American Society for Quality

Transcription

1 American Sciey fr Qualiy Nnparameric Esimain f a Lifeime Disribuin When Censring Times Are Missing Auhr(s): X. Jan Hu, Jerald F. Lawless, Kazuyuki Suzuki Surce: Technmerics, Vl. 4, N. 1 (Feb., 1998), pp Published by: American Saisical Assciain and American Sciey fr Qualiy Sable URL: hp:// Accessed: 7/1/21 18:22 Yur use f he JSTOR archive indicaes yur accepance f JSTOR's Terms and Cndiins f Use, available a hp:// JSTOR's Terms and Cndiins f Use prvides, in par, ha unless yu have bained prir permissin, yu may n dwnlad an enire issue f a jurnal r muliple cpies f aricles, and yu may use cnen in he JSTOR archive nly fr yur persnal, nn-cmmercial use. Please cnac he publisher regarding any furher use f his wrk. Publisher cnac infrmain may be bained a hp:// Each cpy f any par f a JSTOR ransmissin mus cnain he same cpyrigh nice ha appears n he screen r prined page f such ransmissin. JSTOR is a n-fr-prfi service ha helps schlars, researchers, and sudens discver, use, and build upn a wide range f cnen in a rused digial archive. We use infrmain echnlgy and ls increase prduciviy and faciliae new frms f schlarship. Fr mre infrmain abu JSTOR, please cnac suppr@jsr.rg. American Saisical Assciain and American Sciey fr Qualiy are cllabraing wih JSTOR digiize, preserve and exend access Technmerics. hp://

2 Nnparameric Esimain f a Lifeime Disribuin When Censring Times Are Missing X. Jan Hu Deparmen f Bisaisics Harvard Schl f Public Healh Bsn, MA 2115 Jerald F. LAWLESS Deparmen f Saisics and Acuarial Science Universiy f Waerl Waerl, Onari N2L 3G1 Canada (jlawless@sesa.uwaerl.ca) Kazuyuki SUZUKI Deparmen f Cmmunicains and Sysems Engineering Universiy f Elecr-Cmmunicain Chfu-Ciy, Tky 182 Japan We cnsider daases fr which lifeimes assciaed wih he unis in a ppulain are bserved if hey ccur wihin cerain ime inervals bu fr which lenghs f he ime inervals, r censring imes f unfailed unis, are missing. We cnsider nnparameric esimain f he lifeime disribuin fr he ppulain frm such daa; a maximum likelihd esimar and a simple mmen esimar are bained. An example invlving aumbile warrany daa is discussed a sme lengh. KEY WORDS: Incmplee daa; Mmen esimaes; Warrany reprs. There are several cnexs in he analysis f failure ime r lifeime daa in which censring imes fr unfailed unis are missing. The area ha mivaed he curren research cncerns he esimain f failure-ime disribuins r raes frm prduc warrany daa. If under warrany a prduc may experience a cerain ype f even, r "failure," hen we can esimae he disribuin f ime failure (r he inensiy funcin fr recurren evens) ver he warrany perid frm warrany reprs. We ypically have deal wih missing censring imes, hwever, as we nw discuss. Suppse ha Ti is he ime failure fr prduc uni i in a ppulain f M manufacured unis. In sme applicains Ti is measured in calendar ime since he sale ime f he uni. Fr many ypes f prducs he manufacurers d n knw he dae f sale fr ms unis, and herefre he censring ime (i.e., he elapsed ime beween he sale f he iem and when he daa are assembled) fr ms unfailed iems is unknwn. Fr unis ha fail under warrany, he failure ime and he penial censring ime are knwn because he dae f sale is verified as par f he warrany claims prcess. Similar prblems arise when Ti is sme ype f usage, r perainal ime. A familiar example is in cnnecin wih aumbiles, where Ti represens he mileage a failure. If failure daa are clleced up sme curren dae, he censring ime is he minimum f he vehicle's curren mileage and he mileage a which i passes u f he warrany plan. Fr example, fr a w-year/24,-mile warrany, his laer mileage is he lesser f he vehicle's mileage a w years and 24, miles. Because exac mileage accumulain daa are n available fr ms cars, he exac censring imes are in general unknwn. By making he crude bu generally 3 saisfacry assumpin ha mileage accumulain is linear ver ime, hwever, hey may be esimaed fr cars experiencing a failure because he dae f sale, dae f failure, and mileage a failure are all bserved. An applicain invlving aumbile warrany daa is discussed a sme lengh in Secin 5. Suppse ha he lifeime variable T has disribuin funcin F() = Pr(T < ) and ha he ppulain f M unis has independen lifeimes l,...,j generaed frm ha disribuin. There are als censring imes r1,..., TM assciaed wih he unis, and we assume ha he ri's are independen f each her, wih cmmn disribuin func- in G(r) = Pr(Ti < T). The disribuin G(T) is deermined by he randm prcess by which unis are sld and by he randm prcess by which unis accumulae usage ver calendar ime, when T is a usage ime. A reviewer asked wheher finie ppulain assumpins migh be used insead. This is feasible bu mre difficul implemen and, given he large sample sizes ypical f his area, unlikely give resuls ha differ much frm urs. The bserved daa are as fllws: If i < Ti, we bserve i (and pssibly als Ti), bu if, > Ti, we knw nly ha fac and n he value f T, r i. Our bjecive is esimae he disribuin F() frm such daa, aviding any parameric assumpins. Suzuki (1985), Kalbfleisch and Lawless (1988), and Hu and Lawless (1996a) discussed he use f supplemenary fllw-up samples f unfailed unis as a way cmpensae? 1998 American Saisical Assciain and he American Sciey fr Qualiy TECHNOMETRICS, FEBRUARY 1998, VOL. 4, NO. 1

