Bayesian Designing and Analysis of Simple Step-Stress Accelerated Life Test with. Weibull Lifetime Distribution. A thesis presented to.

Transcription

1 Bys Dsgg d lyss of Sml S-Srss lrd f Ts wh Wull fm Dsruo hss rsd o h fuly of h Russ Collg of Egrg d Thology of Oho Uvrsy rl fulfllm of h rqurms for h dgr Msr of S X u Novmr X u. ll Rghs Rsrvd.

2 Ths hss ld Bys Dsgg d lyss of Sml S-Srss lrd f Ts wh Wull fm Dsruo y X U hs rovd for h Drm of dusrl d Sysms Egrg d h Russ Collg of Egrg d Thology y To Yu sss Profssor of dusrl d Sysms Egrg Ds rw D Russ Collg of Egrg d Thology

3 3 BSTRCT U X M.S. Novmr dusrl d Sysms Egrg Bys Dsgg d lyss of Sml S-Srss lrd f Ts wh Wull fm Dsruo 8. Dror of Thss: To Yu Ths hss dvlos Bys mhod o lyz d l sml s-srss lrd lf s wh Ty- sorg. s ssumd h h flur ms h srss lvl follow h Wull dsruo d h sl rmr s log-lr fuo of h srss lvl. Th o sms of modl rmrs oh y M lyss d Bys lyss r orgzd h hss. Th symo ofd rvls d Bys ofd rvls r ddud. Th flu of ror dsruo o h smo s sudd. Th Bys oml l s od y mmzg h rosror vr of sfd low rl of h lf m dsruo h orml srss odo. Mo Crlo smulo lgorhm sd o Gs smlg s dvlod o fd h oml srss hgg m. Th mmum lklhood mhod s lso dvlod for h us of omrso. flu of ror dsruo d sml sz o h oml l s lso dsussd. Th rsuls show h Bys roh hs hoful ol lyzg d dsgg rlly lf s wh hr s ror kowldg d ury for h modl rmrs. rovd: To Yu sss Profssor of dusrl d Sysms Egrg

4 4 CKNOWEDGMENTS Durg h dvlom of my grdu suds Oho Uvrsy svrl rsos d suos ollord drly d drly wh my rsrh. Whou hr suor would mossl for m o fsh my work. Th s why wsh o dd hs so o rogz hr suor. Frs d formos offr my srs grud o my dvsor Dr. To Yu who hs suord d gudd m hroughou my hss wh hs d kowldg. Whou hs ourgm d ffor hs hss would o hv omld. Bsds my dvsor would lk o hk h rs of my hss omm mmrs: Dr. drw Sow Dr. Dvd Koo d Dr. D Shwrh for hr gud sghful omms d suggsos. would lk o hk Slm Z. Rmd for sulyg m d s s my s sudy. Th rml d mk h hss mor rl. Ths work ws rlly suord y Nol S Foudo roj CMM-964. Flly would lk o hk my fmly for hr uodol suor sro d lov.

5 5 TBE OF CONTENTS BSTRCT... 3 CKNOWEDGMENTS... 4 TBE OF CONTENTS... 5 ST OF TBES... 7 ST OF FGURES... 8 CHPTER : NTRODUCTON Movo Bkgroud Rlly lrd lf s S-srss lrd lf s Th umulv osur modl lro modls Mmum lklhood smo ME Bys fr Ojvs... CHPTER : TERTURE REVEW.... lyss of Sml SST..... lyss of sml SST usg M mhod..... Bys lyss of sml SST Dsg of Sml SST Drmo of oml srss hgg m Drmo of oml sml sz... 6 CHPTER 3: PROBEM STTEMENT Noos Bs ssumos Prolm Dsro lyss of sml SST Dsg of sml SST Dvlom of M Mhods... 9

6 6 3.5 Dvlom of Smulo lgorhm vsgo of Prl roh o Sfyg h Pror Dsruo Comrso w M mhod d Bys mhod Ssvy lyss... 3 CHPTER 4: METHODOGES lyss of SST M lyss Bys lyss Oml Dsg of Sml SST Mmum lklhood l Bys l Smulo lgorhm Pror sfo... 4 CHPTER 5: CSE STUDY lyss of Sml SST Dsg of Sml SST... 5 CHPTER 6: CONCUSONS ND FUTURE RESERCH Colusos lyss of SST Dsg of SST Fuur Rsrh REFERENCES... 6 PPENDX : OBSERVED FSHER NFORMTON MTRX PPENDX B: NFERENCE OF MXMUM KEHOOD PN PPENDX C: R CODE FOR M NYSS PPENDX D: WNBUGS CODE FOR BYESN NYSS PPENDX E: R CODE FOR MXMUM KEOOD METHOD OF PNNG PPENDX F: R ND WNBUGS CODE FOR BYESN PNNG... 8

7 7 ST OF TBES Pg Tl 5.: Erml d wh =4 = 6 d = Tl 5.: Posror sss of modl rmrs...45 Tl 5.3: Esm of d y M d Bys mhod...49 Tl 5.4: Pror dsruos for Bys lyss...49 Tl 5.5: Esm of d wh dffr ror dsruo...5 Tl 5.6: Pror dsruos for Bys l...5 Tl 5.7: Oml Bys dsgs...54

8 8 ST OF FGURES Pg Fgur.: Th rlly fuo CDF d PDF of Wull dsruo wh = d= 5... Fgur.: S-srss lrd lf s...3 Fgur 4.: Flow hr of smulo...4 Fgur 5.: Tr los of d...46 Fgur 5.: Glm-Ru ovrg ss of d...47 Fgur 5.3: Dsy los of d...48 Fgur 5.4: Vr vs. mmum lklhood mhod...5 Fgur 5.5: E [Vr ] vs. ssumg Pror d = Fgur 5.6: E [Vr ] vs. ssumg Pror d = Fgur 5.7: E [Vr ] vs. ssumg Pror d = Fgur 5.8: E [Vr ] vs. ssumg Pror d =...56

9 9 CHPTER : NTRODUCTON Ths hr rodus h movo d ojvs of hs sudy s wll s som kgroud kowldg ludg rlly lrd lf s s-srss lrd lf s umulv osur modl lro modl mmum lklhood smo d Bys fr.. Movo Tody s rsg mrk omo d hghr usomr os r drvg mufurrs o dsg d rodu hghly rll rodus []. s mor o ssss d rd h rlly of rodu durg h dsg d dvlom sg us h m-o-mrk s gg shorr d shorr. Rlly ssssm usully dds o rml lf ss o o flur d for lfm lyss. Bus lf ss r sv d h dso md sd o h lf ss ffs h ol lf-yl os hy d o rfully ld d lyzd. Ths hss dsgs d lyzs sml s-srss lrd lf s usg h Bys roh. Th mjor movo of usg Bys roh sd of mmum lklhood mhod s s ly of ororg ror kowldg d ury h modlg d lyss.. Bkgroud.. Rlly ordg o Elg[3] rlly s dfd s h roly h omo or sysm wll rform rqurd fuo for gv rod of m wh

10 usd udr sd org odos. ordr o rss h rlosh w hs roly d h m rod w df h ouous rdom vrl T o h m o flur of h sysm or omo d h h rlly rsd s R = Pr{T }. whr T R d lm R. For gv vlu of R s h roly h h m o flur s grr h or qul o whh s lld h rlly fuo of h flur dsruo. Df F= -R = Pr{T <}. whr F = d lm F. Th F s h roly h flur ours for m whh s kow s h umulv dsruo fuo CDF of h flur dsruo. ohr fuo lld h roly dsy fuo PDF s lso ommoly usd rlly grg o dsr h sh of h flur dsruo. s dfd y df dr f.3 d d whr f d f. Fgur. llusrs hs hr fuos wh h m o flur follows h Wull dsruo wh h sh rmr = d h sl rmr = 5.

