Uncertainty Analysis of Weibull Estimators for Interval-Censored Data

Sameer Vittal, PhD, C.R.E., GE Energy Randolph Phillips, PhD, GE Energy

Key Words: Weibull, Interval-censored, inspection data

SUMMARY

The two-parameter Weibull is one of the most popular statistical distributions used to describe the reliability and time-to-failure characteristics of mechanical systems. This paper deals with a Monte Carlo approach to quantify two major sources of uncertainty in Weibull analysis that are commonly found in industrial applications – parameter error introduced due to extended inspection intervals, and error due to different estimation algorithms for interval-censored data. The problem of extended inspection intervals is important to industry as many components are designed to “fail safe” and failure is detected only upon inspection or maintenance of another item. The objective of this paper is to describe some of the numerical methods and processes used to estimate Interval-Censored Weibull parameters and estimate their bias for a typical industrial application. Some of the estimators considered here include five regression-based methods (Rank-Regression, Probit, Kaplan-Meier, Inspection-Option and Herd-Johnson) and five maximum likelihood estimators (standard MLE, median and mean-bias corrected MLE, mid-point and full interval-censored MLE algorithms)

1 INTRODUCTION

The Weibull distribution is a popular, widely cited statistical distribution used to characterize the reliability of mechanical systems, and has been the subject of intense research since its introduction by Prof. Wallodi Weibull in 1951 [1]. There are numerous articles on estimating the parameters of this versatile distribution [2] and many software programs include different estimation algorithms like Rank-Regression and Maximum-Likelihood Estimation (MLE) methods. While there are several variants of the Weibull distribution [3], the most popular one is the “standard” two-parameter Weibull, whose cumulative density function is

( ) ( )1 expF t t βη = − − . Here, ( )F t is the probability of failure

at time‘t’, and the two parameters are β , which is the slope of the distribution and , which is the “characteristic life” of the model. A typical objective of ‘Weibull analysis” in industry is to obtain the distribution parameters ,β η and their associated uncertainties from field data which is often skewed by measurement error, imperfect coverage and censoring due to inspection and units that haven’t failed. There has been a lot of effort in developing algorithms to estimate the Weibull parameters when the data is either sparse or censored [7] . A

good review of the estimation methods for complete and censored data is found in the textbook by Meeker [6] and the Weibull handbook by Abernathy [5]. There are many cases where it is not possible to estimate the failure time from field data and the Weibull has to be estimated using Bayesian methods [8, 9] or by imputation methods, bootstrapping, etc.

In this paper, we deal with one particular type of data, referred to as “interval-censored” data. Consider a unit which is periodically inspected for failure. At some point ( )t t− ∆ , the unit was functioning normally, however it has failed before the next inspection time t . The “true” time-to-failure was somewhere in the interval ( ),t t t− ∆ and a common assumption is to state the failure time as ‘t’ and estimate the Weibull parameters accordingly. However, this introduces an error in the model, which is a function of a) The inspection length t∆ and b) The type of estimator used.

In Table 1, we consider a random sample of 25 “times-to-failure” drawn from a Weibull distribution with β =2.2 and η =1000 hours. This set is inspected every 200 hours, and 10 inspections are considered. All 25 failures are detected in these inspections. The time-to-failure data (TTF) is shown in column [B] in Table 1. This data is sorted in ascending order in column [C]. The data in the “Observed TTF” columns is the equivalent of field data that most reliability engineers have to deal with – and it is clear that any Weibull parameters estimated from this dataset will be influenced by the impact of inspection intervals. As the inspection interval is increased, we can expect the error in estimating β and η to increase.

The data in Table 1 was analyzed using standard estimation algorithms and estimated values of Beta & Eta are as shown in Table 2. The variability is significant and could adversely impact part reliability warranties, reliability-based maintenance policies, etc.

