Inference on Parameters of a Shifted and Rescaled Wiener Process Based on Nonidentically Distributed Observations

Transcription

1 WDS'09 Proceedgs of Cotruted Ppers, Prt I, 95 0, 009. ISBN MATFYZPRESS Iferece o Prmeters of Shfted d Rescled Weer Process Bsed o Nodetclly Dstruted Oservtos A. Kvtkovčová Chrles Uversty, Fculty of Mthemtcs d Physcs, Prgue, Czech Repulc. Astrct. The pper dels wth ferece out the prmeters d of the rdom process {w t + t; t 0}, where {w t ; t 0} s the Weer process, 0 d > 0. Iferece s sed o the frst tme whe the process reches pre-specfed postve oudry, where the oudry c e dfferet for ech oservto. We focus o estmto d we shortly del wth testg hypotheses out oe prmeter t tme. We wll show tht ether the vrce of estmtors, or the power of tests deped o the choce of the oudres. These chrcterstcs oly deped o the umer of oservtos d o the sum of the oudres. Itroducto Rdom processes provde useful tool for descrg developmet of vrous chrcterstcs my prctcl stutos. Sce t s usully costly d sometmes ufesle to oserve whole developmet of the chrcterstc, the formto out the process s ofte lmted. Perhps most frequetly, the process c oly e oserved set of dscrete tme pots. Altertvely, we c oly oserve the tme pots whe the process reches some oudry. The secod pproch s turl stutos whe rechg gve vlue y the process leds to esly oservle evet. I wht follows, we wll focus o ths type of formto o the uderlyg process. To ot more formto, we c oserve the evet tmes for severl relztos of the process. For some processes, we c s well utlze oservtos of set of evets correspodg to rechg dfferet oudres. If there s o pror reso why the oudres correspodg to susequet evets should e eqully spced, or why ll the relztos should produce the evet t the sme level of the process, t s of terest whether dfferet spcg s somehow reflected the qulty of formto. I wht follows, we wll work wth Weer process wth costt drft rte d stteous vrce. We wll ssume tht sted of oservg the trjectory of the process, we oserve the frst tmes whe the process reches prespecfed set of oudres. We wll explore how uequl spcg of the oudres flueces vrce of estmtors d power of tests of hypotheses out the prmeters of the process. We wll strt wth more detled prolem formulto the followg secto. Lter we wll del wth estmto d shortly wth hypotheses testg. Flly, we wll summrze the results Cocluso d Dscusso. Prolem Formulto Let us cosder rdom process Y t = w t + t, where {w t ; t 0} s Weer process, 0 s drft rte d > 0 s stteous vrce. Isted of the whole trjectory of the process, let us oserve the frst tme whe the process reches oudry pot A 0. Thus, sted of {Y t ; t 0} we oserve X = f {t 0; Y t A}. I most stutos, we eed more th oe oservto to mke relle ferece. A usul pproch s to use depedet detclly dstruted oservtos X,..., X. A ovous wy to ot them s to oserve depedet relztos of the process Y t d the correspodg frst httg tmes of A. Whe delg wth the Weer proces, there s yet other wy of otg the oservtos. We c oserve the tmes Z A, Z A,..., Z A, whe sgle relzto of the process reches the oudres A, A..., A, respectvely, for the frst tme. Let us cosder rdom vrles Z,..., Z such tht Z = Z A d Z = Z A Z A for. It follows from the strog Mrkov property of the Weer process tht L X,..., X = L Z,..., Z. Thus, we c use y of the pproches, or comto thereof, to ot ss for the ferece. I wht follows, we wll use the otto X,..., X for the oservtos, regrdless of the wy of otg them. 95

