Iteatioal Joual of Pefomability Egieeig Vol. 4, No. 3, July 28, pp. 233-24. RAMS Cosultats Pited i Idia The Applicatio of a Maximum Likelihood Appoach to a Acceleated Life Testig with a Udelyig Thee- Paamete Weibull Model DANIEL I. DE SOUZA, JR. Flumiese Fed. Uivesity, Civil Egieeig Dept., Gad. Pogam, Niteói, RJ, Bazil & Noth Flumiese State Uivesity, Idustial Egieeig Dept., Campos, RJ, Bazil. (Received o Novembe 28, 26) Abstact: Acceleated life testig povides timely ifomatio about the life distibutio of poducts, compoets ad mateials. Acceleated testig is attaied by subjectig the poducts o pats to testig coditios much highe tha these poducts o pats ae likely to expeiece ude omal use coditios. The thee-paamete Weibull model is commoly used to epeset the udelyig failue-time distibutio fo electoic poducts, i which the miimum lives of these poducts ae diffeet fom zeo. The stadad maximum likelihood method fo estimatig the paametes of the thee paamete Weibull model ca have poblems sice the egulaity coditios ae ot met (see Muthy et al. []; Blischke [2]; Zaakis ad Kypaisis [3]). I this study, we will develop a acceleated life testig model i which the udelyig samplig distibutio is the theepaamete Weibull model. We will be assumig a liea acceleatio coditio. To estimate the shape, scale ad miimum life of the thee-paamete Weibull model we will use a maximum likelihood appoach fo cesoed failue data. To ovecome the oegulaity poblem esultig fom the above metioed, we will apply a modificatio poposed by Cohe et al. [4]. A example will illustate the applicatio of the poposed acceleated life testig model. Keywods: Acceleated life testig, thee-paamete Weibull model, liea acceleatio, maximum likelihood estimato, acceleated coditios.. Itoductio Acceleatio models allow us to taslate the failue ate obtaied at high stess to what a poduct o sevice is likely to expeiece at much lowe stess, ude omal use coditios. The simplest model assumes a costat (liea) acceleatio effect ove time. Theefoe, if we defie the liea acceleatio facto by AF, we will have: t = AF t a () whee, t is the time to failue ude omal (stadad) stess, ad t a is the time to failue at high stess levels. * Coespodig autho s email: daiel.desouza@hotmail.com 233
234 Daiel I. De Souza, J The cumulative distibutio fuctio at omal testig coditio F (t ) fo a cetai testig time t = t will be give by: P ( T < t ) = F ( t ) = P ( T ) a < t a = F a ( t a ) = t a. AF The: F ( t ) = t a (2) AF The desity fuctio at omal testig coditio f (t ) fo a cetai testig time t will be give by: f ( t ) = d ( ) dt F t = d a ( ) dt F t a = d t t a = f a (3) dt AF AF AF The hazad ate at omal testig coditio h (t ) fo a cetai testig time t will be give by: t f a h ( t ) ( ) = f t AF = AF t = h a (4) R ( t ) t AF AF a AF 2. Thee-Paamete Weibull Distibutio Acceleatio A acceleated life testig appoach i which the udelyig samplig distibutio is the two-paamete Weibull model was addessed befoe by De Souza [5] ad [6]. We will ow develop a acceleated life testig model i which the udelyig samplig distibutio is the thee-paamete Weibull model. A maximum likelihood estimato pocedue will be used to estimate the shape, scale ad miimum life paametes of ou samplig Weibull model. These paametes will be defied as such: = shape paamete ude omal testig coditios; a = shape paamete ude acceleated testig coditios; = scale paamete ude omal testig coditios; a = scale paamete ude acceleated testig coditios; ϕ = miimum life ude omal testig coditios; ϕ a = miimum life ude acceleated testig coditios. The the cumulative distibutio fuctio at acceleated coditio F a (t a ϕ a ) of the thee-paamete Weibull distibutio will be give by: F a ( t a ϕ a ) = t ϕ a exp a a (5) a I geeal, the scale paamete ad the miimum life ca be estimated by usig two diffeet stess levels (tempeatue o cycles o miles, etc.), ad thei atios ca povide the desied value fo the acceleatio facto AF ad AF ϕ. So, we will have: AF = (6) a
