Weibull Reliability Analysis

Transcription

1 Weibull Reliability Analysis = Fritz Scholz ( , 7L-22) Boeing Phantom Works Mathematics & Computing Technology Applied Statistics Weibull Reliability Analysis FWS-5/21-23/2002 1

2 Wallodi Weibull Weibull Reliability Analysis FWS-5/21-23/2002 2

3 Seminal Paper Weibull Reliability Analysis FWS-5/21-23/2002 3

4 The Weibull Distribution Weibull distribution, usefuluncertainty model for wearout failure time T when governed by wearout of weakest subpart material strength T when governed by embedded flaws or weaknesses, It has often been found useful based on empirical data (e.g. Y2K) It is also theoretically founded on the weakest link principle T =min(x 1,...,X n ), with X 1,...,X n statistically independent random strengths or failure times of the n links comprising the whole. The X i must have a natural finite lower endpoint, e.g., linkstrength 0 or subpart time to failure 0. Weibull Reliability Analysis FWS-5/21-23/2002 4

5 Theoretical Basis Under weakconditions Extreme Value Theory shows 1 that for large n P (T t) 1 exp t τ α β for t τ,α > 0,β >0 The above approximation has very much the same spirit as the Central Limit Theorem which under some weakconditions on the X i asserts that the distribution of T = X X n is approximately bell-shaped normal or Gaussian for large n Assuming a Weibull model for T, material strength or cycle time to failure, amounts to treating the above approximation as an equality F (t) =P (T t) =1 exp t τ α β for t τ,α > 0,β >0 1 see: E. Castillo, Extreme Value Theory in Engineering, Academic Press, 1988 S. Coles, An Introduction to Statistical Modeling of Extreme Values, Springer Verlag, 2001 Weibull Reliability Analysis FWS-5/21-23/2002 5

6 Weibull Reproductive Property If X 1,...,X n are statistically independent with X i Weibull(α i,β)then P (min(x 1,...,X n ) >t)=p (X 1 >t) P (X n >t) with α = = n i=1 exp n i=1 α β i Hence T =min(x 1,...,X n ) Weibull(α, β) t α i 1/β β This is similar to the normal reproductive property X 1,...,X n independent normal = n =exp t α β i=1 X i is normal Weibull Reliability Analysis FWS-5/21-23/2002 6

7 Weibull Parameters The Weibull distribution may be controlled by 2 or 3 parameters: the threshold parameter τ T τ with probability 1 τ =0 = 2-parameter Weibull model. the characteristic life or scale parameter α>0 P (T τ + α) =1 exp α β =1 exp( 1) =.632 α regardless of the value β>0 the shape parameter β > 0, usually β 1 Weibull Reliability Analysis FWS-5/21-23/2002 7

8 2-Parameter Weibull Model We focus on analysis using the 2-parameter Weibull model Methods and software tools much better developed Estimation of τ in the 3-parameter Weibull model leads to complications When a 3-parameter Weibull model is assumed, it will be stated explicitly Weibull Reliability Analysis FWS-5/21-23/2002 8

9 Relation of α & β to Statistical Parameters (τ =0) The expectation or mean value of T µ = E(T )= 0 tf(t) dt = αγ(1 + 1/β) with Γ(t) = 0 exp( x) x t 1 dx The variance of T σ 2 = E(T µ) 2 = 0 (t µ) 2 f(t) dt = α 2 [ Γ(1 + 2/β) Γ 2 (1 + 1/β) ] p-quantile t p of T, i.e., by definition P (T t p )=p t p = α [ log(1 p)] 1/β, for p =1 exp( 1) =.632 = t p = α Weibull Reliability Analysis FWS-5/21-23/2002 9

10 Weibull Density The cumulative distribution function F (t) =P (T t) isjust one way to describe the distribution of the random quantity T The density function f(t) is another representation (τ =0) f(t) =F (t) = df (t) dt = β α t α β 1 exp t α β t 0 P (t T t + dt) f(t) dt F (t) = t 0 f(x) dx Weibull Reliability Analysis FWS-5/21-23/

