Delta Method on Bootstrapping of Autoregressive Process. Abstract

Transcription

1 Proceeigs 59h ISI Worl Saisics Cogress 5-30 Augus 03 Hog Kog (Sessio CPS04) p.3959 Dela Meho o Boosrappig of Auoregressive Process Bambag Suprihai Suryo Gurio 3 Sri Haryami 4 Uiversiy of Sriwijaya Palembag Ioesia 34 Uiversiy of Gajah Maa Yogyakara Ioesia Correspoeig auhor: Bambag Suprihai bambags@usri.ac.i Absrac Le T be auoregressive ime series where T is iscree ime a le be he sample ha saisfies he AR() process. Thus he sample follows he relaio is a zero mea whie oise process where wih cosa variace. Le ˆ be he esimaor for parameer. Brockwell a Davis (99) showe ha ˆ N 0. Meaime by p ˆ a some assumpios ca be showe ha he isribuio of coverges o ormal isribuio wih mea 0 a variace as. I boosrap view he key of boosrap ermiology says ha he populaio is o he sample as he sample is o he boosrap samples. Therefore whe we wa o ivesigae he cosisecy of he ieresig boosrap esimaor for sample mea we ivesigae he isribuio of where is boosrap versio of compue coras o. Asympoic heory of he boosrap sample mea is useful from sample boosrap o suy he cosisecy for may oher saisics. Le ˆ be he boosrap esimaor for ˆ. I his paper we ivesigae he cosisecy of ˆ usig ela meho a applyig he resiuals boosrap. We also prese he Moe Carlo simulaios i regar o yiel appare coclusios. Keywors: Boosrap cosisecy ela meho Moe Carlo simulaios ime series. Iroucio Suyig of esimaio of he ukow parameer ivolves: () wha esimaor ˆ shoul be use? () havig choose o use paricular ˆ is his esimaor cosise o he populaio parameer? (3) how accurae is ˆ as a esimaor of rue parameer? The boosrap is a geeral mehoology for aswerig he seco a hir quesios. Cosisecy heory is eee o esure ha he esimaor is cosise o he acual parameer as esire. Cosier he parameer is he populaio mea. The cosise esimaor for is he sample mea ˆ. The cosisecy heory is he eee o i i he cosisecy of boosrap esimaor for mea. Accorig o he boosrap ermiology if we wa o ivesigae he cosisecy of boosrap esimaor for mea we ivesigae he isribuio of a boosrap uer Kolmogorov meric is efie as. The cosisecy of

2 Proceeigs 59h ISI Worl Saisics Cogress 5-30 Augus 03 Hog Kog (Sessio CPS04) p.3960 sup P P. () Bickel a reema (98) a Sigh (98) showe ha () coverges almos surely o 0 as. Meawhile Suprihai e.el (0) complee he resuls by givig ice ilusraios for his case. The cosisecy of boosrap for mea is a worhy ool for suyig he cosisecy of oher saisics. I his paper we suy he cosisecy of boosrap esimaor for parameer of he AR() process. The cosisecy of boosrap esimaor for mea is he applie o suy he cosisecy of boosrap esmae for parameer of he AR() process usig ela meho. We escribe he cosisecy of boosrap esimaes for mea a parameer of he AR() process. Secio reviews he cosisecy of boosrap esimae for mea uer Kolmogorov meric. Secio 3 eal wih he cosisecy of boosrap esimae for parameer of he AR() process usig ela meho. Secio 4 iscuss he resuls of Moe Carlo simulaios ivolve boosrap saar errors a esiy esmaio for mea a parameer of he AR() process. Secio 5 is he las secio briefly escribes he coclusios of he paper.. Cosisecy of Boosrap Esimaor or Mea Le be a raom sample of size from a populaio wih commo isribuio a le T ; be he specifie raom variable or saisic of ieres possibly epeig upo he ukow isribuio. Le eoe he empirical isribuio fucio of i.e. he isribuio puig probabiliy / a each of he pois. The boosrap meho is o approimae he isribuio of T ; uer by ha of T ; uer whrere eoes a boosrappig raom sample of size from. We sar wih efiiio of cosisecy. Le a G be wo isribuio fucios o sample space. Le G be a meric o he space of isribuio o. or i.i. from a a give fucioal T ; le H ( ) P T ; H Boo( ) P T ;. We say ha he boosrap is cosise (srogly) uer for T if H H 0 a. s. Boo Le fucioal T is efie as T ; where a are sample mea a populaio mea respecively. Boosrap versio of T is T ; where Boosrap meho is a evice for esimaig P P Kolmogorov meric which is efie as K is boosrappig sample mea. by. We will ivesigae he cosisecy of boosrap uer G sup ( ) G( ) = sup P P. Some heorems a lemma which are eee o show ha KH H 0 a. s. Boo ake from Hall (99) Serflig (980) a va er Vaar (000) such as Khichie- Kolmogorov Covergece Theorem Berry-Esse Theorem a Zygmu-

