CHAPTER I INTRODUCTION

Transcription

1 CHAPER I INRODCION. Biarai Logiy ailr Modlig Popl rqly dic qipm bhaior i rm o ag ad ag. Commo xampl ar aomobil ad aomobil ir i which modl yar ad accmlad milag ar ally boh icldd i dicio o logiy. L wll rcogizd xampl or which wo marm cal ar qi impora icld acory qipm powr graio machi ad aircra. I ac h o may o h dic rliabiliy pciali dy i maiglly dcribd i rm o wo mar. Yar o ag ad milag ar o h oly wo qaii ha migh dcrib dic logiy. W h rm ag ad hr b h rm ar gric ad may rpr qi dir mar ha draio o owrhip ad diac rald. I h xampl o a aomobil ir ag migh corrpod o accmlad milag ad ag migh b mard a rad lo. E mor complicad mar ch a crr low ad hrmal hiory may b appropria or om igrad circi. h poi i ha dic li i a rorc ha may b b rprd ad or which h compio may b b mard ig a wo or highr dimioal cor ad h qaii ha compri h cor ar pciic o h qipm. Som rliabiliy modl ha rpod o h d o icld boh ag ad ha b dlopd. Howr arly all o h modl ha ha b dlopd ar did i a mar ha prmi hm o b rdcd o igl dimioal modl i which h idpd ariabl i im. Som o h xiig modl ar dicd rhr i Chapr. A rcrd xamiaio o biaria mar o qipm ilizaio ha

2 o b prormd o biaria modl ha o b lly dlopd ad om poially l modl orm ha o b did a all. Or i i hi rarch i o proid a ramwork or biaria dic modlig. W bgi wih h corcio o biaria ailr modl. I rlad o h craio o biaria ailr modl ha ha ad ho ha ha o b dlopd ar xamid. W proid a bri axoomy o biaria dic logiy modl orm ad di om xampl modl o illra ad xplai om o h ky i. h biaria ailr modl play a ky rol i or biaria modlig. h ky moiaig acor i h ac ha xiig biaria ailr modl ar did o ha hy ca b collapd io igl ariabl li im diribio modl dpi h ac ha biaria modl may b mor rprai or dcripi o dic logiy.. Biaria Maiac Modlig Maiac ca igiicaly ac h qaliy o prodc ar hy ha b prodcd. A maiac policy ca b iwd a a combiaio o dir maiac acio ch a ipcio rpair ad/or rplacm ad may b claiid a corrci chdld maiac or pri chdld maiac. A corrci maiac policy icld all chdld maiac acio prormd a a rl o ym ailr o ror h ym o a pciid codiio. Pri maiac icld all chdld maiac acio prormd o rai a ym i a pciid codiio. ally h co o maiac ad ppor i 60 o 75% o h oal li-cycl co o a ym. hror a pri maiac policy i o gra imporac i rdcig hi co ad lowrig h rik o a caarophic brakdow. Mo di o maiac polici corrci or pri cocra o ym wih iaria liim diribio i.. bad o iaria ailr modl.

3 Howr i may pracical iaio ym liim ad maiac polici dpd o ral acor. or xampl i i commo or aomobil o carry a maiac policy or a oil chag ry hr moh or 3000 mil or a rplacm o h camha imig bl ry 60 moh or mil. h cmlai ag o a ym or dir priod o im will ha dir c o h ym ailr bhaior. rhrmor mo ym driora o ha h ad im bw rpair bcom horr ad horr ad h rpair im icra. Som ym lik car airpla ligh blb compr or ay objc ha ca accmla d o iromal c may ag or war or im. A ym may b i a iadqa opraioal codiio ar om amo o im or ag. h ym may rqir a maiac acio o ror i back o a ormal codiio. h prpo i o pr h occrrc o a caarophic ailr drig i opraio ad o rdc h orall maiac co. Howr ry limid lirar ha addrd mlidimioal ailr modl ad hir rlad maiac polici. or modlig biaria maiac polici w ir adap ad xd Hr biaria rwal hory [974a] o acilia h corcio ad aalyi o biaria maiac modl. Som baic rl o h biaria rwal hory ar prd ad applid o h biaria maiac modl. wo yp o biaria maiac policy ar coidrd. h ar corrci maiac CM ad pri maiac PM polici. h corrpodig biaria maiac modl ar dlopd bad o h biaria dic logiy modl. Laplac raorm chiq ar d o aalyz h modl. or mo o h dlopd modl w ar o abl o ir h raorm xplicily. Nw mrical mhod ad algorihm ha may b d o approxima h biaria Laplac ir raorm ar idiid. 3

4 .3 Biaria Aailabiliy Modlig Aailabiliy ha appard o b a appropria mar o h ci o a maiaid ym. I coidr h ailr bhaior ad h c o maiac acio. I h iaria ca aailabiliy i did a h probabiliy ha h ym i cioig aiacorily a ay poi o im Li Hwag ad illma [977]. Biaria aailabiliy ca b did ad xdd dircly rom h iaria aailabiliy. ollowig biaria dic logiy ad maiac modlig w coidr h corcio o biaria dic aailabiliy modl. Bad o h wo yp o maiac polici w dlop h corrpodig biaria aailabiliy modl. h Laplac raorm or h biaria aailabiliy modl ar drid. Dirc irio o h raorm ar o obaid hr..4 Approach o Biaria Modlig h approach o biaria modlig may b orgaizd io i ky or: i Biaria ailr Modlig. Modl a igl-i ym wih biaria logiy. ii Biaria Rwal Modlig. Modl a igl-i ym wih idpd ad idically diribd i.i.d. liim ad immdia/iaao maiac ric. 4

5 iii Biaria Corrci Maiac Modlig. Modl a igl-i ym wih i.i.d. liim ad i.i.d. corrci maiac im. i Biaria Pri Maiac Modlig. Modl a igl-i ym wih i.i.d. liim ad pri maiac a wll a rpair b boh haig diic i.i.d. diribio. Biaria Aailabiliy Modlig. Modl h biaria aailabiliy mar or a igl-i ym wih maiac ric. W dcrib ach modlig ag i daild a ollow. i Biaria ailr Modlig. No ha i hi ag w oc o corcig biaria ailr modl rahr ha maiac polici. Or prpo i o impro iaria ailr modl by dlopig ad aalyzig biaria ailr modl. W di h gral rcr o biaria probabiliy modl o ym ailr. Sral biaria ailr modl ar corcd o rpr h poibl corrlaio rcr o h wo ym agig ariabl: im ad ag. h bhaior o h modl i xamid dr h ario corrlaio rcr. h dlopd modl will b d o aalyz xampl maiac problm. ii Biaria Rwal Modlig. Coidr a ym wih a biaria radom logiy i.. h logiy dpd o h opraig im ad h ag. A aild ym i immdialy ad iaaoly rplacd by a i.i.d. w o. h w am ha h maiac i prc ad h 5

6 ym opraio ca b modld a a biaria rwal proc. ig biaria rwal hory w modl ad aalyz h ym ailr bhaior. iii Biaria Corrci Maiac Modlig. h biaria rwal modl dlopd i ag ii ar xdd o icld rpair im. h maiac im may icld ipcio rpair ad/or rplac im ar amd o b i.i.d. ad h ym ar maiac i amd o b "a good a w." h combid biaria liim-ad-maiac-im proc may b modld a a alraig biaria rwal proc or imply a ordiary biaria rwal proc. o corc hi modl biaria rwal hory may b d i h aalyi o ym ailr bhaior. Hr w coidr rpair im b o pri maiac. i Biaria Pri Maiac Modlig. W modiy ad graliz h biaria corrci maiac modl dlopd i ag iii o rlc h c o pri maiac by dlopig maiac modl wih i.i.d. pri maiac im ad i.i.d. rpair im. ha i w corc biaria pri maiac modl. h modl ar xamid dr a ag rplacm pri maiac policy. Ohr polici ch a groprplacm opporiic rplacm ad/or h combiaio o polici may b modld i a imilar way. Biaria Aailabiliy Modlig. or h dlopd corrci ad pri maiac modl w corc h Laplac raorm or hir corrpodig biaria aailabiliy modl. Som xampl biaria aailabiliy cio ar prd wih h drid biaria ailr modl. h ida o h qaliy o aailabiliy mar i did i rm o biaria aailabiliy modl. 6