2 OVERVIEW OF ESTIMATION METHODS

There are several estimation methods for the Weibull distribution, and they fall into two main categories – those based on minimizing least squares error using regression, and those based on maximizing the likelihood function. We first briefly describe the regression-based estimators used in this study.

The two-parameter Weibull distribution can be linearized by taking logarithms on both sides and simplifying as shown below in Eq (1) which is of the form, Y mX c= + , where the transformed variables are as per Eq (2).

0-7803-9766-5/07/$25.00 ©2007 IEEE 292

( )

( ) ( ) ( )

1 exp

1ln ln ln ln1

tF t

tF t

β

η

β β η

= − −

→ = − −

1)

Table 1 : Example of interval censoring

Summary of results Eta Error Beta Error R^2 Original Parameters 1000 2.200 Rank-Regression 951 5% 1.742 21% 95.1% Probit Analysis 887 11% 1.756 20% 95.6% Kaplan-Meier 937 6% 1.809 18% 96.3% Inspection-Option 1230 -23% 1.465 33% 99.4% Herd-Johnson 987 1% 1.752 20% 95.9% MLE-Standard 948 5% 1.797 18% NA MLE-Mid Point 821 18% 1.520 31% NA MLE-Interval Censor 820 18% 1.517 31% NA

Table 2 : Weibull estimates for censored data using multiple estimators, Interval length = 200 hrs

( ) ( ) ( )1ln ln , ln , , ln1

Y X t m cF t

β β η

= = = = − − (2)

Consider a set of ‘N’ random times to failure (TTF) drawn from a Weibull distribution and then sorted in an

ascending order., i.e. we have

{ } ( )1

1 2

, , ,

1, ,

N

N

t t WEI

t t ti N

β η≤ ≤ ≤

=

… ∼… (3)

Using the method of Rank-Regression, the rank of the ith failure time, ( )iF t , is given by Bernard’s approximate formula is as per Eq (4) below.

( ) 0.30.4i

iF tN

−≈+

(4)

Once the pairs of ( )( ),i iF t t are available, the Weibull coefficients can be estimated from Eq (1) and (2) as described earlier. For interval censored data, the common practice is to assume the failure time was the inspection time at which failure was detected.

An alternate approach is to estimate the failure probabilities using the non-parametric Kaplan-Meier (KM) method. Consider the failure-time set as per Eq (3). Using the KM method, the “reverse rank” is first computed using Eq (5) and the reliability is calculated using a step-wise process as per Eq (6). Finally, the failure probabilities are easily calculated as per Eq (7).

( ) ( )1iKM t N i= + − (5)

( ) ( )( ) ( )1

1iKM i KM i

i

KM tR t R t

KM t − −

= ⋅

(6)

( ) ( )1i KM iF t R t= − (7) Similar to the Kaplan-Meier technique, a method known

as the Herd-Johnson (HJ) approach is also used. This method also requires a set of sorted failure time data as per Eq (3), and a “reverse rank” is estimated for each failure data point using the same approach as KM, as shown in Eq (8). The reliability is estimated using Eq (9), and the corresponding failure probability is as per Eq (10).

( ) ( )1iHJ t N i= + − (8)

( ) ( )( ) ( )11

iHJ i HJ i

i

HJ tR t R t

HJ t −

= ⋅ + (9)

( ) ( )1i HJ iF t R t= − (10) The required Weibull parameters are estimated from a

standard least-squares regression using Eq (10) and Eq (2). A simple approach favored by statisticians when a lot of uncensored data is available is the Probit method. As with all regression-based estimators the first step is to obtain the sorted failure-time dataset as per Eq (3), and the failure probability is calculated quite simply using Eq (11).

