Weibull Reliability Analysis tmlFritz Scholz (425-865-3623, 7L-22)Boeing Phantom WorksMathematics & Computing TechnologyWeibull Reliability Analysis—FWS-5/1999—1

Wallodi WeibullWeibull Reliability Analysis—FWS-5/1999—2

Seminal PaperWeibull Reliability Analysis—FWS-5/1999—3

The Weibull Distribution Weibull distribution, useful uncertainty model for– wearout failure time Twhen governed by wearout of weakest subpart– material strength Twhen 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 principleT min (X1, . . . , Xn) ,with X1, . . . , Xn statistically independent random strengths orfailure times of the n “links” comprising the whole.The Xi must have a natural finite lower endpoint,e.g., link strength 0 or subpart time to failure 0., Weibull Reliability Analysis—FWS-5/1999—4

Theoretical Basis Under weak conditions Extreme Value Theory shows1 that forlarge n β t τfor t τ, α 0, β 0P (T t) 1 exp α The above approximation has very much the same spirit as theCentral Limit Theorem which under some weak conditions onthe Xi asserts that the distribution of T X1 . . . Xn isapproximately bell-shaped normal or Gaussian Assuming a Weibull model for T , material strength or cycle time tofailure, amounts to treating the above approximation as an equality F (t) P (T t) 1 exp 1t τα β for t τ, α 0, β 0see: E. Castillo, Extreme Value Theory in Engineering, Academic Press, 1988Weibull Reliability Analysis—FWS-5/1999—5

Weibull Reproductive PropertyIf X1, . . . , Xn are statistically independentwith Xi Weibull(αi, β) thenP (min(X1, . . . , Xn) t) P (X1 t) · · · P (Xn t) βn t β tX exp exp i 1i 1 αiαi β t exp αnYHence T min(X1, . . . , Xn) Weibull(α, β) with α n X i 1 αi β 1/βWeibull Reliability Analysis—FWS-5/1999—6

Weibull ParametersThe Weibull distribution may be controlled by 2 or 3 parameters: the threshold parameterτT τ with probability 1τ 0 2-parameter Weibull model.the characteristic life or scale parameter βα 1 exp( 1) .632P (T τ α) 1 exp αregardless of the value β 0 the shape parameter α 0β 0,usually β 1Weibull Reliability Analysis—FWS-5/1999—7

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 modelleads to complications When a 3-parameter Weibull model is assumed,it will be stated explicitlyWeibull Reliability Analysis—FWS-5/1999—8

Relation of α & β to Statistical Parameters The expectation or mean value of Tµ E(T ) Z 0t f (t) dt αΓ(1 1/β)with Γ(t) Z 0exp( x) xt 1 dx The variance of Tσ E(T µ) 22Z 0 (t µ) f (t) dt α Γ(1 2/β) Γ (1 1/β)222 p-quantile tp of T , i.e., by definition P (T tp) ptp α [ log(1 p)]1/β , for p 1 exp( 1) .632 tp αWeibull Reliability Analysis—FWS-5/1999—9

Weibull Density The cumulative distribution function F (t) P (T t) is justone way to describe the distribution of the random quantity T The density function f (t) is another representation (τ 0)f (t) F 0(t) β 1dF (t) β t dtα α exp tα β t 0P (t T t dt) f (t) dtF (t) Zt0f (x) dxWeibull Reliability Analysis—FWS-5/1999—10

Weibull Density & Distribution FunctionWeibull density α 10000, β 2.5total area under density 1p1pcumulative distribution function00500010000cycles1500020000Weibull Reliability Analysis—FWS-5/1999—11

Weibull Densities: Effect of ττ 0τ 1000τ 2000probability densityα 1000, β 2.5020004000cyclesWeibull Reliability Analysis—FWS-5/1999—12

Weibull Densities: Effect of αα 1000probability densityτ 0, β 2.50α 2000α 30002000400060008000cyclesWeibull Reliability Analysis—FWS-5/1999—13

Weibull Densities: Effect of ββ 7probability densityβ .5β 4β 1β 2τ 0, α 100001000200030004000cyclesWeibull Reliability Analysis—FWS-5/1999—14

