STEP-STRESS ACCELERATED LIFE TESTS - A PROPORTIONAL HAZARDS BASED NONPARAMETRIC MODEL. Cheng-Hung Hu

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1 Page of 34 STEP-STRESS ACCELERATED LIFE TESTS - A PROPORTIONAL HAZARDS BASED NONPARAMETRIC MODEL Cheng-Hung Hu hu0@purdue.edu; Krannert Graduate School of Management, Purdue Unversty, 403 West State Street, West Lafayette, IN Robert D. Plante pbob@purdue.edu; Krannert Graduate School of Management, Purdue Unversty, 403 West State Street, West Lafayette, IN Jen Tang jtang@purdue.edu; Krannert Graduate School of Management, Purdue Unversty, 403 West State Street, West Lafayette, IN Correspondence to: Jen Tang, jtang@purdue.edu

2 Page of 34 ABSTRACT Usng data from a smple Step-Stress Accelerated Lfe Testng (SSALT), a nonparametrc Proportonal Hazards (PH) model s proposed for obtanng Upper Confdence Bounds (UCBs) for the cumulatve falure probablty of a product under normal use condtons. The approach s nonparametrc n the sense that most of the functons nvolved n the model do not assume any specfc forms, except for certan verfable condtons. Test statstcs are ntroduced to verfy assumptons about the model and to test the Goodness of Ft (GOF) of the proposed model to the data. A numercal example, usng data smulated from the lfetme dstrbuton of an exstng parametrc study on Metal-Oxde-Semconductor capactors, s used to llustrate the proposed methods. Dscussons on how to determnng the optmal stress levels and sample sze are also gven. Keywords: Lfetme dstrbuton; Accelerated lfe test; Step-stress tests; Nonparametrc approach; Proportonal hazards model; Cumulatve exposure; Goodness-of-ft test; Optmal desgn.

3 Page 3 of 34. INTRODUCTION The lfetme dstrbuton of a product s often used to assess the cumulatve lkelhood of product falure up to a certan tme. Such nformaton s mportant to managers when, for example, establshng a warranty perod or when prcng extended warrantes. To obtan an estmate of the lfetme dstrbuton of a product n a tmely manner, Accelerated Lfe Testng (ALT) s wdely used n ndustry. In an ALT, test unts are exposed to relatvely greater stress wth the objectve of ncreasng the lkelhood of observng falures. Inferences about the true lfetme dstrbuton under normal use condtons can then be made based on these observed accelerated falure tmes. Dfferent stress strateges are used when conductng an ALT. For example, n a Parallel Constant-Stress ALT (PCSALT), test unts are dvded nto groups and each group s tested under a dfferent stress level throughout the test. Another strategy s the Step-Stress ALT (SSALT). In a smple SSALT, the test begns by exposng all test unts to a pre-specfed stress level. For those test unts that survve up to a pre-specfed tme, the stress level s changed and held constant untl all unts fal (uncensored test) or untl a pre-specfed test termnaton tme s attaned (censored test). SSALTs are commonly used; for example, n testng dodes, electrcal cable nsulaton, nsulatng flud (Ba and Chun (99)), rear suspenson aft lateral lnk (Lu, Leman, Rudy, and Lee (006)), etc. Possble advantages of usng SSALTs nclude: ()Tme savng: A SSALT can substantally shorten the duraton of the test wthout affectng the accuracy of lfetme dstrbuton estmates (Zhao and Elsayed (005)).

4 Page 4 of 34 () Practcal: Relatve to a PCSALT, a SSALT s more practcal n the sense that fewer test unts are requred (Tseng and Wen (000)). () Adaptable: A SSALT s a flexble test strategy, especally for new products when one presumably has lttle nformaton regardng approprate test stresses. In a SSALT, the later stress levels, as well as the transton tmes, could be dynamcally adjusted as falure nformaton s beng gathered over tme. (v) Economcal: Testng unts at hgher stress levels often ncurs greater costs. Therefore, testng all unts at a lower stress level at the begnnng of a SSALT may lower the expected cost. For example, Tang, Yang, and Xe (004) proposed an optmal step-stress accelerated degradaton plan that requres fewer samples and less test tme, and hence reduces the total test cost, than an exstng degradaton test. Khams (997) studed the SSALT and PCSALT, assumng a Webull lfetme dstrbuton, and concluded that SSALT should be consdered as an alternatve to the PCSALT when there s heavy censorng for the PCSALT at lower stress levels. Ma and Meeker (008) concluded that a SSALT can have a smaller varance for the Maxmum Lkelhood Estmates (MLEs) of the quantles of a lfetme dstrbuton than the PCSALT when the scale parameter of the assumed lfetme dstrbuton (Webull or lognormal) s large or when the falure probablty at the censored tme under hgh stress levels s relatvely small... Parametrc and Nonparametrc Models

5 Page 5 of 34 In a parametrc analyss for a SSALT, one assumes a true lfetme dstrbuton (e.g., exponental, Webull, or lognormal) wth certan unknown parameters and an acceleraton functon to relate these parameters to stress levels (Zhao and Elsayed (005)). Other assumptons such as Nelson s (980) Cumulatve Exposure (CE) assumpton (beng descrbed later) are used for dervng the accelerated lfetme dstrbuton for the data obtaned from a SSALT, to obtan sample estmates of the true lfetme parameters. Tang, Sun, Goh, and Ong (996) proposed a lnear CE model and obtaned the MLEs of the lfetme parameters n a mult-censored SSALT. Balakrshnan and Han (008) and Han and Balakrshnan (00) consdered an exponental SSALT wth competng rsks. For more recent studes regardng parametrc nferental methods of SSALT, see, for example, Balakrshnan, Zhang, and Xe (009), Balakrshnan, Beutner, and Kater (009), Wang (00), and Lee and Pan (00). As to optmally desgnng a SSALT, Ismal and Aly (009) and Srvastava and Mttal (00) consdered the optmum plan for a partal SSALT. Most of the lterature assumed Nelson s CE assumpton wth dfferent lfetme dstrbutons and censorng schemes. For example, Mller and Nelson (983), Ba and Chun (99), Ba and Km (993), Alhadeed and Yang (005), Elsayed and Zhang (005), and Gouno (007). Snce the parametrc approach s not the man focus of the paper, the reader s referred to Nelson (990, 005a, 005b) for comprehensve overvews on theory of parametrc PCSALT and SSALT. Although specfc parametrc lfetme dstrbutons are used for certan types of products, Nelson (990) ponted out that the assumed form of the lfetme dstrbuton s questonable for many 3