( )iiF tN

= (11)

The “Inspection Option” method [5] is also included in this study. The failure-time data is initially processed as per Eq (3), and the median rank (approximated by Bernard’s equation, Eq (4)) is computed for the highest failure-time data point in a given interval. Then, least-squares regression is used to estimate the Weibull parameters. Consider a case where 5 parts have failed in a certain inspection interval. In the rank-regression approach, a rank would be calculated for each of those five failures. Using the “Inspection-Option”

[A] [B] [C] Interval IntervalS.No. TTF-raw TTF-sorted OK Failed

1 1877 46 0 2002 111 75 0 2003 765 111 0 2004 967 205 200 4005 46 277 200 4006 870 322 200 4007 1468 396 200 4008 978 422 400 6009 277 441 400 60010 760 462 400 60011 322 509 400 60012 441 537 400 60013 537 656 600 80014 972 760 600 80015 1412 765 600 80016 422 870 800 100017 1138 967 800 100018 509 972 800 100019 656 978 800 100020 205 1138 1000 120021 462 1199 1000 120022 75 1412 1400 160023 1644 1468 1400 160024 1199 1644 1600 180025 396 1877 1800 2000

Actual TTF Observed TTF

293

approach, only one rank is calculated corresponding to the inspection time and the other four data points are ignored.

We briefly describe iterative methods like the likelihood-based estimators that are preferred due their superior statistical properties. In the standard MLE (Newton-Raphson) method we would like to find the values of Beta and Eta that maximize the likelihood function for the data. The various steps are, 1) the dataset { }1, , nTTF t t= … where the TTF values are

Obtain at the end of the inspection interval. Assume a starting value for ,η β .

2) Compute the log-likelihood function using Eq (12) shown below

( ) ( ) ( ) ( )1

ln 1 ln lnN

ii

ttβ

β β β ηη=

Λ = + − − −

∑ (12)

3) Calculate the first partial derivatives of the likelihood function with respect to the Weibull parameters, as per Eq (13).

1

1

1 ln ln

Ni

i

Ni i i

i

t

t t t

β

β

β βη η η η

β β η η η

=

=

∂Λ = − + ∂ ∂Λ = + − ∂

∑

∑ (13)

4) Compute second partial derivatives (Eq (14)). 2

2 2

2

22

2 2

2

1

1 1ln

1 ln

1 l

i

i i

i i

i

t

t t

t t

t

β

β

β

β

β β βη η η η η η

βη β η η η η η

β β η η

ββ η η η η

∂ Λ = − + ∂ ∂ Λ = − − + ∂ ∂

∂ Λ = − − ∂

∂ Λ = − − ∂ ∂

∑

∑

∑

1n itη η

+

∑

(14)

5) Solve the set of simultaneous equations shown in Eq (15) for ‘a’ and ‘b’

2 2

2

2 2

2

a b

a b

η η βη

β β ηβ

∂Λ ∂ Λ ∂ Λ = − + − ∂ ∂ ∂∂ ∂Λ ∂ Λ ∂ Λ = − + − ∂ ∂ ∂∂

(15)

6) Update the values of Eta and Beta for the next iteration, as per Eq (15)

1

1

k k

k k

ab

η ηβ β

+

+

= += +

(16)

7) Check to see if the values have converged. 1 1,k k k kTol Tolη η β β+ +− > − > (17)

8) If the difference is greater than a user-specified tolerance (Tol), repeat steps (2) through (7) with the updated values of Eta and Beta until Eq (12) is maximized. These are the MLE estimates of the Weibull.

Note: When the sample size is small (N<30), Abernathy

[4] had proposed a correction to the Weibull MLE beta estimate via a correction factor C4.

4

22 2

312

N

CNN

− Γ =

−− Γ

(18)

Where ( )Γ i is the standard gamma function. In the Reduced-Bias-Adjustment (RBA) MLE method, a

biased value of Beta is first obtained from the Likelihood ratio equation described in Eq (12) by solving Eq (19) for Beta. Any standard root-finding algorithm can be used.

( ) ( )1 ln1ln 0ii i i

i i i

tt t t

rβ β

β

− ∗ − − = ∑ ∑ ∑ (19)

Once β is obtained from Eq (19), it is corrected ( )Uβ using the C4 factor from Eq (18), as per Eq (20).

