A Sigle-Idex Cx Mdel Drive by Lévy Subrdiatrs Ruixua Liu Departmet f Ecmics Emry Uiversity Atlata, Gergia 3322 First Versi: September 215 This Versi: February 216 Abstract I prpses a ew durati mdel where the durati utcme is de ed as the rst time a Lévy subrdiatr a stchastic prcess with statiary, idepedet ad -egative icremets crsses a expetial threshld, ad the e ect frm bservable cvariate is actig multiplicatively the latet prcess. Our speci cati is a atural variat f the mixed prprtial hazard mdel frm a stchastic prcess pit f view. Whe the multiplicative e ect is parameterized as a liear idex, ur mdel reduces t a sigle-idex prprtial hazard mdel, where the ukw lik fucti captures the characteristic f the latet prcess. The large sample prperty f a sieve maximum partial likelihd estimatr f the ite dimesial parameter is studied with right cesred data. Keywrds: First Passage Time, Lévy Prcesses, Sieve Partial Likelihd, Semiparametric E ciecy, Sigle-Idex Mdels JEL Cdes: C14; C24; C41 Email: ruixua.liu@emry.edu. This paper is based the rst chapter f my dissertati submitted t Uiversity f Washigt. I wuld like t thak Prfessrs Jaap Abbrig, Yaqi Fa, Jiahua Z. Huag, Rger Klei, ad J Weller fr helpful cmmets.
1 Itrducti Sice the semial wrk f Cx (1972), the prprtial hazard mdel (PH) has uarguably bee the fcal pit i the durati r survival aalysis. PH is semiparametric i ature where the hazard desity fucti f the durati utcme r survival time T with cvariate X is speci ed as: T (tjx) lim P ft T < t + jt t; X = xg (1.1) = (t) exp x ;! 1 depedig a parametric baselie hazard desity fucti (t) ad a liear idex x via the expetial lik fucti. Partial likelihd methd (Cx, 1975) is used t estimate the regressi ce ciet withut estimatig the baselie hazard fucti give right cesred survival data. With the estimatr f i had, the estimati f the baselie cumulative hazard fucti (t) = R t (s) ds readily fllws frm Breslw s prpsal (Breslw, 1972). This estimati prcedure delivers rt- csistet ad asympttic rmal estimatrs fr bth the ite dimesial regressi parameter ad cumulative baselie hazard fucti, see Aderse ad Gill (1982). Mre imprtatly, the estimatrs are semiparametrically e ciet i the sese f Begu, Hall, Huag ad Weller (1983). Despite its theretical elegace, PH cmes with several restrictis. First f all, the cvariates e ect is liear i the cditial (lg-)hazard fucti. Secd, the plai Cx mdel des t icrprate ay ubservable hetergeeity term. Last but t least, PH may t be applicable t structural duratis triggered by stchastic prcesses, which aturally arises frm ptimal stppig time prblems i ecmics. Ideed, develpmets have bee made i the literature extedig the scpe f PH i all three directis. Maitaiig the prprtial hazard structure, the fuctial frm exp x ca be relaxed t a fully parametric versi as i Tibshirai ad Hastie (1987), Fa, Gijbels ad Kig (1997), Che ad Zhu (27) r semiparametric variats i Nielse, Lit ad Bickel (1998), Huag (1999), Fa, Li ad Zhu (26). Itrducig a multiplicative hetergeeity term (als kw as the frailty term) i the cditial hazard desity fucti, e arrives at Lacaster s (1979) mixed prprtial hazard mdel (MPH). Fially, the recet surge f prcess-based durati mdels f Abbrig (212) r Btsaru (213) als brrw several isights frm Cx s rigial cstructi, while beig able t capture ecmic agets ptimizig behavirs i a dyamic setup with certai time-varyig hetergeeity. I this paper, I prpse a ew durati mdel that iherits prmiet features frm all three afremetied braches. Here the durati utcme is de ed as the rst time a Lévy subrdiatr a stchastic prcess with statiary, idepedet ad -egative icremets crsses a expetial threshld, ad the e ect frm bservable cvariate is actig multiplicatively the latet prcess. Such a mdel is f substatial iterest because t ly is it related t ptimal stppig time mdels where agets ptimally time their discrete actis, but it als applies t the sceari i which the termiati f durati is caused by sme gradual ad irreversible accumulati f damage. I fact, I prvide a ui ed framewrk f durati mdellig, where the durati utcme is speci ed as the rst passage time f a latet stchasitic prcess crssig ver a radm threshld, icludig bth traditial hazard-based mdels ad recet prcess-based 1
mdels. Thereafter, ur mdel culd be see as a atural variat f the mixed prprtial hazard mdel frm a stchastic prcess pit f view. Eve thugh the mdel is cstructed i a highly structural way, its empirical ctet is very tractable. Whe the multplicative e ect is parameterized as a liear idex, ur mdel exhibits a sigle-idex structure via the cditial hazard fucti: h T (tjx) = (t) exp i x with a ukw smth lik fucti (). This sigle-idex mdel icrprate the liear e ect while alleviatig the curse f dimesi whe multivariate cvariates are csidered. The restrictive parametric lik fucti such as the idetity map i PH is avided, while greater estimati precisi fr ite dimesial parameter is still pssible, i ctrast t fully parametric estimati. Ntice this ukw lik fucti is t just mtivated frm techical relaxati, but it is related t the characteristics f the latet stchastic prcess. The ctributis f ur paper are maily threefld. First, we preset a ui ed perspective f mdellig durati utcmes. The sythesis is that we culd always view the durati r survival time as the rst passage time f a certai stchastic prcess, bridgig the cceptual gap betwee the existig hazard-based mdels ad the prcess-based mdels. (1.2) Secd, ur ew cstructi f the structural durati mdel drive by Lèvy subrdiatrs culd be see as a atural variat f MPH where this latet prcess replaces the rle f the static frailty term, embeddig time-varyig hetergeeity. The tractability assciated with Lèvy subrdiatrs lead us t ivestigate ecmetric prperties f the sigle-idex PH mdel. The third ctributi f this paper is a cmplete study f the large sample prperties f the estimatr fr frm sieve partial likelihd maximizati. Namely, the sieve maximum partial likelihd estimatr fr is prved t be asympttically rmal with rt- cvergece rate ad it achieves the semiparametric e ciecy bud. Furthermre, I preset the e ciet scre fucti ad ifrmati bud i explicit frms, ad prve the btstrap csistecy with geeral exchagable btstrap weights. We are w i a psiti t describe the ageda fr this paper. Secti 2 ctais a brief survey f the related literature. Secti 3 prvides a ui ed framewrk f the durati mdellig. Thereafter, I preset the sigle-idex Cx mdel drive by Lévy subrdiatrs as well as its ideti ability ad estimati i Secti 4, while Secti 5 ctaiig mai theretical results. A small scale Mte Carl experimet is cducted i Secti 6 examiig the ite sample perfrmace f the btstrapped c dece itervals. We cclude i Secti 7. Techical lemmas ad prfs are cllected i the Appedix. Fr a vectr v, we let v traspse f it ad v 2 = vv i the sequel. 2 Literature Review be the I this secti, we rst review the related literature extedig the scpe f Cx s PH mdel i three categries. The we discuss the existig develpmet f sigle-idex PH mdel. Althugh the baselie hazard fucti is left uspeci ed i Cx mdel, the e ect f cvariates the 2