Failure Rate or Hazard Function A third representation of the Weibull distribution is through thehazard or failure rate function f (t)β t β 1λ(t) 1 F (t) α α λ(t) is increasing t 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) F (t) 1 exp Zt0P (t T t dt)f (t) dt λ(t) dtP (T t)1 F (t)!λ(x) dxandf (t) λ(t) exp Zt0!λ(x) dxWeibull Reliability Analysis—FWS-5/1999—15

Exponential Distribution The exponential distribution is a special case: β 1 & τ 0 t F (t) P (T t) 1 exp αfor t 0 This distribution is useful when parts fail due torandom 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 overloadsWeibull Reliability Analysis—FWS-5/1999—16

Unknown Parameters Typically will not know the Weibull distribution: α, β unknowncβc Will only have sample data estimates α,get estimated Weibull model for failure time distribution double uncertaintyuncertainty 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 appreciatedhow do small sample sizes affect our confidence inestimates and predictions concerning future failure experiences?Weibull Reliability Analysis—FWS-5/1999—17

Estimation Uncertainty0.00004A Weibull Population: Histogram for N 10,000 & Density63.2% 36.8%0.00002characteristic life 30,000shape 2.50.0true modelestimated model from 9 data points 0 20000 400006000080000cyclesWeibull Reliability Analysis—FWS-5/1999—18

Weibull Parameters & Sample Estimatesshape parameter β 463.2%36.8%characteristic life α 30,000p P(T t )t t p p-quantile 253903.02 338604.27 294105.01parameter estimates from three samples of size n 10Weibull Reliability Analysis—FWS-5/1999—19

Generation of Weibull Samples Using the quantile relationship tp α [ log(1 p)]1/βone can generate a Weibull random sample of size n by–generating a random sample U1, . . . , Unfrom a uniform [0, 1] distribution–and computing Ti α [ log(1 Ui)]1/β , i 1, . . . , n.–Then T1, . . . , Tn can be viewed as a random sample of size nfrom a Weibull population or Weibull distributionwith parameters α & β. Simulations are useful in gaining insight on estimation proceduresWeibull Reliability Analysis—FWS-5/1999—20

Graphical Methods Suppose we have a complete Weibull sample of size n:T1, . . . , Tn Sort these values from lowest to highest: T(1) T(2) . . . T(n) Recall that the p-quantile is tp α [ log(1 p)]1/β Compute tp1 . . . tpn for pi (i .5)/n, i 1, . . . , n Plot the points (T(i), tpi ), i 1, . . . , n and expectthese points to cluster around main diagonalWeibull Reliability Analysis—FWS-5/1999—21

Weibull Quantile-Quantile Plot: Known Parameters100005000 0tp15000 050001000015000TWeibull Reliability Analysis—FWS-5/1999—22

Weibull QQ-Plot: Unknown Parameters Previous plot requires knowledge of the unknown parameters α & β Note thatlog (tp) log(α) wp/β , where wp log [ log(1 p)] Expect points log[T(i)], wpi , i 1, . . . , n, to cluster around linewith slope 1/β and intercept log(α) This suggests estimating α & β from a fitted least squares lineWeibull Reliability Analysis—FWS-5/1999—23

Maximum Likelihood Estimation If t1, . . . , tn are the observed sample values one can contemplatethe probability of obtaining such a sample or of values nearby, i.e.,P (T1 [t1 dt/2, t1 dt/2], . . . , Tn [tn dt/2, tn dt/2]) P (T1 [t1 dt/2, t1 dt/2]) · · · P (Tn [tn dt/2, tn dt/2]) fα,β (t1)dt · · · fα,β (tn)dtwhere f (t) fα,β (t) is the Weibull density with parameters (α, β) Maximum likelihood estimation maximizes this probabilityover α & βcc and β maximum likelihood estimates (m.l.e.s) αWeibull Reliability Analysis—FWS-5/1999—24

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 scenarioscensoring and covariates Least squares method applied to QQ-plot is not entirely appropriatetends to be unduly affected by stray observationsnot as versatile to extend to other situationsWeibull Reliability Analysis—FWS-5/1999—25

Weibull Plot: n 20.990.900.632probability.500.200 .100 .010true model, Weibull( 100 , 3 )m.l.e. model, Weibull( 108 , 3.338 )least squares model, Weibull( 107.5 , 3.684 ).0011020501002005001000cycles/hoursWeibull Reliability Analysis—FWS-5/1999—26