6 Page 6 of 34 products (also see Pascula and Montepedra (003)). Moreover, when assessng the ablty of several potental lfetme models, t can be dffcult to assess whch one provdes a better ft to the data. Sgnfcant errors n the extrapolaton of ALT results could occur f the assumed model does not provde a good approxmaton of the actual falure mechansm. Thus, a nonparametrc model absent specfc assumptons regardng lfetme dstrbutons should be consdered. A few research efforts have consdered the nonparametrc approach for analyzng falure tme data under a PCSALT (Schmoyer (986)). In partcular, for a PCSALT wth two parallel constant-stress levels, Schmoyer (988, 99) suggested a nonparametrc estmaton procedure for obtanng an Upper Confdence Bound (UCB) for the cumulatve falure probablty at any gven pont under normal use condtons. Some nonparametrc approaches for SSALT have been proposed, ncludng Shaked and Sngpurwalla (983) and Tyoskn and Krvolapov (996)... Objectve and Organzaton of the Paper Gven the advantages of SSALT and the lack of a nonparametrc model for data analyss, we propose a nonparametrc approach for estmatng the true lfetme Cumulatve Dstrbuton Functon (CDF) based on data obtaned from a smple step-up SSALT, where the stress level s swtched to a hgh level at certan stress change tme. The paper s organzed as follows. In Secton, a Proportonal Hazards (PH) based nonparametrc model s ntroduced for the underlyng lfetme dstrbuton. The CE assumpton of Nelson (980) s also used, n leu of acceleraton functons common to the parametrc approach, to 4

7 Page 7 of 34 derve the lfetme CDF under a smple SSALT. Usng data from ths CDF, we obtan n Secton 3 a nonparametrc UCB for the lfetme CDF under normal use condtons. Secton 4 provdes methods for verfyng the suffcent condtons n the development of the UCB n Secton 3. Secton 5 develops a procedure for assessng the Goodness of Ft (GOF) of the underlyng PH model to the data. A smulated numercal study utlzng the work of Elsayed and Zhang (005) regardng Metal-Oxde-Semconductor capactors s provded n Secton 6 to both llustrate the mplementaton and to emprcally assess the senstvty of the proposed method to varous stress levels and sample sze. Dscussons on how to determne the optmal stress levels and sample sze are also gven. Fnally, Secton 7 summarzes the results and suggests drectons for future research.. A PH-BASED NONPARAMETRIC MODEL In ths secton, a nonparametrc PH model for the lfetme CDF s ntroduced. Then, wth the CE assumpton, we derve the lfetme CDF under a smple SSALT... The PH Model Consstent wth the work of Schmoyer (99), the lfetme CDF for a gven constant-stress s assumed to follow a general PH model. A PH model assumes that the effect of stress s multplcatve n the hazard rate and that the rato of hazard rates under two dfferent stress levels s constant over tme. That s, f a unt s tested under stress level x throughout the test, the hazard rate functon at any gven tme t s λ ( t; x) = g( x) h ( t), t 0 () 5

8 Page 8 of 34 where g( x) the damage rate functon whch s a non-negatve and non-decreasng sgmod functon of stress defned on [0, ) ; h( t) a non-negatve, non-decreasng, and dfferentable functon of tme. A non-negatve and non-decreasng functon s sgmod f there s a pont M [0, ) to the left of whch the functon s convex and to the rght concave. (The cases M = 0 or M =,.e., concave or convex, are admtted). Several commonly used damage rate functons n parametrc models conform to such assumptons (see examples n Schmoyer (988, 99)). Let x 0 be the normal use stress level and x and x ( x 0 x < x ) be two test stress levels, whch are assumed to be pre-determned and fxed. The hazard functons for unts under these two dfferent test stress levels are proportonal snce λ ( t; x ) λ ( t; x ) = g ( x ) g ( x ) s constant for all t 0. The model n () s nonparametrc because we do not specfy any parametrc forms of the functons g( x) and h( t ), except certan verfable condtons to be descrbed later. It follows from () that the lfetme CDF under a constant-stress x ( =, ) s F ( t) = exp( g( x ) h( t)), t 0. ().. Lfetme CDF Under a Smple SSALT Let τ be the stress change tme (from x to x ) n a smple SSALT. To derve the lfetme CDF under the smple SSALT, a model s requred to relate ths CDF to the two CDFs n () under constant x and x, respectvely. For ths purpose, Nelson s (980) CE assumpton s used. Ths 6

9 Page 9 of 34 assumpton assumes that the remanng lfe of a test unt at any gven tme depends only on the cumulatve exposure t has receved up to that tme and the current stress level, regardless of how the exposure was accumulated. Under the CE assumpton, there exsts a tme, denoted by s ( < τ ), such that the cumulatve amount of exposure at τ under stress level x s equvalent to the CE amount at s as f the ALT had been conducted wth stress level x (see Fgure ). Snce the CEs are equvalent, we have F ( s) = F ( τ ), and hence g( x ) h( s) = g( x ) h( τ ). Therefore, the lfetme CDF of a test unt under a smple SSALT s F SS F ( t) = exp( g( x ) h( t)), f t τ ( t) = F ( t τ + s) = exp( g( x ) h( t τ + s)), f t > τ (3) ============ Fgure ============ 3. UCBs OF LIFETIME CDF UNDER NORMAL CONDITIONS A nonparametrc UCB, based on data obtaned from a smple SSALT, s developed for the cumulatve falure probablty under normal use condtons, x 0. The UCB s derved wth the followng addtonal assumpton on g( ) (a method for verfyng ths assumpton wll be proposed n Secton 4): g( x ) g( x ) 0, or x x 0 g( x ) g( x ) x. (4) x 0 0 Denote Pr( t x ) F ( t) and Pr( t SS) F ( t). Then, usng (4) and the fact that Pr( t SS ) SS = Pr( t x ), for t τ, we can prove 7

