New Jersey Institute of Technology Digital Commons @ NJIT Dissertations Electronic Theses and Dissertations Spring 5-31-2010 Variance reduction techniques for estimating quantiles and value- at-risk Fang Chu New Jersey Institute of Technology Follow this and additional works at: https://digitalcommons.njit.edu/dissertations Part of the Databases and Information Systems Commons, and the Management Information Systems Commons Recommended Citation Chu, Fang, "Variance reduction techniques for estimating quantiles and value-at-risk" (2010). Dissertations. 212. https://digitalcommons.njit.edu/dissertations/212 This Dissertation is brought to you for free and open access by the Electronic Theses and Dissertations at Digital Commons @ NJIT. It has been accepted for inclusion in Dissertations by an authorized administrator of Digital Commons @ NJIT. For more information, please contact [email protected]. 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Ch, ., nd M K Confidence Intervals for Quantiles When Applying Variance- Reduction Techniques, 1 t b bttd t jrnl v M lvd Wf, rnt nd Grndprnt ACKOWEGME I dpt rttd t r Mrvn drttn dvr I hv bn nrdbl frtnt t hv n dvr h fll ndrtd rr l nd hlhrtdl hlpd rlz t ntrhp prnt n nrn t b rrhr nd t prtntl n ndpndnt thnr r vrthn hv dn fr r thn I ld l lv t thn ll f tt br fr thr dn nd pprt r Gr Wdr n h dn-pprt t l hd nprd t xplr th ltn ppltn n bn hh vntll dvlpd nt th fr- r f rrh tp r nr Chn h nt nl ntrdd t ntttv fnn th h nlhtnn ltr bt l tvtd t ndt rrh nd pr- rr n th hllnn fld th nth Wtht th vlbl fdb nd nr tn fr r ln Shr r Mhl nd r Clvn th r ld nvr hd bn pltd I ld l t t th pprtnt t thn r Mhl br nd r r W fr ttn rdt rr trtd n th rht ft nd prvdn th fndtn fr bn rrhr hn l t t ll th flt br h brht ndrtndn f nfrtn t t hhr lvl thrh thr pn n rrh nd thn I ld l l t xtnd rttd t ll th ll n th t IS lb A vr pl thn t frnd Y nd Qn Wn fr th pprt nd tn th hv lnt vr r hn lt fr vrthn hn l t t Yn Y Yn W Wn n W Y n n C Yxn brt Kn-E Chn Al Avn Mr lr n Chn xn hn Yn Yn nd n "l pln" bdd fr prvdn h ndd hr nd ntrtnnt n ht ld hv bn thr ht trfl h v lf nll nd t prtntl I ld l t thn f nn Sh r pprt nrnt t ptn nd nvrn lv r ndnbl th bdr pn hh th pt fv r f lf hv bn blt r nldn dvtn nd lv hv vd fr th t dprn nt n lf It ndr hr thfl tht I nd h drv nd n blt t tl hlln hd n I thn prnt nd rndprnt fr thr fth n nd lln t b bt I ntd Al I thn nn prnt fr lln t t thr dhtr vr x thnd l fr th "rd Ct" n Chn nd prvdn th nndn nrnt nd pprt TABLE OF CONTENTS Chapter Page 1 IOUCIO 1 EIEW O ECIQUES O ESIMAIG MEAS 1 1 Crd Mnt Crl 1 rn dtn 1 1 Anttht rt 1 Cntrl rt 15 3 Strtfd Spln 17 Iprtn Spln 19 3 Appltn 1 31 Un Crd Mnt Crl t r n An Optn 3 Un C t r n An Optn 33 Un A t r Kn-t Optn 5 3 Un IS t r Kn-n Optn 7 3 QUAIE ESIMAIO O CUE MOE CAO 3 AAU-GOS EESEAIO WE AYIG S 33 1 rf 3 11 rf f hr 1 3 1 Sffnt Cndtn fr Aptn A 13 rf f hr 1 rf f hr 3 5 QUAIE ESIMAIO USIG IS+SS 3 51 rf 9 511 rf f hr 9 51 rf f hr 5 5 viii AE O COES (Cntnd Chptr Page QUAIE ESIMAIO USIG A 5 1 rf 57 11 rf f hr 57 1 rf f hr 9 7 QUAIE ESIMAIO USIG C 1 71 rf 711 rf f hr 1 EMIICA SUY 7 1 rl trbtn 7 Stht Atvt tr 1 IS fr SA 9 IS+SS fr SA 7 3 A fr SA 7 3 Chn th Sthn rtr 7 n f Eprl lt 7 5 rr rptn nd tn 9 51 rr rptn 9 5 rr tn 1 Mtlb Cd 19 1 A/C/CMC 19 IS nd IS+SS 13 9 COCUIG EMAKS 155 EEECES 157 x IS O AES bl Page 1 Cvr fr IS th c = 1 fr th rl hn p = 95 Avr lf-Wdth fr IS th c = 1 fr th rl hn p = 95 3 Cvr fr IS th c = 5 fr th rl hn p = 95 Avr lf-Wdth fr IS th c = 5 fr th rl hn p -- 95 5 5 Cvr fr IS+SS th c 5 n th vrt rl trbtn th p = 9 hn p = 95 5 Avr lf-Wdth fr IS+SS th c = 5 fr th vrt rl th p = 9 Whn p 95 5 7 Cvr fr IS+SS th c = 5 n th vrt rl trbtn th p = 9 hn p = 95 Avr lf-Wdth fr IS+SS th c = 5 fr th vrt rl th p = 9 Whn p = 95 9 Cvr fr A th = 1 fr th rl hn p = 95 1 Avr lf-Wdth fr A th = 1 fr th rl hn p = 95 7 11 Cvr fr A th 5 fr th rl hn p = 95 7 1 Avr lf-Wdth fr A th = 5 fr th th rl hn p = 95 7 13 Cvr fr C th c = 1 fr th vrt rl th p = 9 hn p = 95 7 1 Avr lf-Wdth fr C th c = 1 fr th rl th p = 9 hn p 95 15 Cvr fr C th c = 5 fr th vrt rl th p = 9 hn p = 95 1 Avr lf-Wdth fr C th c = 5 fr th vrt rl th p = 9 hn p = 95 17 Cvr fr CMC th c = 1 fr th rl hn p = 95 9 1 Avr lf-Wdth fr CMC th c = 1 fr th rl hn p = 95 9 IS O AES (Cntnd bl Page 19 Cvr fr CMC th c = 5 fr th rl hn p = 95 9 Avr lf-Wdth fr CMC th c = 5 fr th rl hn p = 95 9 1 Cvr fr IS th c = 1 fr th SA hn p = 95 9 Avr lf-Wdth fr IS th c = 1 fr th SA hn p = 95 9 3 Cvr fr IS th c = 5 fr th SA hn p = 95 9 Avr lf-Wdth fr IS th c = 5 fr th SA hn p = 95 9 5 Cvr fr IS+SS th c = 5 fr th SA hn p = 95 91 Avr lf-Wdth fr IS+SS th c = 5 fr th SA hn p = 95 91 7 Cvr fr IS+SS th c = 5 fr th SA hn p = 95 91 Avr lf-Wdth fr IS+SS th c = 5 fr th SA hn p = 95 91 9 Cvr fr A th c = 1 fr th SA hn p = 0.95 9 3 Avr lf-Wdth fr A th c = 1 fr th SA hn p = 95 9 31 Cvr fr A th c = 5 fr th SA hn p = 95 9 3 Avr lf-Wdth fr A th c = 5 fr th SA hn p = 95 9 33 Cvr fr C th c = 1 fr th SA hn p = 95 93 3 Avr lf-Wdth fr C th c = 1 fr th SA hn p = 0.95 93 35 Cvr fr C th c = 5 fr th SA hn p = 95 93 3 Avr lf-Wdth fr C th c = 5 fr th SA hn p = 95 93 37 Cvr fr CMC th c = 1 fr th SA hn p = 95 9 3 Avr lf-Wdth fr CMC th c = 1 fr th SA hn p 95 9 39 Cvr fr CMC th c = 5 fr th SA hn p = 95 95 Avr lf-Wdth fr CMC th c = 5 fr th SA hn p = 95 95 xi LIST OF FIGURES Figure Page 11 95-ntl f dtrbtn 3 1 Invrtn th C 3 13 Invrtn th prl C 5 1 Stht tvt ntr Cvr fr IS-nl fr th nrl fr p = 77 3 Cvr fr IS-nl fr th nrl fr p = 95 77 Cvr fr IS+SS fr p = fr th bvrt nrl th p = 5 77 5 Cvr fr IS+SS fr p = 95 fr th bvrt nrl th p 5 7 Cvr fr IS+SS fr p = fr th bvrt nrl th p = 9 7 7 Cvr fr IS+SS fr p = 95 fr th bvrt nrl th p = 9 7 Cvr fr A fr p = fr th nrl 79 9 Cvr fr A fr p = 95 fr th nrl 79 1 Cvr fr C fr p = fr th bvrt nrl th p 5 79 11 Cvr fr C fr p 95 fr th bvrt nrl th p = 5 1 Cvr fr C fr p = fr th bvrt nrl th p = 9 13 Cvr fr C fr p = 95 fr th bvrt nrl th p = 9 1 Cvr fr CMC fr p = fr th nrl 1 15 Cvr fr CMC fr p = 95 fr th nrl 1 1 Cvr fr IS-nl fr p 95 fr th SA 17 Cvr fr IS+SS fr p = 95 fr th SA 1 Cvr fr A fr p = 95 fr th SA x IS O IGUES (Cntnd r Page 19 Cvr fr C fr p = 95 fr th SA 3 Cvr fr CMC fr p = 95 fr th SA 3 CHAPTER 1 INTRODUCTION May sysems a ocesses eii come eaios a oeae i uceai eio- mes Eames icue comue ewoks aaase sysems eecommuicaio e- woks asoaio sysems a iacia makes aom ees suc as aiues o comoes ukow uue emas a ices a aua isases ae iee o ese sysems A sysem esige mus eemie e eomace o a oose sysem eoe imemeig i a a oeao ees o esimae e eec o maageme eci- sios o uue eomace o ee uesa ow suc a sysem o ocess eaes oe oe wi ui a maemaica moe We e suie ocess o sysem icues aomess e moe is oe a socasic ocess iscee-ee socasic simuaio a Moe Cao simuaio (eg aw ( ae oweu oos i sciece a egieeig o suyig e eaio o socasic sys- ems