A COMPARISON OF DEGRADATION AND FAILURE-TIME ANALYSIS METHODS FOR ESTIMATING A TIME-TO-FAILURE DISTRIBUTION

Sttistic Siic 6(1996), 531-546 A COMPARISON OF DEGRADATION AND FAILURE-TIME ANALYSIS METHODS FOR ESTIMATING A TIME-TO-FAILURE DISTRIBUTION C. Joseph Lu, Willim Q. Meeker d Luis A. Escor Ntiol Cheg-Kug Uiversity, Iow Stte Uiversity d Louisi Stte Uiversity Astrct: Degrdtio lysis c e used to ssess reliility whe few or eve o filures re expected i life test. I this pper, we use simple ut useful degrdtio model to compre degrdtio lysis d trditiol filure-time lysis i terms of symptotic efficiecy. The comprisos cosider rge of prcticl testig situtios d provide isight ito the trde-offs etwee these two methods of estimtig the qutiles of the time-to-filure distriutio. We ivestigte the effect tht the umer of ispectios, the mout of mesuremet error, d the qutile of iterest hve o the symptotic vrices of the qutile estimtors. Although mesuremet error c iduce some loss of precisio i degrdtio lysis, our comprisos show tht, except i extreme cses, degrdtio lysis provides more precisio th trditiol filure-time lysis. Key words d phrses: First crossig time, life dt lysis, mesuremet error, reltive efficiecy. 1. Itroductio Trditiol filure-time lysis (FTA) methods for estimtig compoet reliility record oly the time-to-filure (for uits tht fil) or the ruig time (for uits tht do ot fil). I life tests for high reliility compoets there will e few or o filures, mkig reliility ssessmet difficult. Degrdtio lysis (DA) is lterte pproch tht uses sequece of degrdtio mesures to ssess reliility. Lu d Meeker (1993) discuss prticulr pproch to degrdtio lysis. Nelso (1990, chpter 11) discusses other methods for lyzig degrdtio dt, prticulrly with ccelertio. I this pper we use simple, ut physiclly resole, degrdtio model to compre DA d FTA. This degrdtio model implies logorml distriutio for the correspodig time-to-filure distriutio. We use the rtio of the symptotic vrices of estimtors of qutile of the time-to-filure distriutio to compre DA d FTA.

532 C. JOSEPH LU, WILLIAM Q. MEEKER AND LUIS A. ESCOBAR Suzuki, Mki d Yokogw (1993) (SMY) lso compre degrdtio d filure-time lysis, ut use differet ssumptios out how dt ecome ville, focus o differet qutities of iterest, d use somewht differet model. I prticulr, SMY do ot let the ility to oserve degrdtio deped o the level of degrdtio (i some pplictios, uits must e removed from service due to filure or for sfety resos). SMY evlute the ility to estimte me time to filure, while our evlutios re for selected percetiles of the time-to-filure distriutio. SMY del with ccelerted degrdtio model. We re cocered with tests d ifereces uder specified set of coditios. 2. Models 2.1. Degrdtio model We ssume degrdtio model i which the degrdtio level is proportiol to time with degrdtio rte tht is rdom from uit to uit. More specificlly, we ssume tht mesured degrdtio = exp(θ) time exp(ɛ) which implies the followig simple pth model: Y =Θ+x + ɛ, Θ N ( µ Θ,σΘ) 2, ɛ N ( 0,σɛ 2 ), where Y = log(mesured degrdtio), Θ = log(degrdtio rte), x = log(time), ɛ is the mesuremet error, d Θ d ɛ re idepedet. We ssume tht filure occurs whe ctul degrdtio (Θ + x) reches specified criticl level D c,thtis,wheθ+x = D c. 2.2. Time-to-filure model From the degrdtio model i Sectio 2.1 the rdom log time-to-filure c e expressed s = D c Θ N ( D c µ Θ,σΘ) 2. Therefore, the time-to-filure distriutio is logorml with prmeters µ = D c µ Θ d σ = σ Θ. 3. Dt, Estimtio d Asymptotic Vrices I this sectio we defie the ispectio dt used i the compriso, we give estimtors of the model prmeters, d we provide expressios for the symptotic vrices of these estimtors. For oth the DA d FTA methods, we ssume fixed log ispectio times x 1,...,x m. For DA we oserve degrdtio level (plus oise) t ech ispectio. At ech ispectio i FTA we oserve the umer of uits tht hve filed up to the curret ispectio. 3.1. Degrdtio lysis 3.1.1. Degrdtio ispectio dt I some situtios degrdtio will result i sudde ctstrophic filure. Typiclly, however, there is ot exct reltioship etwee the level of degrdtio d the ctstrophic filure. I other situtios, performce degrdes