10 Page 0 of 34 Proposton. Under model (3) and assumpton (4), the followng nequalty holds: 0 x [ ] 0 x t SS Pr( t x ) Pr( ), t 0. (5) Let ˆPr( t SS ) be the emprcal CDF for F ( t ), a conservatve00( α)% UCB of Pr( t SS ) n SS (5) s (Meeker and Escobar (998)) n n ˆPr( t SS) Pr % ( t SS) = α + ( Pr( ˆ, (6) n t SS) + ) F( α ;n Pr( ˆ t SS ) + ;n n Pr( ˆ t SS )) where n s the total number of test unts, and F ( p ; v ; v ) s the p-th percentle of the F dstrbuton wth ( v, v ) degrees of freedom. Then by (5), % 0 t SS 0.05 provdes a 95% UCB for 0 Pr ( ) x x Pr( t x ). 4. VERIFICATION OF ASSUMPTION (4) A method for verfyng assumpton (4) s provded n Secton 4.. To smplfy the notaton, denote c g( x ) g( x ), whch s a constant snce x and x are fxed. Then c (0,). Estmaton procedures for c are gven n Sectons Verfcaton Procedure and Unmodalty Snce the damage rate functon g ( x) s assumed to be sgmod, the functon g( x) x s unmodal (Schmoyer (988)). Non-ncreasng and non-decreasng functons are consdered unmodal functons (wth nfnte mode). Assume that g( x ) g( x ), or equvalently x x c x, (7) x then g( x) x cannot be non-ncreasng over (0, ). Frst, f g( x) x s non-decreasng, then assumpton (4) s true for 0 < x x 0. On the other hand, f g( x) x has a fnte mode, then x has to be to the left of the mode because of (7). Ths mples that g( x) 8 x s non-decreasng n the

11 Page of 34 nterval [0, x ], and hence assumpton (4) s true for 0 < x x 0. Consequently, f we can verfy (7), then (4) s true. Accordng to (3) and the defnton of s, we have ln( Pr( t SS)) = c, for all t [ s, τ ]. (8) ln( Pr( t s + τ SS)) An UCB of the left hand sde of (8) can be obtaned as the rato of a 00 ( α) / % UCB of Pr( t SS ) to a α lower confdence bound of Pr( t s + τ SS). If ths UCB for the left / 00( ) % hand sde of (8) s smaller than x / x, we conclude that (7) s vald up to a certan level of confdence. Schmoyer (99) used a smlar procedure for a PCSALT. However, unlke Schmoyer (99), the range of useful t for a smple SSALT s lmted to the nterval [ s, τ ], where s as well as c are unknown parameters to be estmated. 4.. Modfed Lkelhood Functon of Parameter c We propose a modfed lkelhood functon for obtanng an estmate of c. Ths lkelhood functon wll also be used later for developng a GOF test of the PH model to the data. Suppose that the parameter s s known or well-estmated. Let t, t,..., t, n be the observed falure tmes (under the stress level x ) wthn the nterval [ s, τ ] and t, t,..., t, n be the observed falure tmes (under the stress level x ) after tme τ. Let t s + t τ, j =,..., n, be the shfted j j falure tmes. Under model (3) and Fgure, { t, t,..., t, n } and { t, t,..., t, n } are consdered as two samples from the two CDFs under x and x, respectvely, over the same nterval [ s, τ ]. Note that the PH model n (3) also holds for any monotone transformaton of the falure tmes n the sense 9

12 Page of 34 that the dstrbutonal form n (3) stays the same. Hence, for the estmaton (and hypothess testng) problem about the model unknown parameters to reman nvarant under these tme transformatons, the estmators (and test statstc) wll depend only on the rankng nformaton of t, t,..., t, n and t, t,..., t, n (see Kalbflesch and Prentce (980, Chapter 4) and Schmoyer (99)). The jont rankng nformaton s characterzed by ( a, a,..., a ) 0, where a s the number of t j s such that n t t < t j, + wth t s 0 and t +, for = 0,,..., n., n Let R exp( g( x ) h( t)), whch s the survval probablty of a test tem at tme t as f the t tem had been exposed to the constant stress level x throughout the test. The condtonal lkelhood functon of c s (dervaton n Appendx) n n L( c a, a,..., a, s, n, n ) ( R c) 0 n a!! n = n s n a! n = k = 0 j j = a + ( n + ) c + k, (9) where R s unknown. However, accordng to the CE assumpton, R =. Thus, the emprcal s s R τ survval probablty, R ˆ τ = Pr( ˆ τ SS) s used as an estmate of R s n (9) to obtan an approxmate condtonal lkelhood functon of c Search Procedure for Parameters s and c It s apparent from equatons (8) and (9) that estmaton of s depends on c and vce versa. In ths secton, a search procedure s suggested for the smultaneous estmaton of these two parameters. The proposed jont estmaton procedure s teratve and based on the followng consderatons: (a) For a gven c, an estmate of s ( [0, τ ] ) s that satsfyng (8); 0

13 Page 3 of 34 (b) If s s gven (estmated), the MLE of c can be obtaned va (9). The proposed search procedure s gven as follows: Step. Assume we have an ntal estmate of c, say c (a method for obtanng a good c s provded later n ths secton). We re-parameterze s such that s = ατ, where α (0, ). Let t be the last falure tme such that ˆPr( t SS ) =, for all t t. We seek an estmate of α (and hence of s) such that the followng average squared error s mnmzed: SV ( α, c ) = t τ τ ατ ατ + ( t τ ) ατ τ ατ ( (, α ) ) τ τ ατ c t c dt, f t ( (, α ) ) τ > τ ατ c t c dt, f t (0) where (cf. (8)) ln( Pr( ˆ t SS)) c( t, α), t [ ατ, mn( ατ + t τ, τ )]. ln( Pr( ˆ t ατ + τ SS)) The ratonale for (0) s related to consderaton (a) above. The mnmzng α value s denoted by α and the correspondng s by s = α τ. Step. Usng s = α τ as an estmate of s, we fnd c to maxmze the lkelhood functon (9). The resultng estmate of c s denoted by c. Step 3. Replace c n Step by c and repeat these steps untl the estmated values of ( c, s ) converge. Intal Estmate of c We propose the followng procedure for obtanng a good ntal estmate of c, c. () The procedure begns by obtanng a reasonable estmate of s wthn the nterval (0, τ ). Recall