The Applicatio of a Maximum Likelihood Appoach to a Acceleated Life Testig with a Udelyig Thee-Paamete Weibull Model 235 Also, AF = AF ϕ = AF. AF ϕ = ϕ ϕa (7) Now usig Equatio (2) with t a ϕ a = (t ϕ a )/AF, we obtai the cumulative distibutio fuctio at omal testig coditio F (t ϕ ) fo a cetai testig time t, ad we will have F ( t ϕ ) = t ϕ a a = t ϕ a exp a AF a AF Fom Equatio (7), we have ϕ a = ϕ /AF. The, we obtai ϕ t F ( t ϕ ) = ϕ a t AF a AF = exp AF (8) a AF Sice a AF= ad a = =, Equatio (8) becomes: ϕ t F ( t ϕ ) = ϕ t AF a AF = exp AF (9) Agai, sice R (t ϕ ) = F (t ϕ ), we will have: ϕ t = l R ( t ) + () ϕ AF Equatio (9) tells us that, ude a liea acceleatio assumptio, if the life distibutio at oe stess level is epeseted by a thee-paamete Weibull model, the life distibutio at ay othe stess level is also epeseted by a thee-paamete Weibull model. The shape paamete emais the same while the acceleated scale paamete ad the acceleated miimum life paamete ae multiplied by the acceleatio facto. The equal shape paamete is a ecessay mathematical cosequece of the othe two assumptios; that is, assumig a liea acceleatio model ad a thee-paamete Weibull samplig distibutio. If diffeet stess levels yield data with vey diffeet shape paametes, the eithe the thee-paamete Weibull samplig distibutio is the wog model fo the data o we do ot have a liea acceleatio coditio. The hazad ate of a thee-paamete Weibull samplig distibutio vaies ude acceleatio. Fo a stess failue ate the hazad fuctio h a (t a ϕ a ) will be give by: h a ( t a ϕ a ) = t a ϕa a a Agai, fom Equatio (7) we have ϕ a = ϕ /AF. The, we obtai:
236 Daiel I. De Souza, J ϕ t h a ( t a ϕ a ) = a AF () a a Now, whe we multiply Equatio (), the hazad ate at acceleated testig coditio h a (t a ϕ a ), by the facto /(AF), we will have as a esult the hazad ate at omal testig coditio h (t ϕ ). The, with t a ϕ a = (t ϕ a )/AF ad ϕ a = ϕ /AF we will have: ϕ t ϕ h ( t ϕ ) = t h AF a a AF = AF, o yet: a A a AF ϕ t h ( t ϕ ) = a AF h = a ( ta ϕa ) (2) ( AF) a a (AF) Thee was a liea chage i the hazad ate at acceleated testig coditio h a (t a ϕ a ). Whe h a (t a ϕ a ) is multiplied by the facto /(AF), we will have as a esult the hazad ate at omal testig coditio h (t ϕ ). Oly whe the samplig populatio is expoetial (the shape paamete is equal to ), will the multiplicatio facto be equal to /AF. 3. Maximum Likelihood Estimatio Fo The Weibull Model Fo Cesoed Type II Data (Failue Cesoed) The maximum likelihood estimato fo the shape, scale ad miimum life paametes of a Weibull samplig distibutio fo cesoed Type II data (failue cesoed) will be give by: L ( ; ; ϕ) = k! f ( t i ) [ F( t )] = k! f ( t i ) [ R( t )] ; t >. With f ( t i ) = ( t i ϕ) ( t ϕ e ) i ad R ( ) = ( t ϕ e ) t, we will have: ( ; ; ϕ) L = k! ( t i ϕ) The log likelihood fuctio L = [ ( )] ( ) e ( t ϕ ) t i ϕ e l L ; ; ϕ will be give by: t L = l ( k) + l( ) l( ) + ( ) l ( t i ϕ) i ϕ t ( ) ϕ = i To fid the value of, ad ϕ that maximizes the log likelihood fuctio, we take the, ad ϕ deivatives ad make them equal to zeo. The, applyig some algeba, we will have:
The Applicatio of a Maximum Likelihood Appoach to a Acceleated Life Testig with a Udelyig Thee-Paamete Weibull Model 237 dl d = dl dϕ ( t ) i ϕ dl = + ( )( t ) + ϕ = (3) d + + t l l t i ϕ i ϕ t i ϕ t l ( ) ϕ = i l = (4) ( ) + ( ) = ( ) ( t ϕ) i + ( ) t ϕ i + ( )( t ϕ) = (5) Fom Equatio ( 3) we obtai: ( ) ( )( ) ti ϕ + t ϕ = (6) Notice that, whe =, Equatio (6) educes to the maximum likelihood estimato fo the two-paamete expoetial distibutio. Usig Equatio (6) fo i Equatios (3) ad (4) ad applyig some algeba, Equatios (4) ad (5) educe to: l + ( t i ϕ) t ϕ i + ( ) ( )( t ϕ) ( t ) ( ) ( ) ( ) i ϕ l ti ϕ + t ϕ l( t ϕ) ( t ) i ϕ + ( )( t ϕ) ( ) ( t ϕ) i + ( ϕ) i + ( )( t ϕ) = (7) t = (8) The poblem was educed to the simultaeous solutio of the two iteative Equatios (7) ad (8). The simultaeous solutio of two iteative equatios ca be see as elatively simple whe compaed to the aduous task of solvig thee simultaeous iteative Equatios (3), (4), (5) as outlied by Hate et al. [7]. Eve though this is the
238 Daiel I. De Souza, J peset case, oe possible simplificatio i solvig fo estimates whe all thee paametes ae ukow could be the appoach poposed by Bai [8]: fo example, usig Bai s appoach let us suppose that ˆ ad ˆ epeset the good liea ubiased estimatos (GLUEs) of the shape paamete ad of the scale paamete fo a fixed value of the miimum life ϕ. We could choose a iitial value fo ϕ to obtai the estimatos ˆ ad ˆ, ad the apply these two values i Equatio (8), that is, the maximum likelihood equatio fo the miimum life ϕ. A estimate ϕ ca the be obtaied fom Equatio (8), the the GLUEs of ad of ca be ecalculated fo the ew estimate ϕ, ad a secod estimate fo the miimum life ϕ obtaied fom Equatio (8). Cotiuig this iteatio would lead to appoximate values of the maximum likelihood estimatos. As we ca otice, the advatage of usig the GLUEs i this iteatio is that oly oe equatio must be solved implicitly. The existece of solutios to the above set of Equatios (7) ad (8) has bee fequetly addessed by eseaches as thee ca be moe tha oe solutio o oe at all; see Zaakis [3]. The stadad maximum likelihood method fo estimatig the paametes of the thee paamete Weibull model ca have poblems sice the egulaity coditios ae ot met (see Muty et al. []; Blischke [2]; Zaakis ad Kypaisis [3]). To ovecome this egulaity poblem, oe of the appoaches poposed by Cohe [4] is to eplace Equatio (8) with the equatio E ( ϕ) = ϕ = t Γ + (9) Hee, t is the fist ode statistic i a sample of size. I solvig the maximum likelihood equatios, we will use this appoach poposed by Cohe [4]. Appedix () shows the deivatio of Equatio (9). 4. Example We ae tyig to veify if, fo a thee-paamete Weibull samplig distibutio, a compoet, opeatig ude pedetemied (coect) levels of iceased stess, will have exactly the same failue mechaism as obseved whe used at omal stess levels. I ode to do so, a cetai type of electoic pat was subjected to a acceleated life test, whee 2 of such pats wee cycled with the testig beig tucated at the momet of occuece of the ith failue. Table () below shows the failue time data (hous) obtaied fom the life testig ude acceleated coditios: Table : Failue Times (hous) of Electoic Pats tested ude Acceleated Coditios 493.2 595.4 559.6 46.3 55.3 478.5 597.7 53.2 56. The udelyig samplig distibutio is the thee-paamete Weibull model. Usig the maximum likelihood estimato appoach fo the shape paamete, fo the scale paamete ad fo the miimum life ϕ of the Weibull model fo cesoed Type II data (failue cesoed), we obtai the followig values fo these thee paametes ude acceleated coditios of testig.