7 Sag i o dmora a ramwork i biaria rliabiliy ad maiac modlig. h rl rom ag i o proid a odaio or rhr dy o biaria ad mliaria modl..5 Oli o Rarch I Chapr w proid a lirar riw or biaria rliabiliy modlig biaria rwal hory ad maiac modlig or o-i ym ad mli-i ym. hr i ry limid dy i biaria ailr modlig ad o i biaria maiac modlig ad biaria aailabiliy mar. I Chapr 3 w pr a axoomy o biaria ailr modl cla ad idiy wo cla a or oc. h ochaic cio modl ad corrlad modl ar dlopd ad xamid. Som i rlad o modl ormlaio aalyi ad rhr dy ar dicd. Som xampl mrical calclaio ar proidd. I Chapr 4 w dy a biaria rwal proc ad pr i baic rl bad o Hr biaria rwal horm. Som gralizaio o hi horm ar prd. W obai h rwal cio ad rwal diy or ordiary biaria rwal biaria qai-rwal dlayd biaria rwal ad alraig biaria rwal proc. Biaria Laplac raorm aociad wih h rwal proc ar obaid. I Chapr 5 w pr om xampl or h biaria corrci maiac modl. W apply biaria rwal hory o corrci maiac modlig. Rwal modl ad corrci maiac modl ar obaid wih coidraio o h biaria ailr modl dlopd i Chapr 3. I Chapr 6 w rhr xd h corrci maiac modl o rlc h c o pri maiac polici. A ag rplacm pri maiac i coidrd. Gral rl ar obaid. Exampl o h biaria pri 7

8 maiac modl ar prd i Chapr 7 wih h coidraio o h biaria ailr modl dlopd i Chapr 3. I Chapr 8 w dlop biaria aailabiliy or h biaria corrci ad pri maiac modl. h Laplac raorm or h biaria aailabiliy modl ar obaid. I rlad o h qaliy o aailabiliy mar ar dicd. Exampl or h biaria aailabiliy modl ar prd i Chapr 9. Coclio ad r rarch ar mmarizd i Chapr 0. 8

9 CHAPER II LIERARE REVIEW Chapr II proid a hiorical oriw o om o h xiig lirar o maiac polici biaria rwal proc ad biaria ailr modl. h xiig lirar abo maiac polici do o ra biaria or mliaria maiac i h did hr. Lirar abo aailabiliy mar i h wo-dimioal ca do o xi.. Irodcio I h la w dcad maiac polici ha b a aci ara o rarch. Pirkalla ad Volkr [976] Shri ad Smih [98] Cho ad Parlar [99] ad Mrdock [995] ha prd a daild ry o mch o h xiig work. Nachla [998] Barlow ad Procha [975] ad Grbakh [977] prd maiac ad rplacm modl i a rliabiliy cox. Maiac problm ca b claiid by h complxii o h ym ad hir aociad maiac acio amly igl-i. mli-i ym maiac problm prc. imprc maiac problm ad iaao. oiaao rpair maiac problm. or xampl h igl-i ym maiac modl rpr a igl-i ym wih idpd ad idically diribd i.i.d. liim dr prc maiac polici wih immdia rplacm. ypically i hi modl i.i.d. maiac ric im ar alo amd ad h ym aailabiliy ca b obaid. rhrmor h i.i.d. ampio or boh ym liim ad maiac im ca b gralizd o modl maiac 9

10 problm wih diic maiac ric im. Mli-i maiac problm ollow imilar procdr ad approach. I boh cagori w ca alo coidr dir ym coigraio ad/or rcr o modl mor complicad qipm. Bor h rhr riw o lirar w ir proid om diiio o maiac i bri... Diiio o Maiac Diiio... Bahb-hapd ailr ra BR diribio. A li diribio i aid o b a bahb-hapd ailr ra BR diribio i: dcraig ailr ra DR 0 < i aid o b a coa ailr ra CR <. icraig ailr ra IR < h im iral [0 i calld h arly ailr or ia moraliy priod h iral [ i calld h l or cioal li ad h iral [ i calld h war-o priod. Diiio... Hazard cio. h hazard cio or h ailr ra cio i did a: z.. whr h diy cio xi ad i did a d ad h rliabiliy or rior cio i did a. d 0

11 Diiio..3. IR diribio. Dcraig coa ad icraig ailr ra DR CR ad A li diribio i aid o b a DR diribio i: d z 0 d or 0 <..3 A li diribio i aid o b a CR diribio i: d z 0 d or 0 <..4 A li diribio i aid o b a IR diribio i: d z 0 d or 0 <..5 h cio z i h hazard cio or h ailr ra cio. Diiio..4. Corrci Maiac: prc rpair miimal rpair ad imprc rpair. L z 0 z 0 b h hazard cio or a w ym. h diig z δ z 0 a h hazard cio or a ym ar ric a im. I δ h h ym drgo a prc rpair. I δ z/ z 0 h h ym drgo a miimal rpair. I δ c whr c i a coa ad c > h h ym i imprcly rpaird. Diiio..5. Pri maiac: ag rplacm ad block rplacm. dr a ag rplacm policy a i i i rplacd wih a i.i.d. w o po ailr or a ag a i whichr com ir. dr a block rplacm policy a i i i rplacd wih a i.i.d. w o po ailr ad a ag b b 3b. i i i

12 Diiio..6. Opporiic maiac. Opporiic maiac occr wh i i ha rachd a ag o i ad hr ar ohr i i h ym big rplacd d o ihr corrci maiac CM or pri maiac PM. Diiio..7. Grop maiac. By rplacig grop o aild i iad o rplacig idiidal aild i grop maiac i prormd ihr wh a ixd im iral g i xpird or wh a ixd mbr o i Ng ail whichr com ir.. Biaria Rliabiliy Modl hr ar craily may way o di a biaria rliabiliy modl. I addiio hr ar probably ral alra way o claiy h modl yp. W l ha a iormai claiicaio chm i bad o h rlaiohip bw h wo ariabl. Spciically w diigih bw ho modl or which ag ad ar cioally rlad ad ho i which hy ar corrlad rahr ha cioally dpd. W rhr para h modl i which h wo ariabl ar cioally rlad o h bai o whhr h cio ar drmiiic or ochaic. h modl bad o corrlaio o h wo ariabl may b rhr claiid by whhr 0 or 0. O cor rom a rliabiliy prpci h ca i which ag ad ar idpd i likly o b pracically maigl. Mo o h prioly dlopd biaria rliabiliy modl ra h wo ariabl a cioally rlad. May o h modl o war proc.g. Mrcr [96] ad Lmoi ad Wocr [985] ad ral o ho or cmlai damag.g. Barlow ad Procha [975] ad Birbam ad Sadr [969] porray qipm rliabiliy i

