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A generic method for modeling accelerated life testing data

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Accelerated life testing (ALT) is widely used to expedite failures of a product in a short time period for predicting the product’s reliability under normal operating conditions. The resulting ALT …

Accelerated life testing (ALT) is widely used to expedite failures of a product in a short time period for predicting the product’s reliability under normal operating conditions. The resulting ALT data are often characterized by a probability distribution, such as Weibull, Lognormal, Gamma distribution, along with a life-stress relationship. However, if the selected failure time distribution is not adequate in describing the ALT data, the resulting reliability prediction would be misleading. In this talk, we provide a generic method for modeling ALT data which will assist engineers in dealing with a variety of failure time distributions. The method uses Erlang-Coxian (EC) distributions, which belong to a particular subset of phase-type (PH) distributions, to approximate the underlying failure time distributions arbitrarily closely. To estimate the parameters of such an EC-based ALT model, two statistical inference approaches are proposed. First, a mathematical programming approach is formulated to simultaneously match the moments of the EC-based ALT model to the ALT data collected at all test stress levels. This approach resolves the feasibility issue of the method of moments. In addition, the maximum likelihood estimation (MLE) approach is proposed to handle ALT data with type-I censoring. Numerical examples are provided to illustrate the capability of the generic method in modeling ALT data.

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  • 1. A Generic Method for Modeling  A Generic Method for Modeling Accelerated Life Testing Data:  Statistical Inference and Applications  ( 种通用的加速寿命实验数据建 (一种通用的加速寿命实验数据建 模方法:统计推断和应用) Dr. Haitao Liao  (廖海涛博士) ©2013 ASQ & Presentation Liao ©2013 ASQ & Presentation Liao Presented live on Jan 19th, 2013http://reliabilitycalendar.org/The_Reliability Calendar/Webinars_‐ y_ /_Chinese/Webinars_‐_Chinese.html