3 4 X. JOAN HU, JERALD F. LAWLESS, AND KAZUYUKI SUZUKI fr he missing censring imes. In many circumsances i is pssible esimae he censring-ime disribuin, hwever, and his prvides anher apprach. We presen in his aricle w nnparameric esimain mehds fr he case in which he censring-ime disribuin G(7) is knwn, r a leas esimaed frm her surces. The main assumpins ha we make iniially are (a) he number f prduc unis M in service is knwn, (b) all failures are repred under he daa-cllecing scheme, (c) he censring-ime disribuin is knwn, and (d) censring imes are saisically independen f failure imes. The assumpins are discussed furher in he aricle, and ways handle deparures frm hem are presened. Secin 1 presens nnparameric maximum likelihd and mmen esimars fr F(), assuming ha G(7) is knwn. Secin 2 reprs n a small simulain sudy cmparing he w esimars. Secin 3 cnsiders cases in which G(7) is esimaed, and Secin 4 examines he assumpin f independence f failure and censring imes. Secin 5 presens a deailed example invlving aumbiles. Secin 6 ulines exensin f he mehdlgy deal wih muliple failure mdes and recurren evens, and Secin 7 presens sme cncluding remarks. 1. MAXIMUM LIKELIHOOD AND SIMPLE MOMENT ESTIMATORS A nnparameric mehd f lifeime disribuin esimain was previusly given by Suzuki (1988). Suzuki and Kasashima (1993), hwever, shwed ha mehd be inferir maximum likelihd, s we will n discuss i here. T develp nnparameric esimars, i is cnvenien and cusmary wrk wih discree disribuins; finie-sample esimaes f cninuus cumulaive disribuin funcins F() are discree anyway and may be bained frm he discree-ime framewrk. Thus, we assume ha lifeime T and censring ime T may each ake n values 1, 2,..., and f() = Pr(Ti = ),g(r) = Pr(TJ = r). The crrespnding cumulaive disribuin funcins are F() = f(1) f() and G(T) = g(l) (r). In his secin, we assume ha T, and 1 are independen and ha G(r) is knwn. 1.1 Maximum Likelihd Esimain Wih knwn ppulain size M and censring-ime disribuin G(T), he likelihd funcin based n he prbabiliy f he bserved daa fr he ppulain is f he familiar censred-daa frm (Lawless 1982, chap. 1), I7 f(i) n Pr(Ti > i). (1) i <T i,>ri The difference wih he usual siuain is ha ri is n bserved fr he unfailed unis. Thus, reaing i as a randm variable wih disribuin G(-), Tmax Pr(T, >T ) = 1 - f() 9(T) T=1 <T Tnl ax = 1- f()g(), =l TECHNOMETRICS, FEBRUARY 1998, VOL. 4, NO. 1 where G(7) = Pr(T/ > T) and T-max = SUp{ : G(T) > }. We assume ha Tmax < c, which is unresricive in he presen cnex. The likelihd funcin (1) may be wrien as T ~~- - M-m Tmax Tmax - H f() 1-1 f()g() =l - =l where n is he number f failures a ime ha are bserved and m is he al number f failures bserved. Nice ha Zmx n = m. Using #A represen he number f elemens in se A, we have n = #{i: i < r, = } and m = #{i : i < 7i}. Esimaes f f(), = 1,..., max, can be bained by maximizing (2) wih respec f(1),..., f(max) under he cnsrains f() > and f(1) +. + f(7tmax) = F(Tmax) < 1. Dene he lg-likelihd funcin by Tmax l(f) = E =l Tmax n lg f() + (M -m) lg 1- E f()g() The sluin f he equains 91 n af() f() (M-m)- G() _ I f((s) s=l ( s) =l, (2) = 1,...,Tmax, (3) maximizes l(f) wih n cnsrains n he f()'s. By rewriing (3) as (M - m)g()f() = n { Tmax E fi(s)(s) s=l/ and summing bh sides ver = 1,..., 77max, we see ha EZml f(s)g(s) equals m/m. Insering his in (3), we hen find ha (3) is slved uniquely by fml() -MG() = 1 ' *. Tmax This hen gives he maximum likelihd esimaes, prvided ha fml(1) +- + fml(tmax) < 1. This is virually always he case when F(Tmax) is n clse 1, which is saisfied in ms applicains. In he warrany-reprs cnex, fr example, 7max is he maximum failure ime bservable and n larger han he warrany ime limi, and he prbabiliy ha a uni fails while under warrany is cnsiderably less han 1. If he cnsrain is n me, hen dub may be cas n he validiy f he assumed funcin G(7). When he esimaes (4) sum slighly ver 1, a reasnable apprach is simply rescale hem s ha hey sum 1, bu his may differ slighly frm he maximum likelihd esimar (MLE). T ge he MLE, we need maximize (2) under cnsrains f() > and f(1) f(tmax) = 1 in ha siuain. The nnparameric maximum likelihd esimae f F(), = 1,..., max, is FML() =- ML(1) fml(). Similar he discussin fr he nnparameric MLE f Hu and Lawless (1996a), fr example, argumens can be given esablish he cnsisency and asympic nrmaliy f his esimar. This als fllws frm he nex secin. (4)