2 Iverse Guss Dstruto It s well kow tht the frst httg tme of postve oudry A y shfted d rescled Weer process wth prmeters 0 d s solutely cotuous.s. postve rdom vrle wth desty wth respect to Leesgue mesure gve y f A,, x = A π x 3 exp { A x x } I {x > 0} for ll x R see for exmple Chhkr d Folks [989], Seshdr [993] or Steele [00] for vrous dervtos of the result. If there s o drft volved, ths s specl cse of verse gmm dstruto. Otherwse, the desty s tht of verse Guss rdom vrle. The verse Guss dstruto s well kow uder the prmetrzto IG µ, λ wth the desty f λ,µ x = λ π x 3 exp { } λ x µ µ I {x > 0} for ll x R. x We c esly verfy tht we deed ot for λ = A d µ = A, f > 0. The dstruto s populr oth theoretcl d ppled sttstcs. I theory, t s mportt ecuse of the uderlyg process motvto metoed ove. The pplclty s stressed y the fct tht shfted d rescled Weer process c e used s frst pproxmto to other processes wth pproxmtely costt drft rte d pproxmtely costt stteous vrce see Seshdr [999]. Further, the verse Guss dstruto s the lmtg dstruto of the smple sze some sequetl prolems see Johso et l. [994]. From pplctol pot of vew, t hs severl ppelg propertes s well. The dstruto hs ee recommeded for modellg postve rght skewed dt see Seshdr [999] d Johso et l. [994]. It hs ee show to offer terestg ltertve to logorml, Weull d severl other dstrutos used ths cotext. It hs lso ee used s the uderlyg dstruto geerlzed ler models see McCullgh d Nelder [983]. Further, logy of the ANOVA models hs ee developed for verse Guss rdom vrles. These re kow s lyss of recprocls methods see Seshdr [999] or Chhkr d Folks [989]. A cosderle umer of pplctos comes from survvl lyss, where the uderlyg process pproch hs ee dvocted recetly see Ale d Gjessg [00], Lee d Whtmore [006] or Blk et l. [009] for exmple. Aprt from the process motvto, the verse Guss dstruto offers ppelg shpe of the hzrd rte. Wheres expoetl dstruto c e used to model costt d Weull dstruto to model mootoe hzrd rte, logorml d verse Guss dstrutos model tlly cresg d the decresg hzrd rte. For logorml dstruto, the symptotc hzrd rte decreses to zero, ut for verse Guss dstruto t pproches postve costt determed y the prmeters of the dstruto. Ths s pplcle my stutos see Hwks d Olwell [997]. Aprt from tht, the populrty of the verse Guss dstruto s stressed y ts coveet propertes. Whle offerg vrety of shpes, t hs closed form solutos for my relted prolems d t possesses some coveet propertes smlr to orml dstruto, such s depedece of mxmum lkelhood estmtors of the prmeters d severl others see Chhkr d Folks [989]. Dt Exmples Let us ow cosder exmples of stutos, whe verse Guss dstruto hs ee used to model the dt. Seshdr [999] reports successful usge of the verse Guss dstruto to ft the slug legths ppeles trsportg crohydrtes. The model s ult o the ssumpto tht for sgle slug geerted t the let d movg towrds the outlet of ppele, ts dstce from the let t tme c e descred y shfted d rescled Weer process. Clerly, the whole trjectory of the process cot e oservle. But f some mesuremet sttos re plced log the ppele, the tmes whe the slug reches the sttos c e oserved. Ale d Gjessg [00] llustrted the process pot of vew o survvl dt y modellg the survvl tme of ptets wth crcom of the orophryx. They ssumed tht there s process of dsese progresso volved, whch s ot oservle. A ptet des, s soo s the process reches gve level. They use the verse Guss dstruto to model the survvl tme, ut they ssume tht the drft depeds o severl covrtes mesured t the tme of dgoss. 96