11 R F f 5 5 m o flur Fgur. Th rlly fuo CDF d PDF of Wull dsruo wh = d= 5 ddo o hs hr dsruo fuos ohr fuo lld h flur r or hzrd r fuo s of usd rlly. rovds sous m r of flur h s Pr{ T + }= R- R+..4 Usully w s f..5 R Th s kow s h sous hzrd r or flur r fuo. Th m m o flur MTTF d h md m o flur r frquly usd s msurm of rlly. Th MTTF s h m or d vlu of h flur m whh dfd y

12 MTTF E T f d..6 Gv dsrd rlly s R = = Pr{ T }.7 h s lld h h rl of h lfm dsruo whh ms m % of h oulo wll hv fld. f =.5 h.5 s h md m o flur... lrd lf s Rlly lf sg s ommoly dsgd for lf d lyss whh smls r sd o o flur m d for rlly ssssm []. Hghly rll rodus lk my lro ssors usully hv log lf udr orml oro odos. Thrfor s ulkly h rodus r sd udr h orml oro odos us of m d os lms. Th lrv s o us flur d olld lrd lf s T o roj h rlly udr h orml odos. Durg h T h us r ld udr hghr h orml srss odos.g. volg urr humdy mrur. o sd u h dgrdo ross so h flur formo olld shor rod of m [3]. Th flur formo s h rsformd hrough lro modl o rd h rlly udr h orml oro odos...3 S-srss lrd lf s Thr r wo ys of T: os-srss T d s-srss T. os-srss T h us r ld oly udr o hghr h orml srss lvl

13 3 whl h s srss lrd lf s SST llows svrl srss lvls. h SST h smls r sd hrough mor h o lvl of srss usully wh rsg ordr. Grlly wh h umr of flurs hvs r-sfd umr flur-s srss s or h s hs susd r-sfd m udr o srss m-s srss s h rmg us wll ld udr ohr srss. s wh oly wo srss lvls s lld sml s-srss lrd lf s whl s wh mor h wo srss lvls s lld mull s-srss lrd lf s [4]. f s s dd ul r-sfd m s lld Ty- sorg; f s s dd wh h flurs mou o r-sfd umr s lld Ty- sorg. Fgur. shows h srss rofl of mul-sst wh fv srss lvls d Ty- sorg h s h s sos r-sfd m d ll us h hv o fld y r sord. s d s r rsvly h srss lvls d srss hgg ms. Srss vl Tm Fgur. S-srss lrd lf s

14 4 Th dvg of SST s h rduo of h duro of s us f o ough flur d r olld o srss hghr srss ould s o rs h roly of flur...4 Th umulv osur modl Bus h us wll sd udr mor h o srss lvl SST modl s dd o dsr h ff of hgg srss. ommoly usd modl SST s h umulv osur modl [5] whh ssums h h rmg lf of sms dds oly o h urr umulv fro fld d urr srssrgrdlss of how h fro umuld. For SST wh m srss lvls h umulv osur modl ssums h h flur ms from h SST follow h followg CDF [4]: F F u F3 u F F m m 3.8 whr τ s h m of hgg h srss from h h srss lvl o h +h srss lvl F s h umulv dsruo fuo udr h h srss lvl d u s h soluo of F + u =F τ - τ - + u -.

15 5 Ths sudy wll osdr h sml SST wh wo srss lvls d ssum h h flur m h srss lvl hs h Wull dsruo.. F =.9 whr θ s h sl rmr of h Wull dsruo h h srss lvl d = dos h orml org odo. Bus h flurs r ssumd o usd y h sm flur mhsm ll srss lvls h sh rmr s ssumd o dl ll srss lvls. ordg o h umulv osur modl.8 flur m olld from hs sml SST hs h followg umulv dsruo fuo [6] F F F. h s. τ θ τ τ θ θ τ θ F β β. Th roly dsy fuo of h flur m s h gv y

16 6. f...5 lro modls ordr o roj h rlly orml oro odo usg d olld hghr srss odos lro modl s rqurd. Th lro modls dsr h rlosh w h lfms d h srss odos. ommoly usd lro modl s h log-lr modl. Tkg h modl. s ml h log-lr modl rssd s log.3 whr dos h srss lvl or ror rsformo of. W us hs log-lr modl us s sml d hs usd rlly my los. For ml wh h srss s so h Powr or vrs Powr w usd s h lro modl Cs.4 whr C d r ukow oss d s dos h so srss. Th log=logc + logs. Hr =logs. rrhus w s wdly usd for lf ss wh mrur:

17 7 E.5 Ks whr E rrss h vo rgy s s h solu mrur K d K s Bolzm s os kow hysl os qul o V/ K [7]. Th log =-E /K /s d =/s hs s. ohr ml of h log-lr rlosh s h E-modl usd o sudy h m-dd dlr rkdow of ulr-h g dlrs whr h srss s h lrl fld. Th E-modl s gv s G E E.6 whr d G r ukow mrur-dd oss. E s h ld lrl fld d E d s h fld ov whh rkdow ours mmdly [8]. Th E-modl lso rssd s log-lr fuo log=log + G Ed E. Com formul. d.3 h CDF of h flur m gv rmrs d s rssd s d F Mmum lklhood smo ME Th mmum lklhood M mhod s oulr ssl d lyss o sm h modl rmrs gv h sml d. Th sml SST modl

18 8 osdrd hs sudy hs hr rmrs d. Gv h d = olld from h s h mmum lklhood smors of h hr rmrs r h os h mmz h followg lklhood fuo.8 whr s h lklhood of h h osrvo. ordr o mk h omuo mor ov h logrhm of lklhood fuo lld log-lklhood fuo s mmzd sd of h lklhood fuo. Grlly h MEs r od y kg h frs rl drvvs of h logrhm of h lklhood fuo d sg hs rls qul o zro [3]...7 Bys fr Th mmum lklhood mhod ssums h h rmrs r ukow u fd. Th Bys roh howvr ssums h h rmrs r rdom d urs o h rmrs r dsrd y jo ror dsruo whh s formuld for h flur d r olld d s sd o hsorl d r wh smlr rodus dsg sfos d rs oos. Th ly of ororg ror kowldg h lyss mks h Bys roh vry vlul h rlly lyss us o of h mjor hllgs ssod wh h rlly lyss s h lmd vlly of d. fr o h modl rmrs s md rms of roly sms whh r odol o h osrvd d. O h sml d hv olld h ror dsruo s udd ordg o Bys horm

19 9 f f f.9 whr f s h jo ror dsruo f s h jo osror dsruo d f f ddd.. fr o h rmr s sd o s mrgl osror dsy. For ml h mrgl osror dsy for h rmr β s dfd s f dd f f dd.. f Mull lvls of gro r ssry o o h ormlzg os f d h mrgl osror dss. For omld modls hs gros r of lylly rl d somms v umrl gro o drly od. hs ss Mrkov h Mo Crlo MCMC smulo s h ss wy o g rll rsuls whou vlug grls [9]. MCMC lgorhm h s rulrly usful hgh dmsol rolms s h lrg odol smlg lld Gs smlg. Eh ro of h Gs smlg yls hrough h ukow rmrs drwg sml of o rmr odog o h ls vlus of ll h ohr rmrs. Wh h umr of ros s lrg ough h smls drw o o rmr rgrdd s smuld osrvos from s mrgl osror dsruo. Fuos of h modl rmrs suh s h h rl of h lfm dsruo h orml srss odo dod y lso ovly smld. Posror fr omud usg sml sss. hs sudy w