4 NONPARAMETRIC ESTIMATION OF A LIFETIME DISTRIBUTION A Simple Mmen Esimar We see ha he number f bserved unis wih failures a in (2) is n = ^il I (i =,ri > ), where I(A) is he indicar f even A (i.e., i equals 1 if A is rue and if n). Le ui() = I (i =, Ti > )/G(), = 1, 2,... A simple mmen esimar f f(), fsm () - M il u =l n MG()' =1,..,? Tmax, (5) is bained by ning ha E{n} = MG()f(), = 1,... -max., Nice ha fsm() is he same as he nnparameric MLE fml() in (4) in he curren siuain. Under ur assumpins, n is binmial(m, f()g()) and i is easy see ha fsm() is unbiased wih variance var{fsm ()} f() The sample variance esimar M 1 - f()g()] = 1,...,r M ar{fsm ()} = M2 {() u i=1 n(m - M3G()2 n) ' - U()}2 wih u() = i uim()/m = fsm(), is he same as he cnsisen esimae fr he variance (6) achieved by replacing f() wih is esimae fsm(). Then he esimar fr F(), = 1,..., max, based n fsm(-), is FsM()= E fsm(s), s=l nax (6) = 1, T.,Tmax (7) By ning ha he n's are mulinmial ( = 1,..., we have Tmax), cv{fsm(sl), fsm(s2)}= f [I(sl = s2)-f(s2)g(sl) and cnsisen a esimae fr he variance fsm() as and a cnsisen esimae fr he variance f FsM() as var{fsm ()} M ( A 2 M2 E E [ui(s)-u(s)]l M i=1 s=l ns(m - ns) ne nsls2 s M3G(s)2 S12 M3G(s()G(s2) 1.3 Exensin We generalize he preceding siuain slighly allw he disribuin f T depend n a discree cvariae r grup indicar xi. Suppse ha xi akes n values x?,..., x? and is bservable. This is useful because wih aumbiles, fr example, he censring ime fr a car may depend n wha (8) ime i enered service and ha ime is usually prvided by he dealer. In such cases i is n pssible esimae G(r) wihu cnsidering he sales paern. We dene gk(t) = Pr(T = rtxi = x4) and Gk(T) Pr(17 < T7xi = x). Le Mk = #Pk, wih Pk = {i: Xi = x?,i = 1,...,M}, = 1,...,K. We assume ha he disribuin f Ti des n depend n xi nw. Wih knwn subppulain sizes Mk and censring-ime disribuins Gk(r), k = 1,..., K, he likelihd funcin f he failure-ime disribuin based n he daa available is f f(i) n Pr(T > Tixi) i <T i i >T i Tmax K - max - k --mk fn f()" 1- E f()gk(s) (9) =l k=l s=l where Gk() = Pr(7i > T x = - ),Tmax = maxksup{t : Gk(T) > }, and mk = #{i: xi = x, i < ri}. A nnparameric MLE f f() can be bained similarly he prcedure in Secin 1.1. In his case he maximum likelihd equains reduce fml() = n T )Gk() k=l 1 - = Is:l fml(s)g k(s) -1 1,..., Tmax. (1) There is n clsed frm fr fml() when K > 2. Equain (1) can be used prvide an ierain scheme fr baining he fml()'s. We can, hwever, bain a mmen esimar. The fac ha E{nxi, i = 1,... M} - = MkGk()f(), k=l gives an unbiased esimar f he f(): fesme() = K IEk=l MkGk() - 1,... Tmax, = 1,..., Tmax. (11) This als arises if we apprximae 1 - E x fml(s) Gk(s) in (1) wih (Mk- mk)/mk, k = 1,...,K. Nice ha he hree esimars fsm(') in (5), fml( ) frm (1), and fesm(.) in (11) have he same numerar n bu differen denminars: MG(), EK=l ( - mk)gk)/( - Is=T M ) s Mkk. fml(s)gk(s)), and Lk=1 MkGk(). Because K G() = E Gk() Pr(X - x4) k-1 k--1 Gk()Pr(T > T, X ()Pr(T > TX : 4) 4x)' he difference beween fsm and fesm is due he difference beween Pr(X - x) and is mmen esimar TECHNOMETRICS, FEBRUARY 1998, VOL. 4, NO. 1

5 a 6 X. JOAN HU, JERALD F. LAWLESS, AND KAZUYUKI SUZUKI Mk/M,k = 1,...,K, which is small when M is large enugh; he difference beween fml and fesm is due he difference f he MLE 1 - E'max fml(s)gk(s) and he mmen esimar (Mk - mk)/mk fr Pr(T > TIX -= z), which is als small when Mk is large, k = 1,..., K. Thus, he hree esimars are virually he same fr cases wih large M and Mk's. We have f(s) COV{fESM(sl)fESM(s2)} S = M,s x I(s, K MkGk(Sl) f imk G(SGk( )Gk(S2) 32) - f(2) Mk=l (12) The variance f FESM() = E s= var{fesm () } and can be cnsisenly esimaed by var{fesm()} - fesm(s) is COV{fESM(S1), fesm(s2)} K S =1 S21 {lk=l AMkGk(S1)}2 E Ik=lMkGk(s1)Gk(S2) X I(s1 -s2) - n2 K, (13) {Ek=lMkGk (S2) bained by replacing he f(.) in (12) wih he esimae fesm('). I is easily seen ha M{FESM() - F()} has a limiing nrmal disribuin as M - and ha cnsruc ess r cnfidence inervals i can be reaed as nrmal wih mean and variance esimaed by (13). Esimain f he asympic variance f fml() may be bained hrugh he sandard prcedure fr an MLE. The infrmain marix is nw INFO (f ) =- ( 21 ((14) MF()-Of(s1)Of(82)) where he elemens are 21 _ ns- Of (S)Of() = -I2) f(i )2 XTma KE( Gk(s1)Gk(s2) - (Mk - - mk) max f()g() k=l f-e=1 G }2' 51, 2 = 1,..., Tmax. Thus, an esimae f he asympic variance f FML() = E= fml() may be bained as well. T implemen he prcedure is cmplex when Tmax is large, hwever, because inversin f large marices is invlved. In addiin, when censring is heavy, INFO(f) culd be singular and he mehd cann be applied. The discussin by Hu and Lawless (1996a) fr esimain f asympic variances f nnparameric MLE's applies