A COMPARISON OF DEGRADATION AND FAILURE-TIME ANALYSIS 533 more grcefully. The, service life of uit would ed roud the time tht degrdtio hs cused system performce to rech specified level. I such situtios filure c e defied s some oserved level of system performce or i terms of ctul degrdtio o compoet tht c e mesured, with some degree of mesuremet error. I either cse, it my e ecessry to replce degrdig uit fter degrdtio hs reched specified level, either ecuse of loss of dequte fuctiolity or for sfety resos. I some reliility studies, this will restrict the time of oservtio. I our compriso, we ssume tht there re m pled ispectio times t log(times) x 1,...,x m. The ctul umer m Θi of ispectios o uit is rdom d depeds o the uit s log(degrdtio rte) = Θ i s follows: m Θi = j (j = 2,...,m 1) if the uit fils etwee ispectio times x j d x j+1 (i.e., if Θ i + x j <D c Θ i + x j+1 ), m Θi = m whe the uit survives eyod x m (i.e., Θ i + x m <D c ). Filly, whe Θ i + x 2 >D c, we ssume tht mesuremets re ville o the uit t x 1 d x 2. This would e relistic if we get sigl tthetimethtθ+xexceeds D c (e.g., we detect loss of performce), eve though there is ot ctstrophic filure d we cot oserve Θ + x directly. Hvig two ispectios o ech uit isures tht the prmeters Θ d σ ɛ c e estimted for ech uit. Thus, hve dt Y ij =Θ i + x j + ɛ ij, j =1,...,m Θi, 2 m Θi m, where ɛ ij is the mesuremet error for the jth ispectio o the ith uit. I situtios with lrge mesuremet error, our criterio for stoppig oservtios will ot provide descriptio of the ctul stop-oservtio rule. The descriptio is, however, useful for comprig DA d FTA for situtios whe oservtio is limited y high levels of degrdtio. 3.1.2. Two-stge estimtio of the degrdtio prmeters We use the two-stge estimtio procedure descried i Lu d Meeker (1993) to estimte the model prmeters for DA. I the first stge, lest squres estimtio gives ˆΘ i = ˆσ 2 ɛi = m Θi 1 (Y ij x j ), m Θi j=1 m 1 Θi [ 2. Y ij ( m Θi 1 ˆΘ i + x j ) j=1 By lier model orml theory, the coditiol distriutios of ˆΘ i d ˆσ 2 ɛi,give Θ i,re ˆΘ i N ( Θ i,σ 2 ɛ /m Θ i ) d (mθi 1)ˆσ 2 ɛi /σ2 ɛ χ2 m Θi 1,whereΘ i d m Θi re reliztios of Θ d m Θ, respectively.

534 C. JOSEPH LU, WILLIAM Q. MEEKER AND LUIS A. ESCOBAR We use ˆσ 2ˆΘi =ˆσ ɛi 2 /m Θ i to estimte σ 2ˆΘi =Vr ɛ ( ˆΘ i Θ i ), the vrice due to mesuremet error for relized vlue of Θ i. Tkig the vriility of rdom effects ito ccout, the ucoditiol distriutio of ˆΘ i hs me E ( ˆΘ i )=µ Θ d vrice Vr( ˆΘ i )=σθ 2 + σ2ˆθ, whereσ 2ˆΘ =E(ˆσ 2ˆΘi )=E Θ [E ɛ (ˆσ 2ˆΘi Θ i ) = E Θ (σɛ 2 /m Θi )=σɛ 2 E Θ (1/m Θi ). Note tht expecttios without suscript eed to e tke with respect to the joit distriutio of Θ d ɛ. I Stge 2 of the DA estimtio procedure, we comie ( ˆΘ i, ˆσ 2ˆΘi ), i =1,..., ito ˆµ Θ = i=1 ˆΘi / d ˆσ 2ˆΘ = i=1 ˆσ 2ˆΘi /. The, ecuse E ( ˆΘ i )=µ Θ d E(ˆσ 2ˆΘi )=σ 2ˆΘ, we hve E (ˆµ Θ )=µ Θ d E (ˆσ 2ˆΘ) =σ 2ˆΘ. Let SΘ 2 = i=1 ( ˆΘ i ˆµ Θ ) 2 /( 1) deote the smple vrice of the Stge 1 estimtes of Θ. This smple vrice, E (SΘ 2 )=σ2 Θ +σ2ˆθ, reflects oth the mesuremet error vrice d the uit to uit rdom vlues of Θ. Therefore, we estimte σθ 2 y ˆσ2 Θ = SΘ 2 ˆσ2ˆΘ. If ˆσ Θ 2 < 0, settig ˆσ2 Θ = 0 is specil cse of the pproch used i Lu d Meeker (1993), origilly suggested y Amemiy (1985). 3.1.3. Vrice-covrice mtrix for DA ˆσ [DA Notig tht µ = D c µ Θ d σ = σ Θ,wetkeˆµ [DA = D c ˆµ Θ d =ˆσ Θ. I Appedix A, we show tht the symptotic vrice-covrice mtrix for the DA estimtors c e expressed s ( ˆµ [DA ) Vr = σ2 ˆσ [DA ( V [DA 11 V [DA 12 V [DA 12 V [DA 22 Appedix B gives computtiol expressios for V [DA 11,V [DA 12,dV [DA 22 d shows tht they deped o the stdrdized ispectio times z j =(x j µ )/σ, j =2,...,m 1, d the vriility rtio R σ = σ ɛ /σ Θ, the rtio of mesuremet error vritio versus rdom effect vritio. 3.2. Filure-time lysis 3.2.1. Filure-time dt Becuse DA uses ispectios durig the test, we lso ssume the use of ispectios i FTA. The dt cosist of the umer of oserved filures i ech itervl. We let 1 deote the umer of uits filed efore log time x 1,let j deote the umer of uits filed etwee ispectios t log times x j 1 d x j, j =2,...,m,dlet m+1 deote the umer of uits tht survived to the lst ispectio x m.notetht = m+1 j=1 j. 3.2.2. Mximum likelihood estimtio of model prmeters Detiled discussio of mximum likelihood estimtio for filure-time dt c e foud, for exmple, i Lwless (1982) or Nelso (1982). For the ispectio filure-time dt, the mximum likelihood estimtes ˆµ [FTA ). d ˆσ [FTA re