(lg-)hazard fucti is restricted t be liear. Obviusly, ay misspeci cati f the e ect f cvariates wuld lead t icsistet estimati ad misleadig iferece, which mtivates csiderati f exible parametric r semiparametric methds. Fr a parametric smth risk fucti (x) measurig the cvariates e ect, Tibshirai ad Hastie (1987) iitiate the lcal partial likelihd fr estimatig the parametric Cx mdel. Subsequetly, Fa, Gijbels ad Kig (1997), Che ad Zhu (27) have develped large sample thery fr estimatig the derivative r the di erece f the ukw risk fucti. Fr varius semiparametric sibligs, we refer the readers t the semiparametric mdel with parametric baselie fucti i Nielse, Lit ad Bickel (1997), the partial liear ad additive mdel i Huag (1999), the fuctial ANOVA mdel i Huag, Kperberg, Ste, ad Trug (2), ad the varyig-ce ciet mdel i Fa, Li, ad Zhu (26). The characterizati f idividual hetergeeity is a cetral issue i ecmetric durati aalysis. Igrig the ubservable hetergeeity wuld ievitably bias the estimati ad iterpretati f the e ect frm explaatry variables. Lacaster s (1979) mixed prprtial hazard (MPH) mdel has becme a stadard tlkit allwig researchers t icrprate bth bserved ad ubserved variables it the cditial hazard fucti f the durati utcme. Relaxig the parametric assumpti i Lacaster (1979), Elbers ad Ridder (1982), Heckma ad Sigh (1984) have demstrated the pssibility f idetifyig MPH with a sigle spell f durati i a cmpletely parametric way, further bradeig its scpe. Whe the hetergeeity fllws the gamma distributi as i Lacaster (1979), Ha ad Hausma (199) prpse a estimatr assumig the baselie hazard fucti is piece-wise cstat. Hrwitz (1999) cmes up with the rst parametric estimatr allwig bth baselie hazard ad frailty s distributi t be uspeci ed, see als Che (22). Withut parameterizig the hetergeeity term s distributi fucti, the semiparametric e ciecy bud f MPH fr the ite dimesial parameters i multiplicative e ect ad baselie hazard fucti culd be sigular as rst realized by Hah (1994), which excludes the pssibility f btaiig rt- csistet estimatrs. Ridder ad Wuterse (23) preset su ciet cditis which avid this sigularity. Fr this regular semiparametric mdel, Bearse, Caals-Cerda, ad Rilste (27) have studied semiparametric e ciet estimati. Hausma ad Wuterse (214) prpse a di eret rt- csistet estimatr fr the ite dimesial parameters icrpratig time-varyig cvariates ad discretely bserved duratis. Needless t say, the hazard based durati mdels play dmiat rles i empirical applicatis, but ecmic thery i geeral des t lead t the abve speci cati, which cmplicates the structural iterpretati f varius reduced-frm estimates (va de Berg, 21). A typical durati f iterest t ecmists very fte arises frm slvig a ptimal stppig time prblem, where ptimizig ecmic agets make decisis abut the time at which t switch frm e state t ather, such as i rms etry/exit chice (Dixit, 1989), wrker s uemplymet decisi (Mrtese ad Pissarides, 1994), ad real pti type ivestmet (McDld ad Siegels, 1986). I these examples, the structural durati is related t sme threshld crssig behavir f a latet stchastic prcess (Abbrig, 212). Ideed what is usually 3
disregarded i the traditial apprach t durati r survival aalysis is that the particular durati is the ed pit f sme prcess (Aale, Brga, ad Gjessig, 28; Lee ad Whitmre, 26). Parallel t the traditial hazard-based mdelig strategy, may prcess-based mdels have bee develped by makig use f a speci c parametric sub-class f Lévy prcesses. Fr example, the Brwia mti appears i Lacaster s (1972) strike mdel ad Whitmre s (1979) jb teure mdel, while the gamma prcess has bee utilized i Sigpurwalla (1995), Lawless ad Crwder (24) t mdel the degradati prcess. Abbrig (212) has pieered the study f ideti ability f prcess-based durati mdels, withut makig ay parametric assumpti. Abbrig (212) speci es the durati frmally as rst passage time f a radm threshld by the sample path f a latet stchastic prcess i his Mixed Hittig-Time mdel (MHT). Abbrig (212) has demstrated that t ly MHT icrprates the time-varyig hetergeeity represeted by the latet prcess, but als it arises mre aturally frm the ptimal stppig time prblem where the sluti culd be described by this threshld-crssig behavir. Whe the prcess is chse t be the spectral egative Lévy prcess, the ideti ability f all structural cmpets culd be shw, adapted frm the strategies aalyzig the ideti ability f MPH. S far the ideti cati f MHT is restricted t the uivariate durati mdel, ad the suggested estimati is fully parametric. Btsaru (213) prpses ather uivariate durati mdel makig use f the Lévy subrdiatr, which culd be either viewed as a hazard-based mdel with certai radm hazard desity r a prcess-based mdel with the latet prcess beig a duble stchastic itegral f the Lévy subrdiatr. Ideti cati i Btsaru s (213) radm hazard desity mdel relates t slvig a liear itegral equati. Fr the ideti ed case, Btsaru (213) presets a sieve type estimatr ad prves its csistecy. There is dubt that the sigle-idex structure is e f the mst ppular semiparametric mdellig chice fr cditial mea fucti (Ichimura, 1993), cditial distributi fucti (Klei ad Spady, 1993; Che, 27), r cditial quatile fucti (Ichimura ad Lee, 21) ad its extesi t the Cx mdel has already bee evisied earlier (Ichimura, 1993; Nielse, Lit ad Bickel; Che, 27; Dig ad Na, 211); Hwever, the rigrus large sample thery fr (1.2) has t bee preseted i a etirely satisfactry fashi whe right cesred data is available. Che, Li, ad Wag (1999) are amg the rst t csider the mdel i (1.2) based sliced iverse regressi techique which requires very strg assumptis the cvariates, such as symmetry. Wag (24) suggests t iterate certai kerel based estimatr while assumig sme prelimiary rt- csistet estimatr fr is available befrehad, which is presumably take t be the estimatr i Che, Li, ad Wag (1999). Huag ad Liu (26) have prpsed maximizig sieve partial likelihd with beig replaced by a plymial splie fucti, but the umber f kts is held t be xed. Csequetly, they treat the lik fucti exactly t be sme splie fucti uder csiderati ad derive parametric type asympttics assumig e f the splie fuctis with the prirly determied umber f kts is the true ukw lik fucti. 4