20 us WBUGS [] slzd sofwr kg for mlmg MCMC smulo d Gs smlg..3 Ojvs Ths hss osss of wo ojvs. Th frs o s o lyz h SST wh Bys mhod. Th modl rmrs of h lfm dsruo d r smd. Th sod o s o dsg oml SST l wh Bys roh. Th oml srss hgg m of h SST s drmd o mmz h rso of h smo of h h rl of h lf m dsruo orml srss odo. For oh ojvs rl mls rld o olros rlly r usd o llusr h roosd mhods. ordr o hv hs ojvs svrl sks wr omld. h lyss so h sks r dsrd s: Dvlo Mrkov h Mo Crlo MCMC smulo lgorhm sd o Gs smlg for osror smulo d fr; Dvlo h M sm for uros of omrso; 3 Comr M sm wh Bys sm; 4 Prform ssvy lyss o vsg how hgs ror dsruo wll ff vr d ofd rvl of h modl rmrs. h dsg so h sks r dsrd s: Dvlo h M l for h uros of omrso; vsg rl roh o sfyg h ror dsruo;

21 3 Dvlo h smulo lgorhm o o h r-osror vr of ; 4 Comr M l wh Bys l; 5 Prform ssvy lyss o vsg how hgs h ror dsruo d sml sz wll ff h oml srss hgg o.

22 CHPTER : TERTURE REVEW Ths hr rvws rvous rsrh o lyss d dsg of sml ssrss lrd lf s. Boh mmum lklhood mhod d Bys mhod r rvwd.. lyss of Sml SST Gv h d olld h s h uros of lyzg sml SST s o sm rlly msur suh s h m m o flur or rl of h lfm dsruo udr h orml srss odos. Boh h mmum lklhood mhod d h Bys mhod hv ld h lrur... lyss of sml SST usg M mhod Th mjory of h lrur rld o sml SST lyss usd h M mhod. Th ol dsruo hs svly sudd du o s smly. Xog [] ddud h MEs of rmrs sml SST wh ol dsruo d y- sorg d osrud h ofd rvls C usg vol quy. Xog d J [] sudd smlr rolm wh y- sorg. Th symo ofd rvls of h rmrs r rovdd sd of h mhod whh smlfy h ross of solvg C udr lrg sml sz. Howvr h C udr smll sml sz my o ur ough. Blkrsh l. [3] lso lyzd h sml SST wh ol lfm dsruo d y- sorg. hs rl log-lr lro modl

23 3 s ssumd. Th uhors drvd h MEs of h rmrs hrough h us of odol mom-grg fuos. Thy osrud hr rohs o drmg h Cs for h rmrs: mhod symo mhod d h rmr oosr mhod. Blkrsh l. [4] sudd smlr rolm wh y- sorg. Kr d Blkrsh [6] lyzd h sml SST modl wh Wull lfm dsruo d Ty- whou ssumg y lro modl. Bus h losd form MEs o od h uhors usd h Nwo-Rhso lgorhm o omu h MEs umrlly. Thy lso rovdd symo d oosr ofd rvls for h rmrs of h Wull sml s-srss modl. Mos of h suds h lrur dsussd h m-s srss s. lyss of flur-s srss s ws rord y Xog d Mllk [5] whh ssumd h ol lfm dsruo d h h srss hg ms r rdom vrls. MEs for modl rmrs sd o oh h mrgl d odol lf dsruos wr osdrd... Bys lyss of sml SST Th Bys roh hs o wdly ld o lyz d from SST. d P [6] rsd Bys fr modl for sml SST usg y- sorg. Thy ssumd h h flur ms h srss r olly dsrud wh m h s log-lr fuo of h srss lvl. Thy grd h grg kowldg o h ror dsruo of h rmrs h log-lr

24 4 fuo d hrough Sgl-gmm dsruo ojugo hy drvd h osror dsruo for rmrs of rs. lyg h Bys roh o SST h ssl rso of rmr fr s mrovd d h rqurd umr of smls s rdud. Md [7] lso ld h Bys fr for sml SST ssumg h lfm udr orml oro odo s olly dsrud. Gs smlg ws mlmd hs rl o rform MCMC smulo d osror fr.. Dsg of Sml SST Th mjory of h lrurs rld o lg sml SST md o o h oml srss hgg m som of hm gd h oml lowr srss lvl s wll. fw of lrurs md o drm h oml sml sz. Mos of hs lrurs usd mmum lklhood mhod... Drmo of oml srss hgg m Plg oml sml SST usg M mhod hs h suj of my suds. Gv kow sml sz h ojv s usully o fd h oml srss hgg o h mmzs h symo vr of h ME of quy of rs. Comuo of h symo vr s usully sd o h vrs of h Fshr formo mr. B l. [8] lhdd d Yg [9] [] d Srvsv d Shukl[] followd smlr rodur o fd h oml srss hgg m y mmzg h

25 5 symo vr of h ME of quy of rs suh s m lf or h h qul of h lfm dsruo udr orml odos. Th dffr s h lfm dsruo ssumd. B l. [8] ssumd ol lfm dsruo lhdd d Yg [9] usd Khms-Hggs modl lhdd d Yg [] osdrd h log-orml lfm dsruo d Srvsv d Shukl [] ssumd log-logs lfm dsruo. B d Km [] rd o o h low srss lvl s wll s srss hgg m y mmzg symo vr of ME of sd qul dsg srss. Elsyd d Zhg [3] lso dsgd h low srss lvl d srss hgg o y mmzg symo vr of ME of h rlly rdo dsg srss odos ovr r-sfd rod of m. Co s roorol hzrds modl ws ssumd h r. Wg d Yu [4] rodud ohr smor lld Uformly Mmum Vr Usd Esmors UMVUEs. Th uhor omrd h M Squr Error MSE of UMVUES wh h of MEs d rovd h UMVUEs hd smllr MSE h MEs hd. Th h uhor dsussd h oml s l sd o mmzg vr of UMVUEs of log-m lf. Tg d Yo [5] ld h s- Squrs roh o o omum srss hgg m wh smll sml sz. hs rl h ojv ws o mmz h rsformd ls squr of lf-srss rmrs sd of vr of MEs of hm. Ths mhod drv losd-form of smd rmrs; howvr oly dl wh smll sml sz.

26 6.. Drmo of oml sml sz Sslly h lrgr h sml sz s h r rsuls- mor ur smo- hvd. O h ohr sd from h rodur s o of vw odug s wh lrg sml sz wll k u log m d l. Thrfor s sgf o g ror sml sz o l hs wo sds. Thr r o so my rls o dsgg sml sz of SST s dsgg srss hgg o. Ts l. [6] d Chu l. [7] ddd rodur o osdro ssumg Rylgh lfm dsruo d Wull lfm dsruo rsvly. Chu l. [7] drmd h sml sz sd o mmzg vr of MEs d rovdd ls d fgurs of smlg l udr dffr osumr s d rodur s rsk whh ovly usd whou y omuo.

27 7 CHPTER 3: PROBEM STTEMENT hs hr h s ssumos d rolm r dsrd. Th h susks whh r slshd ordr o hv h ojvs of hs hss r dsrd. 3. Noos : srss or ror rsformo of h srss : sl rmr of h Wull dsruo udr srss : sh rmr of h Wull dsruo : srss hgg m : sorg m : rmrs of log-lr rlo : lfm of m udr SST : h h rl of h flur m dsruo orml org odo F f: umulv dsruo fuo CDF d roly dsy fuo PDF of flur m R: rlly fuo of flur m R=-F 3. Bs ssumos Ths sudy mks h followg s ssumos: dl d dd us r ld h SST. Th flur ms h srss lvl r dsrud ordg o h Wull dsruo.