here. 2. SIMULATION Cmparisn f he nnparameric maximum likelihd and mmen esimars presened in Secin 1 was invesi- TECHNOMETRICS, FEBRUARY 1998, VOL. 4, NO. 1 Sl=l 82=1 gaed hrugh a simulain sudy. Being mivaed by warrany daa, we chse he fllwing simulain seup. Cnsider a prduc wih a ne-year warrany; suppse ha here are M = 4, unis sld wihin a year and he warrany daa have been clleced ver ne and a half years since he firs uni was sld (we ake is sale ime as ). Time is calendar ime, r age f a uni, measured frm he dae f sale. Suppse furher ha he imes f hese unis heir firs failures are independen frm each her, idenically Weibull disribued, and independen f heir sale imes and he sale imes are unifrmly disribued ver he ne-year perid. The censring ime assciaed wih uni i is nw Ti = min(1, xi) year, where xi is is sale ime and Tmax = 1. We generaed sale imes xi and failure imes i,i - 1,..., M, respecively, frm discree apprximains he unifrm disribuin ver (, 1] and he Weibull disribuin (\/ [ f( =- )-- exp {-( wih 6 = 2. and a = 3.95, Discree randm variables were bained hrugh discreizing ime as fllws. We divided he ime perid in inervals, ((k - 1)/12, k/12], k = 1, 2,..., and assigned variables having values in he kh inerval he value k/12. The values = 1,2,... hen crrespnd k/12, k = 1,2,... The values f he parameers were chsen make he simulain realisic. The crrespnding values f m/m fr cases f a = 3.95, 1.85 are abu.5,.2, respecively, which allws us sudy siuains wih heavy and mderaely heavy censring. Frm he simulaed daa, we evaluaed he hree nnparameric esimaes f F() = 1 - exp{-(/a)6}, (, 1]: (1) FSM(), based n (5); (2) FESM(), based n (11) and assuming ha he numbers f unis sld in each quarer f he year (M1, M2, M3, M4) are knwn; (3) FML, he MLE based n (1) fr he same sraified-sample siuain as (2). We used S-Plus (?1988 MahSf, Inc.) fr generaing he randm variables needed and all he cmpuing. The maximum likelihd esimae was evaluaed by using he ierain prcedure based n (1); we k f() (k/12) = f(k/12) and erminaed he ierains when E l fj+l)(k/12) - f(j)(k/12) < 1-4, where f()() is he jh ierae ward fml(). Figure 1, (a)-(d), shws esimaes fr F(.8) in he cases a = 3.95 and ac = 1.85 based n he hree esimars frm 1, simulain repeiins. The crrespnding sample means and sandard errrs are essenially he same. The sample means are very clse he rue values f F(.8) in he w cases, respecively. This indicaes ha here is alms n difference in he hree esimars, and hey all esimae F() well in he siuains we cnsider. We repeaed he preceding simulain wih M = 4 insead, sudy siuains wih sample size fairly small. The crrespnding sample means and sandard errrs f he esimaes fr F(.8) are nce again essenially he same. We presen he esimaes fr case a = 1.85 in Figure 1, (e)-(f).

6 NONPARAMETRIC ESTIMATION OF A LIFE ETIME DISTRIBUTION 7 -.a, _. Ia ' 6O.' e' 2 2, LA cl. w L. d CO).. a FSM(.8) 1.1. alpha=3.95, M=4 FESM(.8) 1.2. alpha=3.95, M=4 (a) (b) d i LL ) r- 6) In, J. -J 6 G N- 1: IF 9 w~~~~~< 6D.. ) I FSM(.8) FESM(.8) 1.4. alpha=1.85, M=4 (c) -3w._a.*~ ~~~~~~~. w 17 (d) 1 d N U- d.. Ua [J LL N 4 46 cd cu d.i... I * FSM(.8) 1.5. alpha=1.85, M=4 FESM(.8) 1.6. alpha=1.85, M=4 (e) (f) c c C/ n w LL, LLJ L.* FSM(.8) 1.7. alpha=1.2, M=4 (9) Figure 1. Esimaes f 8% Quaniles f F(): FSM = FSM (), FSM(.8 ) 1.8. alpha=1.2, M=4 (h) FESM = FESM ), E MLE F = FML() TECHNOMETRICS, FEBRUARY 1998, VOL. 4, NO. 1

7 8 X. JOAN HU, JERALD F. LAWLESS, AND KAZUYUKI SUZUKI Table 1. Sample Means and Sandard Errrs (in brackes) f he Esimaes fr F(.5) and F(.8) Frm 1, Simulain Repeiins: a = 1.2. F(.5) =.2136 F(.8) =.4594 Cases M = 4, M= 4 M = 4, M= 4 FSM.2139 (.66).2127 (.24).469 (.82).4595 (.278) FESM.2139 (.66).2127 (.24).469 (.81).4596 (.276) FML.2135 (.62).216 (.27).463 (.76).4564 (.275) FsM ()= s=l fsm (s), ,Tmax, (16) are bh cnsisen. Behavir f he esimar fsm() depends n hw well G(T) esimaes G(r) and wheher G(T) is relaed he primary daa-ha is, he n's. In his aricle, we assume ha G(T) is independen f he primary daa. The cvariance f he fsm()'s is hen T invesigae he esimars furher, we cnduced he w preceding simulains fr he case a = 1.2 (m/m in his case is abu.5). The sample means and sandard errrs f he esimaes are nw slighly differen and are presened in Table 1. Figure 1, (g)-(h), shws he esimaes fr F(.8) based n FSM() and FESM() in his case wih M = 4, and M = 4, respecively. Table 2 shws sandard errrs f esimaes FSM(.8) and FESM(.8) in differen cases and he sample means f esimaes fr heir sandard deviains based n (8) and (13), respecively. We can see ha (8) and (13) esimae well he variances f FSM() and FESM(), respecively. As discussed in Secin 1.3 and shwn by he simulains, he hree esimars agree clsely, especially in he siuains wih heavy censring. The pracical cnsequence f his is ha i is saisfacry use he easily cmpued mmen esimaes. 