3 A Ui ed Perspective f Duratis Mdellig We prvide a ui ed perspective the structural durati mdel described by the threshld-crssig behavir f sme latet prcess, icludig PH i Cx (1972), MPH i Lacaster (1979), MHT i Abbrig (212), ad a radm hazard desity mdel (RHD) i Btsaru (213). Nw we start with a geeric durati mdel where the durati is related t the threshld crssig behavir f a latet stchastic prcess. Frmally, the structural durati is de ed as the rst passage time f a latet prcess (t) crssig a radm threshld e, with structural prprtial e ect (X) actig the prcess: T if ft : (X) (t) eg : (3.1) I all mdels discussed i the paper, X, (t) ad e are assumed t be mutually idepedet. The rule speci ed i (3.1) resembles the classical biary chice mdel clsely i a dyamic setup. 1 But itead f makig a discrete chice based static cmpariss, it stems frm the ptimal stppig time mdel where aget ptimally chses whe t switch t ather state at time T with cmplete ifrmati, as i de Paula ad Hre (21). Namely, the ecmic aget is facig a utility maximizati prblem where the curret state geerates a pay t the aget with e ad switchig t ather state gives (X) (t), 2 i.e., the durati i (3.1) is the ptimal sluti f ( Z T max e exp ( s) ds + T with the discut rate equal t. Z 1 T (X) (t) exp ( s) ds I spite f the fact that PH r MPH is cstructed frm a regressi mdelig the hazard rate, it is iterestig t ask whether it culd be put it the same framewrk where the durati is speci ed as the rst passage time f a latet prcess. Ideed this is pssible. This alterative pit f view further reveals the restrictis the latet prcess ad threshld impsed by MPH, ad it prvides the lik f ur speci cati t MPH. Example 3.1 (PH) It turs ut that the durati i Cx mdel culd be equivaletly de ed as i (3.1) T if t : exp X (t) e by settig (t) equal t the cumulative baselie hazard fucti (t). Furthermre, the multiplicative e ect is parameterized as exp X, ad the threshld fllws uit expetial distributi, i.e., e Exp (1) : The asserti is easy t see as Pr ft > tjx = xg = Pr e > (t) e x jx = x = exp ) (t) e x : 1 I discrete-time settig, Heckma ad Navarr (27) have cstructed a geeral mixture durati mdel based a latet prcess crssig threshlds. 2 Because ly the cmparis plays a rle frm the aget s perspective, it is t restrictive t let e side t becme time ivariat. (3.2) 5
This alterative view is favred by Sigpurwalla (26) wh has advcated this iterpretati f the cumulative fucti (t) is mre sesible tha merely treatig it as the primitive f a hazard desity. Example 3.2 (MPH) The durati i a parametric MPH culd be als de ed as i (3.1) by T if ft : (X) (t) eg ; (3.3) where the latet prcess is equal t (t) = (t) with the cumulative baselie hazard fucti (t). Agai, the threshld fllws uit expetial distributi, i.e. e Exp (1). I thse tw classical hazard-based mdels, the drivig stchastic prcess is either determiistic (as i PH) r its radmess is pied dw at time zer (the frailty term is static i MPH). I MPH give the realizati f, we will just have a a determiistic ad decreasig tred (t) apprachig the threshld frm belw. The idividual hetergeeity des t evlve ver time alg the etire spell f durati, hece it is certaily desirable t cstruct a mdel with time-varyig hetergeeity. Apparetly, e has t make sme assumptis the uderlyig class fr (t), i rder t maitai a tractable structure. I the sequel, we shall restrict ur atteti t geeral Lévy prcesses. Lévy prcesses cstitute a very rich ad attractive class f stchastic prcesses, icludig the cmmly ecutered Brwia mti, gamma prcess ad stable prcess as special cases. They have attracted csiderable atteti due t the exibility fr a wide variety f mdelig issues i ace, isurace ad egieerig, see Kypriau (26). I the study f durati mdels, parallel t the traditial hazard-based mdelig, may prcess-based mdels have bee develped by makig use f a speci c parametric sub-class f Lévy prcesses. Fr example, Brwia mti appears i Lacaster s (1972) strike mdel ad Whitmre s (1979) jb teure mdel, while gamma prcess has bee utilized i Sigpurwalla (1995), Lawless ad Crwder (24) t mdel the degradati prcess. Abbrig (212) has pieered the study f ideti ability f prcess based durati mdels, withut makig ay parametric assumptis. Befre reviewig his ctributi, it is wrthwhile t itrduce the frmal de iti f a uivariate Lévy prcess ad tw imprtat subclasses. T avid digressis, we refer the readers t Sat (1999) ad Kypriau (26) fr mre detailed discussis. De iti 3.3 A uivariate Lévy prcess L (t) is a right ctiuus stchastic prcess with left limits such that fr every t ad r, the icremet L (t + r) L (t) is idepedet f fl (s) ; s tg ad has the same distributi as L (r). De iti 3.4 A uivariate Lévy subrdiatr L (t) is a Lévy prcess with almst surely decreasig sample path, i.e. fr t s e has L (t) L (s). De iti 3.5 A uivariate spectral egative Lévy prcess L (t) is a Lévy prcess with psitive jumps. The mst remarkable prperty f a Lévy prcess is that eve it is purely de ed i terms f descriptive features abut the sample path, it admits a very ccrete aalytic characterizati via the well-kw 6