Weibull Plot: n 100.990.900.632probability.500.200.100 .010true model, Weibull( 100 , 3 )m.l.e. model, Weibull( 102.8 , 2.981 )least squares model, Weibull( 102.2 , 3.144 ) .0011020501002005001000cycles/hoursWeibull Reliability Analysis—FWS-5/1999—27

Tests of Fit (Graphical) The Weibull plots provide an informal diagnosticfor 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 scattertakes a lot of practice and a good sense for the effectof sample size on the variation in point scatter Formal tests of fit are available for complete samples2and also for some other censored data scenarios2R.B. D’Agostino and M.A. Stephens, Goodness-of-Fit Techniques, Marcel Dekker 1986Weibull Reliability Analysis—FWS-5/1999—28

Formal Goodness-of-Fit Tests Let Fα,b (t) be the fitted Weibull distribution functionb β#{Ti t; i 1,.,n}nd Let Fn (t) be the empirical distribution functiondb and Fn , Compute a discrepancy metric D between Fα,b βddDKS (Fα,b , Fn ) sup F b b (t) Fn (t)b βα,βKolmogorov-SmirnovtdDCvM (Fα,b , Fn ) b βdb , Fn ) DAD (Fα,b βZZ 0 0!2Fα,b (t) Fn (t)b βdfα,Cramer-von Misesb (t) dtb β!2Fα,b (t) Fn (t)b βb (t) dtAnderson-Darlingfα,b βFα,b(t)(1 Fb(t))b βb βα,d The distributions of D, when sampling from a Weibull population,are known and p-values of observed values d of D can be calculatedp P (D d) BCSLIB: HSPFITWeibull Reliability Analysis—FWS-5/1999—29

1.0Kolmogorov-Smirnov Distance0.8n distance0 50100 150200250Weibull Reliability Analysis—FWS-5/1999—30

-2 -3 4.44.8 3.84.21104.6-1 -2-35.0 p(KS) 0.34 4.2 p(KS) 0.86 p(AD) 0.57 4.6p(CvM) 0.56 p(AD) 0.22 3.814.4p(CvM) 0.23-3-2-3 -1-1 -1 3.8 p(AD) 0.32 4.0 p(KS) 0.16 00 4.8p(CvM) 0.21 p(AD) 0.64.4 -14.0 -3 -3-35.0p(KS) 0.72p(CvM) 0.53 14.6 -2 -24.2 -213.8n 101 0-3-2 -1-1 p(AD) 0.69 -2 p(KS) 0.88p(CvM) 0.64 p(AD) 0.66 00 p(CvM) 0.62 p(AD) 0.35 p(KS) 0.880p(CvM) 0.27 p(AD) 0.62 p(KS) 0.28-1p(CvM) 0.541 p(KS) 0.6201Weibull Plots: n 10 Reliability Analysis—FWS-5/1999—31

Weibull Plots: n 20 4.8 3.0 0.017p(AD) 0.023 -3-34.41-2-3 11p(AD) 0.644.6p(KS) 0.04 4.0p(CvM) 0.6 -33.6 4.20 00-3 3.8p(KS) 0.850p(AD) 0.69 4.6-1 p(CvM) 0.634.2-2-2 4.5p(KS) 0.47-2-10p(AD) 0.413.5 -1p(CvM) 0.41 2.5 -2p(KS) 0.465.014.504.0 -13.5 p(CvM) 0.39p(AD) 0.38 n 20 p(KS) 0.46p(AD) 0.048 -3 p(CvM) 0.053-1 p(KS) 0.12 -3 1 -110p(AD) 0.028-2-2p(CvM) 0.027-3 p(KS) 0.13-2-10p(AD) 0.49 -11p(CvM) 0.491 p(KS) 0.554.0 Reliability Analysis—FWS-5/1999—32

-4 3.54.5100-1-1-2-3-44.53.5 3.5 0.181 p(CvM) 0.56 p(AD) 0.62 0p(KS) 0.85110-1-2-34.0-1 p(CvM) 0.22 p(AD) 0.23 3.5p(KS) 0.27 4.0-25.0 -24.0 -33.00 p(CvM) 0.66 p(AD) 0.72 p(AD) 0.52 p(AD) 0.039 p(CvM) 0.035-45.0-14.0p(CvM) 0.49 -23.0p(KS) 0.76p(CvM) 0.23-4 n 50-410 p(KS) 0.88-1 -32.0-2 -31-10p(AD) 0.32 p(AD) 0.12 p(KS) 0.15-4 -3 p(CvM) 0.35-4-4-3-2-10p(AD) 0.43p(KS) 0.22-2 p(CvM) 0.41-31p(KS) 0.71Weibull Plots: n 503.5 Reliability Analysis—FWS-5/1999—33