11 Weibull Density & Distribution Function Weibull density α = 10000, β = 2.5 total area under density = 1 p 1 p 0 cumulative distribution function cycles Weibull Reliability Analysis FWS-5/21-23/

12 Weibull Densities: Effect of τ τ = 0 τ = 1000 τ = 2000 probability density α = 1000, β = cycles Weibull Reliability Analysis FWS-5/21-23/

13 Weibull Densities: Effect of α α = 1000 probability density α = 2000 α = 3000 τ = 0, β = cycles Weibull Reliability Analysis FWS-5/21-23/

14 Weibull Densities: Effect of β β = 7 β =.5 probability density β = 1 β = 2 β = 4 τ = 0, α = cycles Weibull Reliability Analysis FWS-5/21-23/

15 Failure Rate or Hazard Function A third representation of the Weibull distribution is through the hazard or failure rate function λ(t) = f(t) 1 F (t) = β β 1 t = d [ log(1 F (t))] α α dt λ(t) isincreasingt for β>1 (wearout) λ(t) is decreasing t for β<1 λ(t) is constant for β = 1 (exponential distribution) P (t T t + dt T t) = P (t T t + dt) P (T t) f(t) dt 1 F (t) = λ(t) dt F (t) =1 exp ( t 0 λ(x) and f(t) =λ(t)exp ( t dx) 0 λ(x) dx) Weibull Reliability Analysis FWS-5/21-23/

16 Exponential Distribution The exponential distribution is a special case: β =1&τ =0 F (t) =P (T t) =1 exp t for t 0 α This distribution is useful when parts fail due to random external influences and not due to wear out Characterized by the memoryless property, a part that has not failed by time t is as good as new, past stresses without failure are water under the bridge Good for describing lifetimes of electronic components, failures due to external voltage spikes or overloads Weibull Reliability Analysis FWS-5/21-23/

17 Unknown Parameters Typically will not know the Weibull distribution: α, β unknown Will only have sample data = estimates α, β get estimated Weibull model for failure time distribution = double uncertainty uncertainty of failure time & uncertainty of estimated model Samples of failure times are sometimes very small, only 7 fuse pins or 8 ball bearings tested until failure, long lifetimes make destructive testing difficult Variability issues are often not sufficiently appreciated how do small sample sizes affect our confidence in estimates and predictions concerning future failure experiences? Weibull Reliability Analysis FWS-5/21-23/

18 Estimation Uncertainty A Weibull Population: Histogram for N = 10,000 & Density % 36.8% characteristic life = 30,000 shape = 2.5 true model estimated model from 9 data points cycles Weibull Reliability Analysis FWS-5/21-23/

19 Weibull Parameters & Sample Estimates shape parameter β = % 36.8% p=p(t < t ) t = t p p-quantile characteristic life α = 30, parameter estimates from three samples of size n = 10 Weibull Reliability Analysis FWS-5/21-23/

20 Generation of Weibull Samples Using the quantile relationship t p = α [ log(1 p)] 1/β one can generate a Weibull random sample of size n by generating a random sample U 1,...,U n from a uniform [0, 1] distribution and computing T i = α [ log(1 U i )] 1/β, i =1,...,n. Then T 1,...,T n can be viewed as a random sample of size n from a Weibull population or Weibull distribution with parameters α & β. Simulations are useful in gaining insight on estimation procedures Weibull Reliability Analysis FWS-5/21-23/

21 Graphical Methods Suppose we have a complete Weibull sample of size n: T 1,...,T n Sort these values from lowest to highest: T (1) T (2)... T (n) Recall that the p-quantile is t p = α [ log(1 p)] 1/β Compute t p1 <...<t pn for p i =(i.5)/n, i =1,...,n Plot the points (T (i),t pi ), i =1,...,n We expect these points to cluster around main diagonal Weibull Reliability Analysis FWS-5/21-23/

22 Weibull Quantile-Quantile Plot: Known Parameters tp T Weibull Reliability Analysis FWS-5/21-23/

23 Weibull QQ-Plot: Unknown Parameters Previous plot requires knowledge of the unknown parameters α & β Note that log (t p )=log(α)+w p /β, where w p =log[ log(1 p)] Expect points ( ) log[t (i) ],w pi, i =1,...,n, to cluster around line with slope 1/β and intercept log(α) This suggests estimating α & β from a fitted line, by eye or least squares Weibull Reliability Analysis FWS-5/21-23/