28 8 3 Th rlosh of sl rmrs udr dffr srss lvls s log-lr whh rssd s log = +. =. 4 Th sh rmr s dl d s dd of h srss. 5 umulv osur modl s ssumd. 6 u s frs sd udr h lowr srss lvl. f h u hs o fld y h srss lvl s rsd o h hgg m d h s s oud ul flur or h sorg m.. Ty- sorg. Wull dsruo s ssumd us s mor fll h ol dsruo s my rsrhrs ssumd. h T h flurs r rqurd o usd y h sm flur mhsm ll srss lvls ordr o roj h rlly udr orml org odos hrfor h sh rmr s ssumd o dl ll srss lvls [8]. 3.3 Prolm Dsro 3.3. lyss of sml SST hs hss h sml sz h sg srss lvls d r r-sfd. Th frs rolm s lyzg sml s-srss lrd lf s y smg h rmrs of h lfm dsruo from rml d. do h rsfd srss hgg m h ojv s o sm d wh Bys mhod. rml ds olld h Th Flm No & Mrolros Rsrh orory Ts &M Uvrsy s usd o llusr h

29 9 Bys mhod. Th M mhod s lso dvlod d omrd wh h Bys mhod h hss Dsg of sml SST Th sod rolm hs sudy s lookg for h oml srss hgg m suh h h s ury of h smo of h h rl of h lfm dsruo orml srss odos dod y wll od. For h M mhod h ury s msurd y h symo vr of h ME of. For h Bys mhod h ury s howvr msurd y h r-osror vr of. rl ml sd o rml d olld h sm l s usd. 3.4 Dvlom of M Mhods Bus hr s o lrur osdrg SST wh Wull dsruo d y- sorg s wll s log-lr lro fuo. Th M mhods for oh lyzg d lg of h SST r ddud hs sudy. 3.5 Dvlom of Smulo lgorhm Th osror lyss ould rfrrd sos 4 rqurs mull lvls of gro o o f d h mrgl osror dss whh hv o losd form soluos. Mo Crlo smulo lgorhm volvg Gs Smlg wll dvlod hs sudy.

30 3 3.6 vsgo of Prl roh o Sfyg h Pror Dsruo rl ssu rld o h Bys dsg mhod s h sfo of h jo ror dsruo f. O my ssum h h hr rmrs r dd d ssg dd ror o h of hm h s f =fff. 3. lhough h ror gv y 3. s omuol ov my rlly dfful o drly sfy rors for d d o jusfy hr dd. Thrfor hs hss lors rl roh o sfyg h jo ror dsruo for Bys dsg. 3.7 Comrso w M mhod d Bys mhod Comrso w mmum mhod d Bys mhod s orgzd oh lyss so d dsg so. h lyss so rl d s s usd o omr h wo mhods. Th M mhod fds h oml l sd o h symo vr d h oml l s dd of h sml sz. Th sml sz s wll s h ror dsruo s flul o Bys oml l. Th s sudy h hss shows wh hs wo mhods yld smlr rsuls. 3.8 Ssvy lyss Ssvy lyss s rformd h lyss r o vsg how hgs ror dsruo wll ff h vr of smd rmrs. h lg so h ssvy lyss s rformd o vsg how hgs h ror dsruo

31 3 wll ff oml srss hgg m. Eslly h flu of h ror ury msurd y h ror dsruo rg o h oml l s sudd. lso h flu of sml sz o h oml l s sudd.

32 3 CHPTER 4: METHODOGES Th ojvs of hs hss lud lyzg d dsgg oml sml SST usg h Bys roh. For h uros of omrso M mhods wll lso drvd hs hr. 4. lyss of SST 4.. M lyss Thr s o lrur rfrrd o lyzg h SST wh Wull dsruo y- sorg d log-lr lro modl smulously s hs hss ssums. For h uros of omrso wh Bys lyss h o d ofd rvl smo y M mhod r ddud. h umr of flurs h frs sg of SST h umr of flurs for sorg m d h sml sz. From h PDF rlly fuo of h lfm d h log-lr lro fuo h log-lklhood fuo of h flur m rssd s l l log l l 4. whr = s h ordrd flur d. Th dls of h M mhod foud d.

33 33 h frs rl drvvs of h log lklhood wh rsv o h modl rmrs r l l l l l l 4. l 4.3. l 4.4 Th MEs of h modl rmrs d r h soluo of. l l l 4.5 Bus hs s sysm of o-lr quos w ld h Nwo s mhod whh rqurs h vrs of h osrvd Fshr formo Mr whh od s h gv of h sod rl drvvs of h log-lklhood fuo wh rs o h modl rmrs

34 34. l l l l l l l l l F 4.6 Th lms of h formo mr r rovdd d. MTB s usd o solv h quos. Bus h MEs of h modl rmrs r o losd-form rssos s mossl o drv h ofd rvls C hs hss w us h symo C sd. W o h symo vr of h smd modl rmrs â ˆ d ˆ from h symo vr-ovr mr whh s dfd s h vrs mr of h Fshr formo mr.. F Vr Cov Cov Cov Vr Cov Cov Cov Vr 4.7 Th h wo sdd -% C of h modl rmrs â ˆ d ˆ gd from ˆ ˆ ˆ ˆ ˆ ˆ / / / Vr z Vr z Vr z 4.8 Bus

35 35 ] l [ / 4.9 h symo vr of h ME of h h rl orml org odos s gv y T Vr 4. ordg o h dl mhod [3]. 4.. Bys lyss ordg o Bys horm h osror dsruo of h modl rmrs gv d s f f f 4. whr s h lklhood of h d gv y 4. d. dd d f f f 4.3 Th h mrgl osror dsruo of d r rsvly

36 f f dd f f dd 4.5 f f dd. 4.6 Th osror m of d gv r rsvly E E E f d 4.7 f d 4.8 f d 4.9 whh my usd s o sms of h rmrs. Gs smlg s usd o drw rdom sml of h rmrs d from hr ow mrgl osror dsruo f f d f rsvly d h sm h d vlu usg h sml m. Th -% ofd rvls for h rmrs d r rvls U U d U rsvly ssfyg U P U f 4. U P U f 4. U P U f 4. whr h d U r h lowr d ur lm of h rvl rsvly. W my hoos d U s

37 37 P U P 4.3 P U P 4.4. P U P 4.5 g w do Gs smlg o gr rdom smls of h rmrs d from h mrgl osror dsruos d h h lowr lm s h vlu of h /h rl of h sml d h ur lm s h -/h rl of h sml. 4. Oml Dsg of Sml SST 4.. Mmum lklhood l For h uros of omrso h M mhod for dsgg oml SST s dsrd. From h ssumo of umulv osur modl Wull dsruo d log-lr lro modl h CDF of h flur m of s u udr h sml SST s. F 4.6 Th PDF s

38 38. f 4.7 Th roly h s u wll o fl y h sorg m s. R 4.8 Df h dor fuo y flur osrvd fr m f flur osrvd for m f 4.9 d h dor fuo y. sord m f flur osrvd for m f 4.3 Th h log-lklhood fuo s. l l l ; 4.3 Th Fshr formo mr for MEs of od s h os of h gv of h sod rl drvvs of h log-lklhood fuo wh rs o h modl rmrs. Th Fshr formo mr for h smls s rssd s