( )3.54U Cβ β= ⋅ (20)

The values of Eta is finally obtained using Eq (21)

( )1

ii

t tβ βη = ∑ (21)

We have incorporated the Bias correction for both the mean value of beta obtained from a Monte Carlo simulation (MLE-Mean) as well as the median value (MLE-Median) in this paper.

In the Mid-Point MLE method, the approach is identical to the steps described for standard MLE, except the failure time set { }1, , nTTF t t= … is calculated based on the mid-value of the inspection interval, rather then the ending value. E.g. If the part was OK at 400 hrs and had failed at 600 hrs, then it is 500 hours. In some cases [10], we obtain the MLE estimates using the mid-point and take it through one more iteration and update Eta & Beta. This method provides estimates that have good statistical properties, almost as accurate as those obtained using a full interval censored MLE algorithm described next.

From a strict statistical perspective, the complete Interval-Censored MLE method is the most accurate, as the likelihood function reflects the uncertainty in both ends of the inspection interval. The start of the inspection interval is denoted by iL and the end if the interval is iR . The true failure time is

within this interval, i.e. , ,i i it L R∈ . Mathematically, the likelihood function is now shown in Eq (21)

ln exp expi iL Rβ β

η η

Λ = − − − ∑ (22)

The first partial derivates are as per Eq (23). The second partial derivatives can be obtained in closed form, but we found it easier to calculate those using finite-differences.

( )( ) ( )( )

( )( ) ( )( )

β β

β β

β βη η

η η

η ηβη η

− ∂Λ = − ∂ −

∑i i

i i

R Li i

L R

L Re e

e ei

294

( ) ( )( ) ( )( )

( )( ) ( )( )

β β

β β

β βη η

η η

ηη ηβ

β η

− ∂Λ = − ⋅ ∂ − ∑

ln i i

i i

R Li i

i

L R

L RL e e

e e

(23)

The rest of the update process follows the standard MLE algorithm, i.e. we calculate the next updates for Eta & Beta using Eq (15) and Eq (16), and keep iterating until the values of Eta & beta have converged.

100.00 10000.001000.001.00

5.00

10.00

50.00

90.00

99.00

Probability - Weibull

Time, (t)

Unr

elia

bilit

y, F

(t)

WeibullMLE-Interval

W2 MLE - RRM MEDF=25 / S=0

β1=1.5172, η1=819.6505

MLE-Mid

W2 MLE - SRM MEDF=25 / S=0

β2=1.5196, η2=821.3155

MLE-Std

W2 MLE - SRM MEDF=25 / S=0CB[FM]@95.00%2-Sided-B [T2]

β3=1.7965, η3=948.1780 Figure 1: Reliability plots using standard MLE, Mid-Point

MLE and Interval-Censored MLE

0.94

4.00

1.55

2.16

2.78

3.39

550.00 1290.00698.00 846.00 994.00 1142.00

Contour Plot

Eta

Beta

WeibullMLE-Interval

W2 MLE - RRM MEDF=25 / S=0

β1=1.5172, η1=819.6505

MLE-Mid

W2 MLE - SRM MEDF=25 / S=0

β2=1.5196, η2=821.3155

MLE-Std

W2 MLE - SRM MEDF=25 / S=0Level 1 : 95%

β3=1.7965, η3=948.1780 Figure 2: Joint confidence interval plots using standard, Mid-

Point & Interval-Censored MLE

We ran several simulations and noticed that the mid-point MLE method provide results that were very close to that obtained from the full Interval-censored MLE, at a fraction of the computational cost (Figures 1 and 2). For the example data shown in Table 1, we noticed the joint confidence intervals for Eta & Beta as well Reliability almost overlap for the Mid-Point MLE and Full interval-censored MLE algorithms. A similar trend was observed for other Weibull’s as well, and we concluded that for the purposes of simulation, we can use the Mid-Point MLE as a surrogate for the full Interval-Censored MLE algorithm, without any significant loss in accuracy. Note

that this is only to reduce the computational burden for the Monte Carlo runs, and we recommend the full interval-censored MLE algorithm be used for individual data sets (often seen in industry). The next section describes the simulation process used to generate the Weibull’s used in this study