A COMPARISON OF DEGRADATION AND FAILURE-TIME ANALYSIS 535 otied y mximizig the log likelihood of test uits: L = 1 log Φ(z 1 )+ m j=2 [ j log Φ(z j ) Φ(z j 1 ) + m+1 log[1 Φ(z m ), where z j,j =1,...,m, re the stdrdized ispectio times defied t the ed of Sectio 3.1.3. 3.2.3. Vrice-covrice mtrix for FTA The symptotic vrice-covrice mtrix of the FTA mximum likelihood estimtors c e expressed s ( ˆµ [FTA ) ( Vr = σ2 V [FTA ) 11 V [FTA 12. V [FTA 12 V [FTA 22 ˆσ [FTA Meeker (1986) gives expressios for computig V [FTA 11,V [FTA 12,dV [FTA 22 d shows tht these qutities deped oly o the stdrdized ispectio times z j, j =1,...,m. 3.3. Asymptotic vrices of ˆx [DA p d ˆx [FTA p The p qutile of the log time-to-filure distriutio is x p = µ + u p σ where u p =Φ 1 (p) is the stdrd orml p qutile. The DA estimtor of x p is otied from ˆx [DA p =ˆµ [DA + u pˆσ [DA. The symptotic vrices of these estimtors re Vr(ˆx [DA p ) = Vr(ˆµ [DA + u pˆσ [DA ) = Vr(ˆµ [DA )+2u p Cov(ˆµ [DA The correspodig symptotic vrice fctor (VF) is VF [DA (p) = σ 2, ˆσ [DA )+u 2 p Vr(ˆσ[DA ). Vr(ˆx [DA p )=V [DA 11 +2u p V [DA 12 + u 2 pv [DA 22. To compute Vr(ˆx [FTA p )dvf [FTA, we use exctly the sme formuls with FTA sustituted for DA. 3.4. Reltive efficiecy To compre DA d FTA, we use (symptotic) reltive efficiecy (RE) computed s the rtio of the symptotic vrices of the estimted p qutile of time-to-filure distriutio for the DA d FTA methods RE = Vr(ˆx[FTA p ) Vr(ˆx [DA ) = (p) VF[FTA VF [DA (p) = V [FTA 11 +2u p V [FTA 12 + u 2 pv [FTA 22. V [DA 11 +2u p V [DA 12 + u 2 pv [DA 22 p