14 Page 4 of 34 from (8) that c( t, α ), as a functon of t and α, should be constant, for all t ( s, τ ). In other words, the most reasonable estmated value of s ( = ατ ) would be the pont whch generates the most stable quanttes of c( t, α ) for all t ( ατ, τ ). Let c ( α ) be the average of c( t, α ) over t ( ατ, τ ). The followng modfed crteron of (0) provdes a useful means for obtanng a reasonably good ntal value of s or α : SV ( α, c ( α )) t τ = τ ατ ατ + ( t τ ) ατ τ ατ ( (, ) ( )) c t α c α dt, f t τ τ ατ ( (, ) ( )) c t α c α dt, f t τ > τ ατ () () Substtute the estmated s ατ nto (9) to obtan c for Step of the search process. Note that takng logarthmetc transformatons n (0) and () provdes the followng alternatve crtera, whch may be computatonally more convenent: t τ SV ( α, ln( c ) ) = τ ατ ατ + ( t τ ) ατ τ ατ ( ( (, )) ( )) ln c t α ln c dt, f t τ τ ατ ( ( (, )) ( )) ln c t α ln c dt, f t τ > τ ατ () and t τ SV ( α, ln( c ( α )) = τ ατ ατ + ( t τ ) ατ τ ατ ( ( (, )) ( ( )) ln c t α ln c α dt, f t τ τ ατ ( ( (, )) ( ( )) ln c t α ln c α dt, f t τ > τ ατ (3) 4.4. Modfed Lkelhood Functon of c for Type-I Censored SSALT To obtan data n a tmely manner, censorng s common n lfe testng. Consder a smple SSALT whch s termnated at tme η wth some survvng tems. Our search procedure for c

15 Page 5 of 34 developed for the uncensored case n Secton 4.3 can be drectly appled to the censored SSALT; but the condtonal lkelhood functon for c requres some modfcatons, as follows. For a gven s, two scenaros may occur followng a transformaton of the censored tme from η to η s + ( η τ ). Scenaro I: η τ (see Fgure ) As shown n the fgure, after transformng the falure tmes under x, one could stll count the number of transformed falure tmes wthn each nterval ( t, t ), = 0,,..., n, +, to observe the jont ranks, ( a, a,..., a ). Hence, the condtonal lkelhood functon of c n (9) for the uncensored 0 n SSALT can be used drectly for ths scenaro. Scenaro II: η < τ (see Fgure 3) Frst, let t m be the last observed falure tme under x before η. Under ths scenaro, one would not be able to obtan the exact number of transformed falure tmes fallng wthn each nterval ( t, t ), +, for m. Thus, defne a m as the number of survvng tems plus the number of transformed falure tmes under stress x that exceed t m. Instead of ( a, a,..., a ), one would only 0 n observe ( a, a,..., a ) 0 and the condtonal lkelhood functon of c based on ( a, a,..., a ) 0 s m m 0 m m a!! m n m = s m a! m = k = 0 j j= L( c a, a,..., a, m, n, s) ( R c) 3 ================= Fgures and 3 ================= a + ( m + ) c + k. (4) To estmate parameters c and s under type-i censorng, we replace t wth the censored tme η n

16 Page 6 of 34 Step n Secton 4.3 and use the lkelhood functon (4) nstead of (9) n Step f Scenaro II occurs. 5. GOF Test of PH Model In ths secton we frst derve a GOF Lkelhood Rato Test (LRT) for the assumed PH model when the test s uncensored. We then extend the proposed methodology to accommodate Type-I censorng schemes n Secton GOF for Uncensored SSALT Under the PH assumptons, f we have an uncensored SSALT, then the jont ranks (,, K, ) are observed and the constraned lkelhood functon based on the jont ranks a a a 0 n (,, K, ) s provded n (9). Wthout the PH assumptons, the unconstraned condtonal a a a 0 n lkelhood of (,, K ) s (see Schmoyer (99)) a a a 0 n a n n a a, a,..., a. (5) 0 n = 0 n The LRT statstc for the GOF test of the PH model s then the rato of (9) to (5). The exact null dstrbuton of the LRT statstc, when the jont ranks ( a, a,..., a ) are 0 n consdered random, s dffcult to obtan analytcally. However, va smulaton of ( a, a,..., a ), we 0 n can obtan an observed sgnfcance level (.e. p-value) for the current observed value of the LRT statstc. Snce ( a, a,..., a ) s nvarant under any monotone transformaton of the falure tmes, 0 n the dstrbuton of the LRT statstc s therefore ndependent of h( ) n the PH model and depends on g( x ) and g( x ) only through c = g( x ) / g( x ). However, unlke Schmoyer s (99) approach n whch two constant stress levels are appled n parallel over tme t [0, ), the nterval for 4

17 Page 7 of 34 comparson n a smple SSALT s lmted to t ( s, τ ). Therefore, the Monte Carlo smulaton approach of Schmoyer (99) s revsed for the smple SSALT, as follows. () Generate n pseudorandom falure tmes, t, t,..., t n, from the, cˆ Exp() dstrbuton (where ĉ s the fnal estmate of c obtaned from Secton 4) satsfyng ˆPr( s SS)-th percentle of () cˆ Exp t ˆPr( τ SS)-th percentle of cˆ Exp(). () Generate n pseudorandom falure tmes, t, t,..., t n, from Exp (), condtoned on, ˆPr( τ SS)-th percentle of Exp() t j. () Based on the smulated falure tmes from () and (), the values of ( a, a,..., a ) and the 0 n lkelhood rato are obtaned. Repeat () to () many tmes (000 s suffcent). The sgnfcance level (p-value) for the LRT statstc from the data s then the percentle of the observed value of the lkelhood rato relatve to the smulated lkelhood ratos. Hence, the decson s to reject the null hypothess, H : 0 PH model s adequate, at α = 5% sgnfcance level f p-value #{smulated rato the rato from data} GOF Test for Type-I Censored SSALT For a Type-I censored SSALT, the prevous GOF test procedures n Secton 4.4 requre some mnor modfcatons but only for Scenaro II, because under Scenaro I we stll observe the jont ranks ( a, a,..., a ) here. The modfcatons are: Replace n wth m and τ wth η n (), and 0 n replace ( a, a,..., a ) wth ( a, a,..., a ) 0 n (). 0 n m 5