39 39. E E E E E E E E E F 4.3 Th h symo vr-ovr mr for ME s. F Vr Cov Cov Cov Vr Cov Cov Cov Vr 4.33 Th oml hgg o wll h o h mmzs Vr gv y formul 4.. Dls of h M mhod foud d B. 4.. Bys l Th M mhod dsrd h rvous so rqurs kowldg of h ru ukow rmrs vlus lld lg vlus. Nvrhlss h vlus of h modl rmrs r ukow d h uros of odug h s s o sm hm. Tylly lg vlus my od from r wh smlr rodus dsg sfos d grg judgm. Ury h lg vlus lwys ss whh mks h Bys roh rv lrv o h rdol M roh rlly s lg. ordg o Bys horm h osror dsruo of h modl rmrs gv d s

40 4 f f f 4.34 whr s h lklhood of h d gv y ` 4.35 whr s h umr of flurs udr h lowr srss s h umr of us fl for sorg m d ddd f f f 4.36 whh s lso lld h r-osror mrgl dsruo of. lyg mulvr rsformo of rdom vrls h jo osror dsruo of l gv d drvd s. l l l l l l l f f 4.37 Th h mrgl osror dsruo of s. l dd f f 4.38 Th osror m d vr of gv r rsvly d f E 4.39 d

41 Vr E f d. 4.4 Th r-osror vr of dfd s E Vr Vr 4 f d 4.4 dos o dd o d hrfor wll usd s h ojv fuo. Th oml hgg m wll h o o mmz E [Vr ]. For h uros of omrso h ror vr of s lso luld h hss. lyg mulvr rsformo o h jo ror dsruo of d ylds l l l l f l f. 4.4 l l Th ror m d vr of r rsvly d Vr E f d 4.43 E f d Smulo lgorhm Evluo of h r-osror vr of rqurs hgh-dmsol gro d hs o losd-form soluo. Thrfor Mo Crlo gro s usd for h rom vluo of h grl 4.4. For h vlu hs sudy rooss o vlu E [Vr ] ordg o h followg lgorhm:

42 4 lrg umr of ss of s smuld from hr jo ror dsruo. For h s of s of flur ms s smuld from h dsruo fuo F gv y formul.7. Smuld flur ms h r grr h h sorg m om rgh-sord osrvos. Th flur ms r h lyzd usg Gs smlg o o h osror vr of Vr. vrgg Vr ovr ll ss of ylds h E [Vr ]. Fgur 4. shows h flow hr of h smulo lgorhm. F Gs smlg Vr f Vr E [Vr ] Vr Fgur 4. Flow hr of smulo Dffr vlus of r usd o lo h rlosh w d E [Vr ] ordr o dd h oml hgg m Pror sfo Th jo ror dsruo sfd hs hss s sd o h sudy of Sgurwll [8]. sd of ssumg wo Wull sl rmrs r dd Sgurwll [8] dmosrd h h Wull sh rmr d h md mo-flur r dd d h ssgd h dd rors o h sh

43 43 rmr d h md lf. Bus h md lf rssd s sml fuo of h wo Wull rmrs h jo ror dsruo for wo Wull rmrs drvd hrough vr rsformo of rdom vrls. h lg r of hs hss w ssum h w hv dd ror kowldg of h sh rmr d h md lvs wo srss lvls. Dog h md lvs h srss lvls d y M d M h jo ror dsruo rssd s fm M = fm fm f. Bus M / / *l *l 4.45 h jo ror dsruo of d drvd v mulvr rsformo of rdom vrls. f f M M * M M M M / f M M *[ ]l Th jo ror dsruo gv y 4.46 hough umrsom r hs h vru of g ror dsy d o h hs rsoly wll movd [8].

44 44 CHPTER 5: CSE STUDY hs hr ss sudy r orgzd o rform h mhodologs dsrd hr 4 d omr M mhod wh Bys mhod. 5. lyss of Sml SST Th rml d s usd o dmosr h roosd mhod. R s usd o rform M lulo d WBUGS s usd o rform Bys lyss. Th od foud d C d d D rsvly. rml ds olld h Th Flm No & Mrolros Rsrh orory Ts &M Uvrsy s usd h s sudy. Th d s olld l vrom o from ommrl roduo. h rm h orysll ZO mddd hgh-k dv ws sd. Th - ZO mddd hgh-k h flm ws rrd o ml-od-smoduor MOS or whh ws frd o h HF-l -y 5 m -3 wfr for s [9]. Tl 5. dsrs h d od from h rm. Frsly w s uformv rors o h rmrs whh r orml dsruo wh = d = 6 for d d gmm dsruo wh sh rmr quls o d r rmr quls. for. Th rsuls of Bys lyss od from WBUGs r showd Tl 5..

45 45 Tl5. Erml d wh =4 = 6 d =78 : < Tl 5. Posror sss of modl rmrs Nod M sd MC rror.5% Md 97.5% To rv osror dd o h srg o of smulo svrl hs wh ovr-dsrsd srg os o MCMC smulo should ru. Th smulo ovrgs o h rg dsruo wh rs of ll hs r o mg oghr [8]. hs s wo hs wh dffr ls ru smulously o smulo. Eh h ous for 8 ros. Thr r wo mhods o hk ovrg. O s mg r los of h sml vlus vrsus ro. Fgur 5. shows r los for rmrs d rsvly. W rsoly ofd h ovrg hs hvd s h wo hs r o wll md.

46 46 Fgur 5. Tr los of d Ths s forml roh o ovrg dgoss. quv wy of hkg ovrg s sd o lyss of vr. Th Glm-Ru ovrg

47 47 ss R s rodud o dgos ovrg. R s dfd s h ro of h wdh of h rl 8% rvl of h oold hs o h vrg wdh of h 8% rvls wh h dvdul hs. Wh WBUGS smulo ovrgs R should or los o o [8]. Fgur 5. shows Glm-Ru ovrg sss of d d from Fgur 5. h smulo s lvd o hv ovrgd. Fgur 5. Glm-Ru ovrg ss of d Th ury of osror sm s luld rms of Mo Crlo sdrd rror MC rror of h m whh s sm of h dffr w h m of h sml d h ru osror m. ordg o Sglhlr[] h smulo should ru ul h MC rror for h od s lss h 5% of h sml sdrd dvo. hs s hs rul hs hvd. Fgur 5.3 shows krl dsy sms of dsruos of d. rs h dsruo of s qu symmr. Thrfor md vlu sd of m vlu s hos o h o smd vlu for. Th o sms od from M mhod d Bys mhod r show Tl 5.3. Th smd vlus gd from hs wo mhods r los o h ohr. Th w s som

48 48 formv rors whh r showd Tl 5.4. Pror s h s of rors usd h ov smulo. Pror d Pror us ouous uform dsruo o rss h ury h vlus of rmrs. For s U 5 ror for ds h s lvd o w d 5 d ll vlus wh h rvl r qully rol. From Pror o Pror h ror ury s drsg Fgur 5.3 Dsy los of d

49 49 Tl 5.3 Esm of d y M d Bys mhod Mhod M Bys Prmr Esmd vlu Vr 95% C rvl lgh E E Tl 5.4 Pror dsruos for Bys lyss Pror N 6 N 6. Pror U 5 U-5 U 5 Pror U 4 U- U Tl 5.5 shows h sms wh hr ss of ror dsruos. rs h s h ror ury drss h vr of smd rmrs drs. Ths s d us h ror kowldg hs orord wh d d rsd h ury of sm. From Tl 5.5 s lso rsoly o olud h h rvl lgh s rrowr wh hghr formv rors.