24 Maximum Likelihood Estimation If t 1,...,t n are the observed sample values one can contemplate the probability of obtaining such a sample or of values nearby, i.e., P (T 1 [t 1 dt/2,t 1 + dt/2],...,t n [t n dt/2,t n + dt/2]) = P (T 1 [t 1 dt/2,t 1 + dt/2]) P (T n [t n dt/2,t n + dt/2]) f α,β (t 1 )dt f α,β (t n )dt where f(t) =f α,β (t) is the Weibull density with parameters (α, β) Maximum likelihood estimation maximizes this probability over α & β = maximum likelihood estimates (m.l.e.s) α and β = the most likely Weibull model explaining the data Weibull Reliability Analysis FWS-5/21-23/

25 General Remarks on Estimation MLEs tend to be optimal in large samples (lots of theory) Method is very versatile in extending to may other data scenarios censoring and covariates Least squares method applied to QQ-plot is not entirely appropriate tends to be unduly affected by stray observations not as versatile to extend to other situations Weibull Reliability Analysis FWS-5/21-23/

26 Weibull Plot: n =20 probability true model, Weibull( 100, 3 ) m.l.e. model, Weibull( 108, ) least squares model, Weibull( 107.5, ) cycles/hours Weibull Reliability Analysis FWS-5/21-23/

27 Weibull Plot: n = 100 probability true model, Weibull( 100, 3 ) m.l.e. model, Weibull( 102.8, ) least squares model, Weibull( 102.2, ) cycles/hours Weibull Reliability Analysis FWS-5/21-23/

28 Tests of Fit (Graphical) The Weibull plots provide an informal diagnostic for checking the Weibull model assumption The anticipated linearity is based on the Weibull model properties Strong nonlinearity indicates that the model is not Weibull Sorting out nonlinearity from normal statistical point scatter takes a lot of practice and a good sense for the effect of sample size on the variation in point scatter (simulation helps) Formal tests of fit are available for complete samples 2 and also for some other censored data scenarios 2 R.B. D Agostino and M.A. Stephens, Goodness-of-Fit Techniques, Marcel Dekker 1986 Weibull Reliability Analysis FWS-5/21-23/

29 Formal Goodness-of-Fit Tests Let F α, β (t) be the fitted Weibull distribution function Let F n (t) = #{T i t; i=1,...,n} n be the empirical distribution function Compute a discrepancy metric D between F α, β and F n, D KS (F α, β, F n )=sup F α, β (t) F n (t) Kolmogorov-Smirnov t D CvM (F α, β, F n )= ( 0 F α, β (t) F n (t) ) 2 f α, β (t) dt Cramer-von Mises ( D AD (F α, β, F n )= F α, β (t) F n (t) ) 2 0 F α, β (t)(1 F α, β (t)) f α, β (t) dt Anderson-Darling The distributions of D, when sampling from a Weibull population, are known and p-values of observed values d of D can be calculated p = P (D d) = BCSLIB: HSPFIT Weibull Reliability Analysis FWS-5/21-23/

30 Kolmogorov-Smirnov Distance n = 10 K-S distance Weibull Reliability Analysis FWS-5/21-23/

31 Weibull Plots: n =10(solid line is true model) p(ks) = 0.62 p(cvm) = 0.54 p(ad) = 0.62 n = p(ks) = 0.28 p(cvm) = 0.27 p(ad) = p(ks) = 0.88 p(cvm) = 0.62 p(ad) = p(ks) = 0.88 p(cvm) = 0.64 p(ad) = p(ks) = 0.72 p(cvm) = 0.53 p(ad) = p(ks) = 0.16 p(cvm) = 0.21 p(ad) = p(ks) = 0.34 p(cvm) = 0.23 p(ad) = p(ks) = 0.86 p(cvm) = 0.56 p(ad) = Weibull Reliability Analysis FWS-5/21-23/