13 rm o drmiiically did drioraio occrrig a radom poi i im. Impora diig ar o all o h modl ar:. hy oc o a igl ailr mchaim or phomo. h ariabl drmi ailr by a hrhold al ad 3. hy ar limaly rdcd o iaria im modl o h bai o h cioal dpdc. h oc o h modl i o drmi how mch im will lap bor h hrhold i rpad. A irig ad o complly obio obraio i ha whil h war proc ad cmlai damag modl ra a a cio o im h proporioal hazard modl Lmi [995] ra ag a a drmiiic cio o coaria. Bca h modl bad po drmiiic cio ar rdcd o iaria orm hy ha b did xily ad ar o xamid rhr hr. h aalyical mphai wih modl bad o ochaic cio ha alo b hir rdcio o a igl dimio - im. h ochaic war modl ad cmlai damag modl ha ra damag magid a a radom ariabl ar all did i a mar ha prmi oc o rliabiliy i im. h am i r o h ho oi modl Lmoi ad Wocr [986]. E h diio proc modl Cox [96] ha rally ar xprd comprhily i rm o boh ariabl ar aalyzd i rm o ir paag im o a ailr a. ially h im dpd r-rgh irrc modl Kapr ad Lambro [977] ha h am characriic ha hy ar d o obai a diribio i im. I i appropria o o ha hr ar ral papr ha addr biaria ad mliaria rliabiliy modl i a ry dir cox ha h o rad hr. Spciically Marhall ad Olki [967a 967b] dlopd mliaria modl or h rliabiliy o ri ym comprid o o-idpd compo. I h modl ach ariabl corrpod o h ag o o o h compo. I h corcio o h modl Marhall ad Olki proid om l diiio b do o ra biaria or mliaria rliabiliy i h did hr. 3

14 I i Sigprwalla ad Wilo [993] who wr h ir o gg h dy o wo-dimioal rwal proc wih h dlopm o biaria ailr modl idxd by im ad ag. hy ollowd h rl o Sigprwalla [99] o irodc a gric biaria modl or ailr ad o pr a gral xprio or a cla o dii idxd by im ad ag. hy d hi ormla a a bai o imla h diy or a wid rag o pciicaio or h ag proc ad applid h biaria ailr modl o h dy o opimal warray problm by ig Mo Carlo imlaio o h xpcd mbr o rwal mbr o prc rpair ia a biaria warray policy. hy h ormlad a gam horic -p or drmiig opimal warrai ad d dyamic liar modl o orca warray claim o ym idxd by im ad ag. Sigprwalla ad Yogr [993] dlopd amili o mliaria li diribio or a wo-compo ym opraig dr dyamic irom. Coidrig h gamma proc ad h ho-oi proc hy xdd Sigprwalla' [99] rl o a w amily o mliaria li diribio. I appar ha h oly or o dy a r biaria rliabiliy modl wa ha o Eliahbrg Sigprwalla ad Wilo. [997]. hy dlopd modl ig boh drmiiic ad ochaic cio ad obaid l rl. hir oc wa h drmiaio o cary warray rr or prodc ch a aomobil. dr i.i.d. ampio or ym liim ad ag hy d biaria rwal hory o pciy h probabiliy graig cio o h diribio o h mbr o warray claim or a im. By ilizig Mo Carlo imlaio hy h obaid approxima rl o h mbr o warray claim ad hir diy cio or a im dr imprc rpair. I or dlopm o biaria ailr modl w icld hir modl ad how how hir approach ca b xdd o di ohr modl. Sigprwalla ad Wilo [998] propod a ragy or corcig biaria rliabiliy modl ha dcrib ailr i rm o im ad ag i a imilar way o h modl o Eliahbrg Sigprwalla ad Wilo [997]. By ig a addii hazard 4

15 cio hy modl h rlaiohip bw h ariabl ad ra a a imdpd ariabl. hy dcrib h olio o ag by ochaic proc lik h Poio h gamma ad h Marko addii ad dlopd h biaria ailr modl or ag dcribd by h Poio h dobly ochaic Poio ad a compod Poio proc. hy h h modl o ol h warray problm by Bayia aiical dciio hory o obai h opimal warray. W icld ad xd hir approach o h corcio o biaria ailr modl..3 Biaria Rwal horm Chg [95] wa h ir o dy ad xd h claical iaria rwal horm o highr dimio. Hr [974a b] did a rwal proc i wo dimio ad dlopd rhr rl. H dicd h biaria graig cio ad biaria Laplac raorm a h baic ool o graliz h claical hory o iaria rwal proc. H prd a xampl o a biaria xpoial diribio o illra h gral hory ad dlopd xplici xprio o h wodimioal rwal cio a wll a h corrlaio bw h margial iaria rwal coig proc. h rwal horm or h mli-dimioal ca ca b od i Mod [967] Spizr [986] ad Sibach ad V.R. Eawood [996]. h mahmaical dlopm ad applicaio o h claical iaria rwal horm ca b od i Çilar [975] Cox [96] Ro [996] ad Smih [958]. 5

16 .4 Maiac modl or o-i ym A igl-i modl or o-i rpr a complicad ym a a igl iy. h rm "igl-i" or "o-i" ym d hr ma ha h ym or prodc i iwd collcily ad ha oly h prormac o h ym a a whol i coidrd o h prormac o h idiidal compo. Nrhl h ym i a collcio o h compo o compo ac h ym. h ym il ca b compod o o dic or may diimilar dic. A complx ym ca alo b iwd a a o-i ym wh h ailr o ay compo i irprd a ailr o h ir ym. I pracic omim i i diicl o obai h rliabiliy daa or idiidal compo whra daa or h ochaic bhaior o h ir ym i aailabl or air o obai. Valdz-lor ad ldma [989] proid a daild ry o pri maiac modl or ochaically drioraig igl-i ym. I i mor impora o o ha a o-i ym i iwd ad rad a a bildig block or a mli-i ym. h mhod or aalyzig igl-i ym orm h bai or aalyzig mlii ym. or o-i or igl-i ym Barlow ad Procha [964] how ha amig a IR DR icraig dcraig ailr ra i ailr diribio h mbr o ailr i [0 ] i ochaically largr mallr wih a ag rplacm policy ha wih a block rplacm policy. hy alo howd ha h mbr o plad rplacm ad h oal mbr o rmoal i alway ochaically mallr dr a ag policy ha dr a block policy. Cox [96] ggd allowig h ym o rmai iaci i a ailr occr i [ k k or ay k ad or om im 0. Sh 0 [996] propod a modiid block rplacm policy wih wo ariabl ad gral radom miimal rpair co. H xdd Cox' modl ch ha i h i ail i [ k k i i ihr rplacd by a w o or miimally rpaird ad i i ail i [ k k 0 0 i i ihr miimally rpaird or rmai iaci il h 6

17 x plad rplacm. h modl wih wo ariabl i raormd io a modl wih o ariabl o obai h opimal policy..4. Imprc maiac Wag ad Pham [ ] propod a qai-rwal proc ad h dy o i applicaio i imprc maiac. hy did a coig proc {N > 0} ad l X do h irarrial im bw h - ad h o h proc or. I h qc o o-gai radom ariabl { X 3...} i idpd ad X α Z or 3... whr Z ar idpd ad idically diribd ad α > 0 i a coa h h coig proc {N > 0} i aid o b a qai-rwal proc wih paramr α ad ir irarrial im X. hy obaid h opimal imprc block rplacm polici or 0 < α bad o h xpcd maiac co pr rwal cycl ad/or h limiig arag aailabiliy. Brow ad Procha [983] coidrd a imprc rpair modl i which a im i rpaird po ailr. Wih probabiliy p h rpair i a prc rpair ad wih probabiliy - p h rpair i miimal rpair i.. ar rpair h ym i "a bad a old". Block Brog ad Sai [985] xdd h modl o Brow ad Procha o h agdpd imprc rpair modl i which a im i rpaird po ailr. Wih probabiliy p h rpair i a prc rpair ad wih probabiliy q - p h rpair i a miimal rpair whr i h ag a h ailr im ic h la prc rpair o h im i. Lam [988a 988b] coidrd a rricd rplacm modl wih parial rpair ha alo accod or h rpair im. H irodcd h gomric proc which i a qc o idpd o-gai radom ariabl { X 3...} ch ha h diribio cio o X i a x whr a i a poii coa. I a > h i i a dcraig gomric proc i a < i i a icraig gomric 7