Lévy-Khitchie represetati, see Sat (1999). Speci cally, its Laplace trasfrm culd be expressed as E fexp [ zl (t)]g = exp [ t (z)] ; where () is the s-called Lévy-Laplace expet fucti, which is parametric ad time-ivariat. This elegat aalytic characterizati plays a crucial rle i ideti cati prblems f thse durati mdels drive by Lévy prcesses, icludig Abbrig (212) ad Btsaru (213). Example 3.6 (MHT) Mtivated frm the ptimal stppig time prblems, Abbrig (212) starts with a mdel where the structural durati is frmally de ed t be T if t : (X) L ~ (t) e ; (3.4) where e is a arbitrary radm threshld ad ~ L (t) is a spectral egative Lévy prcess. The empirical ctet f MHT is revealed thrugh the cditial Laplace trasfrm L T (jx) f the structural durati T : L T (sjx) E fexp ( st ) jxg = L e ~ (s) (X) ; (3.5) where L e is the Laplace trasfrm f e ad ~ (s) is the largest rt satisfyig ~ (s) = s. Despite that there are quite di eret fuctis (i.e. L T (sjx) ad ~ (s)) appearig i (3.5), the mathematical structure is almst idetical t the e i MPH. Assumig regular variatis f certai fuctis, Abbrig (212) has shw e culd idetify the triple L e ; ; ~ frm L T (sjx) parametrically. Example 3.7 (RHD) Give ur discussi likig the hazard-based mdels t their threshld-crssig behavir ver the uit expetial threshld, it is clear w that the durati i Btsaru s (213) radm hazard desity mdel culd be expressed as T if ft : (X) (t) eg ; where the latet crssig prcess is a duble stchastic itegral f the Lévy subrdiatr, as (t) = R t R s f (u) dl (u) ds with a trasfrmati fucti f (u). subrdiatr, the cditial survival fucti culd be writte as Z t S T (tjx) = exp ( (x) f (u) (t Whe the prcess L (u) is take t be a Lévy u)) du : The parametric ideti cati f (; f; ) is related t slvig a liear Vlterra itegral equati f the rst kid with ukw kerel, see Btsaru (213). 4 A New Durati Mdel Ispired by Abbrig (212) ad Btsaru (213), I cstruct a ew durati mdel uder the framewrk f (3.1), replacig the prduct f the cumulative baselie hazard fucti ad the frailty term (t) i MPH all tgether with a latet stchastic prcess L ( (t)). Here L () is a Lévy subrdiatr ad (t) is 7
a determiistically icreasig time-chage fucti. This time-chage fucti culd be see as a suitable trasfrmati frm certai abstract time scale that the prcess evlves t the rate f ecmic trasactis (Kypriau, 26), which simply turs ut t be the cumulative baselie hazard fucti. Hece the durati i ur mdel becmes T if ft : (X) L ( (t)) eg ; (3.6) where the radm threshld is still assumed t be uit expetially distributed. Ntice the latet Lévy subrdiatr L () has replaced the static frailty term ad its sample path prperty iherits the key feature f a mtic drivig tred i MPH. Ather clse cecti t MPH is that the distributi f frailty term is fte take t be i itely divisible, see Hugaard (1986). Crrespdigly i ur preset mdel, it s well kw the Lévy subrdiatr has -egative i itely divisible distributi (Sat, 1999). The cditial survival fucti f T i (3.6) culd be derived i a straightfrward way by the Lévy- Khitchie represetati: S T (tjx) = exp [ (t) ( (x))] ; (3.7) which leads t the cditial hazard desity fucti beig T (tjx) = (t) ( (x)) ; (3.8) where (t) is the derivative f (t). Our ew mdel cmplemets the traditial hazard-based mdels ad the recet study i Abbrig (212) ad Btsaru (213), with the great advatage that the prprtial hazard structure is maitaied as i (3.8). The target here is t t argue the curret mdel (3.6) is mre geeral tha ay ther existig prpsal, but maily t er a iterestig ad smewhat surprisig alterative which delivers very tractable ecmetric r statistical prperties. Nw we preset sme examples with the latet Lévy subrdiatr restricted t be chse frm speci c parametric sub-classes, which make the fucti () parameterized. Example 4.1 Whe L () is the gamma prcess, ur mdel culd be see as a atural variat f MPH with gamma frailty term as rigially i Lacaster (1979). Oe ice prperty f the gamma frailty mdel is that the cditial distributi f frailty term, give survival util ay time (i.e. cditial Y > t), is als gamma with the rigial shape parameter, see va de Berg (21). Abbrig ad va de Berg (27) prvide ather justi cati, shwig that the distributi f frailty term amg survivrs wuld always cverge t a gamma distributi up suitable rmalizati whe t! 1. I Figure 1, we plt a typical realizati f the rst passage time r structural durati with gamma prcess i ur mdel (1.2). Its cditial survival fucti is S T (tjx) = exp [ (t)] 1 + (x) ; where (; ) stad fr the scale ad shape parameter i the gamma distributi. 8
Isert Figure 1 Here Example 4.2 Whe L () is the stable prcess, ur mdel culd be see as a atural variat f MPH with stable frailty term as rigially i Hugaard (1986). Oe advatage f the stable frailty mdel uder MPH is that e still gets a prprtial hazard mdel whe itegratig ut (see Secti 5.2 i va de Berg, 21), which is t the case fr the gamma frailty mdel. I Figure 2, we plt a typical realizati f the rst passage time r structural durati with stable prcess i ur mdel (1.2). Its cditial survival fucti is S T (tjx) = exp [ (t) (x) ] ; where (; ) represet the stable idex ad pwer parameter i the stable distributi. Cmpared with gamma prcess, the sample path f stable prcess is mre irregular ad exhibits larger jumpig magitude. Isert Figure 2 Here Example 4.3 Whe we take the Levy subrdiatr t be a cmpud Piss prcess, i.e., L (t) = P N(t) i=1 i where the arrival f shcks is gvered by a (hmgeeus) Piss prcess N (t) with hazard rate ad idividual i:i:d:shcks f i g are assumed t -egative. I the absece f cvariate X, ur mdel bils dw t the radm shck mdel studied by Esary, Marshall ad Prscha (1973). Uder this circumstace, the margial survival fucti f T culd be determied explicitly by mdel primitives withut referrig t the Lévy-Khitchie represetati. The margial survival fucti f T is equal t 1X (t) k exp ( Pr ft > tg = k! k= t) P k ; where P k is the prbability f survivig after k shcks: P k = Pr f 1 + ::: + k eg : It is well kw that the cmpud Piss prcess has a piece-wise cstat sample path, ad it is the ly prcess whse Lévy measure is ite (see Sat, 1999). Isert Figure 3 Here 4.1 Ideti cati f the Sigle-Idex Mdel Give the prprtial hazard structure i ur mdel (3.7), disetaglig the e ects frm cvariates ad latet stchastic prcesses bils dw t study the cmpsiti f tw fucti. Oce the multiplicative e ect (x) is parameterized t be exp x as i the Cx regressi, we actually arrive at a sigle-idex prprtial hazard mdel with ukw lik fucti () = lg exp () ; (3.9) 9
