, he erm δ i large eough ad he iegral i A.5 ca rephraed a Iδ := δ U +δ δ U δ a d +δ δ d +δ. A.7 For every fixed δ >, from he Π-variaio of U wih for he umeraor of Iδ properly recaled by aδ cf., Theorem B..9 i de Haa ad Ferreira 6: δ U +δ δ U +δ δ d +δ log d = log + δ log. aδ δ + δ For arbirary mall δ, he laer approache zero. We he apply Cauchy rule o obai lim δ Iδ we recall ha δ implie. Apply Eq.. of Chiag o he umeraor of Iδ o ge δ U +δ δ U +δ 3 +δ δ d +δ 3 lim δ I δ = lim δ { + lim δ δ δ U δ +δ aδ δ U δ +δ aδ d Uδ Uδ } δ. + δ aδ Sice U = a/, he limi become equal o he he limi of a δ/ + δ a δ/ + δ d δ aδ aδ + δ U δ/ + δ U δ/ + δ +δ aδ δ d Uδ Uδ + δ aδ We ca ow ae ay arbirary mall δ maig i order o apply he uiform covergece of a RV ad U Πa o ha he above iegral are eured fiie ad he equal o zero by defiiio. Hece, all he erm are egligible a δ coverge o zero meaig ha lim δ Iδ U / U / q / d log. Coiecy of ˆq/ follow by oig ha q/ q /. Appedix B: Aympoic diribuio of ˆq/. become ull. Therefore, I order o eablih he aympoic diribuio of he propoed eimaor for q/ we eed iigh abou he diribuioal repreeaio obaied from

20 83 ISABEL FRAGA ALVES AND CLÁUDIA NEVES A.. Specifically, if he ail quaile fucio aifie 4., he Theorem.4. of de Haa ad Ferreira 6 acerai ha, for each ε >, up X [θ], U θ θ a + log W θa log θ θ θ /+ε p, provided =, / = o ad A / = O. We have he followig reul cf.,.4.7 of de Haa ad Ferreira 6. B. Propoiio B.. Give.3, uppoe 4. hold. Le =, / = o ad A/ λ R, a. The, for θ ad for each ε > ufficiely mall, up /+ε θ X [θ], U /θ θ a /θ W = o p. Proof. A wih he equaliy righ afer A.3, we have ha R θ := X [θ], U θ a = a { θ θ a X [θ], U θ θ a U θ U } θ θ a. θ Noig ha a a / = a a a a /, for all >, Lemma C. combied wih Remar C.. yield he expaio a a = a a a a q log + o = a + A log + o A, q a B. for all >. Here A RV ad a /a = + o A. Havig e /θ, we hu have from B., he uiform boud i C., ad he ecod equaliy i B., ha θ Rθ = W + A θ log ±ε ε log θ A θ W ε ε θ A θ + op ε + o p ε log A θ uiformly i. Hece, he aumpio ha A/ = O eail ha log/a /θ, wherea ε ε θa /θ virually become o / ε for each ε > arbirarily mall ad uiformly i [θ, ]. The o p -erm,

21 ENDPOINT ESTIMATION IN GUMBEL DOMAIN 83 are uiform i [/θ, ]. Hece he followig repreeaio i valid for ε,, θ Rθ = W + o p / ε. Proof of he Theorem. We have q/ ˆq/ a/ q/ = { X a/ log [], U U / U / q d { J, J, } + q / q = log d log } a / log J 3,. X, U d B.3 By mimicig he ep from A.7 o A.8, we obai for he fir iegral above ha J, := = X [], U / a / R d + d a / a / R d. Hece, Propoiio B., while aumig ha a//q/ = O cf., Remar C. ad applicaio of he uiform boud i C.3 wih a := a + oa ad A := A, imply ha for each ε >, J, = W log d +o p log 3/+ε d + op A. Sice he iegral / W log d coverge o a um of idepede ormal radom variable, hi allow u o coclude ha he fir radom compoe i B.3 i egligible wih high probabiliy becaue J, = O p. We ow have ha J, := X, U a d

22 83 ISABEL FRAGA ALVES AND CLÁUDIA NEVES = a { log log a U, + a/ a +A log ±ε ε d }. log d We have ha a /a = o A ad A = A; hece a/ a log J, = log U, log = log U, log + log A + o. log ± ε ε d + o A Furhermore, aumig ha = i uch ha a/a/, he a/ a log J d, Λ log, where Λ deoe a Gumbel radom variable wih diribuio fucio exp{ e x }, x R cf., A.. If a/a/ coverge o a coa differe ha, he a chage i he cale i performed. The followig alo hold give C.3 ad ha A/ = O: log J d, Λ log. We ur o he bia erm J 3,. By aumpio, J 3, A/ = U / U / A/ q / d log λ, a. Therefore, ice A/ a//q/ cf., Remar C., he deermiiic erm J 3, reder a coribuio o he aympoic bia of q/ a/ log J 3, λ log. Appedix C: Auxiliary Reul Lemma C... Suppoe U Πa. The,