3. EFFECT OF USING AN ESTIMATE OF THE CENSORING-TIME DISTRIBUTION The esimain prcedures in Secin 1 assume ha he censring-ime disribuin G(T) is knwn. In ms pracical siuains, hwever, G(T) is esimaed, r nly rughly knwn. Hu and Lawless (1996b) invesigaed likelihdbased parameric esimain fr his siuain; heir apprach can be exended a nnparameric seing. We fcus here n he exensin f he simple mmen esimar in Secin 1.2; he esimar in Secin 1.3 can be exended similarly. An example is presened in Secin 5. Suppse ha G(r) is a cnsisen esimae f G(r), and fr cnvenience le Gb() = 1- G() dene he esimae f G(). In ha case, he esimaes analgus (5), and FSM() in (7), G()' Gb (), = , Tmaxa (15) COv{ fsm (S), fsm(s2)} E{cv[fsM(s), fsm(s2) G(T)]} + cv{e[fsm(sl)lg(t)], E[fsM(s2) G(T)]} { f(si)g(si)[i(s, Gb(sl)Gb(S2) = 2) - f(s2)g(s2)]} + cy { G(sl)f(si) G(S2)f(s2)} Gwb(s) b e (2) b which can be esimaed by fsm(s1) Gb(S2) [( + 2) - fsm(s2)gb(s2)] (17) fsm(s1)fsm(s2) -- ( ( Gb(si), Gb(s2)}, (18) Gb(S1)Gb(s2) assuming ha an esimae cv{gb(s1), Gb(s2)} is available. The secnd erm in (17) accuns fr variain due G(r) having been esimaed. An esimae fr he variance f FSM() can be bained frm var{fsm()} = - E cv{fsm(sj),fm(s2)}. S1=l S2=l In Secin 5, we will discuss his furher based n he example here. 4. NONINDEPENDENT CENSORING Secin 1 assumes ha censring imes Ti,..., TM are independen f lifeimes T,..., TM. This assumpin may smeimes be quesinable: Fr example, if he lifeime f an aumbile cmpnen depends n bh he age f he car and he number f miles i is driven, hen he fac ha warrany plans have age and mileage limiains (e.g., w years and 24, miles) implies a dependence beween Ti Table 2. Cmparisns f Sandard Errrs (sd) and Average Esimaed Sandard Deviains (sd) f FSM (-8) and FESM (8). FSM (.8) FESM (.8) Cases a = 3.95 ca = 1.85 a = 1.2 ac = 3.95 a = 1.85 a = 1.2 M = 4, sd sd M = 4 sd sd TECHNOMETRICS, FEBRUARY 1998, VOL. 4, NO. 1

8 NONPARAMETRIC ESTIMATION OF A LIFETIME DISTRIBUTION 9 and Ti. Our bjecive here is briefly cnsider he effec f nnindependen censring n he esimar f Secin 1.2. We als presen a versin f he simple mmen esimar fr a special case in his siuain. 4.1 Effec n?sm(') The esimar fsm(') f (5) can be wrien in he frm Nw E{I(i Pr(Ti > ITi f SM() I(i, i >) E i_j 1Z l Pr(T > ), Ti > )} = f()pr(t1 > \Ti = ), s if ) f Pr(7 > ), hen fsm() is biased, wih served, we may UVlvuvrL IIUYLVI~~ILILlli cnsider he esimar fr f(\lx), f(\x?)=?,k lmkgk () LCIIILLVLIVIcl\(IUCI:k 1... Trnax, (21) if an esimae Mk is available. We address his in Secin 5 hrugh he example. This idea may be applied siuains in which he number f prduc unis in service MI is unknwn bu here is an esimae fr i. Tha is, fr example, we cnsider fsm() = n/mg() insead f (5). Similarly, as in Secins 1 and 2, we can als cnsider variance esimain f F(lx?), - 1,...,rnax and k 1,...,K. 5. AN EXAMPLE m ipr(--1 > ) E{fsM()} f() Ti { ZPr >T > T ) (19) ) The exen f he bias may be assessed by hyphesizing mdels fr he dependence f Ti and 7, and in many cases we may find ha he brackeed erm in (19) is clse 1. If i is n, here may be lile mivain esimae he marginal disribuin f(); wha is needed insead is a mdel ha accuns fr he dependence f T and T. Wih aumbiles, his usually means ha a lifeime mdel ha incrpraes bh age and mileage is needed. Lawless, Hu, and Ca (1995) discussed such mdels and indicaed hw es independence f lifeime and censring ime frm aumbile warrany daa. If here is a serius cncern in a pracical siuain abu dependence, hen such mehds shuld be emplyed. 4.2 A Special Case In sme siuains Ti and T are relaed nly hrugh a cvariae (r cvariaes), say xi, such ha Ti and 7T are independen given xi, i = 1,..., M. This was cnsidered in differen cnexs, fr example, by Kalbfleisch and Lawless (1991) and Hu and Lawless (1996b). We exend he mdel f Secin 1.3 slighly deal wih his. As in Secin 1.3, we suppse ha xi akes n values Xk, k,... 1,... K, and is bserved fr every uni i. Then fsm(xk?) - -( MkGk() 1,... max, (2) is an unbiased esimar f f(lx?),k = 1,...,K, where n,k = #{i : i < -i,i =, x x?