Weibull Plots: n 100-4 5.00-20-2p(CvM) 0.263.0 p(CvM) 0.025 p(AD) 0.032 p(KS) 0.0533.0 4.0 p(CvM) 0.58 p(AD) 0.6 5.03.5p(KS) 0.87 0.72p(CvM) 0.45p(AD) 0.430 4.5 4.0-40-23.5 p(AD) 0.43 p(KS) 0.21p(CvM) 0.48 0 p(CvM) 0.3 p(AD) 0.37 p(AD) 0.55 p(KS) 0.61-2-20p(KS) 0.473.0-45.004.0-23.0 n 100-40-2-4 p(AD) 0.72 p(CvM) 0.68-4p(KS) 0.86 -4 -2 p(AD) 0.3 p(CvM) 0.26-4p(KS) 0.58 Reliability Analysis—FWS-5/1999—34

Estimates and Confidence Bounds/Intervals For a target θθ α,θ tp α[ log(1 p)]1/β ,θ β,or θ Pα,β (T t)cc β)one gets corresponding m.l.e θc by replacing (α, β) by (α, Such estimates vary around the target due to sampling variation Capture the estimation uncertainty via confidence bounds For 0 γ 1 get 100γ% lower/upper confidence bounds θcL,γ & θcU,γ P θL,γ θ γcor P θ θU,γ γc For γ .5 get a 100(2γ 1)% confidence interval by [θcL,γ , θcU,γ ] P θL,γ θ θU,γ 2γ 1 γ ?ccWeibull Reliability Analysis—FWS-5/1999—35

Confidence Bounds & Sampling Variation, n 10M: MLE Approximate Method35000B: Bain Exact Method (Tables);true α2500090% confidence intervalsL: Lawless Exact Method (RAP);L BM12345624true βL BM090% confidence intervals8samples12345Weibull Reliability Analysis—FWS-5/1999—36

2500015000true 10-percentile5000095% lower confidence boundsConfidence Bounds & Sampling Variation, n 10L BML: Lawless Exact Method (RAP);123M: MLE Approximate Method450.20L BM0.10true P(T 10,000)0.095% upper confidence boundssamplesB: Bain Exact Method (Tables);12345Weibull Reliability Analysis—FWS-5/1999—37

3500025000 true α 1500095% confidence intervalsConfidence Bounds & Effect of n51020501002005001000 sample size7 456 3 2 true β 195% confidence intervals Weibull Reliability Analysis—FWS-5/1999—38

0.15 true P(T 10000) 0.050.095% upper confidence boundsConfidence Bounds & Effect of n 51020501002005001000 20000 true t .10 10000500095% lower confidence boundssample size Weibull Reliability Analysis—FWS-5/1999—39

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 knownthis shows up when we put n units on test simultaneously andterminate the test after the first r units have failed Interval censoring, inspection data, grouped data:ith unit is only known to have failed between two knowninspection time points, i.e., failure time Ti falls in (si, ei]only bracketing intervals (si, ei), i 1, . . . , n, become knownWeibull Reliability Analysis—FWS-5/1999—40

Incomplete Data or Censored Failure Times (continued) Random right censoring:units are observed until failed or removed from observationdue to other causes (different failure modes, competing risks) Multiple right censoring:units are put into service at different timesand 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 correlatewith the (potential) failure times,i.e., no censoring of anticipated failures All data (censored & uncensored) should enter analysisWeibull Reliability Analysis—FWS-5/1999—41

Type I or Time Censored Data: n yclesWeibull Reliability Analysis—FWS-5/1999—42

Type II or Failure Censored Data: n yclesWeibull Reliability Analysis—FWS-5/1999—43

Interval Censored Data: n 10unit10?9 ](8?7 (6? ](5?( 4]3( ?2?1(020004000 ]6000800010000cyclesWeibull Reliability Analysis—FWS-5/1999—44

