Mathematical Modeling of Optimum 3 Step Stress Accelerated Life Testing for Generalized Pareto Distribution

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1 America Joural of Theoretical ad Applied Statistics 05; 4(: 6-69 Published olie May 8, 05 ( doi: 0.648/j.ajtas ISSN: (Prit; ISSN: (Olie Mathematical Modelig of Optimum Step Stress Accelerated Life Testig for Geeralized Pareto Distributio Sadia Awar *, Saa Shahab, Arif Ul Islam Departmet of Statistics & Operatios Research, Aligarh Muslim Uiversity, Aligarh, U.P, Idia address: (S. Awar, (S. Shahab To cite this article: Sadia Awar, Saa Shahab, Arif Ul Islam. Mathematical Modelig of Optimum Step Stress Accelerated Life Testig for Geeralized Pareto Distributio. America Joural of Theoretical ad Applied Statistics. Vol. 4, No., 05, pp doi: 0.648/j.ajtas Abstract: This article cotais the optimum step stress accelerated life test uder cumulative exposure model. The lifetimes of test uits are assumed to follow a geeralized Pareto distributio. The scale parameter of the used failure time distributio at the costat stress level is assumed to have a log-liear ad quadratic relatioship with the stress. A compariso betwee liear pla ad quadratic pla by maximum likelihood estimators for the differet sample sizes is show i the table. th The optimum test plas is obtaied by miimizig the asymptotic variace of the maximum likelihood estimator of the 00 P percetile of the lifetime distributio at ormal stress coditio for the model parameters. Tables of optimum times of chagig stress level for both plas are also obtaied. Keywords: Accelerated Life Testig, Geeralized Pareto Distributio, Asymptotic Variace, Maximum Likelihood Estimate, Life Stress Relatioship (Liear ad Quadratic, Cumulative Exposure Model. Itroductio Due to the fast developmet i the techology, the maufacturig pla is cotiuously improvig. For this reaso it is difficult to obtai iformatio about lifetime of products or materials with high reliability at the time of testig uder ormal coditios because testig uder ormal operatig coditios require a very log period of time ad eed a extesive umber of uits uder test. So it is geerally very expesive ad impractical to complete reliability testig uder ormal coditios. Hece, to hadle these problems, the study of accelerated life test (ALT has bee developed. ALT makes it possible to quickly obtai iformatio o the life distributio of products by iducig early failure with higher tha ormal stress. There are maily three types of life test methods i accelerated life testig desig. The first method is costat stress ALT, secod is step-stress ALT ad the third is Progressive stress ALT. I step stress accelerated life testig, the test items are subjected to successively higher levels of stress at pre-assiged test times. The level of stress is icreased step by step util all items have failed or the test stops for other reasos. There are usually two types of step stress accelerated life test (SSALT plas, a simple SSALT ad a multiple-step SSALT. I the simple SSALT there is a sigle chage of stress durig the test. Nelso (990 preseted the step stress scheme with cumulative exposure model. Miller ad Nelso itroduced (98 a simple step-stress accelerated life test (SSALT pla i a expoetial cumulative exposure (CE model. Bai et al. (989 preseted optimum simple stepstress accelerated life tests with cesorig. Xiog (999 has studied a expoetial CE model with a threshold parameter i the simple SSALT. Lu et al. (00 cosidered the Weibull CE model with the iverse power law i the simple SSALT. I the multiple-step SSALT chages of stress are more tha oce. Khamis ad Higgis (996 itroduced first the optimum three-step SSALT pla usig quadratic stress-life relatioship assumig that the failure time follows a expoetial distributio usig. Khamis (997 exteded the -step stress work for m-step stress ALT with k stress variables usig the expoetial distributio, assumig complete kowledge of the stress life relatio with multiple stress variables. Khamis ad Higgis (998 proposed a ew model for SSALT as a alterative to the CEM, which is based o a time trasformatio of the expoetial CEM. McSorley et al. (00 has itroduced the performace of parameter-estimates i step stress accelerated life-tests with various sample-sizes. Fard ad Chehua (009 studied the optimum step-stress accelerated life test desig for reliability predictio usig Khamis Higgis (K-H model. Hut ad Xu (0 derived the optimum stress-chagig time for the