50 5 Tl 5.5 Esm of d wh dffr ror dsruo Pror Pror Pror Prmr Esmd rvl Vr 95% C vlu lgh E E E Dsg of Sml SST Th s usd o llusr h Bys lg mhodology s sd o h T rm sudy d Bys d lyss odud y uo[8] whh h rlly of ovl omr-hk hgh-k g dlr flm s sudd. Th hgh-k g dlr s lrv mrl o SO h vod h srous fuo d rlly rolm usd y furhr rdug od hkss. h sudy uo[8] rformd os lrd ss o msur h rkdow of hgh-k flm wh h lrd srsss rgg from 5.5 o 8. MV/m d od h m-o-rkdow d from h ss. Th d wr usd o r ror dsruo of h modl rmrs hs hss. Frsly w ssum vlus of hr rmrs r kow d ly mmum lklhood mhod o drm h oml srss hgg o. Th ojv fuo

51 5 s o mmz h symo vr of h ME of h h rl orml srss lvl h s Vr. h s h srss s h lrl fld srss d w ssum =.5 MV/m =7.7 MV/m =8. MV/m =. = 4 sods d =4. Th lg rmrs of hr rmrs r =3 =- d =.66. R d Mrosof El r usd o solv h rolm. Th od foud d E. Fgur 5.4 shows h lo of Vr vrsus h srss hgg o. Th oml srss hgg o s *=774 sod..e+ V of h ME of E+.8E+.6E+.4E+.E+ E Srss Chgg Tm sods Fgur 5.4 Vr vs. mmum lklhood mhod Th w osdr h suo h h rs vlus of h modl rmrs r ukow d ly h Bys mhod. Th ojv fuo s o mmz h rosror vr of h h rl orml srss lvl h s E [Vr ]. Tl 5.6 summrzs h hr ss of ror dsruos ssumd h s sudy. Th

52 5 ouous uform dsruo s usd o rss h ury h vlus of h rmrs. For ml U5 53 ror for M ds h h md lf h srss lvl S s lvd o w 5 sods d 53 sods d ll vlus wh h rvl r qully rol. Th m vlus of hs hr ss r h sm...66 M 46 sods d M 55sods whh orrsod o h lg vlus usd y mmum lklhood mhod. From ror o ror h ury of h vlus of h modl rmrs s rsg. Tl 5.6 Pror dsruos for Bys l M M Pror U U96 96 U5 53 Pror U U46 46 U49 54 Pror U U U47 56 Fgur d 5.7 show h los of h r-osror vr of vrsus h srss hgg m ssumg h hr ror dsruos Tl 5.6.

53 E [Vr ] E [Vr ] Srss Chgg m sods Fgur 5.5 E [Vr ] vs. ssumg Pror d = Srss Chgg m sods Fgur 5.6 E [Vr ] vs. ssumg Pror d = 4

54 E [Vr ] 54 9.E+9 9E+9 8.9E+9 8.8E+9 8.7E+9 8.6E+9 8.5E+9 8.4E+9 8.3E+9 8.E+9 8.E+9 8E Srss Chgg m sods Fgur 5.7 E [Vr ] vs. ssumg Pror d = 4 Tl 5.7 summrzs h oml srss hgg ms d h orrsodg r-osror vr of. Tl 5.7 Oml Bys dsgs Oml Pror Vr. Oml s. E [Vr ] of Pror Pror Pror ordg o Tl 5.7 s h ror ury rss h oml srss hgg o od y h Bys mhod s losr o h od y h mmum lklhood mhod. From h Bys horm gv y 4.34 osror fr of ully dds o formo from wo sours: h ror

55 55 kowldg d h d v h lklhood fuo. s h ror ury rss h formo from d oms mor flul o h osror fr. Th mmum lklhood mhod uss oly h d formo. Thrfor h lrgr h ror ury h mor smlr rsuls h Bys mhod d mmum lklhood mhod yld. Tl 5.7 lso rss h ror vr of whh s luld oly usg h ror formo d qufs h ror ury h vlu of. Wh w r mor r o h vlus of rmrs h oml E [Vr ] s losr o h ror vr of. Furhrmor h rgs of E [Vr ] Fgur d 5.7 show h E [Vr ] s mos ssv o wh Pror s usd s h sm hg lds o h grs hg h vlu of E [Vr ]. Covrsly E [Vr ] s ls ssv o wh Pror s usd. Ths osrvos d h h rml d dos o rovd muh ddol formo o rdu h ury o wh vry formv ror s ssumd. s h ror ury rss from Pror o Pror h oml E [Vr ] oms mor sgfly lowr h h orrsodg ror vr of. Ths mls h wh h rs of ror ury h formo from d om mor flul o h osror fr. Wh w r rmly r o h vlus of modl rmrs h s h rs vlus r kow h Bys roh suggs h hr s o d o do h lf s. Th s us ll h formo dd ddud from h modl. Th mmum lklhood mhod lwys ylds *=774 o mr wh sml sz s. Howvr h sml sz wll flu h oml srss hgg o

56 E [Vr ] 56 Bys mhod. Fgur 5.8 shows h lo of E [Vr ] vs. ssumg Pror d =. Th oml srss hgg m s 8 sods. Comrd wh h oml srss hgg m ssumg Pror wh = 4 s losr o h oml srss hgg o od from M mhod. Ths ds h wh lrgr sml sz h d s mor flul o h osror fr Srss Chgg m sods Fgur 5.8 E [Vr ] vs. ssumg Pror d =

57 57 CHPTER 6: CONCUSONS ND FUTURE RESERCH Ths hss rss Bys lyss mhod s wll s Bys dsg mhod for Ty- sord sml s-srss lrd lf s wh Wull lf m dsruo d log-lr lro modl. Th rl mls r usd o llusr h roosd mhod d o omr h Bys mhod wh h M mhod. 6. Colusos Th m ojvs d sks dsrd so.3 r omld h hss. Th Bys lyss ws odud o sm h o d ofd rvl of h modl rmrs. WBUGS s usd o do h lyss d h od foud d D. Th Bys oml l s od y mmzg h vlu of h r-osror vr of d h smulo lgorhm s hvd y R d WBUGS. Th od s showd d F. Th M oml l s gd wh h hl of R o do omuos. Th od d E luls h symo vr of h ME of wh = svrl vlus of srss hgg m r hos o lo h rlo w Vr vs. d h oml srss hgg m s od from h lo. Grlly oh M mhod d Bys mhod usd o lyz d dsg sml SST u hy ll o dffr odos. M mhod ld wh: rs vlus of modl rmrs hv kow lrg sml sz od h s

58 58 Bys mhod ld wh: urs o h modl rmrs s ror kowldg o h modl rmrs r vll fw d s vll h s 6.. lyss of SST h lyss so h smulo for Bys lyss hs rovd o ovrgd h hss. Bus h dsy lo of dsruo of sms o symmr h md vlu sd of h m s hos o h smd vlu d omrd wh h M rsul. Th Bys mhod d M mhod yld smlr rsul. s h ror ury drss h Bys mhod ylds h smd vlus wh smllr vr d rrowr ofd rvls. 6.. Dsg of SST Ths hss ssums h ror dsruos of h md lvs wo srss lvls.. M d M d h sh rmrs d ssums hy r dd. Th jo ror dsruo of h modl rmrs d s h ddud v mulvr rsformo from h ror dsruos of M M d. Ovrll h Bys l dds o h ror dsruos d h sml sz. Wh w r mor r o h ror dsruo h sml sz s lss flul o h osror fr v vrs.