}. Ning ha () k= f (lx) Pr(X = x), we have an esimar fr f() and als fr F(), prvided ha Pr(X = x?) is knwn r esimaed, k = 1,..., K. The changing paern f FsM(lxT) =1 fsm(six) when he value f x? varies may help us see hw lifeime is relaed censring ime. If he dependences beween Ti and xi and T and xi can be specified paramerically, we can see hw he dependence f he failure ime and he censring ime affecs he simple mmen esimar frm (19). Parameric mdels als allw us handle cninuus cvariaes. Hu and Lawless (1996b) cnsidered his apprach. Fr a slighly differen siuain in which nly he xi's assciaed wih unis having bserved failures can be b- Sme real warrany daa fr a specific sysem n a paricular car mdel are cnsidered fr illusrain. The daa include warrany claims frm 823 cars amng 8,394 cars prduced during a w-mnh perid. The warrany plan in quesin was fr ne year r 12, miles; he daa here cver up 18 mnhs afer he firs car was sld. The numbers f cars sld wihin he firs, secnd,..., and he sixh hree-mnh perids were 5,699, 1,593, 725, 227, 129, and 21, respecively, amng which 479, 232, 83, 18, 1, and 1 cars had warrany claims recrded. We examine he disribuin f he ime he firs failure (claim) f he cars. Fr illusrain we cnsider "ime" bh as mileage in miles and as age in years (i.e., calendar ime), alhugh fr engineering purpses mileage is mre relevan. Le i and Fi, i = 1,..., M = 8,394, be he firs failure imes and censring imes, respecively, and le Si dene he ime f sale fr car i, where he firs car sld has a sale ime f. Calendar ime will be expressed in years, and he warrany daa herefre will include fllw-up f cars up ime 1.5 years. The censring ime ri fr car i may be described as fllws. Le he age f he car (i.e., ime since he car was sld) when i reaches 12, miles be ai years, and define ui = 12,/ai as he average mileage accumulain rae ver he age inerval (, ai], in miles per year. Then fr he case in which, and ri represen age (i.e., years since sale), ri - min(1.5 - si, 1, a). In he case in which i and T- represen mileage, -i = min(min[1.5- si? l]ui, 12, ). The values f i's are bserved nly fr hse cars wih i < i--ha is, fr he m = 823 cars wih heir warrany claims recrded. The numbers f firs failures experienced in he cars' firs, secnd,..., welfh mnh were 71, 81, 77, 92, 84, 75, 98, 69, 42, 57, 37, and 4, and wihin heir firs, secnd,..., welfh 1, miles were 8, 87, 7, 8, 65, 77, 59, 65, 45, 66, 63, and 66, respecively. The fllwing invesigain analyzes he daa mre hrughly. Alhugh he sale daes si's are knwn fr all 8,394 cars, he values f Ti's are n. If we are willing make he simplifying assumpin ha mileage accumulain is linear ver (, ai], hen ui may be evaluaed fr cars ha fail because he mileage as well as he age a failure is recrded. In his case we wuld hus have he -ri's fr he cars ha fail bu n fr hse ha d n. The simple esimars used here d n require any censring imes, bu hey d require an esimae f heir disribuin, which we nw discuss. TECHNOMETRICS, FEBRUARY 1998, VOL. 4, NO. 1

9 ., - 1 CZ - 13 a) r Q (D X. JOAN HU, JERALD F. LAWLESS, AND KAZUYUKI SUZUKI car i ver he firs year afer sale; his is bviusly an versimplificain bu is saisfacry fr pracical purpses in his case. Sale dae and mileage-accumulain rae can reasnably be assumed independen, and we knw ha he survival funcins f he censring ime are G,i--. - / 12, Ga(T) = I(T <1) Pr(1.5 - si > Tr, U< < when T is age a failure and a Q (5 co 6 c 1 age (days) (a) 2 3 Gm(T) = I(T < 12,) Pr(U min[1.5 - s, 1] > T) ='::...:::::::' ~in ^ when T is mileage a failure. Then we can esimae he survival disribuin f censring ime G(r) in he warrany daabase ppulain by using he empirical disribuf sale daes si(i = 1,..., 8,394) alng w ih he em pirical disribuin f Ui based n he cusmer survey. Figure 2 shws esimaes and apprximae pinwise 95% cnfi-.-=. ^dence inervals fr survival funcins f he censring ime in bh ime-scale cases. The mmen esimae (7) may hus be cmpued and is - - shwn in Figure 3, fr he case in which failure "imes" are measured in miles. Figure 3 als shws apprximae pinwise 95% cnfidence inervals fr he failure-ime disribu in funcin F(), bained as FsM()? 1.96V()1/2, mileage (miles) where V() is he esimaed variance f FSM(). These in- (b) ervals are based n he fac ha, as M increases, he dis- 2. Esimaes and Apprximae 95% Cnfidence Inervals fr ribuin f - Figure [FSM() F()]() 1/2 appraches a sandard Funcins f Censring Time: (a) Time = Age, (b) Time = nrmal disribuin fr a given. Tw ses f cnfidence limis are shwn: Inervals I use he variance esimae (8), which assumes ha G(r) is knwn; inervals II are based Survival Mileage. A cu smer survey f 67 cars f he same ype and n (18) and accun fr he fac ha G(r) has been esiapprxi mae gegraphic lcain as hse in he warrany maed by using he car survey. The secnd se f inervals daabas< e was aken, wherein he apprximae mileages a is cnsiderably wider and prvides a mre valid assessmen ne yea.r afer sale were bained fr each car. We assume f uncerainy. We culd similarly prduce esimaes f he ha mil leage accumulain ccurs a a cnsan rae ui fr failure-ime disribuin in erms f car age. 6.., d cm d I..I LL cc - d ;I. -- I... - C\j - I I I I I I I mileage (miles) Figure 3. Apprximae 95% Cnfidence Inervals f Failure-Time Disribuin (ime = I; - -, Cnfidence Inerval II. mileage):, SM Esimae; ----, Cnfidence Inerval TECHNOMETRICS, FEBRUARY 1998, VOL. 4, NO. 1