3 SIMULATION PROCEDURE

One of the objectives of this paper was to describe a process by which one can choose the estimation method that best fits their needs and this is achieved using a Monte Carlo approach. The first step was to code up the following algorithms in Matlab – Rank Regression, Kaplan-Meier, Herd-Johnson, Probit, Inspection-Option, Standard MLE, MLE with Reduced Bias Adjustment (median and mean), Mid-Point MLE and Interval-censored MLE.

The various steps in the simulation are, 1. Choose an initial, representative value of Eta and Beta

(E.g. Eta = 56000 hrs, Beta = 2) 2. Choose part sample size ‘N’, Monte Carlo runs ‘MC’ and

inspection length ‘L’ (hr). 3. Start Monte Carlo run, MC = 1 4. Generate TTF { } ( )1Data: , , ,nt t WEI β η… ∼ 5. Censor the data, based on inspection length. I.e. we

generate a data set for the beginning and end of the corresponding inspection period

{ }{ }

1

1

, ,

, ,n

n

OK L L

Fail R R

=

=

…

…

6. Using the various estimators described in this paper, calculate an estimate of Eta & Beta for the inspection data set for each estimator.

7. Increase, MC = MC + 1 8. Once the Monte Carlo runs are completed, post-process

the distributions for Eta & Beta. 9. Obtain DPMO and Z-Scores for each type of estimator,

and choose the best one. 10. Increase the inspection length, L, and repeat Steps 3

through 9 until all inspection lengths have been analyzed. 11. Choose a new value of Eta & Beta and repeat steps 3

through 10 until all (Eta, Beta) combinations of interest have been analyzed We noticed that in general, the Eta & Beta samples tend

to be normally distributed. This is to be expected for MLE methods, as the parameter distributions are asymptotically normal as sample sizes tend to infinity. The numerical accuracy of the code was verified by checking the results with those from commercial reliability programs and hand calculations. The error in Weibull parameters tended to decrease (for some methods) with increase in the part sample size ‘N’. We chose a sample size of 100 as it was close to the part counts found in the field (from the author’s experience with industrial gas turbines) for the components of interest. In practice, we would recommend choosing a sample size that is close to the system being studied.

We chose the following approach to compare the various estimators. From a practical perspective, we recognize that for interval censoring it is impossible to obtain highly accurate

295

Weibull estimates – however, we would like a method that consistently produces estimate that lie within an acceptable “tolerance band” of the true value. This is shown in Figure 3.

Figure 3 : Variation in Weibull estimates (Beta) for a simulation run

For a given estimator type, the probability of a defect is the probability that a Weibull parameter (Eta, Beta) estimated from interval censored data falls outside a tolerance band around the “true” part Weibull. The mean & variance of the estimated Eta & Beta’s are obtained from Monte Carlo trials. In this case, the upper, lower bounds are assumed to be +/-

10% of the true value of Eta & Beta. The defect probability, Z-scores and DPMO (defects Per Million Opportunities) are calculated using Eq (24). The same process applies to Eta calculations as well.

[ ] [ ]

( )

LSL USL

1 6

P = Prob < Prob >ˆ ˆ

1

, 10

USL LSLP

Z P DPMO Pβ β

β β β β

β β β βσ σ

−

+

− −= − Φ + Φ

= Φ = ∗

(24)

The Z-scores and DPMO values for a given estimator are

now the measures of performance for that algorithm, and form the basis of comparison. The next section describes some of our results and the guidelines that were proposed to correct for Bias.