59 59 For SST l dsg s h ror ury drss h oml srss hgg m od from M mhod d Bys mhod r losr; h rosror vr of E [Vr ] s mor ssv o h srss hgg m; 3 h r-osror vr of E [Vr ] oms muh mor sgfly lowr h h orrsodg ror vr of Vr whh s gd from h ror dsruo. Wh h ror urs r h sm h srss hgg m od y M mhod s losr o h od y Bys mhod wh lrgr sml sz. O rsg dduo s h h Bys roh would rommd h o lrd lf s s dd wh h rs vlus of h modl rmrs r kow. 6. Fuur Rsrh Ths hss dvlod Bys mhod for h SST wh ssumos of Wull lfm dsruo Ty- sorg d h umulv osur modl. Howvr grlzd o h lyss d dsg of ohr ys of rlly lf ss suh s ol lfm dsruo d Ty- sorg du o h grly d flly of h Bys frmwork. h lg rolm som osrs my osdrd suh s lowr lm o h d umr of flurs for h srss hgg m d lowr lm o h d umr of flurs for sorg m. Th sml sz s lso vl for Bys dsg us h Bys l s lrg sml sz o mrov h ury. Howvr h lrg sml sz

60 6 s rqurs gr os d m. Thus h sml sz k o osdro s wll s h srss hgg m h SST lg h fuur rsrh. Ths my d o gr h s os s wll s h rqurm o h ury of h s rsul. Srss lvl s lso mor o h s. Hgh srss lvl ould fsr h s h rdus h os. Howvr f h srss s oo hgh h d my o rfl urly h flurs udr orml srss. Thrfor h srss lvl drmo ohr suj of dsgg h SST.

61 6 REFERENCES [] W. Q. Mkr d.. Esor Ssl Mhods for Rlly D. Nw York: Joh Wly & Sos 998. [] N. Frd d C. Oml sml s srss lrd lf s dsg for rlly rdo Jourl of Ssl Plg & fr vol My 9. [3] C. E. Elg roduo o Rlly d Mly Egrg. Nw York: MGrw Hll. [4] E. Gouo d N. Blkrsh S-srss lrd lf s dvs rlly N. Blkrsh d C. Rdhkrsh dd Nw York: Elsvr Norh Holld. [5] W.Nlso lrd lf sg- s-srss modl d d lyss EEE Trsos o Rlly vol. 9 o Ju. 98. [6] M. Kr d N. Blkrsh fr for sml s-srss modl wh y- sorg d Wull dsruo lfms EEE Trsos o Rlly vol. 57 o D. 8. [7] F. J. S vo Ergs d h rrhus Equo Quly d Rlly Egrg rol vol [8] W. uo Rlly Chrrzo d Prdo of Hgh K Dlr Th Flm doorl dssro Drm of dusrl Egrg Ts &M Uvrsy Ts US 4.

62 6 [9]. Glm J.B. Crl H.S. Sr d D.B. Ru Bys D lyss d Edo. Nw York: Chm & Hll/CRC 3. [] D. Sglhlr. Thoms N. Bs d D. u WBUGS Usr Mul. 3. [Ol]. vll: h:// [] C. J. Xog frs o sml s-srss modl wh y- sord ol d EEE Trsos o Rlly vol. 47 o Ju [] C. J. Xog d M. J lyss of groud d sord d from s-srss lf s EEE Trsos o Rlly vol. 53 o.. -8 Mr. 4. [3] N. Blkrsh D. Kudu H. K. T. NG d N. K Po d rvl smo for sml s-srss modl wh Ty- sorg Jourl of Quly Thology vol. 39 o J. 7. [4] N. Blkrsh Q. X d D. Kudu E fr for sml s-srss modl from h ol dsruo udr m osr ls of h su of Ssl Mhms vol. 6 o Mr. 9. [5] C. J. Xog d G.. Mllk S-srss lf-sg wh rdom srss-hg ms for ol d EEE Trsos o Rlly vol. 48 o. Ju [6] J. d R. P Bys fr modl for s-srss lrd lf s wh y- sorg Prodgs of h ul Rlly d Mly Symosum 8.

63 63 [7] M. T. Md Bys fr for rlly lrd lf ss usg gs smlg Mrolros Rlly vol. 37 o [8] D. S. B M.S. Km d S. H. Omum sml s-srss lrd lf ss wh sorg EEE Trsos o Rlly vol. 38 o D [9].. lhdd d S.S. Yg Oml sml s-srss l for Khms- Hggs modl EEE Trsos o Rlly vol. 5 o.. -5 Ju.. [] P.W. Srvsv d R. Shukl log-logs s-srss modl EEE Trsos o Rlly vol. 57 o S. 8. [].. lhdd d S.S. Yg oml sml s-srss l for umulv osur modl usg log-orml dsruo EEE Trsos o Rlly vol. 54 o Mr. 5. [] D. S. B d M. S. Km Omum sml s-srss lrd lf ss for Wull dsruo d y sorg Nvl Rsrh ogss vol [3] E.. Elsyd d H. Zhg Dsg of omum sml s-srss lrd lf sg ls R Ddvs Sohs Oros Rsrh T. Doh S. Osk d K. Swk dd Sgor: World Sf Pulshg 7.

64 64 [4] B. X. Wg d K. Yu Oml l for s-srss modl wh rogrssv y- sorg X Shyou Du Xuo/Jourl of Souhws Prolum Uvrsy vol. 9 o Ju. 7. [5] S.. Tg d K.P. Yo ls-squrs roh o lyzg lf-srss rlosh s-srss lrd lf ss EEE Trsos o Rlly vol. 5 o Ju.. [6] T. R. Ts H.H. Ch d W. u Th so of smlg for ssrss ss wh qully-sd rvl sorg shm rol Jourl of Rlly Quly d Sfy Egrg vol. 5 o [7] S. W. Chug Y.S. So d W. Y. Yu smlg ls sd o flur-sord s-srss lrd ss for Wull dsruos Jourl of Quly M Egrg vol. o [8] N. D. Sgurwll Rlly d Rsk: Bys Prsv. Egld: Joh Wly & Sos 6. [9] C.H. Yg Y. Kuo d C.H. Chrg drg d dlr rkdow of orysll z od mddd zroum-dod hfum od hgh-k dlrs for ovoll mmors ld Physs rs vol. 96 o. 9.

65 65 PPENDX : OBSERVED FSHER NFORMTON MTRX Do s h umr of flurs h frs sg of SST s h umr flurs for sorg m d s h sml sz h lklhood fuo of sml d s. ; Th log-lklhood of sml d s. l l log l l l l l l ; l l

66 66 h d. Th frs rl drvvs of h log lklhood wh rsv o h modl rmrs r l l l l l l l. l Th sod rl drvvs of h log lklhood wh rsv o h modl rmrs r l l l l l

67 67 l l l l l l l l l. l Th Fshr formo Mr s. l l l l l l l l l F

68 68 PPENDX B: NFERENCE OF MXMUM KEHOOD PN Df h dor fuo y flur osrvd fr m f flur osrvd for m f d h dor fuo y flur osrvd fr m f flur osrvd for m f Th log-lklhood fuo s ddud s l l l l l l ;

69 69 h d. Th frs rl drvvs of h log lklhood wh rsv o h modl rmrs r l l l l l. Th sod rl drvvs of h log lklhood wh rsv o h modl rmrs r

70 7 l l l l l l l l l. Th lms of h Fshr formo mr for osrvo r h gv d vlus of h sod rl drvvs:

71 7 l l l R F d f d f E R d f d f E R R d f d f d f d f E l l l R R d f d f d f d f F E l l l R d f d f F F d f d f F E