18 proc. h h did h rricd rpair rplacm modl or a drioraig ym i which h cci rial im o h ym orm a gomric proc ad ar ochaically o-icraig whra h coci rpair im ar ailr alo coi a gomric proc b ar ochaically o-dcraig. H coidrd imlaoly wo yp o rplacm ragi: a rplac h ym i ad oly i i oal opraig im aai a crai prdrmid ll b rplac h ym a h Nh ailr. H obaid xplici xprio or h log-r arag co o h ragi. Sadj ad Zckrma [990] coidrd addiioal mooo proc rplacm modl. hy prd wo rplacm polici h policy ad policy N. dr h rplacm policy a ym i rplacd a a im which dpd o h im rom h iallaio or h la rplacm. dr policy N a ym i rplacd a h im o h Nh ailr. hy howd ha h opimal policy blog o h policy cla N or h log-r arag rward cririo. Lam [990] ga a xplici ormla or drmiig h opimal policy N ad i [99] did a rpair rplacm modl or a ochaically drioraig ym. or h xpcd dicod rward ca h howd ha h opimal rplacm policy i o h orm "rplac a h im o h Nh ailr." Lam [99] gralizd hi arlir work [988a 988b] o dy a gral rpair rplacm modl i which wo yp o rplacm polici ar xamid dr h xpcd dicod rward cririo. H howd ha h opimal policy N i a la a good a h opimal policy. Zhag [994] gralizd Lam' work [988a] wih h biaria policy N dr which h ym i rplacd a warig ag or a h im o h Nh ailr whichr occr ir. H coidrd h problm o chooig a opimal rplacm policy N ch ha h log-r arag co pr i im i miimizd. dr om mild codiio h prod ha h opimal policy N i br ha h opimal policy N or h opimal policy. 8

19 Mi [998] compard ym aailabiliy mar o a igl-i ym bad o ochaic ordrig ad claiicaio o liim diribio. h compario rl ar l i drmiig maiac polici or improig or opimizig h ym opraio iral. Mrdock [995] applid rwal hory o dlop a aailabiliy modl or a ii im horizo or a coioly dmadd compo which i maiaid by a ag rplacm pri maiac policy. H howd ha h opimal ag rplacm priod i a iii im horizo do o maximiz arag aailabiliy or all ii al o compo coomic li. h rl i criical i licycl maiac plaig..5 Maiac modl or mli-i ym I may applicaio h opimal maiac acio or o compo o dpd o h a o h ohr compo ad h ym rliabiliy rqirm. h ampio o prc maiac o mli-i ym i o logr alid l w am h rwal o all i or compo. h h imprc o maiac or imprc maiac ad coomic dpdcy or opporiic or grop maiac ar h major i or modlig mli-i ym maiac problm..5. Imprc maiac Shakd ad Shahikmar [986] ad Sh ad Griih [99] xdd h iaria imprc rpair modl o a mliaria imprc rpair modl. hy modld ym wih dpd compo haig pciic mliaria diribio ach o which drgo imprc rpair. Shakd ad Shahikmar [986] coidrd modl o ym comprid o compo haig dpd liim diribio. po ailr h compo i imprcly rpaird il i i prcly rpaird. hy 9

20 h gralizd hir modl o cor applicaio whr mor ha o compo ca ail a h am im. Sh ad Griih [99] coidrd a biaria o-ag-dpd imprc rpair modl i which wo im ar o cio a h am im. po ailr a im drgo a rpair. hy coidrd dir xral orc ha ca h ailr o h wo im..5. Opporiic maiac Dgbo [996] did opporiic ag rplacm polici or a wo-i ym ad prd aailabiliy aalyi ig a d rwal hory approach. h ym aailabiliy cio or h ailr rplacm policy h opporiic ailr rplacm policy parial opporiic ag rplacm policy ad h opporiic ag rplacm modl. H compard ario rplacm polici ad howd ha h ailr rplacm policy proid a highr aailabiliy ha h ohr rplacm polici. Wag [997] did h τ opporiic maiac o k-o-o- ym wih h coidraio o ym rliabiliy rqirm. I hi modl oly miimal rpair ar prormd o aild compo bor im τ ad corrci maiac o all aild compo ar combid wih pri maiac o all cioig b driorad compo ar τ. I h ym ri o im wiho prc maiac i will b bjc o pri maiac a im. H drid h ym aympoic co ra ad aailabiliy wih applicaio o aircra gi maiac..5.3 Grop maiac By rplacig grop o aild i iad o rplacig idiidal aild i maiac co may b rdcd. hi co aig kow a h coomy o cal rl moly rom h qaiy dico or rdcio o maiac -p co pr i. Sh ad Jhag [996] did a wo-pha maiac policy or a grop o 0

21 idical rpairabl i which icorpora miimal rpair orhal rplacm ad dowim co. hy dlopd a modl o calcla h log-r arag co pr i im or a gralizd grop maiac policy. Nachla ad Rao [99] coidr grop maiac or a ri ym. hy dlop a mhod or drmiig a maiac chdl ad gropig compo. h mhod i approxima ad hriic ad i bad o a co modl which i did i rm o compo rliabiliy mar. h modlig ramwork proid a aalyical mhod or aalyzig pri maiac chdlig problm ad lcig opimal or ar opimal ym ll maiac pla..6 Aailabiliy Aailabiliy i ally d a h ci or radi mar or rpairabl ad/or maiaid ym bca o i coidraio o boh rliabiliy ad maiaiabiliy. Barlow ad Procha [975] di aailabiliy o a rpairabl ym a h probabiliy ha h ym i opraig a a pciid im. Blachard [998] gi a qaliai diiio o aailabiliy a a mar o h dgr o a ym which i i h oprabl ad commiabl a a h ar o miio wh h miio i calld or a a kow radom poi i im. Li Hwag ad illma [977] gi a compl ry ad ymaic claiicaio o aailabiliy. h claiicaio o aailabiliy ar ir dpdig o h im iral coidrd: i poi or iaao aailabiliy ii arag aailabiliy iii limiig or ady-a aailabiliy i limiig arag aailabiliy Barlow ad Procha [975] Li Hwag ad illma [977] Nachla [998] cod coidrig h yp o dowim: ihr aailabiliy i achid aailabiliy ii opraioal aailabiliy Blachard [998] Li Hwag ad illma [977]. Mi [998] gi om compario rl o

22 aailabiliy i h ir cagory. h ir claiicaio o aailabiliy will b or oc i hi diraio ad will b xdd o wo dimio. Aailabiliy coidrd i maiac modlig ca b od i Barlow ad Procha [975] or rplacm modl awzi ad Hawk [99] or a R-o-o-N ym wih par ad rpair awzi ad Hawk [990] or a ri ym wih rplacm ad rpair Iyr [99] or imprc rpair modl Mrdock [995] or ag rplacm pri maiac modl Nachla [ ] or pri maiac modl ad Wag ad Pham [996] or imprc maiac modl..7 Rmark o Lirar Riw h lirar riw dcribd abo how ha may apc i rliabiliy modlig ad maiac aalyi ha b coidrd. Som o hm ha coidrd apc o h problm w dy b o o ha addrd h ll ad cohr corcio o biaria ailr rpair ad pri maiac modl.