4 RESULTS AND DISCUSSION

Our results indicate that the quality of the Weibull (best estimates of Eta, Beta) is strongly influenced by the ratio of the inspection interval to the characteristic life, i.e. the ratio of L/eta. Larger inspection intervals result in poorer estimates, and for moderate beta’s (Beta > 4) and high ratio’s of L/eta (> 0.7) all algorithms break down – and any numerical results obtained in this range cannot be trusted. We found that methods like Inspection option, rank regression and even the standard-MLE method have a positive bias, i.e. the predicted results and reliability are optimistic compared to the real values – this is a problem, as we prefer a conservative answer rather than one that provides a false sense of security. The Mid-Point MLE and full interval-censored MLE provided almost identical answers and had the highest accuracy and lowest variability. Probit analysis provided a conservative answer for large inspection intervals and the inspection option worked well only for low L/eta ratios (<0.1).

Table 3 shows an example of the Z-score and DPMO calculations for Beta = 2 and Eta = 56000 hrs. This provides a better way of comparing the performance of the various estimators, and it is clear that overall the Mid-Point MLE method (and Interval-censored MLE) had the lowest “failure” probabilities. For example, there is a 38% chance that the Mid-Point MLE will fail to estimate the Beta within the accepted tolerance. Note that these values are applicable to this case study only – however the method used to evaluate estimators and choose inspection intervals can be applied to a variety of industrial settings. Table 3 also includes Z-scores and DPMO estimates for Eta as well as the B1 life. These charts provide a rough estimate of the error one would expect for a given inspection length. In reality, the true values of eta and beta are rarely known a priori, but they can be estimated with reasonable accuracy using probabilistic engineering tools, and an appropriate inspection plan can now be established.

Guidelines on the appropriate choice of estimation algorithm have also been developed, per Figure 4. The X-axis is the ratio of inspection length to Eta (L/Eta ratio) and the Y-axis is the Weibull slope (beta). For low beta’s (< 2) all methods in the figure seem to work well, with the Probit and Inspection MLE (or it’s surrogate, the Mid-Point MLE) being best. As the slope increases, (beta ~ 6, L/Eta ~ 0.5) the Probit approach works well. Note that Probit is particularly sensitive to the sample size, so it would not be appropriate for small sample sizes. For data that includes a significant fraction of suspensions, we recommend using the Interval-MLE or Mid-Point MLE estimator. It is important to note that this paper deals only with complete data, i.e. all data points are failures.

Preliminary results indicate that conventional approaches (rank-regression or standard MLE) often produce “optimistic” Eta’s and beta’s, i.e. produce non-conservative reliability models. While our numerical results and conclusions are specific to this case study, we have notice a similar trend in

ˆLSL True USLβ β β β

Bias

ˆLSL True USLβ β β β

Bias

Beta CalculationsMethod BetaMu BetaSig BetaBias Pf DPMO ZstRR 2.36063 0.23618 18.032 76.06% 760589 -0.708199MLE 2.41784 0.25455 20.892 81.15% 811550 -0.883622MLE-RBA1 2.37505 0.25005 18.753 76.88% 768787 -0.734856MLE-RBA2 2.34495 0.24688 17.248 73.51% 735086 -0.628268Insp 2.08404 0.17559 4.202 30.74% 307369 0.5033223Probit 2.04727 0.22568 2.363 38.59% 385893 0.290039KM 2.49896 0.27997 24.948 86.35% 863471 -1.09605HJ 2.46043 0.27497 23.021 83.64% 836368 -0.979642MLE-mid 2.01691 0.22725 0.845 38.01% 380130 0.30514Eta CalculationsMethod EtaMu EtaSig EtaBias Pf DPMO ZstRR 66120.31 4282.48 18.072 85.45% 854531 -1.056064MLE 65760.58 4262.46 17.43 83.56% 835649 -0.976731MLE-RBA1 65760.58 4262.46 17.43 83.56% 835649 -0.976731MLE-RBA2 65760.58 4262.46 17.43 83.56% 835649 -0.976731Insp 64987.13 3280.31 16.048 84.91% 849100 -1.032583Probit 54246.78 4143.95 -3.131 21.46% 214624 0.7904805KM 64819.31 4262.77 15.749 77.53% 775299 -0.756413HJ 66080.6 4310.89 18.001 85.08% 850821 -1.039959MLE-mid 56143.51 4334.93 0.256 19.67% 196661 0.8536085B1 life calculationsMethod B1mu B1sig B1Bias Pf DPMO ZstRR 25390.11 3139.58 39.681 95.92% 959154 -1.740946MLE 25818.76 3277.17 42.039 96.42% 964172 -1.8013MLE-RBA1 25389.7 3260.95 39.679 95.38% 953782 -1.682693MLE-RBA2 25083.03 3249.04 37.992 94.50% 944957 -1.597805Insp 22088.04 3037.1 21.515 78.43% 784281 -0.786731Probit 18107.1 3443.62 -0.386 59.77% 597678 -0.247342KM 26224.17 3466.73 44.27 96.60% 966039 -1.82552HJ 26355.66 3469.09 44.993 96.86% 968618 -1.860863MLE-mid 18347.7 3085.25 0.938 55.64% 556354 -0.141731