536 C. JOSEPH LU, WILLIAM Q. MEEKER AND LUIS A. ESCOBAR 4. Compriso d Discussio 4.1. Test pls used i comprisos Meeker (1986) evlutes d compres differet methods of plig ispectios i life tests d suggests tht equl-proility-spcig hs good sttisticl properties d provides coveiet method for comprig ltertive life test pls. Here, we lso use the equl-proility-spcig ispectio times to compre DA d FTA. With the equl-proility-spcig ispectios, the expected umer of uits filed is the sme withi ech ispectio itervl. The equl-proility-spcig stdrdized log ispectio times re specified i terms of z j =Φ 1 [(j/m)p F, j =1,...,m, where P F is the expected proportio of filures y the lst log ispectio time x m. The ctul log ispectio times re relted to the stdrdized ispectio times through x j = µ + z j σ, j = 1,...,m. For purposes of compriso, the equl-proility-spcig ispectios hve the dvtge of eig esy to chrcterize d specify ecuse they deped oly o specifictio of m d P F. The vrice fctor VF [DA (p) is fuctio of P F,m,R σ, d p, d VF [FTA (p) is fuctio of P F,m,dp. Thus the RE is fuctio of P F,m,R σ, d p. We compred the DA d FTA methods y computig d grphig VF [DA (p), VF [FTA (p), d RE for ll comitios of the followig fctors: The time-to-filure distriutio qutiles of iterest: p =.01,.02,...,.99. The expected proportio of filures (i.e., proportio exceedig D c )eforethe lst ispectio: P F =.1,.2,...,.9. The vriility rtio: R σ =.1,.5, 1, 2, 5. The umer of ispectios: m =3, 5, 10, 20, 50, 100. 4.2. Compriso figures Figures 1 to 3 provide the results for suset of these comitios: The first plot i Figure 1 shows VF [FTA (p) versusp with m = 10 ispectios, with seprte lies for differet vlues of P F. This plot, for the logorml distriutio, is similr to the plot give i Meeker d Nelso (1976) of the vrice fctor of the estimted qutile of the Weiull time-to-filure distriutio for cesored dt d cotiuous ispectio. The other plots i this figure show VF [DA (p) versusp with the sme umer of ispectios with vriility rtios R σ =.5, 2, 5.

A COMPARISON OF DEGRADATION AND FAILURE-TIME ANALYSIS 537 t1 t2 d d c f c V2 V4 F V9 t3 t4 d d V9 c c f F s3 s4 Figure 1. Vrice fctors for estimted qutiles. t1 t2 f V2 V4 V6 V9 F t3 t4 f V2 V4 V6 V9 F s3 s4 Figure 2. Degrdtio lysis versus filure-time lysis: R σ =.1,.5d m =10, 100.

538 C. JOSEPH LU, WILLIAM Q. MEEKER AND LUIS A. ESCOBAR t1 t2 c c f V2 V4 V6 V9 F t3 t4 c c F s3 s4 Figure 3. Degrdtio lysis versus filure-time lysis: R σ =2, 5d m =10, 100. Figures 2 d 3 plot RE versus p for severl vlues of P F for ll comitios of vriility rtios R σ =.1,.5, 2, 5, d umer of ispectios m =10, 100. 4.3. Discussio Figure 1 shows, for FTA, tht VF [FTA (p) decreses d the icreses s p icreses. There is less precisio for estimtig qutiles tht re remote from P F. The effect of differet P F o VF [FTA (p) is stroger whe estimtig lrger qutiles; there is more spred o the right-hd side th o the left. This idictes tht precisio drops off rpidly whe extrpoltig ito the upper til of the time-to-filure distriutio. The followig poits c e see most clerly i grphs of VF [FTA (p) versus P F which, to sve spce, re ot show here. With cotiuous ispectio, VF [FTA (p) is strictly decresig s fuctio of P F ecuse s more test uits fil, more iformtio out the time-to-filure distriutio ecomes ville d, hece, we c estimte the qutile of the time-to-filure distriutio more precisely. With oly itervl-iformtio o the time to filure, VF [FTA (p) decreses s P F icreses over most of the rge of P F, ut it is possile for VF [FTA (p) to icrese slightly egiig t some poit fter P F exceeds p. Thisisresultof