23 CHAPER III BIVARIAE AILRE MODELING I hi chapr i rlad o h corcio o biaria rliabiliy modl ad hir applicaio o maiac plaig ar dicd. h diicio bw biaria ailr modl ad modl o ir paag im o a ailr hrhold clarii h moiaio or h dlopm o biaria ailr modl. W idiy wo cla a or oc. h modl cla xamid hr ar ho i which h wo ariabl ar rlad by a ochaic cio ad ho i which h ariabl ar imply corrlad. Exampl o h modl o ach o h wo cla ar did. h gral approach o modl ormlaio i xplaid o ha h radr may corc alra orm. I or iw h yp o biaria ailr modl dcribd hr proid a w way o dy h rliabiliy o qipm or which iaria mar ar icompl. h a w ara o rliabiliy rarch i idiid. h diiio w or may b modiid ad h approach o modl ormlaio w pr may b d o di ohr modl. W rai ral op qio cocrig modl corcio ad aalyi. Boh cocpal diiio ad aalyical mhod warra rhr xploraio. 3. Irodcio I Chapr h dicio o biaria rliabiliy modl proid or iw o how biaria ailr modl hold b claiid. I alo idica h yp o modl ha alrady xi. W l ha h modl ha ha b did ha b ry l i h dy o qipm rliabiliy b ha hr ar problm or which hy ar o wll id. Spciically h diiio o pri maiac pla o h bai o 3

24 boh ad ag rqir a biaria ailr modl. hror i hi chapr w xplor h diiio o biaria ailr modl. W coidr boh h ca o ochaic cioal rlaiohip ad impl corrlaio bw h ag ad ariabl. Or i i hi chapr i o corc biaria ailr modl ad o xami h i rlad o h craio o h modl yp ha ha o y b dlopd. W di om xampl modl ad idiy ral qio ha m b rold i ordr o cra ad aalyz iormai ad l biaria modl. W or xampl modl o illra ad xplai om o h ky i ad w gg dircio or coiig dy. h ky moiaig acor i all o or dicio i h ac ha xiig biaria modl ar did o ha hy ca b collapd io igl ariabl li im diribio modl dpi h ac ha biaria modl may b mor rprai or dcripi o dic logiy. 3. Noaio im o ailr o ailr g cio rlaig ad im α β γ paramr o h cio g π α diy o h paramr α ag ad cio ha drmi h ailr hazard corrlaio coici biaria ailr diy ad diribio cio biaria rliabiliy cio M θ θ mom graig cio or margial diy o a ailr 4

25 codiioal diy o ag gi ag z Z biaria ailr hazard cio biaria cmlai ailr hazard cio z codiioal hazard cio o ag gi ag µ µ ma o h ag ad o h ag diribio adard diaio o h ag ad ag diribio 3.3 Exampl ailr Modl h axoomy o modl yp w dic i Chapr idica ha hr ar wo modl cla ha ar o y wll dlopd ar irig ad may b pracically applicabl. h ar h modl bad o a ochaic cioal rlaiohip bw h wo ariabl ad h modl ha rpr h ariabl a corrlad. W di xampl modl i boh o h cla blow. No ha hr ar ry may cociabl approach o h corcio o h modl ad ha i ordr o iiia h dy o biaria modl w mploy h impl poibl approach hr Sochaic cio h diiio o ailr modl o h bai o ochaic cio rlaig ag ad ar wih h pciicaio o how h ochaic ar o h li ariabl i porrayd. W am ha h im ad o ailr ar rlad by h cio g ad ha h ochaic ar o hi rlaiohip ca b rprd by raig o or mor o h paramr o g a radom ariabl. o illra hi corcio w coidr or xampl orm hr: i g α β 5

26 ii g α β γ iii g α α α i g β whr h orh orm i h logiic modl aalyzd by Eliahbrg Sigprwalla ad Wilo [997]. I ach ca w irodc radom io h cio by raig h paramr α a a radom ariabl haig diribio π α. hi impo radom ariaio o h x o xpricd by ay ag. Coqly boh ag ad ag a ailr ar radom ariabl. h o h diribio π o corc h margial probabiliy diribio o ag i accomplihd ig a raormaio o ariabl. I gral: α dα g π α α 3. d or xampl wih g α β olig or α yild: β α ad dα d o: β π α. 3. Oc h margial diribio o ag i obaid w h corc h joi ailr diy ig h codiioig rlaio: 6

27 3.3 ad h codiioal diy i obaid by ig h wll-kow rlaiohip bw a iaria diy ad i hazard cio: z xp z x dx 0 z xp z x dx 3.4 g 0 g W hi orm pciically o ha w ca oc po h hazard cio i h diiio o h ailr modl. W am ha h codiioal biaria hazard cio o ag gi ag may b ad a: z 3.5 o ha h diiio o h cio ad g drmi h codiioal hazard ad limaly h biaria li diribio. Hr i ordr o oc o h cio g w am ha ad ar impl liar cio. h w ad. dr hi modlig orma h biaria li diribio corrpodig o orm i abo i obaid by corcig: z β x x x β 3.6 ad applyig xprio 3.3 ad 3.4 o obai: 7

28 8 β π β α xp 3.7 h am aalyical approach yild: xp γ β π β γ α 3.8 α π xp 3.9 ad l l l xp β π β β β β β β α 3.0 or ca ii iii ad i rpcily. h dail o how w arri a h orm or h joi diy ar prd i Appdix A. No ha i ca i h diiio o h cio limi h ariabl o [0 ] o h cio may rqir rcalig or om applicaio. Alo i ca i ad ii h orm o h cio g imply a o-zro miimm al or ag. ially obr ha all or modl ar wll did ad rqir oly h pciicaio o h diy α π o b compl biaria li diribio. O h ohr had or ach o hm i i likly ha a clod orm xprio ca b obaid or h margial diribio o ag a ailr.

29 3.3. Corrlaio I may applicaio h wo li ariabl appar o b corrlad rahr ha cioally dpd. h diiio o modl ha ca rpr corrlaio i h li ariabl appar iiially o b omwha implr ha h corcio abo. W imply choo a biaria diribio. Howr i i impora ha h diribio b capabl o accraly rprig qipm bhaior ad i pariclar ha i ha margial diribio ha ar coi wih xpric. W ha lcd hr xampl modl ha appar o hold promi or rprig biaria ailr proc i which h wo ariabl ar corrlad. h ir o h cadida modl i h gralizaio o h biaria xpoial modl did by Bagg ad Nagagaja [996]. I hi modl h rliabiliy cio i: 3. o h corrpodig diy cio i: 4 3. h joi hazard i: 4 z 3.3 ad h margial dii ar: ad 9

30 which ar boh coi xpoial rgardl o h al o. A cod modl ha i a obio choic i h biaria Normal. h diy cio or hi modl i wll kow o b: xp µ µ µ µ π. 3.4 A i alo wll kow h margial dii ar Normal. O ial modl ha w wih o coidr hr i h o ad by Hr [974a] i a qig cox b alo coi wih rliabiliy irpraio: I 0 xp 3.5 whr I i h modiid Bl cio o h ir kid o ordr ad i poii. h margial dii ar ad. 3.4 Modl Aalyi 3.4. Biaria Probabiliy Diribio h arig poi or h aalyi o h biaria ailr modl i h carl diiio ad irpraio o biaria probabilii. hr ar om bl ad omim diicl qio ad cocp ha ari i h applicaio o biaria diribio o rliabiliy. ir or h ag ad ariabl ad rpcily w 30

31 irpr h cmlai ailr probabiliy a h probabiliy ha ailr occr by im ad ag ha i: Pr[ ]. 3.6 O may irpr hi probabiliy a corrpodig o h proporio o h poplaio o dic ha ha logiy cor al a ailr ha do o xcd i ihr cor compo. W mphaiz hi diiio bca o h ac ha or a biaria diribio probabiliy i grally compd or rcagl ch a [ ]. Coqly or ay pciic logiy cor h rag o ag ad al impli ha hr ar or rcagl i h pla or which probabilii may b maiglly calclad. I addiio o h o i Eq. 3.6 h rcagl ar Pr[ ] Pr[ ] ad Pr[ ]. igr 3.4. illra h rcagl. I i o obio b rlai o h cmlai probabiliy h probabilii: ad Pr[ > ] dd Pr[ > ] dd ar rial probabilii. hy corrpod o h proporio o h poplaio ha do o ha logiy cor irior o ihr bca hir ailr ag xcd or hir ailr ag xcd. W do o ha iormai am or h probabilii rprd by Eq. 3.7 ad 3.8 b ha coidrd am ch a margial rial probabilii. 3