18 Page 8 of NUMERICAL EXAMPLE 6.. Underlyng Model and Data Generaton Elsayed and Zhang (005) ntroduced an example n whch a smple SSALT plan was desgned for Metal-Oxde-Semconductor capactors to estmate the relablty dstrbuton at desgned temperature, x = 0 50 o C. In ther parametrc model, the lfetme CDF n (3) s F SS β x exp e ( γ t + γ t ), t 0 f τ ( t) = β x exp e ( γ ( t τ + s) + γ ( t τ + s) ), t > τ 0 f x wth β = 3800, γ = 0.000, and γ = Here g( x) = e and 0 β h( t) t ( γ ) t = γ + 0 n our notaton. The optmal smple SSALT plan gven by Elsayed and Zhang (005) s: o = C, x 45 o x C, and 6.5 = 50 τ = hours. Snce s satsfes F ( τ ) = F ( s), we have s = = 0.4τ and c = e e + = Usng these models and parameters, a total of n = 00 falure tmes (assumed n Elsayed and Zhang (005)) are generated from F ( ) to obtan the correspondng SS emprcal CDF, ˆPr( t SS ), needed n our analyss. 6.. Estmatons of Parameters c and s To estmate s and c usng the teratve search procedures developed n Secton 4.3, we need ntal estmates of these two parameters. All results are obtaned by usng Matlab. Intal Parameter Estmates To obtan c for Step of Secton 4.3, crteron (3) was frst used to obtan a reasonable estmate of α (and hence s). The plot of SV ( α, ln( c ( α ))) n (3) aganst α shows the mnmum 6

19 Page 9 of 34 occurs at α = 0.439, and hence s ατ =5.37. Ths value s then substtuted nto (9) to obtan the frst condtonal MLE of c, c = The estmates of ( c, s) n the followng 3 search teratons are: (0.77, 3.38), (0.68, 0.75), and (0.65, 0.75), respectvely. Recall the actual values of s and c are and 0.64, respectvely. Hence, the search procedure provded reasonable estmates of the parameters. The fnal values, (c, s) = (0.65, 0.75), wll be used n the followng numercal analyss. Senstvty Analyss of the Intal Value of c Because of the random nature of data, we chose the followng ntal values of c to assess the senstvty of the search results to c : c = 0.77 ± 0%(0.77), c = 0.77 ± 0%(0.77), c = 0.77 ± 30%(0.77), and c = %(0.77). The resultng convergent estmates of (c, s) are shown n Table. ============= Table ============= From the results n Table, parameter estmates wth a reasonably good ntal estmate for c tend to converge to the same fnal estmates. However, when the ntal c s substantally based, the search procedure can produce based results (the last two cases n Table ). Ths suggests the mportance of an approprate ntal estmate of c GOF Test of PH Model The GOF test of the PH model for F ( t), =,, s conducted followng the procedure proposed n Secton 5, the parameter estmates used to mplement the test are lsted below: 7

20 Page 0 of 34 (c, s) = (0.65, 0.75), n = 09, n = 3, ˆPr( s SS ) = 0.34, and ˆPr( τ SS) = samples are smulated and the correspondng lkelhood ratos are calculated. The resultng p-value of the observed data s 0.477, and hence the PH model assumpton s not rejected at α = 5% sgnfcance level Data Analyss - Verfcaton of (7) Recall that Proposton s used to construct UCBs for the lfetme CDF at dfferent tmes, provded that (4) s true. Further, a suffcent condton for (4) s provded by (7). Two methods are ntroduced for verfyng (7): one s based on Wlks lkelhood rato statstc and the other s based on the upper bound approach developed n Secton 4. Lkelhood Rato Approach (Method ) To assess whether c x / x n (7) s true, we frst use Wlks LRT statstc, defned by R( c) = L( c) / L( cˆ ), where L( c ) s gven n (9), to obtan a 00( α)% UCB of c. Snce ln( R( c)) s approxmately dstrbuted as a Ch-square wth one degree of freedom, an approxmate ( α ) 00 % lkelhood based confdence nterval for c s the set of all c s satsfyng ln( R( c)) χ ( α), (6) from whch an UCB of c can be obtaned. The plot of ln( R( c)) aganst c gve an approxmate 95% confdence nterval (0.0, 0.8) wth an UCB of c at 0.8, whch s less than the rato x x = Condton (7) and hence (4) s verfed wth 95% confdence. Upper Bound Method of Secton 4. (Method ) 8

21 Page of 34 The upper bound method of Secton 4. uses equaton (8). That s, f we can fnd an UCB of the left sde of (8), usng the emprcal CDF, then we have an UCB for c. More specfcally, usng ths method, any t [ s, τ ] = [0.75, 6.5] could be selected and used to establsh an UCB of c. We chose some values of t n ths nterval and all UCBs obtaned were less than x x = 0.58 (fgure s avalable from authors), suggestng that condton (7), and hence (4), s vald UCBs for CDF Under Normal Use Condtons Snce assumpton (4) s vald, the nonparametrc UCB of P( t x ) can now be obtaned from 0 Proposton. Usng (5) and (6) n Secton 4, we obtan nonparametrc UCBs of Pr( t x ) at t = 00, 0 50, 00, 50, 300, and 350 hours. For each t, the true probablty value, the correspondng estmated nonparametrc UCB, and the relatve dfference (bas) between the two are shown n Table. ============ Table ============ 6.6. Expermental Analyss for Optmal Desgn of SSALT We consder the desgn problem of SSALT under our current model settng. In tradtonal parametrc ALT models, optmal stress levels are normally determned by mnmzng asymptotc varances of unbased estmates of (or functons of) parameters of a gven lfetme dstrbuton. However, ths approach s not possble n a nonparametrc settng because the lfetme dstrbuton s not completely specfed. Therefore, mnmzng the Mean Squared Error (MSE) of estmated UCBs of lfetme CDF would provde a more approprate crteron for the selecton of the stress levels. Hence our approach uses a statstcal expermental desgn wth the followng 4 factoral combnatons 9