32 Weibull Plots: n = p(ks) = 0.55 p(cvm) = 0.49 p(ad) = p(ks) = 0.13 p(cvm) = p(ad) = p(ks) = 0.12 p(cvm) = p(ad) = p(ks) = 0.46 p(cvm) = 0.39 p(ad) = 0.38 n = p(ks) = 0.46 p(cvm) = 0.41 p(ad) = p(ks) = 0.47 p(cvm) = 0.63 p(ad) = p(ks) = 0.85 p(cvm) = 0.6 p(ad) = p(ks) = 0.04 p(cvm) = p(ad) = Weibull Reliability Analysis FWS-5/21-23/

33 Weibull Plots: n = p(ks) = 0.7 p(cvm) = 0.41 p(ad) = p(ks) = 0.22 p(cvm) = 0.35 p(ad) = p(ks) = 0.15 p(cvm) = 0.23 p(ad) = p(ks) = 0.76 p(cvm) = 0.49 p(ad) = 0.52 n = p(ks) = 0.88 p(cvm) = 0.66 p(ad) = p(ks) = 0.27 p(cvm) = 0.22 p(ad) = p(ks) = 0.85 p(cvm) = 0.56 p(ad) = p(ks) = 0.18 p(cvm) = p(ad) = Weibull Reliability Analysis FWS-5/21-23/

34 Weibull Plots: n = p(ks) = 0.58 p(cvm) = 0.26 p(ad) = p(ks) = 0.86 p(cvm) = 0.68 p(ad) = p(ks) = 0.61 p(cvm) = 0.48 p(ad) = p(ks) = 0.21 p(cvm) = 0.26 p(ad) = 0.43 n = p(ks) = 0.47 p(cvm) = 0.3 p(ad) = p(ks) = p(cvm) = p(ad) = p(ks) = 0.87 p(cvm) = 0.58 p(ad) = p(ks) = 0.72 p(cvm) = 0.45 p(ad) = Weibull Reliability Analysis FWS-5/21-23/

35 Estimates and Confidence Bounds/Intervals For a target θ θ = α, θ = β, θ = t p = α[ log(1 p)] 1/β, or θ = P α,β (T t) one gets corresponding m.l.e θ by replacing (α, β) by( α, β) Such estimates θ vary around the target θ due to sampling variation Capture the estimation uncertainty via confidence bounds For 0 <γ<1 get 100γ% lower/upper confidence bounds θ L,γ & θ U,γ P ( θ L,γ θ ) = γ or P ( θ θ ) U,γ = γ For γ>.5 get a 100(2γ 1)% confidence interval by [ θ L,γ, θ U,γ ] P ( θ L,γ θ θ U,γ ) =2γ 1=γ Weibull Reliability Analysis FWS-5/21-23/

36 Confidence Bounds & Sampling Variation, n =10 90% confidence intervals L: Lawless Exact Method (RAP); B: Bain Exact Method (Tables); M: MLE Approximate Method LBM true α samples % confidence intervals LBM true β Weibull Reliability Analysis FWS-5/21-23/

37 Confidence Bounds & Sampling Variation, n =10 95% lower confidence bounds LBM true 10-percentile L: Lawless Exact Method (RAP); B: Bain Exact Method (Tables); M: MLE Approximate Method samples % upper confidence bounds LBM true P(T < 10,000) Weibull Reliability Analysis FWS-5/21-23/

38 Confidence Bounds & Effect of n 95% confidence intervals true α sample size 95% confidence intervals true β Weibull Reliability Analysis FWS-5/21-23/

39 Confidence Bounds & Effect of n 95% upper confidence bounds true P(T < 10000) sample size 95% lower confidence bounds true t.10 Weibull Reliability Analysis FWS-5/21-23/

40 Incomplete Data or Censored Failure Times Type I censoring or time censoring: units are tested until failure or until a prespecified time has elapsed Type II censoring or failure censoring: only the r lowest values of the total sample of size n become known this shows up when we put n units on test at the same time and terminate the test after the first r units have failed Interval censoring, inspection data, grouped data: i th unit is only known to have failed between two known inspection time points, i.e., failure time T i falls in (s i,e i ] only bracketing intervals (s i,e i ), i =1,...,n, become known Weibull Reliability Analysis FWS-5/21-23/