as i (1.2). The ideti cati f all mdel primitives i (3.7) culd be achieved withut ivkig the ideti cati-at-limit strategy, i ctrast with Elbers ad Ridder (1982), Heckma ad Siger (1984), r Abbrig (212). It is well-kw the ite dimesial parameter is ly ideti ed upt scale i the sigle-idex mdel (Ichimura, 1993), s we adpt the fllwig cveti by xig the rst elemet t be 1 (see Ichimura ad Lee, 21). We partiti the cvariate vectr X = X 1 ; X 2 with a uivariate cmpet X 1 ; ad stack with e 1 as e = 1;. Mrever, we dete X = X 1 + X2 with the true parameter. The regularity cditis belw required fr ideti cati are fairly weak. Assumpti (I1) The supprt f X is a cvex set with at least e iterir pit. Assumpti (I2) The supprt f X ctais a empty pe iterval. Assumpti (I3) Assume E [ (X )] =. Prpsiti 4.4 Let (x) = exp x e i (3.7) ad assume Assumptis (I1)-(I3), the the triple (; ; ) is ideti able. Our speci cati regardig i hw the cvariates eter it the mdel is ly e f may pssible chices. Ideed, e culd csider a parametric multiplicative fucti as (x) = 1 (x 1 ) d (x d ), ad the prblem bils dw t the ideti cati f the trasfrmati mdel (Hrwitz, 1999; Chiappri, Kmujer, ad Kristese, 215) r the geeralized accelerated failure time mdel (Ridder 199; Abbrig ad Ridder, 215). T be csistet with the asympttic aalysis i sequal, we shall fcus the sigle-idex mdel. 4.2 Sieve Maximum Partial Likelihd Estimati I practice, survival data is fte right cesred due t termiati f the study r early withdrawal frm the study, s we bserve the radm sample csistig f i:i:d data f Z i = (V i ; i ; X i ) where V i = mi (T i ; C i ) ad i = I [T i C i ]. The lgarithm f partial likelihd fucti (eglectig terms idepedet f parameters f iterest) is l (; ) = 1 X i=1 i 8 < : X i1 + Xi2 lg @ X h exp k:v k V i k2i1 9 = X k1 + X A ; : Sice the smth lik fucti is ukw, it is atural t replace it by a apprximatig B-splie fucti 3 () (Schumaker, 1981). Let T K = ft 1 ; :::; t K g be a set f partiti pits f [a; b] with K = O ( ) ad max j jt j t j 1 j = O ( ) fr sme cstat 2 (; 1=2). Let S (T K ; K ; p) be the space f plymial splies f rder p 1, the there exists a set f B-splie basis fuctis fb j ; 1 j q g with q = K + p s.t. fr ay s 2 S (T K ; K ; p) we ca write s () = P q j=1 jb j () : Nw with () = P q j=1 jb j () substitutig, we are maximizig the fllwig criteri fucti i terms f parameters (; ): 8 l (; ) = 1 X < i : i=1 q X j=1 j B j X i1 + Xi2 2 lg @ X exp 4 k:v k V i 3 Other sieve basis culd ptetially wrk as well, see thse preseted i Che (27). q X j=1 j B j X k23 19 = k1 + X 5A ; : 1
A Newt-Raphs algrithm r ay gradiet-based search algrithm ca be applied t slve fr the scre equatis t get estimatrs b; b. Give the ideti cati restricti f, we will ceter the estimatr as fllws. Let b () = q X j=1 b j B j () ad () = The resultig estimatr f is a cetered versi ad de ed t be b () b () () ; P i=1 i b () P i=1 : i s it satis es P i=1 b i X i1 + X b i2 =, see a similar prcedure i Huag (1999). We refer readers t Huag ad Liu (26) fr detailed descriptis f the cmputati ad simulati results. 5 Theretical Results It is clear that ur cstructi results i a sigle-idex Cx mdel where the ukw lik fucti relates t the Lévy expet fucti f the uderlyig Lévy subrdiatr. I fact, this lik fucti de ed by (3.9) is i itely rder di eretialble, because the Lévy expet fucti () pssesses derivatives f ay ite rder with alteratig sigs. Frmally, () is a Berstei fucti, see Schillig, Sg ad Vdracek (212). Frm w, we fcus a mre geeral class f mdels where the lik fucti () is regularly smth, hece it is ly required t be di eretiable upt a ite rder t be speci ed later. A clse examiati f the exisitig literature semiparametric estimati thery reveals the velty ad techicality assciated with (1.2). parameter (; (; )) where (x; ) = I ctrast with Huag (1999), we are dealig with the budled x e i the preset sceari. This termilgy is used by Huag ad Weller (1997) referrig t statistical mdels where the parameter f iterest ad uisace parameter are budled tgether. Geeral thery fr hadlig semiparametric estimati with budled parameter has bee develped by Dig ad Na (211) fr sme criteri fucti which culd be writte as the sample average f i:i:d: bservatis. I ur estimati, the sample criteri fucti is chse t be the lgarithm f partial likelihd fucti, which is either a average f i:i:d: terms as i Ichimura ad Lee (21), Dig ad Na (211), r weakly depedet as i Che (27) due t the cutig ver the at-risk set, see Aderse ad Gill (1982). Cmpared with Na ad Weller (213), we have t apprximate the addtial ukw fucti () by splie sieves. I fact, Dig ad Na (211) lists the extesi t the sigle-idex Cx mdel as a challegig pe prblem. successful marriage f the afremetied papers. All that said, ur large sampel thery fr (1.2) is ideed a Nw we itrduce sme ecessary tatis. Let P be the empirical measure f Z i (V i ; i ; X i ) ad let P be the assciated prbability measure. Let P be the subprbability empirical measure f Z i whe i = 1, with its ppulati versi P. The liear fuctial tati is cveiet, as P f = R fdp = 1 P i=1 if (Z i ). Recall that we let the true parameters be deted as ( ; ; ), ad X = X 1 ; X 2 ad X = X 1 + X2. Furthermre, stack with e 1 as e = 1;. The fllwig 11
stadard cutig prcess tatis i Aderse ad Gill (1982) wuld facilitate subsequet presetati, s N i (t) = I [V i t; i = 1], Y i (t) = I [V i > t] ad M i (t) = N i (t) Z t h Y i (u) exp X i1 + X i2 i (u) du; (3.1) are the crrespdig cutig prcess, at-risk prcess ad assciated martigale respectively. The fllwig imprtat bservati by Sasiei (1992a) wuld appear repeatedly i the remaiig secti E [K (t) g (Z)] = E [g (Z) jv = t; = 1] (3.11) where The regularity assumptis are listed belw. h i Y (t) exp X i1 + X i2 K (t) = E Y (t) exp X i1 + X i2 : (3.12) Assumpti (A1) We have i:i:d: bservatis f Z i (V i ; i ; X i ). The durati utcme T ad cesrig time C are cditially idepedet give cvariates X. Assumpti (A2) The ite dimesial parameter space B is a cmpact subset f R d ad the true parameter is a iterir pit f B. The cvariate X has buded supprt ad fr ay 6= we have Pr X2 6= X2 >. EX2 2 is a strictly psitive de ite matrix. Assumpti (A3) The trucati time < 1 satis es () = R (t) dt < 1; mrever Pr ( = 1jX) > ad Pr (T > jx) > almst surely. Assumpti (A4) Let < c 1 < c 2 < 1 be tw cstats. The jit sub-desity f satis es c 1 f t; x e ; = 1 < c 2 fr all t; x e 2 [; ] [a; b]. Assumpti (A5) Fr tw arbitrary uivariate fuctis f 1 ; f 2, we have t; x e ; = 1 f 1 (V ) + f 2 (X ) = ; a:s:-p if ad ly if f 1 (V ) = f 2 (X ) = ; a:s:-p Assumpti (A6) Let p dete the cllecti f buded fuctis [a; b] with buded derivatives (j), j = 1; :::; k ad the k-th derivative (k) satis es the fllwig Lipschitz cditi: (k) (s) (k) (t) L js tj fr 8s; t 2 [a; b] where k is a psitive iteger ad 2 (; 1] s.t. p = k + 3 ad L < 1. The true ukw lik fucti () 2 p. Remark 5.1 Assumptis (A1)-(A3) are stadard fr derivig large sample prperties f estimatrs i PH r its semiparametric variats, see Aderse ad Gill (1982), Huag (1999), Dig ad Na (211), Na ad Weller (213). The cditial idepedece assumpti i (A1) esures the cesrig mechaism des t a ect the ideti cati. Fr the depedet cesrig ad partial ideti cati aalysis, we refer readers t Fa ad Liu (213). Assumptis (A4)-(A5) are eeded t esure the prjectis ad ifrmati bud are well de ed while e csiders the prjecti t the sumspace f taget sets, see Sasiei 12