3 Proceeigs 59h ISI Worl Saisics Cogress 5-30 Augus 03 Hog Kog (Sessio CPS04) p.396 Marcikiewicz SLLN. The cosisecy of H Boo uer Kolmogorov meric have show by Sigh (98) a DasGupa (008). The cru resul is ha a.s. Suprihai e.al (0) give ice simulaios for his resul. 3. Cosisecy of Boosrap Esimae or Parameer of AR() Process Usig Dela Meho The ela meho cosiss of usig a Taylor epasio o approimae a raom by he polyomial T i T. This vecor of he form T meho is useful o euce he limi law of T from ha of T. This meho is also vali i boosrap view which is give i he followig heorem. k m Theorem (Dela Meho or Boosrap) Le : be a measurable map efie a coiously iffereiable i a eighborhoo of. Le ˆ be raom vecors akig heir values i he omai of ha coverge almos surely o. If ˆ T a ˆ ˆ T coiioally almos surely he boh ˆ T a ˆ ˆ T coiioally almos surely. Le is he populaio mea a he is he sample mea. The Kolmogorov SLLN assers ha a.s. a N 0. The resulig of Secio shows ha N 0 s. Base o he cosisecy of he boosrap for he sample mea we ivesigae he cosisecy of he boosrap esimae for parameer of AR() process usig ela meho. Le { } be ime series aa which saisfies he AR () process i.e. if { } follows he equaio where { } be raom variable sequece of whie oise wih mea 0 a variace. The process is saioary if. The comprehesive iscussios for ime series ca be fou i Wei (990) a Brockwell a Davis (99). or he AR() process from Yule-Walker equaio we obai he esimae for is ˆ ˆ where ˆ be he lag sample auocorrelaio ˆ. () Accorig o Wei (990) a Brockwell a Davis (99) he esimae of saar ˆ error of parameer is se(θ) =. Meawhile he boosrap versio of saar error was irouce by Efro B. a Tibshirai R. (986). I Secio 4 we emosrae resuls of Moe Carlo simulaios cosis he wo of saar errors a give brief commes. rom () we ca see ha ˆ

4 Proceeigs 59h ISI Worl Saisics Cogress 5-30 Augus 03 Hog Kog (Sessio CPS04) p.396 Brockwell a Davis (99) have show ha ˆ is cosise esimaor of rue a. s parameer. Kolmogorov SLLN assers ha E. Sice is iepee of he E = 0. Hece a. s 0. ially () is approimae by ~. Thus for we obai ˆ ~. We see ha ~ equals o for he fucio. Sice is coious a hece is measurable. Meaime he boosrap versio of ˆ eoe by ˆ ca be obaie as follows [see e.g. Efro a Tibshirai (986) a Bose (988)]:. Defie he resiuals ˆ ˆ for 3.. A boosrap sample is creae by samplig wih 3 replaceme from he resiuals. Leig as a iiial boosrap sample a ˆ ially afer ceerig he boosrap ime series replace by i where we obai he boosrap esimaor he sample. ˆ ˆ i.e. i is. Usig he plug-i priciple compue from Aalog wih he previous iscussio we obai he boosrap versio for couerpar of ~ ha is measurable map ~. Thus accorig o Theorem we coclue ha ~ coverges o ~ coiioally almos surely. urhermore ~ ~ T a for we obai