A COMPARISON OF DEGRADATION AND FAILURE-TIME ANALYSIS 539 the limited iformtio from the discrete ispectio dt. For exmple, i the extreme cse with oly m = 2 ispectios, s the lst ispectio time ecomes lrge, the resultig dt from tht ispectio is uimportt to estimte smll qutile. This is the reso tht there is some crossig of lies i the plot of VF [FTA (p) versusp i Figure 1. The plots i Figure 1 lso shows, for DA, tht As with the FTA plots, the plots of VF [DA (p) versusp lso hve U shpe. VF [DA (p) icreses s P F icreses. This geerl ehvior is opposite to tht for the FTA. This is ecuse icresig P F is equivlet to icresig the test legth d, hece, with costt m, reducig the expected umer of ispectios efore the ctul degrdtio crosses D c. The effect is smller whe the mesuremet error rtio R σ is smll. Figures 2 to 3, which focus o the RE of DA versus FTA, show tht RE usully decreses s P F icreses. This is ecuse, s explied erlier, icresig the test legth with the sme umer of ispectios geerlly provides more iformtio for FTA, ut less iformtio for DA. Especilly whe R σ is smll, RE is much lrger whe estimtig qutiles i the upper til of the time-to-filure distriutio, prticulrly for smller P F. This shows the dvtge of DA over FTA whe extrpoltig ito the upper til of the time-to-filure distriutio. From Figure 2, whe the vriility rtio is smll (e.g., R σ =.1 or.5), the umer of ispectios hs little effect o RE. With lrge R σ, RE drops well elow 1 for some comitios of P F d p. I Figure 3, however, we see tht the umer of ispectios c compeste for lrge vlue of R σ (e.g., R σ = 5). With high mesuremet error vriility, for my comitios of P F d p, DA c provide etter precisio th FTA oly if the umer of ispectios is lrge eough. Meeker (1986) shows tht icresig m eyod 10 hs little effect o the precisio of FTA estimtes. Figure 3 lso shows tht RE egis to decrese with p, prticulrly for lrge p. This is ecuse, s see i Figure 1, VF [DA (p) icreseswithp more rpidly th VF [FTA (p) forlrgep. 5. Cocludig Remrks I this pper we compre DA d FTA methods i terms of symptotic vrice fctors d reltive efficiecy of the estimtors of qutiles of timeto-filure distriutio. From the figures d discussio, we c summrize the compriso s follows As the P F icreses, there re differet effects for FTA d DA. For FTA the expected proportio of filures icreses with P F, icresig precisio. For the DA, however, the expected umer of ispectios efore filure decreses s P F icreses, cusig decrese i precisio.

540 C. JOSEPH LU, WILLIAM Q. MEEKER AND LUIS A. ESCOBAR Eve with lrge mesuremet error vriility, DA performs etter th the FTA, provided there re eough ispectios to compeste for the mesuremet error vrice. The results show tht DA is especilly etter suited th the FTA to mke iferece o the qutiles tht re lrger th P F. The dvtges of DA do ot come etirely for free. For FTA we eed to specify time-to-filure distriutio. For DA we eed to specify degrdtio model, implyig time-to-filure distriutio. I either cse dt c e used to ssess the dequcy of the model withi the rge of the dt. I terms of sttisticl efficiecy, DA hs its iggest dvtges whe estimtig qutiles of filure proilities eyod the rge of the dt. With either FTA or DA, there will e potetil for susttil model error (or is) i estimtes tht extrpolte eyod the rge of the dt. Becuse DA offers more precisio i these estimtes we could, outside the rge of the dt, expect less roustess with the use of idequte model. O the other hd, degrdtio modelig should, wheever possile, e tied closely to the physics of filure, providig more cofidece for degrdtio models th is typicl i the commoly used curve-fittig techiques of FTA (which re, of course, dequte for iterpoltive ifereces). Although workig with differet dt d model ssumptios, our coclusios re cosistet with those of Suzuki, Mki, d Yokogw (1993). Also, their results, sed o model with terms llowig for fixed-effect curvture i the degrdtio pths, suggest tht similr coclusios would e otied if our setup were exteded to iclude terms for curvture i the degrdtio pths. I this pper we hve provided evlutios of symptotic vrice fctors for rge of degrdtio testig situtios, givig geerl picture of the effect tht the vrious fctors hve o estimtio precisio. To swer specific questios out the desig of such experimets (i.e., legth of test d smple size) it is useful to do specific computtios of symptotic vrices. Also, t the expese of hvig to use more computer time, symptotic evlutios c e supplemeted with Mote Crlo simultio to llow evlutios without hvig to rely o symptotic pproximtios. I our comprisos we hve ot ccouted for the fct tht there my e differet costs for otiig time-to-filure d degrdtio dt. I my (ut ot ll) situtios, degrdtio mesuremets result i more expese. The cost vries, depedig o the situtio. Ofte the lrger prt of the cost is i metrology reserch or eeded cpitl equipmet eeded to mke mesuremets. I my other situtios it ivolves pistkig disssemly d mesuremet. I my electroic tests, however, the cost of tkig mesuremets is extremely low ecuse the mesuremet work is doe y computer recordig of the sigls tht directly iput from test equipmet.