  • 2. ASQ Reliability Division  ASQ Reliability Division Chinese Webinar Series Chinese Webinar Series One of the monthly webinars  One of the monthly webinars on topics of interest to  reliability engineers. To view recorded webinar (available to ASQ Reliability  ( y Division members only) visit asq.org/reliability To sign up for the free and available to anyone live webinars  To sign up for the free and available to anyone live webinars visit reliabilitycalendar.org and select English Webinars to  find links to register for upcoming eventshttp://reliabilitycalendar.org/The_Reliability Calendar/Webinars_‐ y_ /_Chinese/Webinars_‐_Chinese.html
  • 3. A Generic Method for ModelingAccelerated Life Testing Data: Statistical Inference and Applications Dr. Haitao Liao Associate Professor Department of Systems and Industrial Engineering, The University of Arizona, Tucson, AZ 85721
  • 4. 一种通用的加速寿命试验数据建模方 统计推断和应用 法:统计推断和应用 廖海涛 博士 副教授 系统和工业工程系, 亚利桑那大学, 图森, 亚利桑那州
  • 5. Outline• Background and motivation• Erlang-Coxian-Based (EC-based) accelerated life testing (ALT) models• Statistical inference methods• Numerical examples• Conclusions 3
  • 6. Needs for ALT• Technological advances in materials science, design, and manufacturing• Limited time for R&D• Highly reliable products 4
  • 7. Basic Idea X :Failure time Life pdf(Z0) = ? Z0 :Normal operating condition Zi :Accelerated condition pdf(Z1) X X pdf(Z2) X X X X pdf(Z3) X X X X X X Z0 Z1 Z2 Z3 Stress Z ∞ R(t ; Z , Θ) = ∫ f (τ ; Z , Θ)dτ t(see Nelson, 1990; Meeker & Escobar, 1998; Elsayed, Liao & Wang, 2006) 5
  • 8. ALT Models• Widely used ALT models – Accelerated failure time (AFT) models Cdf: ‫ܨ = ܼ ;ݐ ܨ‬଴ (‫,)ݐ)ߠ ;ܼ(ݎ‬ Hazard function: ℎ ‫ = ܼ ;ݐ‬ℎ଴ ‫; ߠ ;ܼ ݎ ݐ ߠ ;ܼ ݎ‬ – Proportional hazards model and extensions ௥ ௓;ఏ Cdf: ‫ܨ − 1 − 1 = ܼ ;ݐ ܨ‬଴ ‫ݐ‬ , Hazard function: ℎ ‫ = ܼ ;ݐ‬ℎ଴ ‫; ߠ ;ܼ ݎ ݐ‬ where ‫ܨ‬଴ (∙): baseline cdf; ℎ଴ (∙): baseline hazard rate; ‫ :)ߠ ;ܼ(ݎ‬deterministic function of Z, e.g., ݁‫.)∙(݌ݔ‬ 6
  • 9. Probability Distributions Widely Used in ALT Models Exponential Weibull OthersDegroot & Goel (1979), Nelson (1980), Lognormal:Lawless & Singhal (1980), Meeker (1984), Kielpinski & Nelson (1975)Miller & Nelson (1983), Bhattacharyya & Soejoeti (1989), Liao & Elsayed (2010)Bai et al. (1989), Madi (1993), Inverse Gaussian:Bai & Chung (1992), Tang et al. (1996), Doksum & Hóyland (1992),Xiong (1998). Wang & Kececioglu (2000) Onar & Padgett (2000) • Challenges facing engineers – Selection of underlying failure time distribution – Effective way to do statistical inference for reliability prediction 7
  • 10. Motivation and Our Contributions• Question: Can we use a collection of probability distributions to approximate the widely used or complex distributions arbitrarily closely to circumvent distribution selection?• Contributions of this work – Assist practitioners in developing ALT models using a collection of versatile distributions – Develop methods to enable adaptive adjustment of model structure to achieve the best fit 8