2 64 Sadia Awar et al.: Mathematical Modelig of Optimum Step Stress Accelerated Life Testig for Geeralized Pareto Distributio geeralized K-H model by assumig that the lifetime of a test uit follows a Weibull distributio for type-i cesored data. Alhadeed ad Yag (005 cosidered a simple SSALT pla for optimal time of chagig stress level usig the Log- Normal distributio.. The Model ad Test Procedure.. Geeralized Pareto (GP Distributio The geeralized Pareto (GP distributio is also kow as the Pareto of secod kid with two parameters or Lomax distributio. This distributio has bee widely used i the field of reliability modelig ad life testig. Bader Al- Zahrai (0 preseted the maximum likelihood estimatio for geeralized Pareto distributio uder progressive cesorig with biomial removals. The life times of the test items is said to have the geeralized Pareto distributio if it has the probability desity fuctio (PDF ( ( ( α f y = α y y > 0, α, > 0 ( where is the scale parameter ad αis the shape parameter of the distributio. Now, the cumulative distributio fuctio is give by ( ( α F y = y y > 0, α, > 0 ( The survival fuctio of the GP distributio is give by ( ( α F y = y y > 0, α, > 0 ( Ad the correspodig hazard rate is give by ( h y = α ( y.. SSALT with Cumulative Exposure Model Accordig to the cumulative exposure model, the CDF i SSALT for -step is give by ( ( ( τ F t 0 t < τ F( t = F t τ S τ t < τ F t S τ t < with S the solutio of F ( S F ( τ of F ( S F ( τ τ S (4 = ad S the solutio =. O solvig we get S τ = τ τ τ = ad S ( Hece Geeralized cumulative exposure model for step stress is as follow ( t α 0 t < τ α F ( t = ( t τ τ τ t < τ α ( t τ ( τ τ τ τ t < The correspodig PDF for the cumulative exposure model is give as above by equatio (5 α 0 t < τ ( α f ( t = α ( t τ τ τ t < τ ( α α ( t τ ( τ τ τ τ t < ( ( α t.. Assumptios. Testig is performed at the three stress levels S, S ad S, where S > S > S.. The lifetime of the test uits follow a geeralized Pareto distributio uder ay stress.. The parameter α is idepedet of time ad stress. 4. The scale parameter i at stress level i, i =,, is a log liear fuctio of stress give by (i or (ii (5 log i = a bs i (i log i a bs i cs i = (ii where, a, b ad c are ukow parameters o the ature of the product ad the test method. A radom sample of idetical uits are iitially placed o low stress S ad ru util pre-specified time τ whe the stress is chaged to high stress S for those remaiig uits that have ot failed. The test is cotiued util pre-specified time τ whe stress is chaged to S, ad cotiued util all remaiig uits fail. Likelihood Fuctio I order to obtai the MLE of the model parameters, lett ij, i=,,,,,, i be the observed failure time of a test uit j uder the stress level i, where i deotes the umber of uits failed at the stress level i. The likelihood fuctio is give by ( ( ( α { ( } ( α L t, α,,, α tj α tj τ τ = { ( ( } ( α α tj τ τ τ τ (6

3 America Joural of Theoretical ad Applied Statistics 05; 4(: Log likelihood fuctio of equatio (6 ca be writte as l = logα log log log ( α log ( t j log ( tj τ τ log ( tj τ ( τ τ τ 4. Optimum Liear Step-Stress Test Usig the relatio log i = a bsi, i =,,, the log likelihood fuctio becomes ( ( ( l = logα a bs a bs a bs a bs a bs a bs ( α log ( t je log ( tj τ e τe a bs a bs a bs log ( tj τ e ( τ τ e τe (7 l l MLEs of a adbare obtaied by solvig the equatios = 0 ad = 0. a tj ( tj τ τ ( tj τ ( τ τ τ j ( j j ( j (8 = = τ τ ( j τ ( τ τ τ = ( α a t t t l = S S S St j S ( tj τ Sτ S ( tj τ S ( τ τ Sτ ( α j ( tj j ( tj τ = = τ ( tj τ ( τ τ τ (9 t j ( tj τ τ ( tj τ ( τ τ τ (0 ( j ( j τ τ ( j τ ( τ τ τ l = ( α a t t t ( j τ τ ( τ ( j τ ( j l St j S t S S S S S t = ( α ( tj t τ τ ( j τ ( τ τ τ ( τ ( j τ ( S S SS τ ( τ τ ( S S SS ( tj τ ( τ τ ( tj τ ( τ τ τ S t S S S S S S t ( St j S ( tj τ Sτ S ( tj τ S ( τ τ Sτ ( ( j ( j τ τ ( j τ ( τ τ τ l = ( α a t t t The Fisher iformatio matrix is obtaied by takig the egative secod partial derivatives of the log-likelihood fuctio. Fisher Iformatio matrix is give by