41 Incomplete Data or Censored Failure Times (continued) Random right censoring: units are observed until failed or removed from observation due to other causes (different failure modes, competing risks) Multiple right censoring: units are put into service at different times and times to failure or censoring are observed Data can also combine several of the above censoring phenomena It is important that the censoring mechanism should not correlate with the (potential) failure times, i.e., no censoring of anticipated failures All data (censored & uncensored) should enter analysis otherwise bias results Weibull Reliability Analysis FWS-5/21-23/

42 Type I or Time Censored Data: n =10 unit 10 X 9 8?? 7 X 6 X 5? 4 X 3 X 2 1?? cycles Weibull Reliability Analysis FWS-5/21-23/

43 Type II or Failure Censored Data: n =10 unit ???? 6 X 5 X 4 X 3 X 2 X 1 X cycles Weibull Reliability Analysis FWS-5/21-23/

44 Interval Censored Data: n =10 unit 10? 9 ( ] ( ] ( ] (? ( ] (???? cycles Weibull Reliability Analysis FWS-5/21-23/

45 Multiply Right Censored Data: n =10 unit ???? 6 X 5 X 4 X 3 X 2? 1 X cycles Weibull Reliability Analysis FWS-5/21-23/

46 Nonparametric MLEs of CDF For complete, type I & II censored data the nonparametric MLE is F (t) = #{T i t; i =1,...,n}, n <t< for complete sample, <t t 0 for type I censored (at t 0 )sample, <t T (r) for type II censored (at T (r), r th smallest failure time) For multiply right censored data with failures at t 1 <...<t k the nonparametric MLE (Kaplan-Meier or Product Limit Estimator) is F (t) =1 [ (1 ˆp 1 ) δ 1(t) (1 ˆp k ) δ k(t) ], δ i (t) = 1fort i t 0fort i >t where ˆp i = d i /n i, n i = # units known to be at risk just prior to t i and d i = # units that failed at t i The nonparametric MLE for interval censored data is complicated Weibull Reliability Analysis FWS-5/21-23/

47 Empirical CDF (complete samples) empirical and true and Weibull mle distribution functions n = n = n = n = Weibull Reliability Analysis FWS-5/21-23/

48 Nonparametric MLE for Type I Censored Data nonparametric mle of CDF Weibull mle of CDF and true CDF... n = n = n = n = Weibull Reliability Analysis FWS-5/21-23/

49 Nonparametric MLE for Type II Censored Data nonparametric mle of CDF Weibull mle of CDF and true CDF... n = n = n = n = Weibull Reliability Analysis FWS-5/21-23/

50 Kaplan-Meier Estimates (multiply right censored samples) Kaplan-Meier CDF Weibull mle for CDF true CDF n = n = n = n = Weibull Reliability Analysis FWS-5/21-23/

51 Nonparametric MLE for Interval Censored Data CDF superimposed is the true Weibull(10000,2.5) that generated the 1,000 interval censored data cases inspection points roughly 3000 cycles apart cycles Weibull Reliability Analysis FWS-5/21-23/

52 Nonparametric MLE for Interval Censored Data CDF superimposed is the true Weibull(10000,2.5) that generated the 10,000 interval censored data cases inspection intervals randomly generated from same Weibull cycles Weibull Reliability Analysis FWS-5/21-23/

53 Application to Y2K Questionnaire Return Data probability of questionnaire return cases estimated.99-quantile days Weibull Reliability Analysis FWS-5/21-23/

54 Plotting Positions for Weibull Plots (Censored Case) For complete samples we plotted (log[t (i) ], log [ log(1 p i )]) with p i = i.5 n = 1 2 [ F (T (i) )+ F (T (i 1) ) ] = 1 2 i n + i 1 n For type I & II & multiply right censored samples we plot in analogy (log[t (i)], log [ log(1 p i )]), where T (1) <...<T (k) are the k distinct observed failure times and p i = 1 2 [ F (T (i))+ F (T (i 1)) ] and F (t) is the nonparametric MLE of F (t) for the censored sample Weibull Reliability Analysis FWS-5/21-23/