Beta CalculationsMethod BetaMu BetaSig BetaBias Pf DPMO ZstRR 2.36063 0.23618 18.032 76.06% 760589 -0.708199MLE 2.41784 0.25455 20.892 81.15% 811550 -0.883622MLE-RBA1 2.37505 0.25005 18.753 76.88% 768787 -0.734856MLE-RBA2 2.34495 0.24688 17.248 73.51% 735086 -0.628268Insp 2.08404 0.17559 4.202 30.74% 307369 0.5033223Probit 2.04727 0.22568 2.363 38.59% 385893 0.290039KM 2.49896 0.27997 24.948 86.35% 863471 -1.09605HJ 2.46043 0.27497 23.021 83.64% 836368 -0.979642MLE-mid 2.01691 0.22725 0.845 38.01% 380130 0.30514Eta CalculationsMethod EtaMu EtaSig EtaBias Pf DPMO ZstRR 66120.31 4282.48 18.072 85.45% 854531 -1.056064MLE 65760.58 4262.46 17.43 83.56% 835649 -0.976731MLE-RBA1 65760.58 4262.46 17.43 83.56% 835649 -0.976731MLE-RBA2 65760.58 4262.46 17.43 83.56% 835649 -0.976731Insp 64987.13 3280.31 16.048 84.91% 849100 -1.032583Probit 54246.78 4143.95 -3.131 21.46% 214624 0.7904805KM 64819.31 4262.77 15.749 77.53% 775299 -0.756413HJ 66080.6 4310.89 18.001 85.08% 850821 -1.039959MLE-mid 56143.51 4334.93 0.256 19.67% 196661 0.8536085B1 life calculationsMethod B1mu B1sig B1Bias Pf DPMO ZstRR 25390.11 3139.58 39.681 95.92% 959154 -1.740946MLE 25818.76 3277.17 42.039 96.42% 964172 -1.8013MLE-RBA1 25389.7 3260.95 39.679 95.38% 953782 -1.682693MLE-RBA2 25083.03 3249.04 37.992 94.50% 944957 -1.597805Insp 22088.04 3037.1 21.515 78.43% 784281 -0.786731Probit 18107.1 3443.62 -0.386 59.77% 597678 -0.247342KM 26224.17 3466.73 44.27 96.60% 966039 -1.82552HJ 26355.66 3469.09 44.993 96.86% 968618 -1.860863MLE-mid 18347.7 3085.25 0.938 55.64% 556354 -0.141731 Table 3: Comparison of Various Weibull Estimators

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other Weibull’s derived from industrial data. To minimize this error, we recommend using the methodology described in this paper to establish inspection thresholds for part life validation, and recommend using the inspection-censored or Mid-Point maximum likelihood estimation algorithm whenever possible. We have also found that a “Guidelines Chart” similar to the one shown in Figure (4) is helpful in communicating the best algorithm for a given inspection interval.