(1992b). Assumpti (A6) impses smthess restricti f the ukw lik fucti i rder t utilize the parametric estimati. Ntice that ur pstulated mdel drive by Lévy subrdiatrs leads t a lik fucti pssessig derivatives f ay rder, hece it satis es the Assumpti (A6) autmatically. 5.1 Semiparametric Ifrmati Bud I this secti, we calculate the semiparametric ifrmati bud fr the estimati f. We refer readers t Bickel, Klaasse, Ritv, ad Weller (1993) fr a authritative ad bk-legth treatmet f the semiparametric ifrmati bud fr parameters i i ite dimesial mdels. I the stadard sigle-idex mdel via the cditial mmet restrictis, the calculati f the e ciet scre fucti is de by prjectig t a sigle taget space (Ichimura, 1993; Klei ad Spady, 1993; Newey ad Stker, 1993). I ctrast, we have tw uisace parametric cmpets i the mdel. Thereafter, we have t csider the prjecti t a sum-space f tw -rthgal taget spaces (see Bickel, Klaasse, Ritv, ad Weller, 1993). The iterated prjecti preseted belw culd be viewed as a variat f the classical result f Frisch ad Waugh (1933) t btai the least-squares estimates fr a sub-vectr, which states the estimated regressi parameter f iterest is algebraically equal t ru the least-squares regressi f e set f residuals agaist the ther. Recall the true parameters are deted as ( ; ; ), ad X = X 1 ; X 2 ad X = X 1 + X 2. First, te that the lg-likelihd fucti fr a sample f size e is h i X e + (V ) exp X e (V ) ; drppig the terms which d t ivlve parameters f iterest. Csider a parametric smth submdel f () : 2 Rg ad f () : 2 Rg that rus thrugh the true mdel, i.e., () = () ad () = (). Mrever, we let a () = @ @ lg () ad h () = @ @ lg () represet pssbile directis that ca apprach the true mdel. Thereafter fllwig Sasiei (1992ab), the scre vectr fr regressi parameter is X e X 2 _l (Z) = Z = _ X e X 2 dm (t) ; h i exp X e (V ) _ X e X 2 where M (t) is the atural martigale i (3.1). Likewise, we have the fllwig tw scre peratrs fr the parametric cmpets: where a () = @ @ lg () ad h () = @ @ lg () are pssible directis apprachig () ad () respectively frm sme idex sets. Z _l h (Z) = h X e dm (t) ; Z _l a (Z) = a (t) dm (t) : The atural Hilbert spaces where thse fuctis a () ad h () sit i 13
are L 2;V = a : E a 2 (V ) < 1 ; L 2;X = h : E [h (X )] = ; E h 2 (X ) < 1 : Hece the tw taget sets are A = _l a : a 2 L 2;V ; H = _l h : h 2 L 2;X : Uder this circumstace, there are tw parametric cmpets whse taget sets are t rthgal. We shall rely the techiques i Sasiei (1992ab) t d the e ciet scre fucti fr the regressi parameter. Therem 5.2 The e ciet scre fr estimatig i the sigle-idex Cx mdel is Z h i _l (Z) = _ (X ) X 2 a (t) h (X ) dm (t) where a () ad h () are the uique fuctis miimizig Here they take the fllwig frms with ad h () satis es E _ (X ) X 2 a (V ) h (X ) 2 : h i a (t) = E _ (X ) X 2 h (X ) jv = t; = 1 h i E _ (X ) X 2 h (X ) E _ (X ) X 2 h (X ) jv = t; = 1 jx = x ; = 1 = ; a.s. w.r.t P X : Mrever, the semiparametric ifrmati bud fr estimatig is I ( ) = E h _l (Z) 2i : Prf. Let dete the prjecti peratr. The e ciet scre fucti is f the rthgal prjecti f _ l t the clsure f the uisace sum-space A + H, hece we eed t d the least favrable directi (a ; h ) such that _ l _ l a _ l h is rthgal t the sumspace A + H. Assumptis (A3)-(A5) tgether with Prpsiti 1 i Sasiei (1992b) guaratee this prjecti is well-de ed. Furthermre, Prpsiti 3 i Sasiei (1992b) pits ut a explicit way t calculate this rthgal prjecti as h _l j (A + H )?i h h = _l j (A )?i h j H j (A )?ii : 14
Hece, we shall rst elimiate the hazard fucti by prjectig the scres l ad l t the taget space fr the hazard the ad prjectig the residual f the prjecti f l t the subspace geerated by the residual scre f the prjecti f l. By Therem 1 f Sasiei (1992b), the tw residual scres are Z K = D X (X ; t) dm (t) ; ad Z Kh = Dh (X ; t) dm (t) ; where ad D X (x ; t) = _ h i (x ) x 2 E _ (x ) x 2 jv = t; = 1 ; Dh (x ; t) = h (x ) E [h (x ) jv = t; = 1] Thus, the least favrable directi h miimizig E kk Khk 2 satis es E [(K Kh ) Kh] = (3.13) fr all h 2 L 2;W. By Lemma 1 i Sasiei (1992b), (3.13) is equivalet as D (D X Dh ) = a.s. where D is the adjit f peratr D. Nw equatis (ii) ad (iii) frm Lemma 3 i Sasiei (1992b) give us h i D D X = E _ (X ) X 2 E _ (X ) X 2 jv = t; = 1 jx = x ; = 1 D Dh = E fh (X ) E [h (X ) jv = t; = 1] jx = x ; = 1g Whece it is straightfrward t see the e ciet directi h () satis es the fllwig cditi: h i E _ (X ) X 2 h (X ) E _ (X ) X 2 h (X ) jv = t; = 1 jx = x ; = 1 = ; a.s. P X : Mrever, the least favrable directi f the crrespdig hazard fucti is h i a (t) = E _ (X ) X 2 h (X ) jv = t; = 1 : 5.2 Large Sample Prperties The fllwig tatis are stadard i derivig large sample prperties i Cx regressi, see Aderse ad Gill (1982), Huag (1999): S (t; ) = 1 X i=1 h Y i (t) exp 15 i Xi ;
h h S (t; ) = E Y (t) exp S 1 (t; ) [h] = 1 S 1 (t; ) [h] = E X i=1 h Y i (t) exp h h Y (t) exp ii X ; i Xi h (X i ) ; i i X h (X) ; with a real-valued fucti h () depedig x. Furthermre, fr u = (t; x; ) de e The sample criteri fucti is M () = 1 s (u; ) [h] = h (x) s (u; ) [h] = h (x) X i=1 S 1 (t; ) [h] S (t; ) [h] S 1 (t; ) [h] S (t; ) [h] I [V i ] i Xi lg S (V i ; ) : I the sequel, we shall mit writig ut I [V i ] withut cfusi. The ppulati criteri fucti is Ntice _ (x; ) = @ @ x = _ x x. M () = P Xi lg S (V i ; ) : The rst result ccers the csistecy ad rates f cvergece f parameter = (; (; )) i terms f the fllwig metric d ( 1 ; 2 ) = j 1 2 j + k 1 (; 1 ) 2 (; 2 )k where k 1 (; 1 ) 2 (; 2 )k 2 = Z h 1 x 1 2 x 2 i 2 dfx (x) : p Let = S (T K ; K ; p), ad dete H p = (; ) : (x; ) = x ; 2 p ; x 2 X ; 2 B ad p = B H p. The prf f Therem 5.3 is relegated t the Appedix. Therem 5.3 Let K = O ( 1 ) where satis es the restricti 4p 3 < < 1 2p. The give Assumptis A1-A6, we have (i) d b ;! p ad its rate f cvergece is d b ; = O p mi(p;(1 )=2) : Ntice fr = 1 2p+1 we get the ptimal cvergece rate as O p Therem 5.4 Give Assumptis (A1)-(A6), we have p b = I ( ) 1 1 p =) N X i=1 ; I ( ) 1 ; with the e ciet scre _ l ad ifrmati I ( ) i Therem 5.2. p 2p+1 : _l (Z i ) + p (1) 16