55 Weibull Plot, Multiply Right Censored Sample n =50 probability true model, Weibull( 100, 3 ) m.l.e. model, Weibull( 96.27, ) n = 50 least squares model, Weibull( 96.3, ) n = cycles/hours Weibull Reliability Analysis FWS-5/21-23/

56 Software: MLE s & Confidence Bounds for Censored Data For complete & type II censored data RAP computes exact coverage confidence bounds for α, β, t p,andp (T t). WEIBREG and commercial software (Weibull++, WeibullSMITH, common statistical packages) compute approximate confidence bounds for above targets, based on large sample m.l.e. theory RAP & WEIBREG are Boeing code, runs within DOS mode of Windows95 (interface clumsy, but job gets done), contact me Boeing has a site license for Weibull++ Version 4 in the Puget Sound area (with upgrade pressure) contact David Twigg (425) , david.twigg@pss.boeing.com There is a Weibull++ Version 6 out Weibull Reliability Analysis FWS-5/21-23/

57 Other Software S-Plus Statistical data analysis environment ( $2000 for PC licence) R is a GNU version of S-plus, freely available either from: for lots of operating systems or inside Boeing for Windows toolkit/optional.html#l048 Both S-Plus and R have Terry Therneau s survival analysis package built in or as add-on. Using it takes some sophistication. My web-based Weibull analysis tool is based on R Also freely available is the very user friendly package SPLIDA by Bill Meeker which can be added to S-Plus 4.5 or S-Plus Weibull Reliability Analysis FWS-5/21-23/

58 Software in Progress (Needs Pull!) The current version of web-based Weibull analysis tool and also most other versions based on large sample approximations for the MLE s have some defect for extreme sample with very few failures: Confidence bounds for quantiles t p are not always monotone increasing in p, which does not make sense. Also quantile and probability bounds are not inverses to each other. I have an updated version of WEIBREG based on a new method that avoids this problem. It also includes a bootstrap version of the new monotone method. However, it is not yet released. Quantile and probability bounds are true inverses of each other. The plan is to update the web based tool and to also create web based versions of RAP and ABVAL to bypass the legacy issue. Weibull Reliability Analysis FWS-5/21-23/

59 Weibull Plot With Confidence Bound alpha = [ 82.52, ] 95 % beta = [ 1.352, ] 95 % n = 30, r = Weibull Reliability Analysis FWS-5/21-23/

60 Weibull Plot With Confidence Bound (Variation) alpha = [ 77.4, ] 95 % beta = [ 2.15, ] 95 % n = 30, r = Weibull Reliability Analysis FWS-5/21-23/

61 Problem Weibull Plot probability α^ = 38600, 95 % conf. interval ( 30690, ) β^ = 6.088, 95 % conf. interval ( 1.987, ) n = 5, r = 2 m.l.e. 95 % q confidence bounds 95 % p confidence bounds cycles x 1000 Data: , , 25500, , Weibull Reliability Analysis FWS-5/21-23/

62 Weibull Regression Model & Analysis Recall t p = α[ log(1 p)] 1/β or log(t p )=log(α)+w p /β with w p =log[ log(1 p)] Often we deal with failure data collected under different conditions different part types different environmental conditions different part users Linear regression model for log(α) (multiplicative on α) log(α i )=b 1 Z i b p Z ip typically Z i1 1 where Z i1,...,z ip are known covariates for i th unit Now we have p + 1 unknown parameters β,b 1,...,b p parameter estimates & confidence bounds using MLEs Weibull Reliability Analysis FWS-5/21-23/

63 Accelerated Life Testing: Inverse Power Law Units last too long under normal usage conditions Increase stress to accelerate failure time Increase voltage and accelerate life via inverse power law this means with T (Volt) = T (Volt U ) Volt Volt U α(volt) = α (Volt U ) c, where usually c<0 Volt Volt U or log [α(volt)] = log [α (Volt U )] + c log (Volt) c log (Volt U ) = b 1 + b 2 Z b 1 =log[α (Volt U )] c log (Volt U ), b 2 = c, and Z = log (Volt) c Weibull Reliability Analysis FWS-5/21-23/