32 Pr[ < > ] Pr[ > > ] Pr[ < < ] Pr[ > < ] igr 3.4. Biarai Probabiliy Diribio 3

33 A rhr poi ha i rahr bl i h ac ha h rliabiliy a do o icld h probabilii rprd by Eq. 3.7 ad 3.8. h rliabiliy a logiy cor al corrpod o h proporio o h poplaio or which ailr ag xcd ad ailr ag xcd. hror h rliabiliy cio corrpodig o i: Pr[ ] dd 3.9 Bca i do o icld h probabilii rprd by xprio 3.7 ad 3.8 w call hi h rliabiliy rahr ha h rior cio. h appar paradox i h diiio o ad ari rom diicio i poi o obraio. Wh coidrig h diribio all poii ald logiy cor ca poially occr ad acro a poplaio o dic all do occr. Rlai o h diribio h cmlai probabiliy a do o icld dic or which ihr xcd or xcd. O h ohr had all copi o a dic poplaio ha ha achid a logiy o will ha logiy cor a ailr ha li wihi h rcagl [ < < ] o a h rcagl corrpodig o h margial rial probabilii ar o accibl. h compaio o biaria probabilii i raoably clar. or ay rcagl ay [ ] i h pla h probabiliy o obrig a ailr a a poi icldd i h rcagl i: Pr[ ]. 3.0 A l pcial ca o hi xprio appli o h rliabiliy cio ha may b rprd by: 33

34 34 ] Pr[ < <. 3. Obr ha hi xprio may alo b d o comp cmlai probabilii i ca i which h rliabiliy cio i air ha h diribio cio o aalyz Hazard cio Vry o h ir qio ha ollow h diiio o a probabiliy modl or dic ailr i ha o h idiy ad bhaior o h aociad hazard cio. hi i bca h corcio ad characrizaio o h hazard cio aociad wih a li or logiy diribio i a ky apc o qipm rliabiliy modlig ad aalyi. or a biaria ailr diribio a rr o ir pricipl yild: [ ] z > > Pr lim 0 0 [ ] [ ] > > Pr Pr lim 0 0 lim which i a ry appalig rl.

35 By aalogy wih h iaria ca o may wa o corc biaria ailr modl gi h biaria hazard. or h iaria ca hi approach may b accomplihd gi h ollowig rlaiohip bw ad z: z l or z. 3.3 oraly or h biaria ca h abo rlaiohip do o xi xcp or h ca whr ad ar amd o b idpd which i likly o b pracically maigl. Narally h x qio i whhr or o h hazard cio i icraig. Applyig h Barlow ad Procha [975] diiio o MIR mliaria icraig ailr ra o h biaria li diribio impli ha a diribio i MIR i ad oly i: 3.4 i o-icraig i. h am am appli o h margial diribio ha i i m alo b h ca ha: ad 3.5 ar o-icraig i ad rpcily. h applicaio o h codiio i illrad lar i hi papr. 35

36 3.4.3 Mom A rhr qio i how o comp h ma ad ohr dcripi mar or a biaria logiy diribio. h awr i ha a wih iaria diribio o bgi by corcig h mom graig cio or Laplac raorm ad h obai mom a cci driai o h mom graig cio. h mom graig cio or h biaria ailr diribio i: θ θ θ θ [ ] dd M θ θ E ad h mom o h diribio ar obaid a: k k k E [ ] M θ θ ad [ ] θ k θ 0 θ 0 k E M θ θ. 3.7 θ k θ 0 θ 0 Exampl o h o h xprio ar prd i h x cio Rwal cio h ial i o criical imporac o biaria ailr modlig i how coolio ar corcd ad how biaria rwal cio ar did ad irprd. oraly h coolio horm ha b how o xd dircly o h biaria ca Hr [974a] o ha: k k dd O h ohr had h diiio ad irpraio o h aociad coig proc ad h biaria rwal cio i l obio ad may dpd po h applicaio. 36

37 3.5 Exampl Calclaio o Biaria ailr Modl h gral cocp did abo ca b illrad by applicaio o h xampl modl. O cor xcp or h biaria xpoial diribio o xprio 3. o o h abo biaria modl ha clod orm xprio or hir cmlai probabilii. h w m mrical mhod o comp probabilii rliabiliy al ad mom ad o xami h bhaior o h hazard cio. A xampl o h aalyi o h mhod coidr or h ochaic cio modl h dii o Eq. 3.7 ad 3.0 ad or h corrlaio modl h biaria xpoial o Eq. 3. ad h biaria Normal i Eq I h ca o h ochaic cio modl am π i a gai xpoial diy o h orm: α π α α c c. 3.9 h or h modl o Eliahbrg Sigprwalla ad Wilo [997] Eq. 3.0 arbirarily ig h paramr o b 0 6 β c 000 yild h ollowig al or h cmlai diribio cio a how i abl All h mrical rl prd i hi cio ar obaid by ig h mrical igraio cio i Mahmaica 4.0 Wolram Rarch

38 abl 3.5. CD Val or Eq Corrpodig rliabiliy al ar how i abl abl 3.5. Rliabiliy Val or Eq h hazard al ar how i abl

39 abl Hazard Val or Eq A irig apc o hi diribio i ha h o h hazard cio bhaior i icocli bca h hazard i icraig i ag ad dcraig ad h icraig i ag. A a bai o compario o ha wh 0 6 β c 0.75 h liar ochaic cio o Eq. 3.7 yild h ollowig cmlai ailr probabilii a how i abl abl CD Val or Eq h corrpodig rliabiliy al ar how i abl

40 abl Rliabiliy Val or Eq h al o hazard cio ar how i abl or hi modl h hazard cio i icraig i boh ariabl. abl Hazard Val or Eq h biaria xpoial Eq. 3. i mch air o aalyz bca i ca b igrad i clod orm. h rliabiliy i ad i qaio 3. ad h cmlai probabilii ar gi by: 40

41 ig paramr al o ad 0.6 w obai h ollowig al o cmlai ailr probabilii a how i abl abl CD Val or Eq h rliabiliy al ar how i abl abl Rliabiliy Val or Eq

42 I addiio or h ad paramr al h hazard cio ha boh icraig ad dcraig bhaior. h al o hazard cio ar how i abl abl Hazard Val or Eq ig h biaria xpoial o Hr [974] Eq. 3.5 ad ig h paramr o b ad 0.6 a a bai o compario w obai h al o cmlai ailr probabilii a how i abl abl CD Val or Eq

43 h rliabiliy al ar how i abl abl 3.5. Rliabiliy Val or Eq h al o hazard cio ar how i abl abl 3.5. Hazard Val or Eq or h ad paramr al h hazard cio bhaior i icocli bca h hazard i dcraig icraig ad h dcraig ad icraig i ag ad dcraig icraig ad h dcraig i ag. 43

44 or h biaria ormal Eq. 3.4 wih paramr al µ 3000 µ ad 0.6 w obai h ollowig al o cmlai ailr probabilii a how i abl abl CD Val or Eq h rliabiliy al ar how i abl abl Rliabiliy Val or Eq h al o hazard cio ar how i abl