3 Nodetclly Dstruted Oservtos Whe reflectg o the expermetl settg, oe of turlly rsg questos s whether the oservtos hve to e detclly dstruted. I terms of mesuremet sttos log ppele, we c sk whether t would e more coveet to plce the mesuremet sttos oequdsttly. Or, terms of survvl tme of ccer ptets, t mght e resole to ssume tht ech ptet hs hs or her ow crtcl level of the uderlyg dsese process. To ccommodte ths settg, let us cosder oegtve costts A,..., A d oservtos X,..., X, such tht X s the frst httg tme of the rrer A y Y t. For > 0 we the hve depedet oservtos such tht X IG A, A. We c esly verfy tht X A IG, A, whch c e rewrtte s IG µ, λ 0 ω, where µ =, λ 0 =, d ω = A. For = 0 we hve depedet odetclly dstruted verse gmm vrles. For y 0 d > 0 the desty of the rdom vector X,..., X s gve y fx = exp A x x j log + j= A A πx 3 I {x > 0,..., x > 0}. It s cler from tht the desty of the rdom vector X,..., X forms expoetl fmly. I wht follows, we wll use two dfferet prmetrztos of the expoetl fmly. A ovous prmetrzto s s follows. A fx = exp x θ T x x j j= T x θ log + A κθ A 3 I {x > 0,..., x > 0} πx } {{ } hx wth θ = θ, θ =, d Θ = 0, [0,. Thus, κθ = θ θ A log θ, T X = A X s suffcet sttstcs for θ d T X = j= X j s suffcet sttstcs for θ. To troduce secod prmetrzto, let us cosder costt 0 > 0. The c e see s fx = exp A + 0 x j x j= θ T x Thus, ths prmetrzto, θ = θ, θ = x j j= T x, 0 0 } {{ } θ A πx 3 A log + κθ I {x > 0,..., x > 0} } {{ } hx 3 4 d Θ = 0, R. The κθ = A X + 0 j= X j s suffcet sttstcs for log θ θ θ + 0 θ A, T X = θ d T X = j= X j s suffcet sttstcs for θ. We c otce tht for 0 = 0 the form of the prmetrzto s the sme s 3 ut the turl prmeter spce s dfferet. Thus, the ssumpto o 0 > 0 s mportt. 97

4 A Ufyg Frmework for 0 Sce the verse Guss dstruto s well kow, extesve lterture s ccessle summrzg ts propertes see for exmple Chhkr d Folks [989], Seshdr [993], or Seshdr [999]. My of these propertes hold lso for the cse whe = 0 d the dstruto of the frst httg tme s verse gmm. We wll ow lst severl propertes, whch wll e used further the text. Lemm. For the frst httg tme X of postve oudry A y rescled Weer process wth oegtve drft rte we hve If > 0 the E X = A E X = A + A A d vrx =. 3 d vr X = A A 4. Proof. Sttemets d for > 0 c e foud Johso et l. [994]. If = 0 the X hs gmm dstruto d t c e esly show tht ts cumults stsfy Sttemet fter susttutg = 0. Lemm. Let X,..., X e the frst httg tmes of oegtve pots A,..., A y depedet relztos of shfted d rescled Weer process wth prmeters 0 d > 0. The dstruto of j= X j s the sme s the dstruto of the frst httg tme of A y shfted d rescled Weer process wth prmeters d. Proof. See Kvtkovčová [008] for exmple. For > 0 d A = A for ll the result s kow s reproduclty property of the verse Guss dstruto see Johso et l. [994]. Estmto Let us study the mxmum lkelhood estmtors of the process prmeters d depedet frst httg tmes X,..., X of the pots A,..., A respectvely. sed o Theorem 3. Mxmum lkelhood estmtors â d of the prmeters d re gve y â = A j= X j d = A X A j= X. 5 j Proof. Let us frst relze tht the estmtors gve y 5 re oth well defed d postve.s. For â ths mmedtely follows from ts