64 Accelerated Life Testing: Arrhenius Model Another way of accelerating failure in processes involving chemical reaction rates is to increase the temperature Arrhenius proposed the following acceleration model with temperature measured in Kelvin (tempk =temp C ) κ T (temp) = T (temp U )exp temp κ temp U or log [α(temp)] = log [α(temp U )] + κ temp κ temp U with b 1 =log[α(temp U )] = b 1 + b 2 Z κ temp U, b 2 = κ, and Z =temp 1 Weibull Reliability Analysis FWS-5/21-23/

65 Pooling Data With Different α s Suppose we have three groups of failure data T 1,...,T n1 W( α 1,β), T n1 +1,...,T n1 +n 2 W( α 2,β), T n1 +n 2 +1,...,T n1 +n 2 +n 3 W( α 3,β) We can analyze the whole data set of N = n 1 + n 2 + n 3 values jointly using the following model for α j and dummy covariates Z 1,j, Z 2,j & Z 3,j log(α j )=b 1 Z 1,j + b 2 Z 2,j + b 3 Z 3,j, j =1, 2,...,N where b 1 =log( α 1 ), b 2 =log( α 2 ) log( α 1 ) and b 3 =log( α 3 ) log( α 1 ) Z 1,j =1forj =1,...,N, Z 2,j =1forj = n 1 +1,...,n 1 + n 2, Z 2,j =0else and Z 3,j =1forj = n 1 + n 2 +1,...,N & Z 3,j =0else. Advantage: Smaller estimation error all data are used in estimating the (assumed) common β Weibull Reliability Analysis FWS-5/21-23/

66 Pooling Data (continued) log(α 1 ) = b 1 1+b 2 0+b 3 0=log( α 1 ).. log(α n1 ) = b 1 1+b 2 0+b 3 0=log( α 1 ).. log(α n1 +1) = b 1 1+b 2 1+b 3 0=log( α 1 ) + [log( α 2 ) log( α 1 )] = log( α 2 ).. log(α n1 +n 2 ) = b 1 1+b 2 1+b 3 0=log( α 1 ) + [log( α 2 ) log( α 1 )] = log( α 2 ).. log(α n1 +n 2 +1) =b 1 1+b 2 0+b 3 1=log( α 1 ) + [log( α 3 ) log( α 1 )] = log( α 3 ).. log(α N ) = b 1 1+b 2 0+b 3 1=log( α 1 ) + [log( α 3 ) log( α 1 )] = log( α 3 ) Weibull Reliability Analysis FWS-5/21-23/

67 Accelerated Life Testing: Model & Data thousand cycles Voltage Weibull Reliability Analysis FWS-5/21-23/

68 Accelerated Life Testing: Model, Data & MLEs thousand cycles true line mle line mle for.01-quantile 95% lower bound for.01-quantile Voltage Weibull Reliability Analysis FWS-5/21-23/

69 Analysis for Data from Exponential Distribution T E(θ) Exponential with mean θ, i.e., E(θ) =W(α = θ, β =1) It suffices to get confidence bounds for θ since all other quantities of interest are explicit and monotone functions of θ F θ (t 0 )=P θ (T t 0 )=1 exp( t 0 /θ) R θ (t 0 )=P θ (T>t 0 )=exp( t 0 /θ) and t p (θ) =θ [ log(1 p)] Weibull Reliability Analysis FWS-5/21-23/

70 Complete & Type II Censored Exponential Samples T E(θ) Exponential with mean θ, E(θ) =W(α = θ, β =1) For complete exponential samples T 1,...,T n E(θ) or for a type II censored sample of this type, i.e., with observed r lowest values T (1)... T (r),1 r n, one gets 100γ% lower confidence bounds for θ by computing ˆθ L,γ = 2 TTT, χ 2 2r,γ where TTT = T (1) + + T (r) +(n r)t (r) = Total Time on Test, and χ 2 2r,γ = γ-quantile of the χ 2 2r distribution get this in Excel via = GAMMAINV(γ,r,1) or = CHIINV(1 γ,2r)/2 These bounds have exact confidence coverage properties Weibull Reliability Analysis FWS-5/21-23/