The Applicatio of a Maximum Likelihood Appoach to a Acceleated Life Testig with a Udelyig Thee-Paamete Weibull Model 239 a = = = 9.98; a = 495. hous; ϕ a = 86.4 hous A secod sample is obtaied at a omal stess level. Oce agai, 2 electoic pats wee cycled with the testig beig tucated at the momet of occuece of the ith failue. Table 2: Failue Times (hous) of Electoic Pats tested ude Nomal Coditios 4,754.9 5,36.4 5,739. 4,585. 4,43.2 4,955.9 5,78.9 5,382.5 5,86.7 Agai usig the maximum likelihood estimato appoach peseted i this pape, we obtai the followig values fo the shape paamete, fo the scale paamete ad fo the miimum life ϕ of the thee-paamete Weibull samplig distibutio ude omal coditios of testig: = a = = 9.22 = 4,754.7 hous; ϕ = 828. hous Usig Equatio (6), we will have: AF = 4,754.7 = = 9.635 9.6 a 495. ϕ Usig Equatio (7), we will have: AF ϕ = 828. = = 9.5833 9.6 ϕa 86.4 The, as we expected, AF = AF ϕ = AF = 9.6, ad = a = 9.2 As we ca obseve i this example, ude a liea acceleatio assumptio, the electoic pat, opeatig ude pedetemied (coect) levels of iceased stess, has exactly the same failue mechaism as obseved whe used at omal stess levels. That is, sice the life distibutio at oe stess level is epeseted by the thee-paamete Weibull model, the life distibutio at ay othe stess level is also epeseted by a theepaamete Weibull model. As we ca see i this example, the shape paamete value emais the same while the scale paamete ad the miimum life ae multiplied by the acceleatio facto. As we emembe, the equal shape paamete is a ecessay mathematical cosequece to the othe two assumptios; assumig a liea acceleatio model ad a thee-paamete Weibull samplig distibutio. 5. Coclusios Acceleated life testig povides timely ifomatio about the life distibutio of poducts, compoets ad mateials. Acceleated testig is obtaied by subjectig the poducts o pats to testig coditios much highe that these poducts o pats ae likely to expeiece ude omal use coditios. I this study, we developed a acceleated life testig model i which the udelyig samplig distibutio is the thee-paamete Weibull model. The miimum life is cosideed diffeet fom zeo. We assumed a liea acceleatio coditio. A maximum likelihood estimatio pocedue fo cesoed failue data was used to estimate the shape, scale ad miimum life of the thee-paamete Weibull model. I solvig the maximum likelihood equatios, we used the appoach poposed by Cohe [4]. As we ca veify i the peseted example, the shape paamete value emaied the same while the scale paamete ad the miimum life paamete ae multiplied by the acceleatio facto. As we would expect, the acceleatio facto fo the scale paamete
24 Daiel I. De Souza, J is equal to the acceleated facto fo the miimum life ϕ. The equal shape paamete is a ecessay mathematical cosequece to the othe two assumptios; assumig a liea acceleatio model ad a thee-paamete Weibull samplig distibutio. Refeeces [] Muthy, D. N. P., M. Xie ad R. Hag, R, Weibull Models, Wiley Seies i Pobability ad Statistics, Joh Wiley & Sos Ic., New Jesey, USA, 24. [2] Blischke, W. R., O o-egula estimatio II. Estimatio of the Locatio Paamete of the Gamma ad Weibull Distibutios, Com. i Statistics., U.S.A., No. 