  • 11. Phase-Type Distributions Infinitesimal generator matrix: 0 ૙ ࡿ : Subgenerator matrix consisting of the transition rates ‫ =ۿ‬૙ among transient states; ࡿ૙ = −ࡿ૚ where ૚ = (1,1, … , 1)ᇱ ࡿ ࡿ Distribution function: Cdf: ‫ ࣊ − 1 = ݐ ܨ‬exp ‫ ࡿݐ‬૚ pdf: ݂ ‫࣊ = ݐ‬exp ‫ࡿ ࡿݐ‬૙ , 1-π1 λ λ λ −ߣ ߣ 0 ࣊ = (πଵ , 0,0) , ࡿ = 0 −ߣ ߣ . 3 Absorbing π1 1 2 0 0 −ߣ state CTMC for a 3-phase Erlang distribution1-π1 (1-p1)λ1 (1-p2)λ2 ࣊ = (πଵ , 0,0) , p1λ 1 p2λ 2 λ3 Absorbing −ߣଵ ‫݌‬ଵ ߣଵ 0π1 1 2 3 state ࡿ = 0 −ߣଶ ‫݌‬ଶ ߣଶ . 0 0 −ߣଷ CTMC for a 3-phase Coxian distribution 9
  • 12. Rationale for the Use of PH DistributionsDefinition 1. Let ‫ ܺ[ܧ‬௟ ] be the lth moment of random variable X ீwith distribution G. The normalized lth moment ݉௞ of X for l = 2,3 ீ ா[௑ మ ] ீ ா[௑ య ]is defined as: ݉ଶ = and ݉ଷ = . (ா[௑])మ ா ௑ ா[௑ మ ]Definition 2. A distribution G is well represented by a distributionF if F and G agree on their first three moments. A distribution G is in PH3 (i.e., well represented by a PH distribution) if and only if its normalized moments satisfy ீ ீ ݉ଷ > ݉ଶ > 1. Since any nonnegative distribution G ீ ீ satisfies ݉ଷ ≥ ݉ଶ ≥ 1, almost all the nonnegative distributions are in PH3. 10
  • 13. Capability of Phase-Type Distributions Weibull Probability Plot 1 F(x) 0.96 0.5 0.90 0 0.75 0 500 1000 1500 2000 2500 Time -3 0.50 x 10Probability 1 pdf 0.25 0.5 0 0 500 1000 1500 2000 2500 0.10 Time -3 x 10 Hazard rate 0.05 2 1 3 10 0 Data 0 500 1000 1500 2000 2500 Time Weibull (η=1200, β=1.5) MLE fit using the 3-phase E (λ=0.0026344) L-LKV=76.9062 11
  • 14. Capability of Phase-Type Distributions Weibull Probability Plot 1 F(x) 0.96 0.5 0.90 0 0.75 0 500 1000 1500 2000 2500 Time -3 0.50 x 10Probability 1 pdf 0.25 0.5 0 0 500 1000 1500 2000 2500 0.10 Time -3 x 10 Hazard rate 0.05 2 1 3 10 0 Data 0 500 1000 1500 2000 2500 Time Weibull (η=1200, β=1.5) MLE fit using the 3-phase C (λ1=0.00176, λ2=0.00176, λ3=41.20788, p1=1, p2=1) L-LKV=77.7919 12
  • 15. Capability of Phase-Type Distributions -4 x 10 8 Weibull Probability Plot 6 E(3, 0.0026344) 0.96 Weibull(1200, 1.5) 0.90 pdf 4 0.75 2 0.50 0Probability 0 500 1000 1500 2000 2500 3000 3500 4000 Time 0.25 -4 x 10 8 C(0.00176, 0.00176, 41.20788, 1, 1) 0.10 6 Weibull(1200, 1.5) pdf 0.05 4 2 3 10 0 Data 0 500 1000 1500 2000 2500 3000 3500 4000 Time Weibull (η=1200, β=1.5) Comparison 13