10 NONPARAMETRIC ESTIMATION OF A LIFETIME DISTRIBUTION 11 We remark ha an alernaive apprach is sraify cars accrding heir ime f sale and hen use he apprach in Secin 1.3. This prduces an esimae f F() ha is indisinguishable frm ha in Figure 3 in his case, which was based n he unsraified daa. I is pssible here ha failure may be relaed bh age (ime since sale) and mileage. T invesigae his we frmed a cvariae x based n mileage-accumulain raes, as fllws. We divided mileage raes in five classes-(, 6,], (6,, 12,1, (12,, 18,], (18,, 24,], and (24,, c) miles per year-and le x = k dene he kh class (k = 1,..., 5). The numbers f failures fr cars in he five classes are 92, 266, 245, 19, and 111, respecively. Frm he cusmer survey f 67 cars and he car sales daa, we esimaed he censring-ime disribuins Gk(r) = Pr(7T < T-lx = k), k = 1,...,5, hrugh Gak (T) = Pr(min[.5 - s, 1] > ) Pr (U < fr he age case and Gl,k(T) = I(Tr 12,) 12 xi= x JPr(min[l.5-s, 1] > r dpr(u < ulxi- k) fr he mileage case; frm he survey daa alne, we esimaed MIk by AI Pr(X1 = k). The numbers f cars frm he survey sample falling in he five grups are 96, 271, 148, 53, and 39, respecively. Finally, we impued a value f ui, and hus xi, fr each car ha experienced a failure under warrany by dividing he mileage a failure by he age a failure. We hen esimae Fk() = Pr(Ti < lx, = k) as Fk()- = nib,lk,=i AkGbk(s) c. d d CN ; LO q - 9 y = 1,... Tmax,k, (a) age (days) "~...-~-2., ,.. - l (b) mileage (miles) Figure 4. Esimaes f Failure-Time Disribuins Wih Differen Usage Raes: (a) Time = Age, (b) Time = Mileage. k where ns.k = #{i i = s, x = k,tr > s},lk MlPr(X, = k) wih I = 8,394, and Tmax,k = sup{ : Gb,k(T) > }, k = 1,..., 5. Esimaes f Fk(), k 1,..., 5, are presened in Figure 4, (a) and (b), fr he cases in which failure ime is measured as car age and car mileage, respecively. Bearing in mind ha he esimaes are n very precise, in par because he esimaes f G,(rT) and Pr(Xi = k) are based n raher small samples, Figure 4 suggess ha failure imes measured in miles d n depend much n he mileage-accumulain rae bu ha failure imes measured as car age d. This suggess ha mileage is he mre relevan ime scale fr his ype f failure. Lawless e al. (1995) reached a similar cnclusin by using parameric mdels fr failure ha incrprae bh age and mileage as facrs. 6. RECURRENT EVENTS AND MULTIPLE FAILURE MODES Prducs under warrany are usually repairable sysems in which here are muliple ypes f failure ha may ccur mre han nce. The prblem discussed in his aricle can be sudied in his brader cnex, and mehds based n maximum likelihd and n mmen esimain may be develped. We will merely menin he main ideas, which are discussed elsewhere. The mdeling f recurren evens fen uses Pissn r renewal prcesses (Ascher and Feingld 1984; Lawless 1995). Mre generally, he mean and rae funcins fr he recurren evens r failures are f ineres. They are defined as fllws: Le Ni() dene he number f evens ccurring n uni i ver he ime inerval (,]. Then A() E{Ni()} is called he mean funcin and A() = da()/d is called he rae (r rae f ccurrence) funcin. If he recurren evens fllw a Pissn prcess, hen A() is als he inensiy funcin. In he case f recurren evens he "censring" ime Ti refers he ime perid (, ri] ver which uni i is bserved. Hu and Lawless (1996a) discussed maximum likelihd esimain under a Pissn mdel when censring imes are missing fr unis n experiencing any failures. They als presened a mmen esimar fr A() ha is analgus he nes given fr failure-ime disribuins in Secin 1.2 and 1.3 and is f exacly he same frm as (5), I nsm( ) MG{ SMG() = 1,.., Tmax, where nw, hwever, n is he al number f recurren evens bserved a ime acrss all prduc unis. Hu and Lawless (1996a) gave variance esimaes fr ASM() and AsMI() = AsNI() As\s() and discussed heir prperies. Muliple failure mdes may als be deal wih. Fr simpliciy we cnsider w mdes, A and B, and he case f failure imes; recurren evens can als be cnsidered. Le TA and T1B represen he imes failure f mdes A and B, respecively, le fa() = Pr(TA = ) and fb() = Pr(T~ = ) dene he marginal prbabiliy funcins, and le fab(s, ) = Pr(T' = s, TB = ) dene he jin prbabiliy funcin f TA and TB. Under he assumpin f TECHNOMETRICS, FEBRUARY 1998, VOL. 4, NO. 1