45 abl Hazard Val or Eq or h ad paramr al h hazard cio bhaior i icocli bca h hazard i icraig ad h dcraig i ag ad icraig ad h dcraig i ag. h prpo o h xampl i o illra h ac ha cmlai probabilii rliabiliy al ad hazard al ca b coily compd. h modl ca b mad o cor ay appropria cal by adjig h paramr or rirprig h ariabl. h al mrad alo illra h ac ha <. 3.3 h x apc o h aalyi o h biaria ailr modl i h drmiaio o h diribio mom. or h biaria Normal diribio o Eq. 3.4 h aalyi i raigh orward a h mom graig cio i kow o b Mood Graybill ad Bo [974]: M θ θθ θ θ θ xp θµ θ µ

46 ad a aicipad applicaio o Eq. 3.7 yild h ma cor E [ ] µ µ Corrpodig rl occr or highr mom.. A a cod xampl o ha h mom graig cio or h biaria xpoial ca b drid a: M θ θ 4 θ θ θ θ θ θ θ θ ad a a rl h ma cor i [ ] E or h ohr xampl diribio h corcio o h mom graig cio i coidrably mor diicl. I ac or h modl i Eq ad 3.0 clod orm xprio or h mom graig cio do o xi. I ach o ho ca aalyi o mom i b prormd by mrically compig h mom or h margial diribio. or xampl or h modl o Eliahbrg Sigprwalla ad Wilo [997] wih h am paramr a ho d abo mrical compaio o h xpcaio cor yild [ ] E. Similarly h liar ochaic cio modl ha xpcaio cor o oraly a dirc mrical approach will oly work or h corrlad cio modl i h corcio o coolio ad rwal cio. I appar ha coolio or h ochaic cio modl will rqir h o ri xpaio ad mrical approximaio ch a ho did or h iaria Wibll diribio by Lomicki [966]. E wih a approach o compig coolio h diiio o h rwal cio i o raighorward. Hr [974] ar wih margial rwal coig proc ad ak h miimm o ho o obai h biaria coig proc. ha approach appar appropria or h qig applicaio ha Hr di b do 46

47 o appar o coorm o h o qipm rliabiliy ad maiac aalyi. W gg ha h coig proc b did or h ag-ag pla o ha k i irprd a h cmlai probabiliy ha a h kh rwal h logiy cor do o xcd a ag o or a ag o. Wih hi irpraio h coig proc N i wll did ad w ca h coioal "im-rqcy" daliy o a probabilii ch a: k [ N k] Pr h w hold b abl o hi probabiliy am o obai rwal rl ha ar prd i Chapr 4. I Chapr 5 w dlop corrci maiac modl by ig h rl o biaria rwal hory. I Chapr 6 w coidr a ag rplacm pri maiac policy ad dlop biaria pri maiac modl. Exampl or h pri maiac modl ar prd i Chapr 7. h w di a aailabiliy mar or qipm ha ha a biaria logiy mar. hi i prd i Chapr 8 wih i xampl prd i Chapr 9. 47

48 CHAPER IV BIVARIAE RENEWAL MODELING I hi chapr a biaria rwal hory dlopd by Hr [974a b] i prd. Bad o Hr' work w propo a ordiary biaria rwal hory. W alo xd Wag ad Pham [996] iaria qai-rwal hory o h biaria ca. Baic rl or h wo yp o proc ad om gralizaio o ordiary biaria rwal hory ar alo obaid. 4. Noaio X S V h h im o ailr h h o ailr h biaria radom cor h im o h h rwal h ag o h h rwal Y h biaria radom cor o boh im ad o h h rwal S V biaria ailr diy ad diribio cio h -old coolio o G h coolio o... ad N h mbr o rwal by im ad ag M h biaria rwal cio or 48

49 m h biaria rwal diy cio or L { } biaria Laplac raorm or L { } biaria Laplac raorm or { } LS biaria Laplac-Silj raorm or 4. Biaria Rwal Proc L { X { }...} b a qc o idpd ad idically diribd o-gai biaria radom cor wih h commo joi diribio cio j.d.. P{ } ad o aoid riialii am ha 00 P{ 0 0} <. Lig i i i i i Y Y X S V i 4. i ollow ha Y S V i h biaria mar o boh im ad o h h rwal a illrad i igr 4... A h mbr o by im ad ag will qal h larg al o or which h h rwal occr bor or a im ad ag i ollow ha h mbr o rwal by im ad ag i gi by N N p{ : 0 S V } p{ : 0 Y }. 4. No ha { N > 0 > 0} i a coig proc. L ad do h irarrial im ad ag rpcily bw h ad h o hi proc. 49

50 ag V3 Y3 V 3 Y V Y Y0 3 S S S3 im igr 4.. A biaria rwal proc. Diiio 4.. L h qc o o-gai biaria radom cor { X i i...} b idpd ad idically diribd. h coig proc { > 0 > 0} i N calld a biaria rwal coig proc. or a xampl o a biaria rwal proc ppo ha hr i a iii pply o w ir who ag-milag cor { X i i...} ar idpd ad idically diribd. No ha ag liim ad milag ag may b corrlad or a gi ir. Sppo alo ha h ir ar d o a a im ad wh o ail w immdialy rplac i wih a w o. dr h codiio h coig proc { N > 0 > 0} i a biaria rwal coig proc whr N rpr h mbr o ir ha ha aild by im ad by oal milag. 50

51 4.3 Diribio o N or a Biaria Rwal Proc h diribio o N ca b obaid by ir oig h impora rlaiohip ha h mbr o rwal by im ad ag i grar ha or qal o i ad oly i h h rwal occr bor or a im ad ag. ha i { N } { S V } { Y }. 4.3 rom Eq. 4.3 P{ N } P{ N } P{ N } 4.4 P{ Y } P{ Y }. Sic h radom cor X i i... ar idpd ad idically diribd wih a commo joi diribio P{ } i ollow ha Y i diribd a h -old coolio o wih il. h rom Eq. 4.4 w obai h ollowig horm. horm 4.3. or 0 ad 0 P{ N }. 4.5 Proo. S Hr [974] p

52 A impora pciic ralizaio o Eq. 4.5 i: 0 P{ N 0}. 4.6 No ha i Eq.4.6 h probabiliy ha h mbr o biaria rwal bor i zro i o qal o h rliabiliy. I ac < a w ha alrady how i Chapr 3. L M E[ N ] 4.7 rpr h biaria rwal cio or biaria ma-al cio. h ollowig horm illra h rlaiohip bw M ad. horm 4.3. or 0 ad 0 M. 4.8 Proo. L N I whr Hc I i h h rwal occrrd i h rcagl[0 ] [0 ] 0 ohrwi. E[ N ] E[ I ] 5

53 53 ] [ I E } { I P } { V S P } { Y P whr h irchag o xpcaio ad mmaio i jiid by h o-gaiiy o h I. raormig M io h rcri orm c. Nachla [998] p.99: M j j 0 0 j j y x d y x 0 0 y x d y x j j 0 0 y x d y x M. Hc 0 0 y x d y x M M. 4.9

54 Eq. 4.9 i calld h igral qaio o biaria rwal hory. Amig ha i abolly coio h h driai o M dod by m i calld h biaria rwal diy cio m M m x y x y dxdy ad m rpr h probabiliy o a rwal a ay poi i h im-ag pla. No ha i w l h ma ad ariac o ad b µ rpcily h µ ad m ~ φ 4. whr φ i h biaria ormal diy wih ma µ ad corrlaio coici. µ ariac 4.4 Biaria Qai-Rwal Proc Rwal hory proid a damal ool or corcig impl maiac modl. h impliciy com rom h i.i.d. ampio o h im bw cci. h ar maiac h qipm i amd o b "a good a w." Howr i pracic ar maiac h liim o h qipm will bcom horr ad horr whil i maiac im may bcom logr ad logr. I ordr o 54