  • 16. Capability of Phase-Type Distributions Weibull Probability Plot 1 F(x) 0.99 0.5 0.96 0.90 0.75 0 0 1000 2000 3000 4000 0.50 Time -3 x 10Probability 0.25 1 pdf 0.10 0.5 0.05 0 0 1000 2000 3000 4000 0.02 Time 0.01 -3 x 10 Hazard rate 2 0.003 1 2 3 10 10 Data 0 0 1000 2000 3000 4000 Time Mixture of Two Weibull : Weibull (η=1200, β=1.5), P=0.7 MLE fit using the 3-phase E (λ=0.00207) Weibull (η=2000, β=3), 1−P=0.3 L-LKV=807.531787 14
  • 17. Capability of Phase-Type Distributions Weibull Probability Plot 1 F(x) 0.99 0.5 0.96 0.90 0 0.75 0 1000 2000 3000 4000 0.50 Time -3 x 10Probability 0.25 1 pdf 0.10 0.5 0.05 0 0 1000 2000 3000 4000 0.02 Time 0.01 -3 x 10 Hazard rate 2 0.003 1 2 3 10 10 0 Data 0 1000 2000 3000 4000 Time Mixture of Two Weibull : MLE fit using the 3-phase Weibull (η=1200, β=1.5), P=0.7 C (λ1=0.00207, λ2=0.00203, λ3=0.00203, p1=0.88539, p2=1) Weibull (η=2000, β=3), 1−P=0.3 L-LKV=802.37 15
  • 18. Capability of Phase-Type Distributions -4 x 10 6 Weibull Probability Plot E(3, 0.00207) Weibull(1200, 1.5), p=0.7 0.99 4 Weibull(2000, 3), p=0.3 0.96 pdf 0.90 0.75 2 0.50Probability 0.25 0 0 500 1000 1500 2000 2500 3000 3500 4000 0.10 Time -4 x 10 0.05 6 C(0.00207, 0.00203, 0.88539, 1) 0.02 Weibull(1200, 1.5), p=0.7 0.01 4 Weibull(2000, 3), p=0.3 0.003 pdf 2 2 3 10 10 Data 0 0 500 1000 1500 2000 2500 3000 3500 4000 Time Mixture of Two Weibull : Weibull (η=1200, β=1.5), P=0.7 Comparison Weibull (η=2000, β=3), 1−P=0.3 16
  • 19. Combination of Erlang & Coxian (1-pc)λ21-π1 λ1 λ1 λ1 pcλ 2 λ3 Absorbing 1 2 k-2 k-1 k π1 state Erlang: E(k-2, λ1) Coxian: C(λ2, λ3, pc) CTMC for a k-phase Erlang-Coxian (EC) distribution  − λ1 λ1 0 ⋯  Usefulness:Subgenerator    0 −λ1 λ1 0 0  Can represent probabilitymatrix:  ⋮ 0 −λ1 ⋱ ⋱ k-2  ⋮ distributions with all   S= 0 ⋱ λ1 0  ranges of variability using  ⋮ ⋱ −λ1 λ1 0  only a small number of   phases.  0 0 − λ2 p c λ2      ⋯ ⋯ 0 − λ3   17
  • 20. Capability of EC Distributions Weibull Probability Plot 1 F(x) 0.99 0.5 0.96 0.90 0 0.75 0 1000 2000 3000 4000 0.50 Time -3 x 10Probability 0.25 1 pdf 0.10 0.5 0.05 0 0 1000 2000 3000 4000 0.02 Time -3 0.01 x 10 Hazard rate 2 0.003 1 2 3 10 10 0 Data 0 1000 2000 3000 4000 Time Mixture of Two Weibull : MLE fit using the 5-phase Weibull (η=1200, β=1.5), P=0.7 EC (λ1=0.0022136, λ2=872.34395, λ3=585.79145, p1=0.00007) Weibull (η=2000, β=3), 1−P=0.3 L-LKV=807.53189 18