5 Proceeigs 59h ISI Worl Saisics Cogress 5-30 Augus 03 Hog Kog (Sessio CPS04) p.3963 ˆ T where T is a ormal isribuio wih zero mea a variace 4 ˆ wih a 4 are seco a fourh momes repecively. 4. Resuls of Moe Carlo Simulaios The simulaio is couce usig S-Pus a he sample is he 50 ime series aa of echage rae of US ollar compare o Ioesia rupiah. Daa is ake from auhorize websie of Bak Ioesia i.e. hp:// for fify ays of rasacios o March a April 00. Suprihai e. al. (0) has ieifie ha he ime series aa saisfies he AR() procces such ha he aa follows he equaio 3 50 where ~ WN 0. The simulaio yiels he esimaor for parameer is ˆ = wih saar error To prouce a goo approimaio Efro a Tibshirai (986) a Daviso a Hikley (006) sugges o use he umber of resamples a leas B = 50. Boosrap versio of saar errror usig boosrap samples of size B = a 000 yielig as presee i Table. Table Esimaes for Saar Errors of ˆ for Various B B se ˆ ˆ rom Table we ca see ha he values of boosrap saar errors e o ecrease i erm of size of B icrease a close o he value of 0999 (acual saar error). These resuls show ha he boosrap gives a goo esimae. Meaime he hisogram a esiy esimae of ˆ are presee i igure. rom igure we ca see ha he resulig hisogram close relae o he ormal esiy. Of course his resul agree o he resul of reema (985) a Bose (988) eha.boo igure Hisogram a Desiy Esimae of Boosrap Esimaor ˆ

6 Proceeigs 59h ISI Worl Saisics Cogress 5-30 Augus 03 Hog Kog (Sessio CPS04) p Coclusios A umber of pois arise from he suy of Secio 3 a 4 amogs which we sae as follows.. Cosier a AR() process wih Yule-Walker esimaor ˆ = ˆ is a cosise esimaor for rue parameer. By usig he ela meho we have show ha ~ is also a cosise esimaor for ~ where ˆ ~ for. Moreover we obai ha ~ ~ N a for he cru resul is ha ˆ ˆ N where N is a ormal isribuio.. Resulig of Moe Carlo simulaios show ha he boosrap essimaors are goo approimaios as represee by heir saar errors a plo of esiies esimaio. Refereces Bickel P. J. a reema D. A. (98) Some asympoic heory for he boosrap A. Sais Bose A. (988) Egeworh correcio by boosrap i auoregressios A. Sais Brockwell P. J. a Davis R. A. (99) Time Series: Theory a Mehos Spriger New York. DasGupa A. (008) Asympoic Theory of Saisics a Probabiliy Spriger New York. Daviso A. C. a Hikley D. V. (006) Boosrap Mehos a Their Applicaio Cambrige Uiversiy Press Cambrige. Efro B. a Tibshirai R. (986) Boosrap mehos for saar errors cofiece iervals a ohers measures of saisical accuracy Saisical Sciece reema D. A. (985) O boosrappig wo-sage leas-squares esimaes i saioary liear moels A. Sais Hall P. (99) The Boosrap a Egeworh Epasio Spriger-Verlag New York. Serflig R. J. (980) Approimaio Theorems of Mahemaical Saisics Joh Wiley & Sos New York. Sigh K. (98) O he asympoic accuracy of Efro s boosrap A. Sais Suprihai B. Gurio S. a Haryami S. (0) Cosisecy of he boosrap esimaor for mea uer kolmogorov meric a is implemeaio o ela meho Proc. of The 6 h Seams-GMU Coferece. va er Vaar A. W. (000) Asympoic Saisics Cambrige Uiversiy Press Cambrige. Wei W. W. S. (990) Time Series Aalysis: Uivariae a Mulivariae Mehos Aiso Wesley Califoria.