A COMPARISON OF DEGRADATION AND FAILURE-TIME ANALYSIS 541 Ackowledgemets We would like to thk Professors Ysuo Amemiy, Steve Vrdem, the Editor Jeff Wu, d two oymous referees for helpful commets o erlier versio of this pper. Appedix A. Expected Vlues, Vrices d Covrices for the Two Stge Estimtors I this ppedix we derive the symptotic expected vlues d vricecovrice mtrices of the two stge estimtors ˆµ [DA, ˆσ [DA. Recll tht ˆµ [DA = D c ˆµ Θ d (ˆσ [DA ) 2 = ˆσ Θ 2 = S2 Θ ˆσ2ˆΘ, where ˆµ Θ = i=1 ˆΘi /, SΘ 2 = i=1 ( ˆΘ i ˆµ Θ ) 2 /( 1), d ˆσ 2ˆΘ = i=1 ˆσ 2ˆΘi /. For y fixed i, ˆΘi d ˆσ 2ˆΘi re idepedet. Also, uits i d k (i k) re idepedet, which implies, for exmple, tht Cov(Θ i, Θ k )=Cov(ˆΘ i, ˆΘ k )=Cov(ˆΘ i, ˆσ 2ˆΘk )=0. A.1. Expected vlues First, we show tht ˆµ [DA, (ˆσ [DA ) 2 re uised estimtes of µ,σ 2. E(ˆµ [DA )=D c E(ˆΘ i )=D c E Θ [E ɛ ( ˆΘ i Θ i ) =D c E Θ (Θ i )=D c µ Θ =µ, [ E (ˆσ [DA ) 2 =E(SΘ) E(ˆσ 2 2ˆΘ )=σ2 Θ + σ 2ˆΘ σ2ˆθ =σ2 Θ =σ 2. Usig Tylor series expsio, up to terms of order 1,ˆσ [DA σθ 2 )/(2σ Θ). = σ Θ +(ˆσ 2 Θ A.2. Vrice-covrice mtrices I this sectio we show tht, up to terms of order 1, ( ˆµ [DA ) [ Cov = σ2 Θ V [DA 11 V [DA 12, V [DA 12 V [DA 22 ˆσ [DA where V [DA 11 =1+ σ2ˆθ σθ 2 (1) V [DA 12 = 1 σθ 3 Cov Θ (Θ i,σ 2ˆΘ ) (2) i V [DA 22 = 1 {(σ 2 2σΘ 4 Θ + σ2ˆθ) 2 +Vr Θ (σ 2ˆΘi )+2Cov Θ [(Θ i µ Θ ) 2,σ 2ˆΘi ( σ 4ˆΘi )} +E Θ. (3) m Θi 1

542 C. JOSEPH LU, WILLIAM Q. MEEKER AND LUIS A. ESCOBAR Note tht σ 2ˆΘi =Vr ɛ ( ˆΘ i Θ i )=σɛ 2 /m Θi, which is rdom vrile ecuse m Θi depeds o the rdom Θ. The limitig expressios for V [DA 11,V [DA 12,dV [DA 22,sσ 0(whichhppes whe m or σ ɛ 0) re V [DA 11 =1,V [DA 12 =0,dV [DA 22 =1/2. As 2ˆΘ expected, these limitig vlues re equl to the vlues for the correspodig compoets V [FTA 11,V [FTA 12,dV [FTA 22 i the filure-time lysis cse whe P F =1 (o right cesorig) d m = (cotiuous ispectio). To derive the V [DA ij, we mde repeted use of the followig results: Let W, R, S e give rdom vriles tht hve fiite secod momets. The (see, Serle, Csell d McCulloch (1992, pge 461)): Cov(W, R) = Cov[E(W S), E(R S) + E [Cov(W, R S) (4) Vr(W ) = Vr[E(W S) + E[Vr(W S) (5) If W hs me µ W,vriceσW 2, d fiite third momet, the Cov(W, W 2 ) =E[(W µ W ) 3 +2σW 2 µ W. Toshowthis,expdthethirdmometothe right-hd side d simplify. The, whe W is symmetriclly distriuted, E[(W µ W ) 3 =0dCov(W, W 2 )=2σW 2 µ W. Derivtio of Vr(ˆµ [DA ) Vr(ˆµ [DA ) = Vr(D c ˆµ Θ )=Vr(ˆµ Θ )= 1 Vr( ˆΘ i ) = 1 { Vr Θ [E ɛ ( ˆΘ i Θ i ) + E Θ [Vr ɛ ( ˆΘ } i Θ i ) = 1 { } Vr Θ (Θ i )+E Θ (σ 2ˆΘi ) = 1 (σ2 Θ + σ2ˆθ) = σ2 Θ V [DA 11. Derivtio of Cov(ˆµ [DA, ˆσ [DA ) Usig the delt method pproximtio, for lrge, oegets Cov(ˆµ [DA, ˆσ [DA )=Cov(D c ˆµ Θ, ˆσ Θ )= Cov(ˆµ Θ, ˆσ Θ ) 1 Cov(ˆµ Θ, ˆσ Θ 2 2σ ) Θ = 1 Cov(ˆµ Θ,SΘ 2 ˆσ 2ˆΘ) 2σ Θ = 1 [ Cov(ˆµ Θ,S 2σ Θ) 2 Cov(ˆµ Θ, ˆσ. 2ˆΘ ) (6) Θ To simplify the expressio o the right-hd side, oserve tht