  • 21. EC-Based ALT Models Cdf: ‫ܨ = ܼ ;ݐ ܨ‬଴ ‫ ࣊ − 1 = ݐ ߠ ;ܼ ݎ‬exp ‫ ࡿݐ ߠ ;ܼ ݎ‬૚, ‫.0 ≥ ݐ‬ pdf: ݂ ‫ ࣊ ߠ ;ܼ ݎ = ܼ ;ݐ‬exp ‫ࡿ ࡿݐ ߠ ;ܼ ݎ‬૙ , ‫.0 ≥ ݐ‬ Hazard function:ℎ ‫ ࣊ ߠ ;ܼ ݎ = ܼ ;ݐ‬exp{‫ࡿ}ࡿݐ ߠ ;ܼ ݎ‬଴ /(࣊ exp{‫}ࡿݐ ߠ ;ܼ ݎ‬૚) , ‫.0 ≥ ݐ‬  − λ1 λ1 0 ⋯     0 − λ1 λ1 0 0   ⋮ 0 − λ1 ⋱ ⋱ k-2  ⋮ Subgenerator   S = 0 ⋱ λ1 0  matrix:  ⋮ ⋱ −λ1 λ1 0     0 0 − λ2 p c λ2      ⋯ ⋯ 0 − λ3   19
  • 22. Statistical Inference MethodsMethod 1 (Moment-matching) Min k ni 1 (−1) (r ( Z i ;θ )) 1! S 1 − 1 −1 ni −1 ∑t j =1 ij S.T. ni ≤ ε i ,1 , 1 ni ∑t j =1 ij ni 1 (−1) (r ( Z i ;θ )) 2! S 1 − 2 −2 ni −2 ∑t j =1 2 ij ni ≤ ε i ,2 , 1 ni ∑t j =1 2 ij ni 1 (−1) (r ( Z i ;θ )) 3! S 1 − 3 −3 ni −3 ∑t j =1 3 ij ni ≤ ε i ,3 , i = 1,..., M 1 ni ∑t j =1 3 ij where ε i ,l are the pre-specified levels of tolerance for each stress level Zi. 20
  • 23. Statistical Inference Methods Method 2 (Moment-matching) (P1 ) min k k M 3 (SP1 ) min g k = ∑∑ ( hi ,l si ,l + li ,l ε i ,l ) θ ,λ1 , λ2 ,λ3 , pc i =1 l =1 { si ,l ,ε i ,l },i∈{1,..., M },l∈{1,2,3} ni 1 −(r ( Zi ;θ )) 1!π S 1 − ni −1 −1 ∑t j =1 ij Subject to ni = ε i ,1 − si ,1 , for i = 1,..., M , [1st moment] 1 ∑ tij ni j =1ɶE : the pre-specified 1 ni overall tolerance. (r ( Zi ;θ )) 2!π S 1 −−2 ni −2 ∑t j =1 2 ij ni = ε i ,2 − si ,2 , for i = 1,..., M , [2nd moment] 1 ∑ tij2 ni j =1 ni 1 −(r ( Zi ;θ )) 3!π S 1 −−3 ni −3 ∑t j =1 3 ij = ε i ,3 − si ,3 , for i = 1,..., M , [3rd moment] 1 ni 3 ∑ tij ni j =1 ε i ,1 ≥ 0, ε i ,2 ≥ 0, ε i ,3 ≥ 0, si ,1 ≥ 0, si ,2 ≥ 0, si ,3 ≥ 0, for i = 1,..., M , λ1 ≥ 0, λ2 ≥ 0, λ3 ≥ 0, pc ≥ 0, Subject to g k ≤ Ε, k ∈ {3, 4,...}, ɶ 21
  • 24. Statistical Inference Methods Method 3 (Maximum likelihood estimation) For ALT data with type-I censoring, the log-likelihood function is: ݈ ݇, ߣଵ , ߣଶ , ߣଷ , ‫݌‬௖ , ߠ = ∑ெ ∑௡೔ [δ௜௝ log ݂(‫ݐ‬௜௝ ; ܼ௜ ) +(1 − δ௜௝ )log ܴ(‫ݐ‬௜௝ ; ܼ௜ ) ] ௜ୀଵ ௝ୀଵ ௠ ௡೔ = ෍ ෍ δ௜௝ log ‫ܼ ݎ‬௜ ; ߠ ૈ exp ‫ܼ ݎ‬௜ ; ߠ ‫ݐ‬௜௝ ࡿ ࡿ૙ ௜ୀଵ ௝ୀଵ +(1 − δ௜௝ )log ૈ exp ‫ܼ ݎ‬௜ ; ߠ ‫ݐ‬௜௝ ࡿ ૚ . where δ௜௝ = {1, if ‫ݐ‬௜௝ is a failure time; 0, otherwise}. AIC = 2k p − 2ln( L( Data | Θ k p ))How to determine k p is the samethe number of k p is the number of parameters inphases needed: for all k ≥ 3. Θ k p = [k , λ1 , λ2 , λ3 , pc ,θ ] 22
  • 25. Construction of Confidence Intervals Nonparametric bootstrapResample with replacement from Data(B samples each of size n) ˆ ˆ F ( t ; Z0 ) ˆ F ( t ;Z0 ) ˆ ( t ; Z )) w , F ( t ; Z ) + (1− F ( t ; Z )) w  Data  F (t ;Z0 )+ (1− F 0 ɶ ˆ 0 ˆ 0 ɶ ෠ ‫)ݐ( ܨ‬ where Data*1 F(t) ( ) ( ) T ෠ ∗ ∂F ( t ; Z 0 ) ∂F ( t ; Z 0 ) ˆ ˆ ‫ܨ‬ ‫ݐ‬ 1 se Fˆ ( t ;Z0 ) = ∂Θ ˆ ΣΘ ˆˆ ∂Θ ˆ Data*2 w = exp{clogit ( F * ) ˆ se Fˆ ( t ;Z0 ) /[ F (t ; Z 0 )(1 − F (t ; Z 0 ))]}, ˆ ˆ ෠∗ ɶ (1−α / 2 ) ‫ݐ ܨ‬ 2 w = exp{clogit ( F * ) se Fˆ ( t ;Z0 ) /[ F (t ; Z 0 )(1 − F (t ; Z 0 ))]}, ɶ ˆ ˆ ˆ ( α / 2) Clogit ( F * ) = (logit(F j* (t ; Z 0 )) − logit(F (t ; Z 0 ))) / se logit(Fˆ * ( t ;Z ˆ ˆ ˆ Data*B j j 0 )) ෠∗ ‫ݐ ܨ‬ ‫ܤ‬ 23