23 ENDPOINT ESTIMATION IN GUMBEL DOMAIN 833 i here exi a poiive fucio a aifyig a a, a, uch ha for ay ε > here exi = ε uch ha, for,,, ], U U log a ε maxε, ε ; ii a RV ad for ay ε > here exi = ε uch ha, for,,, ], a a ε maxε, ε.. Suppoe a > i a lowly varyig fucio, iegrable over fiie ierval of R + uch ha a d < for every >. The a, a, ad lim a a d =. Proof. Par.i of he Lemma come from de Haa ad Ferreira 6 cf., Propoiio B..7, while ii i a reul from Dree 998 cf., Propoiio B.. of de Haa ad Ferreira 6. The ecod par follow from Karamaa heorem for regularly varyig fucio cf., Theorem B..5 of de Haa ad Ferreira 6. Lemma C.. Uder A ad B, ax/a lim = log x, x >. a/q Proof. The uderlyig aumpio ha U Πa eail q ax a a q Ux U ax = Ux U a a q ax = Ux U log x a + o. Furhermore, accordig o.3, q Ux U = x a d a d = + + o x x a d a d C. + o. By aig he limi of he laer erm whe, we ge from Cauchy rule ogeher wih he fudameal heorem of iegral calculu ha x lim x a d a d ax ax = lim ax a = lim a.

24 834 ISABEL FRAGA ALVES AND CLÁUDIA NEVES Afer C., we ge q ax a a = log x + x x d a a d ax a + o ax = log x + log x a + o. I addiio o he ecod order codiio 4., Theorem.3.6 of de Haa ad Ferreira 6 acerai he exiece of fucio a ad A aifyig, a, A A ad a /a = o A, wih he propery ha for ay ε >, here exi = ε uch ha for all, x, Ux U/a log x A log x ε maxx ε, x ε, C. a x/a log x A ε maxxε, x ε. C.3 Remar C.. Relaio C.3, combied wih Lemma C., acerai ha a /q = ca, wih c becaue ρ = γ = cf., Eq. B.3.4 ad Remar B.3.5 i de Haa ad Ferreira 6. Hece he aumpio i hi paper ha he fucio q ca be redefied i order ha a/q A i aified. Referece Balema, A. A. ad de Haa, L Reidual life ime a grea age. A. Probab., Cai, J. J., de Haa, L., ad Zhou, C. 3. Bia correcio i exreme value aiic wih idex aroud zero. Exreme 6, 73-. Chiag, A. C.. Eleme of Dyamic Opimizaio. Wavelad Pre. de Haa, L. 97. O Regular Variaio ad i Applicaio o he Wea Covergece of Sample Exreme. Mahemaich Cerum Amerdam. de Haa, L. ad Ferreira, A. 6. Exreme Value Theory: A Iroducio. Spriger. Deer, A., Eimahl, J. ad de Haa, L A mome eimaor for he idex of a exreme value diribuio. A. Sai. 7, Dree, H O mooh aiical ail fucioal. Scad. J. Saic. 5, 87-. Eimahl, J. H. J. ad Magu, J. R. 8. Record i Ahleic hrough Exreme-Value Theory. J. Amer. Sai. Aoc. 3, Fal, M Some be parameer eimae for diribuio wih fiie edpoi. Saiic 7, 5-5. Fiher, R. A. ad Tippe, L. H. C. 98. Limiig form of he frequecy diribuio of he large ad malle member of a ample. Cambridge Philoophical Sociey. Mahemaical Proceedig 4, 8-9. Fraga Alve, I., de Haa, L. ad Neve, C. 3. How far ca Ma go? I Advace i Theoreical ad Applied Saiic, Torelli, N., Peari, F., ad Bar-He, A., edior, Spriger, Berli Heidelberg.

25 ENDPOINT ESTIMATION IN GUMBEL DOMAIN 835 Garde, L., Girard, S. ad Guillou, A.. Weibull ail-diribuio reviied: A ew loo a ome ail eimaor. J. Sai. Pla. Iferece 4, Girard, S., Guillou, A. ad Supfler, G.. Eimaig a edpoi wih high-order mome. TEST, Gedeo, B. V Sur la diribuio limie du erme maximum d ue érie aléaoire. Aal. Mah. 44, Goegebeur, Y., Beirla, J. ad De We, T.. Geeralized erel eimaor for he Weibull ail coefficie. Comm. Sai. Theory Mehod 39, Hall, P. 98. O eimaig he edpoi of a diribuio. A. Sai., Hall, P. ad Wag, J. Z Eimaig he ed-poi of a probabiliy diribuio uig miimum-diace mehod. Beroulli 5, Li, D. ad Peg, L. 9. Doe bia reducio wih exeral eimaor of ecod order parameer wor for edpoi? J. Sai. Pla. Iferece 39, Li, D., Peg, L. ad Xu, X.. Bia reducio for edpoi eimaio. Exreme 44, Li, Z. ad Peg, L.. Boorappig edpoi. Sahyā 74, 6-4. Neve, C. ad Pereira, A.. Deecig fiiee i he righ edpoi of ligh-ailed diribuio. Sai. Probab. Le. 8, Scholz, F. W. 3. Saiical exreme value aalyi of ANC axiway ceerlie deviaio for 747 aircraf. FAA/Boeig Cooperaive Reearch ad Developme Agreeme - CRDA-64. hp:// ANC_747.pdf CEAUL ad DEIO, Faculy of Sciece Uiveriy of Libo, Libo -38, Porugal. iabel.alve@fc.ul.p Deparameo de Maemáica, Uiveridade de Aveiro, Campu Uiveriario Saiago, Porugal. claudia.eve@ua.p Received July 3; acceped Jauary 4