4 66 Sadia Awar et al.: Mathematical Modelig of Optimum Step Stress Accelerated Life Testig for Geeralized Pareto Distributio The log of the 00 th l l a a F = l l a the desig stress S0is give by p percetile of the lifetime ( ( ˆ P α ξ ( S0 = log ( tp ( S0 = a bs0 log i t S at p 0 Where ( S ( S ( S ˆ ξ ˆ ˆ 0 ξ 0 ξ 0 H = aˆ ˆ ˆ α ad F is the asymptotic fisher iformatio matrix. Therefore, the optimal times τ ad τ to chage stress levels uder differet values ofstresses ad sample sizes will be obtaied umerically usig equatio ( which AV ˆ ξ S. ( miimizes ( 0 5. Optimum Quadratic Step-Stress Test For the quadratic model, usig the log liear relatioship The Asymptotic variace is give by ( ( ( ˆ ξ ( = log p ( AV S AV t S 0 0 ˆ ( P AV aˆ α = bs0 log HF ˆ = H i ( i i i log = a bs cs, i =,,. (4 For this quadratic model, the testig ca be doe usig a - step, step-stress ALT. Thus, the log likelihood fuctio (7 ca be exteded to the quadratic model after substitutig the value of ifrom equatio (4 give by ( ( l = logα a bs cs a bs cs a bs cs ( a bs cs ( α log tje ( j τ log a bs cs a bs cs t e τe a bs cs ( j τ ( τ τ log t e e a bs cs l MLEs of a, b ad c are obtaied by solvig the equatios = 0, = 0 ad = 0. a c a bs cs τ e t j ( t j τ τ ( tj τ ( τ τ τ (6 ( j ( j τ τ ( j τ ( τ τ τ = ( α a t t t } (5 l = S S S St j S ( t j τ Sτ S ( tj τ S ( τ τ Sτ ( α j ( t j = ( tj τ τ ( tj τ ( τ τ τ (7 l = S S S c St j S ( tj τ S τ S ( tj τ S ( τ τ S τ ( α j ( tj j ( tj τ = = τ ( tj τ ( τ τ τ (8

5 America Joural of Theoretical ad Applied Statistics 05; 4(: t j ( tj τ τ ( tj τ ( τ τ τ (9 ( j ( j τ τ ( j τ ( τ τ τ l = ( α a t t t ( j τ τ ( τ ( j τ ( j l St j S t S S S S S t = ( α ( tj t τ τ ( j τ ( τ τ τ ( τ ( j τ ( S S SS τ ( τ τ ( S S SS ( tj τ ( τ τ ( tj τ ( τ τ τ S t S S S S S S t (0 ( j τ τ ( τ ( j τ ( j 4 l St j S t S S S S S t = ( α c ( t j t τ τ ( j τ ( τ τ τ ( τ ( j τ ( S S S S τ ( τ τ ( S S SS ( tj τ ( τ τ ( tj τ ( τ τ τ S t S S S S S S t ( St j S ( tj τ Sτ S ( tj τ S ( τ τ Sτ ( ( j ( j τ τ ( j τ ( τ τ τ l = ( α a t t t St j S ( tj τ S τ S ( tj τ S ( τ τ S τ ( ( j ( j ( j ( l = ( α α c t t τ τ t τ τ τ τ ( j τ τ ( τ ( j τ ( j l St j S t S S S S S S S t = ( α c ( tj t τ τ ( j τ ( τ τ τ ( τ ( j τ ( S S S S SS τ ( τ τ ( S S SS SS ( tj τ ( τ τ ( tj τ ( τ τ τ S t S S S S S S S S t (4 Fisher Iformatio matrix is give by l l l a a a c l l l F = a c l l l c a c b c Here the Optimum criterio is to obtai optimum stress chage timeτ ad τ.the log of the 00P th percetile of the lifetime ( t S at the desig stress S 0 is give as p 0 ˆ ξ ( S0 = log ( tp ( S0 = a bs0 cs0 log α i ( P The asymptotic variace at the desig stress S 0 is the give by