Multiply Right Censored Data: n yclesWeibull Reliability Analysis—FWS-5/1999—45

Nonparametric MLEs of CDF For complete, type I & II censored data the nonparametric MLE is#{Ti t; i 1, . . . , n}dF (t) ,n t for complete sample, t t0 for type I censored (at t0) sample, t T(r) for type II censored (at T(r), rth smallest failure time) For multiply right censored data with failures at t?1 . . . t?k thenonparametric MLE (Kaplan-Meier or Product Limit Estimator) is ?"# 1 for t tdδ1 (t)δk (t)iF (t) 1 (1 p̂1)· · · (1 p̂k ), δi(t) 0 for t?i twhere p̂i di/ni, ni # units known to be at risk just prior to t?iand di # units that failed at t?i The nonparametric MLE for interval censored data is complicatedWeibull Reliability Analysis—FWS-5/1999—46

0.40.8empirical and trueand Weibull mledistribution functions0.0n 1015000250000500015000250000.40.0n 50050001500025000n 1000.00.40.850000.80n 300.00.40.8Empirical CDF (complete samples)050001500025000Weibull Reliability Analysis—FWS-5/1999—47

.1500025000. .05000.15000n 30250000.4. . . .0500015000n 50250000. mle of CDFWeibull mle of CDFand true CDF0.40.8Nonparametric MLE for Type I Censored Data. . . .05000.15000n 10025000Weibull Reliability Analysis—FWS-5/1999—48

.1500025000. . .05000n 30.15000250000.4. . . . . . .0500015000n 50250000. mle of CDFWeibull mle of CDFand true CDF0.40.8Nonparametric MLE for Type II Censored Data. . . . . .0500015000.n 10025000Weibull Reliability Analysis—FWS-5/1999—49

. .n 10. .1500025000n 30. . . . . . . . . .0500015000250000.4. . . . . . . .05000.15000n 50250000. . CDFWeibull mle for CDFtrue CDF0.40.8Kaplan-Meier Estimates (multiply right censored samples). . . . . . .05000.15000n 10025000Weibull Reliability Analysis—FWS-5/1999—50

1.0Nonparametric MLE for Interval Censored Datasuperimposed is the true Weibull(10000,2.5)inspection points roughly 3000 cycles apart0.40.20.0CDF0.60.8that generated the 1,000 interval censored data cases050001000015000cyclesWeibull Reliability Analysis—FWS-5/1999—51

1.0Nonparametric MLE for Interval Censored Datasuperimposed is the true Weibull(10000,2.5)inspection intervals randomly generated from same Weibull0.00.20.4CDF0.60.8that generated the 10,000 interval censored data cases050001000015000cyclesWeibull Reliability Analysis—FWS-5/1999—52 casesestimated .99-quantile0.0probability of questionnaire returnApplication to Y2K Questionnaire Return Data050100150200250300daysWeibull Reliability Analysis—FWS-5/1999—53

Plotting Positions for Weibull Plots (Censored Case) For complete samples we plotted (log[T(i)], log [ log(1 pi)]) with i .5 1 d1 i i 1 d pi F (T(i)) F (T(i 1)) n22 nn For type I & II & multiply right censored samples?], log [ log(1 pi)]), wherewe plot in analogy (log[T(i)? . . . T(k)are the k distinct observed failure times andT(1)#1 "d ?d?pi F (T(i)) F (T(i 1))2dand F(t) is the nonparametric MLE of F (t) for the censored sampleWeibull Reliability Analysis—FWS-5/1999—54

Weibull Plot, Multiply Right Censored Sample n 50.990.900.632probability.500.200.100 .010true model, Weibull( 100 , 3 )m.l.e. model, Weibull( 96.27 , 2.641 ) n 50least squares model, Weibull( 96.3 , 2.541 ) n 50.0011020501002005001000cycles/hoursWeibull Reliability Analysis—FWS-5/1999—55

Confidence Bounds for Censored Data For complete & type II censored data RAP computes exact coverageconfidence bounds for α, β, tp, and P (T t). WEIBREG and commercial software (Weibull , WeibullSMITH,common statistical packages) compute approximate confidencebounds for above targets, based on large sample m.l.e. theory RAP & WEIBREG are Boeing code, runs within DOS modeof Windows95 (interface clumsy, but job gets done), contact me Boeing has a site license for Weibull Version 4in the Puget Sound areacontact David Twigg (425) 717-1221, [email protected] There is a Weibull Version 5 outWeibull Reliability Analysis—FWS-5/1999—56