costructo. For let us cosder sequece of oegtve rel umers { } d sequece of postve rel umers {x }. The the equlty x j= x j, whch s the Cuchy Schwrz equlty j= xj x c e wrtte s for vectors x,..., x d x,...,. x It s ow cler tht equlty holds wth prolty zero. It ow rems to show tht the estmtors gve y 5 mxmze the lkelhood. Ths s esly cheved y dfferettg the log lkelhood fucto. Ths result s le wth Seshdr [999], where the mxmum lkelhood estmtors of the prmeters µ d λ 0 sed o depedet oservtos Y IG µ, λ 0 ω re gve y ˆµ = ω Y ω d ˆλ 0 = ω ˆµ. 6 Y Seshdr [999] further derved the jot dstruto of the rdom vrles ˆµ d λ ˆλ 0 0, where ˆµ d ˆλ 0 re gve y 6. It s strghtforwrd to verfy tht the proof of the sttemet does ot use the fct tht > 0. Thus, the jot dstruto for the cse whe = 0 c e derved the sme wy. Usg ths result, the sttemet of Lemm, d pplyg the trsformto theorem, we ot the followg. 98

5 Theorem 4. Mxmum lkelhood estmtors â d of the prmeters d gve y 5 re depedet. Further, χ d â s cotuous rdom vrle wth the desty gve y fx = A { x π exp A } x x I {x > 0} for ll x R. Let us ow explore propertes of the estmtors. Theorem 5. For gve y 5 we hve E =. Thus, s symptotclly used estmtor of. = s used estmtor of. vr = 4 d vr = 4. v Both d re wekly cosstet estmtors of. For â gve y 5 we hve v E â = +. Thus, f lm A k= A k = the â s symptotclly used estmtor of. v ã = â v vr â = A + s used estmtor of. A d vr ã = + A A. A v If lm k= A k = the oth â d ã re wekly cosstet estmtors of. Proof. By Theorem 4 we hve tht L = χ. Thus, E = d vr. Sttemets, d ow esly follow. Applyg Lemms d we ot tht E j= Xj = A A3 vrâ + A vr = A A d vr = j= Xj =. Further, sce â d re depedet y Theorem 4, we hve tht vr ã = A4 +. Ths shows Sttemets v, v d v. A Cosstecy of ll estmtors follows from symptotc usedess d vrce pprochg zero for lrge. Theorem 6. Let lm k= A k =. The the mxmum lkelhood estmtors gve y 5 re symptotclly orml for lrge. Specfclly, A 0 â d N, Proof. We c rewrte s 4. Let us cosder rdom vrle Z χ. Applyg the cetrl lmt theorem we ot tht the symptotc dstruto of Z for lrge s stdrd orml. By Theorem 4 t ow esly follows tht the symptotc dstruto of for lrge s N 0, 4. A. Let us cosder r- j= Xj For > 0 we hve A â = A dom vrle Z IGµ, λ. It s kow see Seshdr [999] tht for λ µ λ µ the dstruto of µ Z pproches stdrd orml dstruto. By Lemm we hve j= X j IGµ, λ, A where µ = d λ = A. It ow esly follows tht the symptotc dstruto of A â for lrge A s N 0,. For = 0 Theorem 5 v d v mples tht oth expectto d vrce of A â coverge to zero for lrge. To complete the proof t rems to relze tht the mxmum lkelhood estmtors re depedet for ech fxed y Theorem 4. 99

6 Hypotheses Testg Sce we re delg wth expoetl fmles of dstrutos, we c pply methodology for most powerful tests costructo from Lehm [986] for testg hypotheses out oe prmeter of the process t tme. We wll cosder tests of the hypothess θ θ 0 gst the ltertve θ > θ 0, where θ stds for or d θ 0 s sutle costt. Tests of the hypohess out the prmeter c e sed o the mxmum lkelhood estmtors of the prmeter. Let us ssume ukow vlue of the prmeter. The estmtor gve y 5 s mootoe fucto of T X d t s depedet wth T X gve y 3. Thus, c e used s test sttstc for uformly most powerful used test. If we ssume tht the vlue of the prmeter s kow, the mxmum lkelhood estmtor clculted uder ths ssumpto c e used s test sttstc of uformly most powerful test. Ths