22 Page of 34 of stress levels ( x, x ) : o o (55 C, 50 C), o o (55 C, 00 C), o o (00 C,50 C), and o o (00 C, 00 C). The low accelerated stress level s set at x = 55 o C, whch s close to the normal level, x = 0 50 o C. We used the same stress change tme as Elsayed and Zhang (005): τ = 6.5 hours. For each stress combnaton, we smulated 00 smple SSALTs, each wth a sample of 00 test unts. The averages and standard errors of the 00 smulated estmated 95% UCBs (Equatons (5) and (6)) at varous t are gven n Table 3. Usng the MSE of our estmated UCB as the objectve functon, we conclude that x should be kept as close to x 0 as possble before the stress change tme τ. After τ, the second stress level, x, should also be kept as low as possble. Ths s true for any t. Further, as a part of the expermental analyss, we had graphed the man and nteracton effects of the frst and second stress level. Results suggest that the nteracton effect between the two stress levels on the MSE s neglgble. ============ Table 3 ============ In addton, we have consdered another desgn varable, namely, the sample sze. A smlar smulaton study wth several dfferent sample szes {50, 00, 00, 300, and 500} was conducted usng o o ( x, x, τ ) = (45 C, 50 C, 6.5). The results are shown n Table 4. Both the mean and standard error of the estmated UCB at a gven tme appear to decrease as the sample sze ncreases. That s, the estmated UCB becomes tghter and more precse as the sample sze ncreases. In addton, before the stress change tme, the standard error at a gven tme follows a typcal nverse square-root type of relatonshp to the sample sze (We also smulated a case wth even lower stress levels, ths 0

23 Page 3 of 34 relatonshp holds). The sample standard error becomes stable once the sample sze reaches 00, whch suggests a mnmum sample sze f one would lke to obtan a more precse (smaller varance) estmated UCB of the lfetme CDF at use condtons. ============ Table 4 ============ 7. SUMMARY In ths paper we propose a nonparametrc PH based model, combned wth Nelson s (980) CE assumpton, for estmatng the UCBs of the lfetme dstrbuton under normal use condtons, va a smple SSALT. Procedures are also developed for testng the model assumptons. A parametrc smple SSALT model was used to llustrate the proposed procedure. The results mply that the performance of the estmated UCBs was senstve to the choces of the stress levels, x and x. As expected, the closer these stress levels are to the normal stress level, the better the bounds performed. Such observatons suggest that the desgn of a smple SSALT has crtcal effects on the nferences and should thereby be consdered wth care. Notce that the proposed procedures are not applcable to another type of SSALT, the step-down SSALT, n whch the stress level decreases over tme. The step-down SSALT has mportant applcatons n practce because t can reveal certan lurkng degradaton/falure modes that are ndependent of the man observed falure mode at hgh stress or less accelerated by the stress varables beng used (hence the CE assumpton s not satsfed). Thus, the development of nonparametrc analytc models for step-down SSALTs should be consdered. Some other areas for future research

24 Page 4 of 34 nclude () extendng ths research from the PH model to the Accelerated Falure Tme (AFT) model, and () the nonparametrc analytcal procedure for products that provdes degradaton data. REFERENCES Alhadeed, A. A. and Yang, S. S. (005) Optmal smple step-stress plan for cumulatve exposure model usng log-normal dstrbuton. IEEE Transacton on Relablty, 54, Ba, D. S. and Chun, Y. R. (99) Optmum smple step-stress accelerated lfe-tests wth competng causes of falure. IEEE Transacton on Relablty, 40, Ba, D. S. and Km, M. S. (993) Optmum smple step-stress accelerated lfe tests for Webull dstrbuton and type I censorng. Naval Research Logstcs, 40, Balakrshnan, N., Beutner, E., and Kater, M. (009) Order Restrcted Inference for Exponental Step-Stress Models. IEEE Transactons on Relablty, 58(), 3-4. Balakrshnan, N. and Han, D. (008) Exact Inference for a smple step-stress model wth competng rsks for falure from exponental dstrbuton under Type-II censorng. Journal of Statstcal Plannng and Inference, 38, Balakrshnan, N., Zhang, L., and Xe, Q. (009) Inference for a smple step-stress model wth type-i censorng and lognormally dstrbuted lfetmes. Communcatons n Statstcs-Theory and Methods, 38, Elsayed, E. A. and Zhang, H. (005) Desgn of optmum smple step-stress accelerated lfe testng

25 Page 5 of 34 plans. Proceedngs of 005 Internatonal Workshop on Recent Advances n Stochastc Operatons Research, Canmore, Canada. Gouno, E. (007) Optmum step-stress for temperature accelerated lfe testng. Qualty and Relablty Engneerng Internatonal, 3, Han, D. and Balakrshnan, N. (00) Inference for a smple step-stress model wth competng rsks for falure from the exponental dstrbuton under tme constrant. Computatonal Statstcs and Data Analyss, 54, Ismal, A. A. and Aly, H. M. (009) Optmal plannng of falure-step stress partally accelerated lfe tests under type-ii censorng. Internatonal Journal of Mathematcal Analyss, 3, Kalbflesch, J. D. and Prentce, R. L. (980) The Statstcal Analyss of Falure Tme Data, Wley, New York, NY. Khams, I. H. (997) Comparson between constant and step-stress tests for Webull models. Internatonal Journal of Qualty and Relablty Management, 4, Lee, J. and Pan, R. (00) Analyzng step-stress accelerated lfe tesng data usng generalzed lnear models. IIE Transactons, 4, Lu, M.W., Leman, J. E., Rudy, R. J., and Lee, Y. L. (006) Step-stress accelerated test. Internatonal Journal of Materals and Product Technology, 5, Ma, H. and Meeker, W. Q. (008) Optmum step-stress accelerated lfe test plans for log-locaton-scale dstrbutons. Naval Research Logstcs, 55,