To summarize, this paper attempts to (1) Quantify bias in Weibull parameters for datasets obtained from large inspection intervals, using a Monte Carlo approach and (2) we describe a simple process for estimating errors associated with different inspection methods. This process can be customized to any industry, as needed. It is hoped that this approach will help the

reliability community in developing optimal inspection test plans and in minimizing the error involved.

Disclaimer: The views expressed in this paper are those of the authors only and do not necessarily reflect the views of the General Electric Company. The authors gratefully acknowledge Dr. Robert Abernathy’s work in motivating this study, and thank Dr. Abhinanda Sarkar for introducing us to the Mid-Point MLE approach. Weibull++ and Matlab are trademarks of Reliasoft Corporation and Mathworks, Inc. respectively.

REFERENCES

1. Weibull, W., “A Statistical Distribution Function of Wide Applicability”, ASME Journal of Applied Mechanics, 1951

2. Nelson, W., Applied Life Data Analysis, John Wiley & Sons, 1982

3. Murthy, D.N.P., Xie, M., and Jiang, R., Weibull Models, Wiley-Interscience, 2003

4. Abernathy, R., The New Weibull Handbook, 4th edition, 2002

5. Abernathy, R., et al., Weibull Analysis Handbook, US Air Force Wright Aeronautical Laboratories, Technical

Report: AFWAL-TR-83-2079, 1983. 6. Meeker, W., and Escobar, L., Statistical Analysis of

Reliability Data, John Wiley & Sons, 1996 7. Technical Report TR4, “Sampling Procedures and Tables

for Life and Reliability Testing based on the Weibull Distribution”, US Dept. of Defense, 1962

8. Coolen, F.P.A., et al., “Bayesian Reliability Demonstration for failure-free Periods”, Reliability Engineering and System Safety, Vol. 88, pp 81-91 (2005)

9. Mostafa B., and Celeux, G., “Bayesian estimation of a Weibull distribution in a highly censored and small sample setting”, INRIA Technical Report No. 2993, October 1996

10. Sarkar, A., Chief Scientist, GE Global Research, India. Personal correspondence with the author regarding Mid-Point MLE methods, 2002

BIOGRAPHIES

Sameer Vittal, Ph.D., C.R.E. General Electric Company – Energy Division Gas Turbine Technology Center – Maildrop 256D 300 Garlington Road Greenville, SC 29602-0648, USA

e-mail: sameer.vittal@ge.com

Dr Sameer Vittal has over ten years experience in mechanical design, optimization, risk analysis, reliability engineering and condition-based maintenance. He currently works for the GE Energy Services Technology – Predictive Life Programs department, developing condition-based maintenance models and algorithms for gas turbines. He has a bachelor’s degree in Mechanical Engineering from Bangalore University, India, and a Master’s degree and PhD in Mechanical Engineering from Rensselaer Polytechnic Institute, USA. He is also a Certified Reliability Engineer from the ASQ. His research interests include condition based maintenance, probabilistic design, structural health analysis and actuarial science. Randolph Phillips., PhD General Electric Company – Energy Division Gas Turbine Technology Center – Maildrop 256D 300 Garlington Road Greenville, SC 29602-0648, USA

e-mail: randy.phillips@ps.ge.com

Dr Randolph Phillips is currently the chief engineer for Reliability Engineering, at GE Energy. Prior to this, he held senior management roles in Reliability Engineering in various GE divisions and was a research scientist at the GE Global Research Center. He obtained his PhD in Electrical Engineering from the University of Illinois – Urbana and has worked in the areas of controls engineering, system simulation, probabilistic optimization and reliability engineering. His research interests include probabilistic methods for asset life management, reliability engineering and condition based maintenance.

T/Eta = 0.1

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T/Eta = 0.1

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Figure 4: Guidelines for Choosing Optimal Estimators

(for the case study described in this paper)

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