6 68 Sadia Awar et al.: Mathematical Modelig of Optimum Step Stress Accelerated Life Testig for Geeralized Pareto Distributio ( ˆ ξ ( 0 AV S = HF H (5 ˆ ξ ( S0 ˆ ξ ( S0 ˆ ξ ( S0 ˆ ξ ( S0 Where H = ad aˆ ˆ cˆ ˆ α F is the fisher iformatio matrix Therefore, the optimal times τad τ will be obtaied umerically usig equatio (5 which miimizes AV ˆ ξ S. ( ( 0 Table. The Maximum likelihood estimate ad Mea Absolute Error for α= Parameter Quadratic case Liear case Estimate MAE Estimate MAE ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ e ˆ ˆ Simulatio Study A umerical study was coducted i order to ivestigate the existece of the optimal stress chage poits ad to evaluate them as a fuctio of varyig parameters. Simulatios are performed to ivestigate the performaces of the MLEs through their mea absolute error (MAE for both relatioships. Compariso betwee both plas is show by calculatig efficiecies. Table preset the Maximum likelihood estimates for =0, 60, 80, 00 ad 0 ad their respective Mea absolute Error for both Quadratic ad Liear relatioship. Table presets the results of optimal desig of step-stress ALT for differet sized samples ad fially Table Compoud Liear Test-Pla Efficiecies. Table. The results of optimal desig of step-stress ALT for differet sized sample. a=0.69 b=.5 a=0.69 b=.5 c=.8 AV τ τ AV τ τ e Table. Compoud Liear Test Pla Efficiecies Efficiecies S S S Opt Liear Opt Quadratic Coclusio The preset article deals with parameter estimatio of geeralized Paretodistributio uder step stress ALT pla. The objective is to pla a test that achieves the best reliability estimates. Two types of relatioship are assumed betwee scale parameter ad Stress. Oe liks scale parameter liearly with stress while other have quadratic relatioship. Compariso betwee both is show by calculatig estimates ad their respective error. Efficiecies for both plas are calculated for differet level of stress. Apart from that the results of optimal desig of step-stress ALT for differet sample size is show. The performace of step-stress testig plas ad model assumptios are geerally evaluated by the properties of the maximum likelihood estimates of model parameters. Estimates of quadratic are more stable with relatively small Mea absolute error as sample size icreases. Maximum likelihood estimators are cosistet ad asymptotically ormally distributed. I short, it may be cocluded that the preset step stress ALT pla works well

7 America Joural of Theoretical ad Applied Statistics 05; 4(: ad has a good choice to be cosidered i the field of accelerated life testig. Refereces [] Alhadeed, A. A. ad Yag, S. S. (005. Optimal Simple Step- Stress Pla for Cumulative Exposure Model Usig Log- Normal Distributio. IEEE Trasactio o Reliability, 54(, [] Bai, D. S., Kim, M.S. ad Lee, S. H. (989. Optimum Simple Step-Stress Accelerated Life Tests with Cesorig. IEEE Trasactios o Reliability, 8(5, [] Bader-Al Zahrai (0. Maximum Likelihood Estimatio for Geeralized Pareto Distributio uder Progressive Cesorig with Biomial Removals. Ope Joural of Statistics, Scietific Research,, [4] Fard, N. ad Li, C. (009. Optimal Simple Step Stress Accelerated Life Test Desig for Reliability Predictio, Joural of Statistical Plaig ad Iferece, 9(5, [5] Hut, S. ad Xu, X. (0. Optimum Desig for Accelerated Life Testig with Simple Step- Stress Plas. Iteratioal Joural of per formability Egieerig, 8(5, [6] Khamis, I. H. ad Higgis J. J. (996. Optimum -Step Step- Stress Tests. IEEE Trasactios o Reliability, 45(, [7] Khamis, I. H. (997. Optimum M-Step, Step-Stress Desig with K Stress Variables. Commuicatiosi Statistics, Simulatio ad Computatio, 6(4, 0-. [8] Khamis I.H., ad Higgis J.J. (998., A ew model for stepstress testig, IEEE Trasactios o Reliability, 47, [9] Lu, M.W. & Rudy, R.J. (00. Step-stress accelerated test, Iteratioal Joural of Materials & Product Techology, 7 (5-6, [0] McSorley, E.O., Lu, J.C. & Li, C.S. (00. Performace of parameter-estimates i step stress accelerated life-tests with various sample-sizes, IEEE Trasactios o Reliability, 5, (, [] Miller, R., ad Nelso, W. B. (98. Optimum Simple Step- Stress Plas for Accelerated Life Testig. IEEE Trasactios o Reliability, R-(, [] Nelso, W. (990, Accelerated Testig, Statistical Models, Test Plas ad Data Aalysis. Joh Wiley & Sos. [] Xiog, C. (999 Step stress model with threshold parameter, Joural of Statistical Computatio ad Simulatio, 6,