A COMPARISON OF DEGRADATION AND FAILURE-TIME ANALYSIS 543 Cov(ˆµ Θ,SΘ [ˆµ 2 ) = Cov 1 ( ) Θ, ˆΘ 2 i 1 ˆµ2 Θ i=1 1 [ Cov(ˆµ Θ, ˆΘ 2 i ) Cov(ˆµ Θ, ˆµ 2 Θ) i=1 = 1 [ 1 Cov( ˆΘ i, ˆΘ 2 i ) Cov(ˆµ Θ, ˆµ 2 Θ ) i=1 = 1 [Cov( ˆΘ i, ˆΘ 2 i ) Cov(ˆµ Θ, ˆµ 2 Θ ) Equtio (7) follows from 3 Cov Θ(Θ i,σ 2ˆΘi ). (7) Cov( ˆΘ i, ˆΘ [ 2 i ) = E ( ˆΘ i µ Θ ) 3 +2µ Θ (σθ 2 + σ 2ˆΘ) = 3Cov Θ (Θ i,σ 2ˆΘi )+2µ Θ (σθ 2 + σ2ˆθ) d the pproximtio (otied y igorig terms of order 2 ) Cov(ˆµ Θ, ˆµ 2 Θ) = E[(ˆµ Θ µ Θ ) 3 + 2 (σ2 Θ + σ 2ˆΘ)µ Θ Now, = 1 2 E [( ˆΘ i µ Θ ) 3 + 2 (σ2 Θ + σ2ˆθ)µ Θ 2 (σ2 Θ + σ 2ˆΘ )µ Θ. ( 1 Cov(ˆµ Θ, ˆσ 2ˆΘ ) = Cov ˆΘ i, 1 ) ˆσ 2ˆΘ = 1 i i=1 i=1 2 Cov( ˆΘ i, ˆσ Θ 2 i ) i=1 = 1 Cov Θ(Θ i,σ 2ˆΘ ) (8) i which holds ecuse Cov( ˆΘ i, ˆσ 2ˆΘi )=Cov Θ [E ɛ ( ˆΘ i Θ i ), E ɛ (ˆσ 2ˆΘi Θ i )+E Θ [Cov ɛ ( ˆΘ i, ˆσ 2ˆΘ Θ i ) = Cov Θ (Θ i,σ 2ˆΘ ). Sustitutig (7) d (8) ito (6) d simplifyig gives i i Cov(ˆµ [DA, ˆσ [DA )= σ2 [ Θ 1 σθ 3 Cov Θ (Θ i,σ 2ˆΘ ) = σ2 Θ i V [DA 12.