  • 26. Construction of Confidence IntervalsUse of large sample approximation  ˆ F (t;Zˆ0 ) ˆ , F (t;Z )+(1−F (t;Z ))/ w  ˆ F (t ;Z0 )  F (t;Z0 )+(1−F (t;Z0 ))w ˆ 0 ˆ 0  where w = exp{z(1−α / 2) se Fˆ ( t ;Z0 ) /[ F (t ; Z 0 )(1 − F (t ; Z 0 ))]} ˆ ˆ ( ) ( ) T ∂F ( t ; Z 0 ) ˆ ∂F ( t ; Z 0 ) ˆ se Fˆ ( t ;Z0 ) = ∂Θ ˆ ΣΘ ˆˆ ∂Θ ˆ 24
  • 27. Numerical Examples• Inverse-Power-Law-Weibull ALT model• Arrhenius-Lognormal ALT model• Case study on reliability prediction for miniature lamps 25
  • 28. 1: Inverse-Power-Law-Weibull ALT Model Table 1. Complete ALT data generated from F (t ; Zi ) = 1 − exp(−(t / (1200Zi−2 ))1.5 ) Sample moments Stress level Failure times in hours 1st 2nd 3rd 473.9 531.5 624.9 Z1 = 1 724.4 856.8 1198.7 1138.8 1613532.5 2645501778.0 1367.3 1686.6 1706.3 2217.3 52.5 103.1 112.5 Z2 = 2 120.3 230.1 231.1 195.1 45529.7 11687107.9 241.0 259.8 265.0 335.6 21.1 34.6 46.9 Z3 = 3 77.1 81.1 87.6 94.6 97.9 130.8 80.9 7858.6 844024.8 137.5 26
  • 29. 1: Inverse-Power-Law-Weibull ALT Model ఈభ Assuming: ‫ݎ‬ ܼ௜ ; ߠ = [ߙ଴ , ߙଵ ] = ߙ଴ ܼ௜ [λ1 , λ2 , λ3 , pc , α 0 , α1 ] = ˆ ˆ ˆ ˆ ˆ ˆ h1,1 = h2,1 = h3,1 = l1,1 = l2,1 = l3,1 = 1 [18.980,17.718, 23.366,0.542, 0.000209, 2.442] h1,2 = h2,2 = h3,2 = l1,2 = l2,2 = l3,2 = 0.5 k = 5, g5 = 0.3394 < Ε = 0.35 * ɶ h1,3 = h2,3 = h3,3 = l1,3 = l2,3 = l3,3 = 0.25 Table 2. Deviations of the resulting EC-based ALT model in the first three moments 1 0.9 Percentage deviations from the 0.8 Stress level sample moments 0.7 1st 2nd 3rdEmpirical Cdf F(x) Z1 = 1[Kaplan-Meier] 0.6 0.5 0.09% -1.01% 0.24% Z2 = 2 0.4 Z1=1 0.3 Z2=2 7.48% 18.72% 41.28% Z3=3 Z3 = 3 0.2 0.1 -3.72% -5.10% 0.26% 0 0 500 1000 1500 2000 2500 Failure Time 27
  • 30. 2: Arrhenius-Lognormal ALT ModelTable 3. Complete ALT data generated from F(t; Zi ) = Φ( (ln t − ln(50) − 5/ Zi )/1.5) Sample momentsStress level Failure times in hours 1st 2nd 3rd 2267.9 4758.9 10341.6 11700.3 17300.1 24762.7 Z1 = 1 1.147×105 4.155×1010 1.784×1016 79694.3 105720.3 45017.2 445783.8 84.1 376.2 403.9 584.5 915.8 1642.1 Z2 = 2 2.817×103 1.862×107 1.601×1011 2435.5 5259.2 5689.4 10774.8 102.5 125.0 168.7 Z3 = 3 287.1 334.9 338.1 1.349×103 5.488×106 2.991×1010 775.4 1967.7 2971.1 6420.2 28