72 7. R d f d f E Th Fshr formo mr for h smls s gv y. E E E E E E E E E F Th symo vr-ovr mr for ME s dfd s h vrs mr of h Fshr formo mr. F Vr Cov Cov Cov Vr Cov Cov Cov Vr Bus ] l [ / h symo vr of h ME of h h rl orml org odo s h gv y T Vr ordg o h dl mhod whr

73 73 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ] l l[ ˆ ˆ ˆ

74 74 PPENDX C: R CODE FOR M NYSS = = =.8468 < < =7. =7.9 u=6 =78 =5 =38 =4 =. =5 B<-+* B<-+* <-u/b+/b-u/b B3<-*- BB<-*-*+**log*-+*s= BB<-*logs= C<-u/B+/B-u/B BB<--/-sumBB-sumBB--*C*logCs= <-*-*+** <-* <--sum-sum-*-*c <-*-*+*** <-u*-u*b3*-/u*b3+-u 3<-*-*-*-u*/B-*/B+u*/B 4<-*-*u*/B+*/B-u*/B <--sum+-*sum-sum3-sum4--**- *C-*-u*/B-*/B+u*/B--**C- *u*/b+*/b-u*/b B<-*-*+**log*-+*s=* B<-*-*+* B3<-**logs=

75 75 B4<- B<--+sumB+sumB+sumB3+sumB4+**C*logCs=+-*C B<-u*-*B3/u*B3+-u B<-*-*+**log*-+*s=** B3<-*-*+** B4<-*-*logs=*-u*/B-*/B+u*/B B5<--*-u*/B-*/B+u*/B B<--*--*+sumB+sumB+sumB3-sumB4-sumB5-**C-*logCs=*-u*/B-*/B+u*/B-*C-*-u*/B-*/B+u*/B <-*-*+*** <-*-*-u*/b-*/b+u*/b <--sum+sum+*-*c-*-u*/b- */B+u*/B F<-mr-BB-B-B-B---B--33 NVF<-solvF =+**-log-s=/ <--*log-log-s=s=/ <- B<-* V<-mrB3 TV<-V Vr<-V%*%NVF%*%TV

76 76 PPENDX D: WNBUGS CODE FOR BYESN NYSS modl{ C <- for :{ zros[] <- h[] <- -log.[] + C [] <- su-[] [] <- s-[] om[] <- log - *logh + -*log[] - ow[]/h om[] <- log - *logh + -*logu**-+[]- u - owu/h+[]/h-u/h om3[] <- -owu/h+/h-u/h log.[] <- []*[]*om[] + []*-[]*om[] + -[]*om3[] zros[] ~ dosh[] } <- h* ow -log- / #ror dsruo ~ dgmm. ~ dorm. ~ dorm. h <-+* h <-+* h <-+* }

77 PPENDX E: R CODE FOR MXMUM KEOOD METHOD OF PNNG =3 =- =.66 =6.5 =8. =4 =4 =. =.5 u= B<-+* B<-+* <-fuo {u/b+/b-u/b} B3<-*- 77 F<-fuo {--/B} f<-fuo {/B*-*-/B} f<-fuo {/B*B3*u+-u-*-B3*u+-u/B} F<-fuo {--B3*u+-u/B} R<--B3*u+-u/B <-fuo {*logs=--**/b*-*- /B} C<-gru$vlu <-fuo u/b+/b-u/b*logu/b+/b- u/bs=*/b*b3*u+-u-*-b3*u+u/b C<-gru$vlu E<-/B*C+C+--B3*u+u/B/+=*log=s=*R E<- /B*C+C+F=/+=*log=s=*R <-fuo */B*-*-/B D<-gru $vlu

78 78 <-fuo u/b+/b-u/b*/b*b3*u+-u- *-B3*u+-u/B D<-gru $vlu E<-/B*D+*D+**R <-fuo -u*/b-*/b+u*/b <-fuo u*/b+*/b-u*/b BB<-fuo -u/u*b3+-u*/b*b3*u+-u-*- B3*u+-u/B E<-grBBu $vlu BB<-fuo u/b+/b-u/b-*-u*/b- */B+u*/B*/B*B3*u+-u-*-B3*u+u/B E<-grBBu $vlu BB3<-fuo u/b+/b-u/b-*u*/b+*/b- u*/b*/b*b3*u+-u-*-b3*u+-u/b E3<-grBB3u $vlu EBB<-*/B*D--*u*-*B3*E+*- *E+*E3+*-*-**R+*- **R <-fuo {*logs=--**/b*-*- /B} F<-gru $vlu <-fuo u/b+/b-u/b*logu/b+/b- u/bs=*/b*b3*u+-u-*-b3*u+u/b F<-gru $vlu E<-F=-/B*F-/B*D-*F-D- *=*log=s=*r-=*r B<-fuo /u*b3+-u*/b*b3*u+-u-*- B3*u+-u/B G<-grBu $vlu B<-fuo u/b+/b-u/b-**logu/b+/b- u/bs=+*-u*/b-*/b+u*/b*/b*b3*u+u-*-b3*u+-u/b G<-grBu $vlu EB<-*Fu-*/B*F-/B*D+*F-Fu-u*- *B3*G+G+-**logs=+**R

79 79 <-fuo u/b+/b-u/b-*-u*/b- */B+u*/B*/B*B3*u+-u-*-B3*u+u/B H<-gru $vlu EB<-*/B*D-*H-*-**R F<-*mrEEEBEEEBEBEBEBB33 NVF<-solvF =+**-log-s=/ <--*log-log-s=s=/ <- B<-* V<-mrB3 TV<-V Vr<-V%*%NVF%*%TV Vr

80 8 PPENDX F: R ND WNBUGS CODE FOR BYESN PNNG R Cod: smsstu<-fuohh u u{ F_u <- - -u/h <-mr for : { f u[]<=f_u{ [] <- h*-log-u[]/ } ls { rm <-h*-log-u[]/ [] <- rm + u - u*h/h } } <- sor } <-4 <-. <-4 M_l=946 M_u=346 M_l=47 M_u=56 _l=.64 _u=.68 <-.5 <-7.7 <-8. < <-lgh Ev<-rry sml<- r<-8 s.sd for :sml{ M <-rufm_l M_u M <-rufm_l M_u <-ruf_l _u h<-m / log/ h<-m / log/

81 8 for j :{ u<-[j] <-smsstuhhuu <-sum<=u <-sum<= d<ls"""""""""""u""""""""""m_l""m_u""m_l""m_u"" _l""_u" rmrs<-"" s<-lslsm=46m=55=.66 SST.sm<ugsdsrmrsmodl.fl="C:/WullSST.".hs=.r=r.ur=4.h= ugs.drory="c:/wbugs4" DC=FSE h.ugssst.sm sd_ <- sd m_ <- m dh.ugs Ev[j]<-Ev[j]+sd_ Ev[j]<-Ev[j]+sd_/m_ r rj flush.osol } rev flush.osol } Ev <- Ev / sml r rev WBUGS od: modl{ C <- for :{ zros[] <- h[] <- -log.[] + C log.[] <- log - *logh + -*log[] - ow[]/h zros[] ~ dosh[] } for +:{ zros[] <-

82 h[] <- -log.[] + C log.[] <- log - *logh + -*logu*h/h+[]-u - owu/h+[]/h-u/h zros[] ~ dosh[] } for +:{ zros[] <- h[] <- -log.[] + C log.[] <- -owu/h+/h-u/h zros[] ~ dosh[] } } <- +* * ow -log- / h<-m / ow log / h<-m / ow log / <- logh-logh/- <- logh-* #ror dsruo ~ duf_l _u M ~ dufm_l M_u M ~ dufm_l M_u 8