26 Page 6 of 34 Meeker, W. Q. and Escobar, A. L. (998) Statstcal Methods for Relablty Data, Wley, New York, NY. Mller, R. and Nelson, W. (983) Optmum smple step-stress plans for accelerated lfe testng. IEEE Transacton on Relablty, 3, Nelson, W. (980) Accelerated lfe testng-step stress models and data analyses. IEEE Transactons on Relablty, 9(), Nelson, W. B. (990) Accelerated Testng-Statstcal Models, Test Plans, and Data Analyses, Wley, New York, NY. Nelson, W. B. (005a) A bblography of accelerated test plans. IEEE Transactons on Relablty, 54, Nelson, W. B. (005b) A bblography of accelerated test plans part II-References. IEEE Transactons on Relablty, 54, Pascual, F. G. and Montepedra, G. (003) Model-robust test plans wth applcatons n accelerated lfe testng. Technometrcs, 45, Schmoyer, R. L. (986) A dstrbuton-free analyss for accelerated lfe testng at several levels of a sngle stress. Technometrcs, 8, Schmoyer, R. L. (988) Lnear nterpolaton wth a nonparametrc accelerated falure tme model. Journal of the Amercan Statstcal Assocaton, 83, Schmoyer, R. L. (99) Nonparametrc analyss for two-level sngle-stress accelerated lfe tests. 4

27 Page 7 of 34 Technometrcs, 33, Shaked, M. and Sngpurwalla, N. D. (983) Inference for step-stress accelerated lfe tests. Journal of Statstcal Plannng and Inference, 7, Srvastava, P. W. and Mttal, N. (00) Optmum step-stress partally accelerated lfe tests for the truncated logstc dstrbuton wth censorng. Appled Mathematcal Modellng, 34, Tang, L. C., Sun, Y. S., Goh, T. N., and Ong, H. L. (996) Analyss of step-stress accelerated-lfe-test data: A new approach. IEEE Transactons on Relablty, 45, Tang, L. C., Yang, G. Y., and Xe, M. (004) Plannng of step-stress accelerated degradaton tests. Proceedngs of the Annual Relablty and Mantanablty Symposum, Tseng, S. T. and Wen, Z. C. (000) Step-stress accelerated degradaton analyss for hghly relable products. Journal of Qualty Technology, 3, Tyoskn, O. I. and Krvolapov, S. Y. (996) Nonparametrc model for step-stress accelerated lfe testng. IEEE Transactons on Relablty, 45, Wang, B. X. (00) Interval estmaton for exponental progressve Type-II censored step-stress accelerated lfe-testng. Journal of Statstcal Plannng and Inference, 40, Zhao, W. and Elsayed, E. A. (005) A General accelerated lfe model for step-stress testng. IIE Transactons, 37,

28 Page 8 of 34 Appendx. Dervaton the Condtonal Lkelhood Functon n (9) We frst consder the followng case where, under both stress levels, data are left truncated at s but there s no rght censorng tme for both stress levels. Recall that t, t,..., t n are the observed, g( x ) h( τ ) falure tmes (under the stress level x ) wthn the nterval [ s, τ ] and R = R = e s the s τ survvng probablty under stress x at tme τ. Smlar to Schmoyer (99), we can derve the condtonal lkelhood functon of parameter c = g( x ) g( x ) (condtonal on s, n, and n ), based on observatons on (,,..., ) a a a 0 n. Let { s t t... t }, n Ω = <. Then Pr( a, a,..., a s, n, n ) 0 n = Pr( a, a,..., a s, n, n, t, t,..., t ) df ( t ) Ω 0 n, n = n = n! a0 a a n n F ( t ) F ( s) F ( t ) F ( t ) F ( t ), +,, n n a,..., a R R R Ω 0 n s = s s n = g( x ) h ( t ) e dt g ( x ) h( t ) g ( x ) h( t ) Let v = e, the jont dstrbuton above can be rewrtten as n n n n n a0 a a n c n! c ( R ) ( ) ( ). { 0... s v v v vn v dv a } 0,..., a vn vn v R s n R + s = = Defne w j s by v R w s j j= =, w [0,] for all j, then the functon above becomes j n n n n a j + c( n + ) n n + n a j=! ( s ) ( ) ( ) a0,..., a n R s = 0, n c R w w dw whch can be smplfed to (9). Note that, n a smple SSALT, the falure tmes under x are rght censored at τ. Now, when there s censorng under x but not x, the jont ranks ( a, a,..., a ) 0 n are stll observed and hence, accordng to Schmoyer (99), we could stll use the condtonal lkelhood functon of parameter c as above. 6

29 Page 9 of 34 CDF F (t) F SS (t) F (t) F (τ )=F (s) s τ Tme Fgure. Lfetme CDF under Smple SSALT wth Cumulatve Exposure Assumpton x Tme s t t t 3K t,n x a 0 a a a n s t t t 3 t,n τ η Tme Fgure. Jont Ranks ( a, a, Ka n ) for Scenaro I under Type-I Censorng x Tme s t t t K t 3,n x a 0 a a s t t t 3 t m η t,n τ a m Tme Fgure 3. Jont Ranks ( a, a, Ka ) m for Scenaro II under Type-I Censorng

30 Page 30 of 34 Table. Senstvty Analyss of (c, s) to the Intal Value c c Fnal Estmates 0.77 ( 0%) ( c, s ) = (0.65,0.75) 0.77 (+ 0%) ( c, s ) = (0.65,0.75) 0.77 ( 0%) ( c, s ) = (0.65,0.75) 0.77 (+ 0%) ( c, s ) = (0.65,0.75) 0.77 ( 30%) ( c, s ) = (0.65,0.75) 0.77 (+ 30%) ( c, s ) = (0.,44.38) 0.77 (+ 00%) ( c, s ) = (0.300,68.00) Table. Estmated UCBs of CDFs at Dfferent Tmes Under o o ( x, x ) (45 C, 50 C) =. Stress Levels ( x, x ) Tme True CDF Estmated 95% Upper bound Bas (45 o C,50 o C) Stress change tme s τ = 6.5hrs and n = 00.