follows from the fct tht t s mootoe fucto of the suffcet sttstc for the prmeter, whle s ssumed to e kow. Sutle multples of the test sttstcs c e show to hve χ dstruto. The result for ukow follows from Theorem 4, whle for kow, t follows from pplcto of the Trsformto Theorem. I oth cses, the crtcl vlue d the power of the test deped oly o the umer of oservtos. Uformly most powerful tests of hypotheses out, whle s ssumed to e kow, c e sed o the sttstc j= X j, whch s mootoe fucto of T X gve y 3. Sce, y Lemm, ts dstruto oly depeds o A, the sme holds for the crtcl vlue d the power of the test. Costructo of uformly most powerful used tests of hypotheses out, whle s ot ssumed to e kow, re complcted y the fct tht the prmeters d re ot seprted 3. Thus, we eed to hve θ 0 = 0 o the oudry of the hypotheszed prmeter spce. Becuse of tht, t s ecessry to se the tests o the prmetrzto gve y 3 f = 0 s of terest, d o the prmetrzto gve y 4 otherwse. I oth cses, usg methods logous to those ppled Chhkr d Folks [976] t c e show tht the crtcl vlue d the power of the test deped oth o A d o. Cocluso d Dscusso We hve cosdered estmto d testg hypotheses out the prmeters d of shfted d rescled Weer process sed o the frst httg tme of prespecfed oudry pot y the process. The oudry ws llowed to dffer for ech oservto. We hve show tht the selecto of the oudres flueces ether the vrce of the cosdered estmtors, or the power of the cosdered tests. Qulty of ferece cocerg the prmeter oly depeds o the umer of oservtos. For the prmeter, the qulty depeds o the sum of the cosdered oudres, d, f estmto of hs to e performed order to ot results o, the umer of oservtos s reflected the qulty of ferece s well. Ackowledgmets. I would lke to thk doc. RNDr. Del Hluk, Ph.D. for vlule suggestos d commets, whch helped me durg my work o the project. I would lso lke to thk the referees for creful redg of the muscrpt, ledg to ts mprovemets. The project ws supported y the Czech Grt Agecy uder Cotrct 0/08/0486. Refereces Ale, O. O. d Gjessg, H. K., Uderstdg the Shpe of the Hzrd Rte: A Process Pot of Vew, Sttstcl Scece, 6, 4, 00. Blk, J., Desmod, A. F., d McNchols, P. D., Revew d Implemetto of Cure Models Bsed o Frst Httg Tmes for Weer Process, Lfetme Dt Alyss, 5, 47 76, 009. Hwks, D. M. d Olwell, D. H., Iverse Guss Cumultve Sum Cotrol Chrts for Locto d Shpe, The Sttstc, 46, , 997. Chhkr, R. S. d Folks, J. L., Optmum Test Procedures for the Me of Frst Pssge Tme Dstruto Brow Moto wth Postve Drft Iverse Guss Dstruto, Techometrcs, 8, 89 93, 976. Chhkr, R. S. d Folks, J. L., The Iverse Guss Dstruto. Theory, Methodology, d Applctos, Mrcel Dekker, New York, 989. Johso, N. L., Kotz, S., d Blkrsh, N., Cotuous Uvrte Dstrutos. Volume, Wley, New York, 994. Kvtkovčová, A., Sttstcl Iferece for Rdom Processes, Mster Thess, Chrles Uversty, Prgue, 008. Lee, M-L. T. d Whtmore, G. A., Threshold Regresso for Survvl Alyss: Modelg Evet Tmes y Stochstc Process Rechg Boudry, Sttstcl Scece,, 50 53,

7 Lehm, E. L., Testg Sttstcl Hypotheses, Sprger, New York, 986. McCullgh, P. d Nelder, J. A., Geerlzed Ler Models, Chpm d Hll, Lodo, 983. Seshdr, V., The Iverse Guss Dstruto: A Cse Study Expoetl Fmles, Oxford Uversty Press, New York, 993. Seshdr, V., The Iverse Guss Dstruto: Sttstcl Theory d Applctos, Sprger, New York, 999. Steele, J. M., Stochstc Clculus d Fcl Applctos, Sprger, New York, 00. 0