3, pp. 9-29, 974. [3] Zaakis, S. H. ad J. Kypaisis,, A Review of Maximum Likelihood Estimatio Methods fo the Thee Paamete Weibull Distibutio, Joual of Statistical Computatio ad Simulatio, U.S.A., No. 25, pp. 53-73, 986. [4] Cohe, A. C., B. J. Whitte ad Y. Dig, Modified Momet Estimatio fo the Thee-Paamete Weibull Distibutio, Joual of Quality Techology, U.S.A., No. 6, pp.59-67, 964 [5] De Souza Daiel I., Acceleated Life Testig Models, Poceedigs of the ORSNZ99 Cofeece, Hamilto, Uivesity of Waikato, NZ, Vol., pp. 245-254, 999. [6] De Souza, Daiel I., Physical Acceleatio Life Models, Aais del XIII Cogeso Chileo de Igeieia Electica., Satiago: Uivesidad de Satiago de Chile, Chile, Vol., pp. 9-4, 999. [7] Hate, H. et al., Maximum Likelihood Estimatio of the Paametes of Gamma ad Weibull Populatios fom Complete ad fom Cesoed Samples, Techometics, U.S.A., No. 7, pp. 639-643; eatum, 5 (973), pp. 43, 965. [8] Bai, Lee J., Statistical Aalysis of Reliability ad Life-Testig Models, Theoy ad Methods, Macel Dekke Ic., New Yok, N.Y., U.S.A., 978. Daiel I. De Souza J. is a Pofesso of Life Testig at the Flumiese Fedeal Uivesity i Niteói, Bazil, whee he has taught fo the last 35 yeas. I this capacity he seved as the Gaduate Coodiato of the Civil Egieeig Depatmet ad as Reseach Diecto of the School of Egieeig. He is also a visitig full pofesso at the Noth Flumiese State Uivesity i Campos, Bazil, whee he teaches ad seved as the Gaduate Coodiato of the Poductio ad Civil Egieeig Depatmet. He has bee thee times at Uivesity of Floida, Gaiesville, FL, USA, as a eseach schola, whee he taught each time the couse Idustial Quality Cotol ad wote seveal techical aticles ad a Idustial Quality Cotol wokbook. He also did some eseach at Pesylvaia State Uivesity, USA. He eceived his B.S. i Idustial Metallugical Egieeig fom Flumiese Fedeal Uivesity i Bazil, a M.S. i Opeatios Reseach fom Floida Istitute of Techology, USA, ad a Ph.D. i Idustial Egieeig fom Waye State Uivesity, MI, USA. His eseach iteests iclude life testig ad Weibull ad Ivese Weibull eliability estimatio. His publicatios have appeaed i IIE Tasactios, ASQC Tasactios, Elsevie Sciece Poceedigs, Balkema Poceedigs, Comadem Pocedigs (UK), ESREL Poceedigs ad i seveal Bazilia jouals. Appedix - Detemiig a Iitial Estimate to the Miimum Life ϕ The pdf of t, the fist failue time, will be give by
The Applicatio of a Maximum Likelihood Appoach to a Acceleated Life Testig with a Udelyig Thee-Paamete Weibull Model 24 f ( t ) = [ F( t )] f ( t ), o, sice F ( t ) = R ( t ) f ( t ) = [ R( t )] Fo the thee-paamete Weibull samplig distibutio, we will have: f ( t ) t f ( t ) = exp ϕ The expected value of t is give by E ( t ) = t t ϕ exp ϕ d t, we will have: du Makig U = ; d u = d t ; d t = ; t = U + ϕ t ϕ As t ; U ; Now, as t ϕ; U. The: E ( t ) = ( U + ϕ) whee ϕ U e d u = U U e d u = [ ] ϕ U U e d u + ϕ e = ϕ[ ] = ϕ U e d u ; U I solvig the itegal U e d z Z d u, let Z= U ; d u = ; U = As U ; Z ; Now, as U ; Z. The: E ( t ) = Z z e z d z + ϕ = Z e d z + ϕ. Fially, E ( t ) = t = Γ + + ϕ The expected value of t is give by E ( t ) = Γ + + ϕ, which idicates that ϕ ca be estimated by E ( ϕ) = ϕ = t Γ + (9)