The prf f Therem 5.4 is adapted frm Therem 2.1 i Dig ad Na (211). Fr earlier results withut budled parameter issue, we refer the readers t Huag (1999) fr a similar expsiti. Prf. We will prve the fllwig claims separately i Appedix. First f all, tw estimatig equatis hld: P s ; b h i _ (; ) = p 1=2 ; (3.14) P s ; b [h ] = p 1=2 ; Therefre P s ; b [ em ] = p 1=2 ; (3.15) fr the least favrable directi em = _ (; ) fllwig prcesses P s ; b h i _ (; ) P s ; b h i _ (; ) h. Mrever, the stchastic equictiuity hlds fr the h i s (; ) _ (; ) h i s (; ) _ (; ) = p 1=2 ; P s ; b P s [h ] s (; ) [h ] ; b [h ] s (; ) [h ] = p 1=2 ; leadig t P s ; b P s [ em ] s (; ) [ em ] ; b [ em ] s (; ) [ em ] = p 1=2 : Csiderig (3.15), e arrives at P s ; b [ em ] s (; ) [ em ] = P fs (; ) [ em ]g + p 1=2 : Nw we take the fllwig expasi as i Lemma 5.4 f Huag (1999): P s ; b [ em ] s (; ) [ em ] = P fs (; ) [ em ] s (; ) [ em ]g b h P s (; ) [ em ] s (; ) b i = P fs (; ) [ em ]g 2 b + p 1=2 ; where the al equality abve fllws frm the fact that em is the e ciet directi. Thus cmbiig all the results abve, we arrive at p P fs (; ) [ em ]g 2 b = p P fs (; ) [ em ]g + p (1) : 17
Thus, we get p P fs (; ) [ em ]g = 1 p = 1 p + 1 p X Z i=1 X Z i=1 X Z i=1 Xi X i Xi X i 2 h 4 S 1 (t; ) _ (; ) S (t; ) h Xi h Xi h i S 1 (t; ) [ em ] S (t; ) S 1 (t; ) [ em ] S (t; ) S 1 (t; ) [ em ] S (t; ) dm i (t) dm i (t) 3 5 dm i (t) the secd term is egligible by Leglart s iequality as i Aders ad Gill (1982): Hece 1 X i=1 Z S1 (t; ) [ em ] S (t; ) 2 S 1 (t; ) [ em ] Y i (t) exp [ (X i ; )] d (t) = p (1) : S (t; ) als tice that p P fs (; ) [ em ]g = 1 p X Z i=1 Xi X i I the ed, the desired cclusi fllws: h Xi h S 1 (t; ) _ (; ) S (t; ) p P fs (; ) [ em ]g = 1 p X i=1 h i S 1 (t; ) [ em ] dm i (t) + p (1) S (t; ) = a (t) l (V i ; X i ; i ) + p (1) : 5.3 Btstrap Csistecy It is almst uiversal amg semiparametric mdels that the cstructi f a valid c dece set f ite dimesial parameters requires pluggig i additial parametric estimates, which cmplicates the iferece prblems i practice. Cheg ad Huag (21) prve the btstrap csistecy fr a geeric semiparametric M-estimatr with exchagable weights. They have treated the stadard Cx regressi as e demstratig example, but made e simplifyig assumpti that s (; ) directly appears i the estimatig equati istead f s (; ), pertaiig t their i:i:d: assumpti the idividual criteri fucti. I this secti, we prpse a vel btstrap prcedure csistet with the partial likelihd structure ispired by Na ad Weller (213). We establish the btstrap csistecy with geeral exchagable btstrap weights (W i ) i=1. Seekig fr mre geeral weightig scheme ther tha Efr s multimial weights is particularly imprtat, as the later prcedure fte gives t may dies applied t cesred data. De e the weighted empirical prcess as P f = 1 P i=1 W if (X i ) :We assume the fllwig requiremets the btstrap weight. 18
R 1 Assumpti (W1) The vectr W = (W 1 ; :::; W ) is exchagable fr all. Assumpti (W2) W i fr all, i ad P i=1 W i = fr all. Assumpti (W3) Fr sme psitive cstat C < 1, lim sup!1 kw 1 k 2;1 C, where kw 1 k 2;1 = p Pr (W1 u)du. Assumpti (W4) lim!1 lim sup!1 sup t t 2 Pr (W 1 > t) =. Assumpti (W5) (1=) P i=1 (W i 1) 2! p 1. The fllwig quatities serve as the btstrap aalg f S ; S 1, ad s S (t; ) = 1 X h W i Y i (t) exp X i e i ; S 1 (t; ) [h] = 1 i=1 X i=1 h W i Y i (t) exp X e i i h (X i ) ; s S1 (t; ) [h] (t; ) [h] = h (x) S (t; ) [h]: S the btstrapped estimates b ; b are de ed by maximizig 8 2 l (; ) = 1 X < Xq W i i j B j X i1 + X : i2 lg @ X W k exp 4 i=1 j=1 k:v k V i q X j=1 j B j X k23 19 = k1 + X 5A ; : Befre statig the theretical results fr the btstrapped quatities, we shuld be explicit abut the uderlyig prbability space ad surce f radmess. The fllwig set f de itis ad tatis are adapted frm Cheg ad Huag (21). We have the prduct prbability space (Z 1 W; A 1 ; P ZW ) fr the jit radmess frm bserved data ad btstrap weights. Furthermre, the btstrap weights are idepedet frm the sample bservatis, i.e., P ZW = P Z P W. De iti 5.5 Fr a real-valued radm variable, we de e (i) = PW (1) ; i P Z ay "; > P Z PW jz (j j > ") >! : prbability if fr (ii) We de e = O PW (1) ; i P Z prbability if fr ay > there exists a M s.t. P Z PW jz (j j > M) >! : Therem 5.6 Suppse Assumptis (A1)-(A6) ad (W1)-(W5) hld. Fr the btstrapped estimatr b with exchage weights W = (W 1 ; :::; W ), we have that b b = OPW 1=2 i P Z prbability. Furthermre, p sup P WjZ b b x x2r d cdtial bservatis Z almst surely. P N ; I ( ) 1 x! ; 19
A direct csequece f the last therem is the validity f the btstrap percetile type c dece iterval, i.e., lim P ZW ( 2 BC ()) = 1 :!1 where the btstrapped c dece set is de ed t be BC () = [c (=2) ; c (1 =2)] ; with the th quatile f btstrap distributributi c () = if : P WjZ b b, which culd be easily simulated 4. 6 Mte Carl Results As fr the mea square errr f the sieve maximum partial likehd estimati i ite sample, we refer readers t the simulati results i Secti 4 f Huag ad Liu (26). I this secti we examie the ite sample cverage prperties f ur btstrap percetile c dece itervals. We csider tw mdels geerated by a gamma prcess ( = 1=2 ad = 1) ad a stable prcess ( = 1=2 ad = 1) respectively. We wuld als examie a sigle idex mdel with lik fucti () = si () as i the simulati study f Huag ad Liu (26), eve thugh this is t drive by ay Lévy subrdiatr. Referrig t their reduced-frm cditial hazard fuctis, we have Mdel 1: T (tjx) = (t) lg 1 + exp (X 1 + 2X 2 :5X 3 ) ; 2 Mdel 2: T (tjx) = (t) exp (X 1 + 2X 2 :5X 3 ) ; 2 Mdel 3: T (tjx) = (t) exp [si (X 1 + 2X 2 :5X 3 )] : I all three mdels, e = ( 1 ; 2 ; 3 ) = (1; 2; :5) csistet with the scale rmalizati we chse earlier. We geerate X 2, X 3 fllwig idepedet rmal distributi whse margial distributis have mea equal t 1 ad variace equal t 4 as i Huag ad Liu (26), ad let X 1 U [ 1; 1]. The cesrig variable C is simulated idepedetly frm the rest f the data, fllwig the uit expetial distributi. We als set the baselie cumulative hazard fucti as the idetity map as i Huag ad Liu (26), i.e., (t) = t. Regardig a few tuig parameters we have t pick i the simulati, we chse a third rder squally spaced B-splie fucti with umber f kts equal t 2 1=7, which is certaily ad-hc but csistet with the ptimal rate f a third rder di eretiable fucti. Furthermre, we let the btstrap replicati umber t be 2 ad we adpt the Bayesia btstrap (see Cheg ad Huag, 21) i which case all data pits wuld be assiged with psitive weights. Sice the partial likelihd fucti eeds t be maximized fr each geerated btstrap weight, we ly reprt the size prperties via 1, simulatis, tabulated belw fr ( 2 ; 3 ). 4 The hybrid type btstrap c dece iterval i Cheg ad Huag (21) is valid as well. We ly ivestigate the ite perfrmace f percetile type versi i simulati fr space restrictis. 2