71 Multiply Right Censored Exponential Data Here we define TTT = T T n as the total time on test but now T i is either the observed failure time of the i th part or the observed right censoring time of the i th part The number r of observed failures is random here, 0 r n Approximate 100γ% lower confidence bounds for θ are computed as ˆθ L,γ = 2 TTT χ 2 2r+2,γ This also holds for r = 0, i.e., no failures at all over exposure period Weibull Reliability Analysis FWS-5/21-23/

72 Weibull Shortcut Methods for Known β T W(α, β) (Weibull) T β E(θ) Exponential with mean θ = α β As 100γ% lower confidence bound for α compute ˆα L,γ = 2 TTT(β) χ 2 f,γ where for complete or type II censored samples we take f =2r and TTT = T β (1) + + T β (r) 1/β +(n r)t β (r) and for multiply right censored data we take f =2r +2and TTT = T β T β n. Sensitivity should be studied by repeating analyses for several β s Weibull Reliability Analysis FWS-5/21-23/

73 A- and B-Allowables, ABVAL Program My first Weibull workled to the program ABVAL (1983) supporting Cecil Parsons and Ron Zabora w.r.t. MIL-HDBK-5 ABVAL computes A- and B-Allowables based on samples from the 3-parameter Weibull distribution. The A-Allowable is a 95% lower confidence bound for 99% of the population, i.e., for the.01-quantile. The B-Allowable is a 95% lower confidence bound for 90% of the population, i.e., for the.10-quantile. Hybrid approach of using Mann-Fertig threshold estimate combined with maximum likelihood estimates for α and β, tooklot of simulation and tuning and is very specialized in its capabilities It is now a method in MIL-HDBK-5, I still maintain software Weibull Reliability Analysis FWS-5/21-23/

74 Selected References Abernethy, R.B. (1996), The New Weibull Handbook, 2nd edition, Reliability Analysis Center Bain, L.J. (1978), Statistical Analysis of Reliability and Life-Testing Models, Marcel Dekker, New York Jeng, S.-L. and Meeker, W.Q. (2000), Comparisons of approximate confidence procedures for type I censored data, J. Amer. Statist. Assoc. 42, Kececioglu, D. (1993/4), Reliability & Life Testing Handbook Vol. 1 & 2, Prentice Hall PTR, Upper Saddle River, NJ. Lawless, J.F. (1982), Statistical Models and Methods for Lifetime Data, John Wiley & Sons, New York Mann, N.R., Schafer, R.E., and Singpurwalla, N.D. (1974), Methods for Statistical Analysis of Reliability and Life Data, John Wiley & Sons, New York Weibull Reliability Analysis FWS-5/21-23/

75 References (Continued) Meeker, W.Q. and Escobar, L.A. (1995), Teaching about approximate confidence regions based on maximum likelihood estimation, The American Statistician = Meeker, W.Q. and Escobar, L.A. (1998), Statistical Methods for Reliability Data, John Wiley & Sons, New York Meeker, W.Q. and Hahn, G.J. (1985), Volume 10: How to Plan an Accelerated Life Test Some Practical Guidelines, AmericanSociety for Quality Control Robinson, J.A. (1983), Bootstrap confidence intervals in location-scale models with progressive censoring, Technometrics Nelson, W. (1982), Applied Life Data Analysis, John Wiley & Sons, New York Weibull Reliability Analysis FWS-5/21-23/

76 References (Continued) Nelson, W. (1985), Weibull analysis of reliability data with few or no failures, Journal of Quality Technology 17, Nelson, W. (1990), Accelerated Testing, Statistical Models, Test Plans and Data Analyses, John Wiley & Sons, New York Therneau, T.M. and Grambsch, P.M. (2000), Modeling Survival Data, Extending the Cox Model, Springer Verlag, New York. Scholz, F.W. (1994), Weibull and Gumbel distribution exact confidence bounds. BCSTECH (RAP) Scholz, F.W. (1996) Maximum likelihood estimation for type I censored Weibull data including covariates. ISSTECH Scholz, F.W. (1997), Confidence bounds for type I censored Weibull data including covariates. SSGTECH (WEIBREG) The Boeing Company, P.O. Box 3707, MS-7L-22, Seattle WA Weibull Reliability Analysis FWS-5/21-23/