55 rwal hory o modl a gradal drioraio Wag ad Pham [ ] dlopd a iaria qai-rwal proc i which h i.i.d. ampio i rdcd o am oly ha h cci irarrial im ar idpd. h qai-rwal proc i alo calld a gomric proc by Lam Yh [988b]. I hi cio w xd hir rl o a biaria qai-rwal proc ad pr om baic rl. Diiio 4.4. L { X...} b a ochaic proc ch ha ad do h irarrial im ad ag rpcily bw h ad h o a coig proc { > 0 > 0}. h h coig proc i aid o b a N biaria qai-rwal proc i ad oly i: ζ ζ X whr ar i.i.d. radom cor ad ζ ad ζ ar o-gai cio. I word a biaria qai-rwal proc i a ochaic proc whr h cci irarrial ar idpd ad ar dcraig po ach rwal by a racio ζ or ad icraig by a racio ζ or or boh ζ ad ζ < or ζ or ad ζ or or boh ζ ad ζ >. h or boh ζ ad ζ qal o h biaria qai-rwal proc i h ordiary biaria rwal proc. I olig maiac problm h biaria qai-rwal proc ca b d o modl a biaria ochaic maiac proc wih boh ζ ad ζ <. or boh ζ ad ζ > biaria 55

56 qai-rwal proc ca b applid o modl a rliabiliy growh proc i prodc dlopm ad/or a br-i program. No ha h xio o h biaria ca i o h oly xio o h iaria qai-rwal proc ha w ca. W alo gg ha mor gral orm or boh ζ ad ζ ca b did or h iaria ad biaria ca b ha w will o pr h hr. I w l ζ ζ ζ ζ whr ζ ad ζ ar o-gai coa ad am ha h joi probabiliy diy cio pd cmlai diy cio cd rial cio ad hazard cio h o h radom cor X ar ad z rpcily. h i ollow ha h pd cd h ma ad ariac o radom cor X or ar gi by: ζ ζ ζ ζ 4.3 ζ ζ 4.4 ζ ζ 4.5 z z ζ ζ ζ ζ E Var [ X ] ζ E[ X ] 4.6 ζ 4.7 X ζ Var X ζ. 4.8 Bca h o-gaiiy o X ad h ac X i o idically 0 w cocld ha [ ] µ 0 E. X 56

57 horm 4.4. I blog o MIR MDR h i alo MIR MDR or Proo. Sppo ha h hazard cio o X i diriabl wih rpc o ad. rom Eq. 4.6 h driai o h hazard cio o X i gi by: z z ζ ζ ζ ζ. 4.9 rom Eq. 4.9 w ha i z i icraig dcraig h z i alo icraig dcraig. hi compl h proo. 4.5 Diribio o N or a Biaria Qai-Rwal Proc Coidr a biaria qai-rwal proc. h diribio o N ca b obaid by ir oig h impora rlaiohip ha h mbr o rwal by im i grar ha or qal o i ad oly i h h qai-rwal occr bor or a im ad ag. ha i { N } { S V } { Y }. 4.0 whr Y ad i i ζ i i S V X i i i i Y ζ

58 I ollow ha Y i h biaria mar o boh im ad o h h qai-rwal a i a ordiary biaria rwal proc. Sic h mbr o by im ad ag will qal h larg al o or which h h qai-rwal occr bor or a im ad ag w ha ha h mbr o qai-rwal by im i gi by N N p{ : 0 S V } p{ : 0 Y }. 4. rom Eq. 4.0 w obai P{ N } P{ N } P{ N } 4.3 P{ Y } P{ Y } G G whr G h coolio o h irarrial diribio... i.. G L. Sic h radom ariabl X... ar idpdly diribd wih diribio P{ X } P{ ζ ζ } i ollow ha Y i diribd a G h coolio o h irarrial diribio.... hror h ollowig horm i obaid: horm 4.5. or 0 ad 0 P{ N } G G

59 Proo. Gi abo. A impora pciic ralizaio o Eq. 4.4 i: P{ N 0 0} G G. 4.5 L M G E[ N ] 4.6 whr M G rpr h qai-rwal cio or ma-al cio. h ollowig horm illra h rlaiohip bw M G ad G. horm 4.5. or 0 0 ad 0 M G G. 4.7 Proo. h proo o hi horm i imilar a o horm Hr w pr a horr proo a ollow. M E[ N ] G 0 0 P{ N } [ G G ] by horm 4.5. G. 59

60 60 raormig M G io h rcri orm: G G M G G j j G 0 0 j j y x d y x G 0 0 y x d y x G j j 0 0 y x d y x M G. Hc 0 0 y x d y x M M G G. 4.8 Eq. 4.8 i calld h igral qaio o qai-rwal hory. Amig ha i abolly coio h h driai o M G dod by m G i calld h qai-rwal diy cio G G g M d d m 4.9 G dxdy y x y x m 0 0 ad m G rpr h probabiliy o a qai rwal a ay poi i h im-ag pla.

61 4.6 Biaria Laplac raorm I biaria rwal hory h logiy radom ariabl rpr h irarrial iral ad oly am o-gai al. h biaria Laplac raorm o a arbirary cio φ i did a: { } φ xy L φ x y dxdy 0 0 φ or a joi diribio cio h Laplac-Silj raorm i did a: LS { } d dd L { } Am ha h joi diribio cio i adqaly coio ad diriabl i.. x y dydx 0 < <. 0 0 h ollowig horm idica h rlaiohip bw h Laplac raorm o ad i diy cio. horm 4.6. L ad b h Laplac raorm o ad i diy cio. h

62 6 Proo. { } { } dd dydx y x dydx y x L L x y dddydx y x dydx y x dd dd dd dd x y x y x y 0 0 dydx y x x x x x x x x x 0 0 dydx y x y x 0 0 dydx y x y x rom Eq ad 4.3 i w wri { } dd M M L M 0 0 h i ollow ha { } { } dydx y x y x M L M L M 0 0 dydxdd y x y x M dydxdd y x y x M dydx y x dd y x M y x x y y x 0 0.

63 L w x ad z y o dw d ad dz d or w [ 0 ad z [ 0 h M wz M w z dzdw xy 0 0 x y M x y dydx x y dydx M Eq ca b old o yild h ollowig horm: horm 4.6. L M b h Laplac raorm o. h M ad 4.34 M [ ] M or M M M h corrpodig rl or m ad ar how i h ollowig horm. horm L m b h Laplac raorm o. h m ad 4.36 m 63

64 m m I i impora ha kowldg o h biaria rwal cio M impli compl kowldg o all apc o h biaria rwal proc Hr [974] pp h rl o horm 4.6. ad corrpod o h iaria rwal hory Cox [96] pp.46 which ar ry appalig. or h biaria qai-rwal proc wih ζ ζ obai h ollowig rl or h Laplac-Silj raorm. ζ ζ w horm L b h Laplac-Silj raorm or h diribio or X. L M G b h Laplac-Silj raorm or. L M G m G b h Laplac raorm or. h m G { } ζ ζ LS d 0 0 ζ ζ 4.38 M G { M } LS G ζ ζ L ζ ζ 4.39 ad 64

65 m G g ζ ζ ζ L ζ Dlayd or Modiid Biaria Rwal Proc Coidr a biaria rwal proc whr h ir rwal ha a dir diribio ha bq rwal. Sppo ha w ar obrig a biaria rwal proc a om poi > 0. I a rwal do o occr a h h joi diribio o h im ad ag w m wai il h ir obrd rwal will o b h am a h rmaiig irarrial diribio. L { X...} b a qc o idpd o-gai biaria radom cor wih X haig joi diribio ad joi diribio or i >. L 0 i i i i i X i haig commo Y 00 Y X S V or... h mbr o rwal by im ad ag i did by i ND p{ : 0 S V } p{ : 0 Y }. 4.4 Diiio 4.7. L h qc o o-gai biaria radom cor { X i i...} b idpdly diribd wih X haig joi diribio ad X i haig commo joi diribio i... h coig proc { ND > 0 > 0} i calld a dlayd or modiid biaria rwal coig proc. 65