Weibull Plot With Confidence Bound0.999alpha 102.4 [ 82.52 , 143.8 ] 95 %beta 2.524 [ 1.352 , 3.696 ] 95 %n 30 , r 130.9900.9000.500 0.200 0.100 0.0100.00112351020501002005001000Weibull Reliability Analysis—FWS-5/1999—57

Weibull Plot With Confidence Bound (Variation)0.999alpha 89.79 [ 77.4 , 109.4 ] 95 %beta 3.674 [ 2.15 , 5.198 ] 95 %n 30 , r 130.9900.9000.5000.200 0.100 0.0100.00112351020501002005001000Weibull Reliability Analysis—FWS-5/1999—58

Weibull Regression Model & Analysis Recall tp α[ log(1 p)]1/β orlog(tp) log(α) wp/βwith wp log[ log(1 p)] Often we deal with failure data collected under different conditions– different part types– different environmental conditions– different part users Regression model on log(α) (multiplicative on α)log(αi) b1Zi1 . . . bpZipwhere Zi1, . . . , Zip are known covariates for ith unit Now we have p 1 unknown parameters β, b1, . . . , bp parameter estimates & confidence bounds using MLEsWeibull Reliability Analysis—FWS-5/1999—59

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 cVolt ,T (Volt) T (VoltU ) where usually c 0VoltUthis means cVolt α(Volt) α (VoltU ) VoltUorlog [α(Volt)] log [α (VoltU )] c log (Volt) c log (VoltU ) b1 b2Zwithb1 log [α (VoltU )] c log (VoltU ) , b2 c, and Z log (Volt)Weibull Reliability Analysis—FWS-5/1999—60

Accelerated Life Testing: Arrhenius Model Another way of accelerating failure in processes involvingchemical reaction rates is to increase the temperature Arrhenius proposed the following acceleration model withtemperature measured in Kelvin (tempK temp C 273.15) κκ T (temp) T (tempU ) exp temp tempUorκκlog [α(temp)] log [α(tempU )] temp tempU b1 b2 Zwithκb1 log [α(tempU )] ,tempUb2 κ,and Z temp 1Weibull Reliability Analysis—FWS-5/1999—61

Pooling Data With Different α’s Suppose we have three groups of failure dataT1, . . . , Tn1 W(α̃1, β),Tn1 1, . . . , Tn1 n2 W(α̃2, β),Tn1 n2 1, . . . , Tn1 n2 n3 W(α̃3, β) We can analyze the whole data set of N n1 n2 n3 values jointlyusing the following model for αj and dummy covariates Z1,j , Z2,j & Z3,jlog(αj ) b1Z1,j b2Z2,j b3Z3,j ,where b1 log(α̃1),Z1,j 1for j 1, . . . , N ,and Z3,j 1j 1, 2, . . . , Nb2 log(α̃2) log(α̃1),Z2,j 1and b3 log(α̃3) log(α̃1)for j n1 1, . . . , n1 n2 ,for j n1 n2 1, . . . , N ,& Z3,j 0& Z2,j 0else,else. Advantage: Smaller estimation errorWeibull Reliability Analysis—FWS-5/1999—62

Pooling Data (continued)log(α1) .log(αn1 ) .log(αn1 1) .log(αn1 n2 ) .log(αn1 n2 1) .log(αN ) b1 · 1 b2 · 0 b3 · 0 .b1 · 1 b2 · 0 b3 · 0 .b1 · 1 b2 · 1 b3 · 0 .b1 · 1 b2 · 1 b3 · 0 .b1 · 1 b2 · 0 b3 · 1 .b1 · 1 b2 · 0 b3 · 1 log(α̃1)log(α̃1)log(α̃1) [log(α̃2) log(α̃1)] log(α̃2)log(α̃1) [log(α̃2) log(α̃1)] log(α̃2)log(α̃1) [log(α̃3) log(α̃1)] log(α̃3)log(α̃1) [log(α̃3) log(α̃1)] log(α̃3)Weibull Reliability Analysis—FWS-5/1999—63

Accelerated Life Testing: Model & Data200thousand cycles1005020 10 5 2150100150200250300350400450500VoltageWeibull Reliability Analysis—FWS-5/1999—64

Accelerated Life Testing: Model, Data & MLEs200thousand cycles10050true linemle line20 10mle for .01-quantile5 95% lower bound for Weibull Reliability Analysis—FWS-5/1999—65