11 12 X. JOAN HU, JERALD F. LAWLESS, AND KAZUYUKI SUZUKI independen censring imes 7T, he fllwing are unbiased esimaes f fa() and fb(): fa() A() MG() B( nb() MG() where na() = it I(T-A =, i > ) and nb() = E i I(TB =, Ti > ), and nce again we assume ha G(r) = Pr(T7 > T) is knwn. I is als pssible give a simple mmen esimar f fab(s, ): rany daa, he mileage-accumulain (r usage) rae fulfills his funcin. If censring imes are available fr unis ha fail, hen inferences abu he lifeime disribuin may be bained by cnsidering he disribuin f i given ha i < ri fr failed unis. This gives he runcaed daa-likelihd funcin LT=n.^ i <ri f f (i) (25) nab(s,) ) ABG(S, MG(s v )' (24) where nab(s, ) = M1 I(T-A = s, TB =, ri > s V ) and s V denes he maximum f s and. In applicains in which he prbabiliy f a failure f any given mde is fairly small ver he bservain perid, hwever, he prbabiliy f geing failures n w r mre mdes is usually very small, s (24) may n be very precise. In many siuains i may be adequae simply cnsider he differen failure mdes separaely, in which case he esimaes (23) are all ha are needed. Variance esimaes are hen given by he expressins fr fsm() in Secin 3.2. If, hwever, we wish gain insigh in hw failure imes fr differen mdes are relaed, (24) can be used. If his is imprecise be useful, hen ne can adp a parameric mdel ge mre precise (bu mdel-dependen) esimaes. The preceding discussin f muliple failure mdes assumes ha, when a failure f ne ype ccurs, i des n preclude failures f her ypes. In sme siuains he failure mdes may be cmpeing s ha his des happen. 7. COMMENTS AND RECOMMENDATIONS When censring imes are missing, sandard mehds f esimaing lifeime disribuins are n available. If he censring-ime disribuin G(T) is knwn r esimaed frm addiinal daa, hwever, hen eiher maximum likelihd r mmen esimain may be used bain nnparameric esimaes. The mehds in his aricle depend n he validiy f he assumed G(r), and i is impran in pracice be cnfiden ha G(r) is suiable. The use f he easily cmpued simple esimars FSM and FESM is enirely saisfacry in pracice, as demnsraed by ur simulain resuls. We als recmmend he use f cnfidence limis fr he lifeime disribuin ha accun fr uncerainy in G(r). When G(r) is esimaed, he mehd f Secin 3 can be emplyed. If sandard errrs fr he esimae f G(7) are n available, we recmmend varying G(r) in a sensible way arund he esimae and examining he range f cnfidence limis bained. The esimaes in Secin 3 als require ha he censring imes be independen f lifeimes. This can be a prblem fr sme applicains. As shwn in Secins 4 and 5, we can fen handle dependen censring by using a cvariae x such ha lifeimes and censring imes are rughly independen cndiinal n x. In he case f aumbile war- TECHNOMETRICS, FEBRUARY 1998, VOL. 4, NO. 1 insead f (1). I is well knwn ha fr parameric mdels f(; ) he likelihd (25) gives much less precise esimaes f han d mehds ha use infrmain abu he censring imes fr unfailed unis (Kalbfleisch and Lawless 1988; Hu and Lawless 1996b). Thus, he use f (1) esimae wih a parameric mdel wuld be much preferred (25). The same hlds rue fr nnparameric esimain f f(); Kalbfleisch and Lawless (1991) discussed nnparameric esimain based n (25), bu he mehds f his aricle are be preferred. The price, f curse, is ha a gd esimae f he censring ime disribuin G(r) mus be bained. Finally, in cnexs such as manufacuring, ne may smeimes wish make esimaes fr a finie ppulain f unis. The esimaes f f() given here can be used d his. In principle, finie-ppulain crrecins variance esimaes can be made, bu given he large size f he ypical ppulains, i makes n pracical difference if hese are ignred. ACKNOWLEDGMENTS This research was suppred in par by grans he secnd auhr frm General Mrs Canada, he Manufacuring Research Crprain f Onari, and he Naural Sciences and Engineering Research Cuncil f Canada (NSERC), and by a fellwship he firs auhr frm NSERC. We hank K. Adachi, T. Kasashima, J. MacKay, w referees, and an assciae edir fr helpful cmmens. [Received June Revised March 1997.] REFERENCES Ascher, H., and Feingld, H. (1984), Repairable Sysem Reliabiliy, New Yrk: Marcel Dekker. Hu, X. J., and Lawless, J. F. (1996a), "Esimain f Rae and Mean Funcins Frm Truncaed Recurren Even Daa," Jurnal f he American Saisical Assciain, 91, (1996b), "Esimain Frm Truncaed Lifeime Daa Wih Supplemenary Infrmain n Cvariaes and Censring Times," Bimerika, 83, Kalbfleisch, J. D., and Lawless, J. F. (1988), "Esimain f Reliabiliy Frm Field Perfrmance Sudies" (wih discussin), Technmerics, 3, (1991), "Regressin Mdels fr Righ Truncaed Daa Wih Applicains AIDs Incubain Times and Repring Lags," Saisica Sinica, 1, Lawless, J. F. (1982), Saisical Mdels and Mehds fr Lifeime Daa, New Yrk: Wiley. (1995), "The Analysis f Recurren Evens fr Muliple Subjecs," Applied Saisics, 44,

12 NONPARAMETRIC ESTIMATION OF A LIFETIME DISTRIBUTION 13 Lawless, J. F., Hu, X. J., and Ca, J. (1995),"Mehds fr he Esimain f Failure Disribuins and Raes Frm Aumbile Warrany Daa," Lifeime Daa Analysis, 1, Suzuki, K. (1985),"Esimain f Lifeime Parameers Frm Incmplee Field Daa," Technmerics, 27, (1988),"Analysis f Field Lifeime Daa Drawn Frm Differen Observain Perids," Hinshisu (Qualiy), 18, 4-14 (in Japanese). Suzuki, K., and Kasashima, T. (1993), "Esimain f Lifeime Disribuin Frm Incmplee Field Daa Wih Differen Observainal Perids," Technical Repr UEC-CAS-93-2, Universiy f Elecr- Cmmunicains, Japan, Dep. f Cmmunicains and Sysems Engineering. TECHNOMETRICS, FEBRUARY 1998, VOL. 4, NO. 1