544 C. JOSEPH LU, WILLIAM Q. MEEKER AND LUIS A. ESCOBAR Derivtio of Vr(ˆσ [DA ) Now, where Vr(ˆσ [DA ) = Vr(ˆσ Θ ) 1 E [( ˆΘ i µ Θ ) 4 4σ 2 Θ Vr(ˆσ Θ 2 )= 1 4σΘ 2 Vr(SΘ 2 ˆσ2ˆΘ) = 1 [ 4σΘ 2 Vr(SΘ 2 )+Vr(ˆσ2ˆΘ) 2Cov(SΘ 2, ˆσ2ˆΘ). (9) ( ) Vr SΘ 2 1 { [ E ( ˆΘ i µ Θ ) 4 (σθ 2 + σ2ˆθ) 2}, (10) = 3(σ 2 Θ + σ2ˆθ) 2 +3Vr Θ (σ 2ˆΘi )+6Cov Θ (Θ 2 i,σ2ˆθi ) 12µ Θ Cov Θ (Θ i,σ 2ˆΘi ). The pproximtio i (10) igores terms of order lower th 1 d it follows from the vrice of the momet sttistics (see Kedll d Sturt (1987, pge 322, equtio 10.9)). Usig (5), it is esy to see tht Vr(ˆσ 2ˆΘ) = 1 [ σ 4ˆΘi Vr Θ (σ 2ˆΘi )+2E Θ ( m Θi 1 ). (11) By usig the delt method we oti, for lrge, Cov(ˆµ 2 Θ, ˆσ2ˆΘ) Cov(2µ Θˆµ Θ, ˆσ 2ˆΘ) =2µ Θ Cov(ˆµ Θ, ˆσ 2ˆΘ) =(2/)µ Θ Cov Θ (Θ i,σ 2ˆΘi ). Also, we c compute Cov( ˆΘ 2 i, ˆσ 2ˆΘi ) = Cov Θ [E ɛ ( ˆΘ [ 2 i Θ i ), E ɛ (ˆσ i 2 Θ i ) +E Θ Cov ɛ ( ˆΘ 2 i, ˆσ i Θ i ) = Cov Θ (Θ 2 i + σ2ˆθi,σ 2ˆΘi )=Cov Θ (Θ 2 i,σ2ˆθi )+Vr Θ (σ 2ˆΘi ). From these results it follows tht Cov(SΘ, 2 ˆσ 2ˆΘ) [ 1 = Cov 1 ( ˆΘ 2 i ˆµ 2 Θ), ˆσ 2ˆΘ i=1 1 [ Cov( ˆΘ 2 i, ˆσ2ˆΘ) Cov(ˆµ 2 Θ, ˆσ2ˆΘ) i=1 = 1 [ Cov( ˆΘ 2 i, ˆσ 2ˆΘi ) Cov(ˆµ 2 Θ, ˆσ 2ˆΘ) 1 [ Cov Θ (Θ 2 i,σ2ˆθi )+Vr Θ (σ 2ˆΘi ) 2µ Θ Cov Θ (Θ i,σ 2ˆΘi ). (12) After sustitutig (10)-(12) ito (9) d simplifyig, we oti Vr(ˆσ [DA ) = (σθ 2 [DA /)V 22.

A COMPARISON OF DEGRADATION AND FAILURE-TIME ANALYSIS 545 B. Computtiol Formuls for the Fctors V [DA ij Oserve tht (1)-(3) c e expressed s follows: V [DA 11 = 1+ R2 σ m E Θ(r Θi ), (13) V [DA 12 = R2 σ m Cov Θ(Z i,r Θi ), (14) V [DA 22 = 1 [( 1+ R2 ) σ 2 ( R 2 ) + σ 2VrΘ (r Θi ) 2 m m +2 R2 σ m Cov Θ(Z 2 i,r Θ i )+ ( R 2 ) σ 2E ( r 2 ) Θ i Θ, (15) m m Θi 1 where R σ = σ ɛ /σ Θ, r Θi = m/m Θi,dZ i = (Θ i µ Θ )/σ Θ =( i µ )/σ is the stdrdized log time-to-filure of the ith uit. For exmple, usig σ 2ˆΘi = σɛ 2 /m Θi,oegets V [DA 12 = 1 σθ 3 Cov Θ (Θ i,σ 2ˆΘi )= σ2 ɛ mσ 2 Θ ( Cov Θ Θ i µ Θ m ), = R2 σ σ Θ m Θi m Cov Θ(Z i,r Θi ). All expected vlues, covrices, d vrices o the right-hd side of (13)-(15) c e otied from E Θ [r 2 Θ i /(m Θi 1) d expecttios of the form E Θ (Z l i rk Θ i ), where l =0, 1, 2dk =0, 1, 2. For exmple, Cov Θ (Z i,r Θi )=E Θ (Z i r Θi ) E Θ (Z i )E Θ (r Θi )d E Θ ( r 2 Θ i m Θi 1 ) = E Θ (Z l ir k Θ i ) = m ( m j j=2 m j=2 ( m j ) 2 1 φ(z)dz (16) j 1 A j ) k A j z l φ(z)dz, (17) where φ( ) is the stdrd orml pdf, A 2 = {z z 2 }, A j = {z j 1 <z z j } for j =3,...,m 1, A m = {z >z m 1 },dz 2,...,z m 1 re stdrdized log ispectio times defied y z j =(x j µ )/σ. All of the itegrls eeded for (16) d (17) c e computed esily from the formuls i the followig tle. Colums 3 d 4 of this tle were otied y sustitutig the stdrd orml desity d itegrtig y prts. j φ(z)dz zφ(z)dz z 2 φ(z)dz A j A j A j 2 Φ(z 2) φ(z 2) z 2φ(z 2)+Φ(z 2) 3,...,m 1 Φ(z j) Φ(z j 1) [φ(z j) φ(z j 1) [z jφ(z j) z j 1φ(z j 1) +[Φ(z j) Φ(z j 1) m 1 Φ(z m 1) φ(z m 1) z m 1φ(z m 1)+[1 Φ(z m 1)

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