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 quantitiesof interest are explicit and monotone functions of θFθ (t0) Pθ (T t0) 1 exp( t0/θ)Rθ (t0) Pθ (T t0) exp( t0/θ)andtp(θ) θ [ log(1 p)]Weibull Reliability Analysis—FWS-5/1999—66

Complete & Type II Censored Exponential Samples T E(θ) Exponential with mean θ, E(θ) W(α θ, β 1) For complete exponential samples T1, . . . , Tn 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 computingθ̂L,γ2 TTT ,χ22r,γwhere T T T T(1) · · · T(r) (n r)T(r) Total Time on Test,and χ22r,γ γ-quantile of the χ22r distributionget this in Excel via GAMMAINV(γ, r, 1) or CHIINV(1 γ, 2r)/2 These bounds have exact confidence coverage propertiesWeibull Reliability Analysis—FWS-5/1999—67

Multiply Right Censored Exponential Data Here we define T T T T1 . . . Tn as the total time on testbut now Ti is either the observed failure time of the ith partor the observed right censoring time of the ith partThe number r of observed failures is random here, 0 r n Approximate 100γ% lower confidence bounds for θ are computed asθ̂L,γ2 TTT 2χ2r 2,γ This also holds for r 0, i.e., no failures at all over exposure periodWeibull Reliability Analysis—FWS-5/1999—68

Weibull Shortcut Methods for Known β T W(α, β) (Weibull) T β E(θ) Exponential with mean θ αβ As 100γ% lower confidence bound for α compute α̂L,γ 1/β2 T T T (β) 2χf,γ where for complete or type II censored samples we take f 2r andβββT T T T(1) · · · T(r) (n r)T(r)and for multiply right censored data we take f 2r 2 andT T T T1β · · · Tnβ . Sensitivity should be studied by repeating analyses for several β’sWeibull Reliability Analysis—FWS-5/1999—69

A- and B-Allowables, ABVAL Program My first Weibull work led 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 samplesfrom the 3-parameter Weibull distribution.– The A-Allowable is a 95% lower confidence boundfor 99% of the population, i.e., for the .01-quantile.– The B-Allowable is a 95% lower confidence boundfor 90% of the population, i.e., for the .10-quantile. Hybrid approach of using Mann-Fertig threshold estimatecombined with maximum likelihood estimates for α and β,took lot of simulation and tuning and is very specialized in itscapabilities It is now a method in MIL-HDBK-5, I still maintain softwareWeibull Reliability Analysis—FWS-5/1999—70

Selected References Abernethy, R.B. (1996), The New Weibull Handbook, 2nd edition,Reliability Analysis Center 800-526-4802 Bain, L.J. (1978), Statistical Analysis of Reliability and Life-TestingModels, Marcel Dekker, New York 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 LifetimeData, John Wiley & Sons, New York Mann, N.R., Schafer, R.E., and Singpurwalla, N.D. (1974), Methodsfor Statistical Analysis of Reliability and Life Data, John Wiley &Sons, New York Meeker, W.Q. and Escobar, L.A. (1998), Statistical Methods forReliability Data, John Wiley & Sons, New YorkWeibull Reliability Analysis—FWS-5/1999—71

Meeker, W.Q. and Hahn, G.J. (1985), Volume 10: How to Plan anAccelerated Life Test—Some Practical Guidelines, American Societyfor Quality Control Nelson, W. (1982), Applied Life Data Analysis, John Wiley & Sons,New York— (1985), “Weibull analysis of reliability data with few or nofailures,” Journal of Quality Technology 17, 140-146— (1990), Accelerated Testing, Statistical Models, Test Plans andData Analyses, John Wiley & Sons, New York Scholz, F.W. (1994), “Weibull and Gumbel distribution exactconfidence bounds.” BCSTECH-94-019 (RAP)— (1996) “Maximum likelihood estimation for type I censoredWeibull data including covariates.” ISSTECH-96-022— (1997), “Confidence bounds for type I censored Weibull dataincluding covariates.” SSGTECH-97-025 (WEIBREG)The Boeing Company, P.O. Box 3707, MS-7L-22, Seattle WA 98124-2207Weibull Reliability Analysis—FWS-5/1999—72

We focus on analysis using the 2-parameter Weibull model Methods and software tools much better developed Estimation of in the 3-parameter Weibull model leads to complications When a 3-parameter Weibull model is assumed, it will be stated explicitly Weibull Re