Mdel 1 Mdel 2 Mdel 3 Sample Size = 5 = 1 = 5 = 1 = 5 = 1 BC fr 2.929.944.966.952.967.959 BC fr 3.933.947.94.945.962.955 Table 1: Size Prperties f Btstrapped CIs The abve table c rms the ice cverage prperties f ur btstrapped c dece itervals fr a mderate sample size, hece we recmmed the btstrap iferece fr future empirical applicatis. 7 Cclusi I this paper, we preset a systematic study f durati mdels characterized by threshld-crssig behavir f latet Lévy subrdiatrs. With its distributial prperty ad decreasig sample path, the Lévy subrdiatr culd be see as the atural variat f the static hetergeeity terms i MPH frm a prcess pit f view, ad it resembles the ti f usage ad wear-ut e ect (Sigpurwalla, 1995; Aale, Brga, ad Gjessig, 28) clsely. Hece, besides thse ptimal stppig time prblems i ecmics, ur mdel als applies t the sceari ccerig the life/failure time i bilgy, medical sciece, ad egieerig, where the traditial hazard-based mdels dmiate. Furthermre, it is trasparet hw the structural parameters f theretical mdels ca be related t reduced-frm parameters, ad all mdel primitives culd be pit ideti ed uder semiparametric speci catis. I spite f the sphisticated cstructi, ur mdels exhibit very tractable mathematical structures, iducig exible semiparametric estimati prcedures with rigrusly established large sample prperties. I a cmpai paper, Liu (215) csiders a cmpetig risks mdel where the depedece is geerated via cuplig tw Lévy subrdiatrs, ad btais semiparametric ideti cati ad estimati. Numerius extesis are pssible. First, e culd icrprate edgeus ad time-varyig cvariates by mdelig additial stchastic prcesses explicitly. Reault, va de Heijde, ad Werker (214) preset a structural mdel fr trasacti time (durati) ad asset prices (assciated marks) drive by multivariate Brwia mti. I their mdel, successive passage times f e latet Gaussia cmpet relative t radm budaries de e duratis ad the ther crrelated cmpets geerate the marks. It is bth iterestig ad challegig t search fr ther prcesses which lead t tractable structure. Secd, it is desirable t relax the parametric assumpti the threshld variable ad t csider a mre geeral Lévy prcess, i.e., the spectral psitive Lévy prcess (Sat, 1999). Last but t least, it is wrthwhile t study the empirical ctet f stchastic game mdels drive by a Lévy prcess, see Chapter 11 i Kypriau (26). 8 Appedix We will rst prve the ideti cati result. After itrducig several lemmas istrumetal t the mai results, we prve the claimed csistecy ad rates f cvergece. We let c dete a geeric ite psitive 21
cstat, whse value may chage frm lie t lie. We shall itrduce additial subscripts if mre tha e appear i the display. Mrever, A. B meas there exists a psitive cstat C which des t deped s.t. A CB. The fllwig lemma is a mdi cati f Therem 1 i Li ad Kulasekera (27) with the additial uivariate X 1. Recall we have partitied X as X = X 1 ; X 2. Lemma A.1 The supprt X f m () is a buded cvex set with at least e iterir pit, ad m () is a cstat ctiuus fucti X. If m (x) = x e = ' x e ; fr all x 2 X fr sme ctiuus fuctis ad ', the = ad ' Prf. It su cies t shw that =. Recall that e = cmpsiti fuctis: e () = e where kk is the stadard Euclidea distace. If 1;, e =, ad e' () = ' e, x e = ' x e ; e gets e x = e' x, fr all x 2 X, x 1 + x 2jx 2 X. 1; ad the fllwig tw with = = e e ad = e= kek. Nw we are ready t apply the argumets i Li ad Kulasekera (27), as bth ad have rm equal t 1 ad their rst crdiates are psitive. Suppse 6=, the bviusly 6= hlds as well. By the stated assumptis, there exists a sphere B (ex; r) X fr sme ex s.t. m () is cstat it. Fr all t 2 ( r; r) we have ex + t 2 X as > = 1. S e ex + t e' ex + t = e' = e ex + t = e' (ex + t) Because the rst -zer cmpets f ad are psitive, we get 6= ex + t ; (A.1) = e (ex + t) : carry ut the fllwig iterati based (A.1): e z > + t = e' ex + t = e ex + t 2 = = e ex + t e > = = e ex ;, hece < 1. We culd where we have used the ctiuity f e. This idicates m () is a cstat fucti B (ex; r), leadig t a ctridicati. Thereafter we get =, e = kek, resultig i = as desired. Prf. (f Ideti cati) The mmet restricti i (I3) allws us t idetify (t) frm the cdtial cumulative hazard fucti T (tjx) rst, because lg (t) = E [lg T (tjx)]. Fr the sigle idex structure, we shall apply the previus lemma t m (x) = x e, where () = lg exp () with the Lèvy-Laplace 22
expet fucti. belgs t the class f Berstei fucti whse derivative is cmpletely mte (see Schillig, Sg ad Vdracek, 212), thus () is real aalytic. Give Assumpti (I1), we get the ideti cati f as () is ideed a cstat ctiuus fucti. Fially, the variati f X alg a pe iterval leads t ideti cati f the aalytic fucti () by Prpsiti 1 i Abbrig ad va de Berg (23). Nw we preset several preparatry lemmas. The rst tw essetially ccer the smthess f the mdel, ad wuld be used i determiig the rate f cvergece. Lemma A.2 Uder Assumptis (A1)-(A6), fr a small " > we have sup V ar [m (; ) m (; )] C" 2 : d(; )";2 p Prf. By de iti f the idividual criteri fucti, we have E [m (; ) m (; )] 2 h = E X e i X e i h. E X e i. j j 2 + k (; ) (; )k 2 : i 2 [lg S (V i ; ) lg S (V i ; )] X i e i 2 + E [S (V i ; ) S (V i ; )] 2 Lemma A.3 Uder Assumptis (A1)-(A6), fr a small " > we have if [M ( ) M ()] C" 2 : d(; )";2 p Prf. Here we write M () = M (; (; )) t highlight the ature f the budled parameter case. Fr ay small " > ; ad the pit s.t. d (; ) ", we csider the quadratic expasi fr M ( ; (; )) M (; (; )) = ( ) M ( ) + 2 ( ) M [ ] + M [ ; ] + d 2 (; ) ; with thse secd rder derivatives calculated fllwig the rules i Dig ad Na (211). A straightfrward cmputati reveals that 82 X >< M = P 6 4 e P Y (t) e 2 X e + _ X 3 2 e 2 X e i 7 >: P hy (t) e ) (X e 8 X >< e P Y (t) e _ 1 9 2 X e X > = + P B @ i C P hy (t) e ) >: (X e A >; = P E hk (t) _ i h i 2 2 X e X2 2 E K (t) _ X e X 2 ; 5 X2 2 9 >= >; 23