  • 31. 2: Arrhenius-Lognormal ALT ModelAssuming:‫ܼ ݎ‬௜ ; ߠ = [ߙ଴ , ߙଵ ] = ߙ଴ exp(−ߙଵ ܼ௜ିଵ ) [λ1 , λ2 , λ3 , pc , α 0 , α1 ] = ˆ ˆ ˆ ˆ ˆ ˆh1,1 = h2,1 = h3,1 = l1,1 = l2,1 = l3,1 = 1 [0.255,145835.454, 2637.416,0.024, 0.080,6.665]h1,2 = h2,2 = h3,2 = l1,2 = l2,2 = l3,2 = 0.5 k = 7, g 7 = 1.542 < Ε * ɶh1,3 = h2,3 = h3,3 = l1,3 = l2,3 = l3,3 = 0.25 Table 4. Deviations of the resulting EC-based ALT model in the first three moments Percentage deviations from the sample Stress level moments 1st 2nd 3rd Z1 = 1 0.00% -57.75% -81.19% Z2 = 2 45.46% 20.16% -4.53% Z3 = 3 0.00% -55.79% -81.76% 29
  • 32. Case Study• The ALT data reported by Liao and Elsayed (2010) 1 0.9 0.8 0.7 5V 3.5 V 0.6 2Vcdf: F(t) 0.5 Standardize the stress levels by defining 0.4 ܼ௜ = [ܸ௜ −ܸ଴ ]/[ܸு −ܸ଴ ], 0.3 where ܸ଴ = 2 volts and ܸு = 5 volts. ܼଵ = 1, ܼଶ = 0.5, and ܼଷ = 0. 0.2 0.1 0 The life-stress relationship is : ఈ ‫ܼ ݎ‬௜ ; ߠ = exp(ߙ଴ ܼ௜ భ ). 0 200 400 600 800 1000 Failure Time 30
  • 33. Case Study (Cont.)• Results obtained via MLE: Values of k MLEs of parameters Log-likelihood ݈݊‫ܮ‬ ߙ଴ = 2.9091; ߙଵ = 0.5762; 3: E(1, λ1) ߣଵ = 0.0026; ߣଶ = 0.0026; -518.5038 & C(λ2, λ3, pc) λ ߣଷ = 0.0003; ‫݌‬௖ = 0.6730; ߙ଴ = 2.8807; ߙଵ = 0.5730; 4: E(2, λ1) ߣଵ = 0.0045; ߣଶ = 0.0045; -516.4058 & C(λ2, λ3, pc) λ ߣଷ = 0.0003; ‫݌‬௖ = 0.6980; ࢻ૙ = ૛. ૡ૚ૡ૛; ࢻ૚ = ૙. ૞૟ૢ૜; 5: E(3, λ1) ࣅ૚ = ૙. ૙૙૟ૡ; ࣅ૛ = ૙. ૙૙૟ૡ; -515.4942 & C(λ2, λ3, pc) λ ࣅ૜ = ૙. ૙૙૙૝; ࢖ࢉ = ૙. ૠ૛૟ૢ; ߙ଴ = 2.7463; ߙଵ = 0.5747; 6: E(4, λ1) ߣଵ = 0.0097; ߣଶ = 0.0097; -515.0856 & C(λ2, λ3, pc) λ ߣଷ = 0.0004; ‫݌‬௖ = 0.7545; 31
  • 34. Case Study (Cont.) • Reliability prediction 1 1 0.9 0 0.8 0.7 ecdf (5V) -1 log(-log(survival)) ecdf (3.5V) 0.6 ecdf (2V) cdf (5V, EC model)Cdf 0.5 cdf (3.5V, EC model) -2 cdf (2V, EC model) 0.4 -3 0.3 0.2 -4 0.1 0 -5 0 200 400 600 800 1000 -4 -3 -2 -1 0 1 Failure Time log (residual) Statistical fittings for the three Cox-Snell residual plot test stress levels 32
  • 35. Case Study (Cont.)• Reliability prediction 1 Reliability 0.5 0 0 1000 2000 3000 4000 5000 Time (hours) -3 x 10 1 pdf 0.5 0 0 1000 2000 3000 4000 5000 Time (hours) -3 x 10 hazard rate 1 0.5 0 0 1000 2000 3000 4000 5000 Time (hours) Reliability prediction for the normal operating condition (2V) 33
  • 36. Conclusions• PH distributions are versatile in modeling different distributions• EC-based ALT models are generic and can be used as powerful tools for modeling ALT data• Two statistical inference methods are developed to facilitate the implementation of EC-based ALT models 34
  • 37. Acknowledgment• This work is supported in part by the U.S. National Science Foundation under grants CMMI-1238304 and CMMI-1238301. 35

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