31 Page 3 of 34 Table 3. The Averages and Varances of Estmated UCBs of CDFs at Dfferent Tmes Under Dfferent Stress Levels Combnatons. Stress Levels ( x, x ) Tme True CDF (55 o C,50 o C) (55 o C,00 o C) (00 o C,50 o C) (00 o C,50 o C) Estmated 95% Upper bound Average (Standard error) Bas MSE (0.0337) ( ) (0.0430) ( ) ( ) (0.0898) (0.0355) ( ) (0.0398) ( ) (0.0897) (0.0556) (0.0897) (0.047) (0.084) ( ) ( ) (0.089) (0.0333) (0.05) ( ) (0.0096) (0.0654) ( ) Each of combnaton uses 00 replcatons each wth sample sze 00. The stress change tme s τ = 6.5hrs.

32 Page 3 of 34 Table 4. Samplng Statstcs of Estmated 95% UCBs of CDFs at Dfferent Tmes and Dfferent Sample Szes. Sample sze Tme Average of estmated upper bounds Standard error of estmated upper bounds The expermental settngs are o o ( x, x, ) (45 C, 50 C, 6.5) τ =.

33 Page 33 of 34 Bographes Cheng-Hung Hu s a Ph. D degree canddate of the Krannert Management School, Purdue Unversty. He receved hs B.S. and M.S. degrees n Appled Mathematcs and Statstcs, respectvely, from the Natonal Tsng-Hua Unversty, Tawan, R.O.C. He commenced hs research n the feld of relablty engneerng n 003. Hs master thess focused on nonparametrc analyss of step-stress accelerated tests. Hs current research nterest s n accelerated lfe/degradaton testng, partcularly on plannng a test usng step-stress loadngs. Robert Plante James Brooke Henderson Unversty Professor of Management at the Krannert School of Management, where hs research nterests nclude the development of state-of-the-art statstcal qualty control and mprovement models. Hs efforts have focused on the followng classes of problems: () robust product/process desgn; () screenng procedures for process control and mprovement; () statstcal/process/dynamc process control models; and (v) specalzed process mprovement problems. He has more than 50 research publcatons n these areas whch have appeared n numerous journals, ncludng Operatons Research, Management Scence, Decson Scences, Journal of the Amercan Statstcal Assocaton, Internatonal Journal of Producton Research, The Accountng Revew, Audtng: Journal of Practce and Theory, Naval Research Logstcs, IIE Transactons, Technometrcs, Producton and Operatons Management, Computers & Operatons Research, Journal of Qualty Technology, European Journal of Operatonal Research, Manufacturng and Servce Operatons Management, Journal of the Operatonal Research, Informaton Systems and Operatonal Research, and Journal of Busness and Economc Statstcs. Jen Tang s Professor of Management at the Krannert School of Management, where hs current research nterests nclude the areas of appled statstcs, statstcal qualty control, and relablty analyss. He has publshed varous nternal techncal reports/references for Bell Communcatons Research (formerly part of Bell Labs). Hs journal publcatons have appeared n Bometrka, Statstcs and Probablty Letters, The Australan Journal of Statstcs, Internatonal Journal of Producton Research, IIE Transactons, Journal of Statstcal Computaton and Smulaton, Management Scences, Naval Research Logstcs, IEEE Transactons on Relablty, Journal of Qualty Technology, Journal of Amercan Statstcal Assocaton, Manufacturng and Servce Operatons Management, Producton and Operatons Management, European Journal of Operatonal Research, Statstca Snca, and Technometrcs.

34 Page 34 of 34 Manuscrpt # UIIE-996 COPYRIGHT RELEASE My work enttled, " STEP.STRESS ACCELERATED LIF'E TESTS. A PROPORTIONAL HAZARDS BASED NONPARAMETRIC MODEL lhas not been prevously copyrghted. Copyrght, and all rghts assocated wth copyrght, to the above work s hereby hansferred to the Insttute of Industral Engneers, accordng to the provsons of the I 976 revson of the Copyrght Law, Ttle 7 of the Unted States Code. Ttle 7 Ttle l7 of the U.S. Code grants copyrght holders the exclusve rghts to do qnd to authorze any of thefollowng: I. To reproduce the copyrghtedwork n copes or phonorecords;. To prepare dervatveworks based upon the copyrghtedwork; 3. To dstrbute copes ofphonorecords ofthe copyrghted work to the publc by sale or other transfer of ownershp, or by rental, lease, or lendng. 4. In the case of lterary, muscal, dramatc and choreogrøphc worlts, pantommes, and moton pctures and other audovsual worlrs, to perþrm the copyrghtedwork publcly; and 5. In the cqse of lterary, muscal, dramatc and choreographc works, pantommes, and pctoral, graphc or sculptural worlø, ncludng the ndvdual mages of a moton pcture or other audovsual work, to dspløy the copyrghted work publcly. / has been prevously copyrghts. I hereby grant permsson to the Insttute oflndustral Engneers to reprnt the above referenced work n the IIE Transactons. Vwe represent and warrant that Vwe are the sole and exclusve owners of all rghts granted hereunder or that Vwe are authorzed to grant rghts on the owner's behalf. Chens-Hune Hu Name of Copyrght Owner Robert D. Plante Name of Copyrght Owner Jen Tang Name of Copyrght Owner Sgnature (\. _,4* ln ùrgnaíure S- 3o - â"tl Date t-)l-0 l Date f-7 -Àôtl Date One author's sgnature on behalf of all s suffcent. Scan the sgned form, create pdf fle, and upload to Manuscrpt Central wth fnal manuscrpt fles: