QUALITY AND RELIABILITY ENGINEERING INTERNATIONALQual. Reliab. Engng. Int. 2000; 16: 2326

WEIBULL PREDICTION OF A FUTURE NUMBER OF FAILURES

WAYNE NELSON739 Huntingdon Drive, Schenectady, NY 12309, USA

SUMMARYThis paper presents simple new prediction limits for the number of failures that will be observed in a futureinspection of a sample of units. The past data consist of the cumulative number of failures in a previous inspectionof the same sample of units. Life of such units is modelled with a Weibull distribution with a given shape parametervalue. Copyright 2000 John Wiley & Sons, Ltd.

KEY WORDS: Weibull distribution; life data; prediction limits

1. INTRODUCTIONMotivation

The new prediction limits presented here weremotivated by the following application. Certainnuclear power plants contain steam generators, whichare large heat exchangers, each with 10,000 to20,000 stainless steel tubes. In service the tubesdevelop defects, which are detected during periodicinspections; then the defective tubes are plugged totake them out of service. To help manage such asteam generator, a utility uses a prediction of the addednumber of tubes that will need plugging by a specifiedfuture time. The prediction helps the utility planfor (1) future plugging effort, (2) repair of pluggedtubes (restoring them to service), (3) reduced powerproduction and (4) retirement of the steam generator.The statistical uncertainty of such a prediction isimportant information, as it indicates how reliable theprediction is.

Prediction

Prediction is concerned with predicting the futurevalue of a random quantity (observation or statistic),such as the random number of tubes found defective ina future inspection of a steam generator. A predictioninterval encloses the random quantity with a specifiedhigh probability. In contrast, estimation is concerned

Correspondence to: W. Nelson, 739 Huntingdon Drive, Schenec-tady, NY 12309, USA.Presentation of this work at the 1995 Joint Statistical Meetingsreceived an Honorable Mention Outstanding Presentation Award(third among 60 contributed papers) from the Section on Physicaland Engineering Sciences of the American Statistical Association.

with estimating fixed population parameters, suchas the mean or a percentile. A confidence intervalencloses such a fixed parameter value with a highprobability (confidence). Nelson [1] gives examplesof predictions and prediction intervals for reliabilityapplications. Hahn and Meeker [2] survey variousprediction intervals.

Overview

Section 2 states the prediction problem and presentsa statistical model for it. Section 3 provides aprediction and three approximate prediction intervalsfor a future number of failures. Section 4 discussessome features of the model and prediction problem.

2. MODEL

Trinomial

Suppose that N sample units start service at time 0,and that by time t the observed cumulative numberof failures is X. We seek a prediction and predictioninterval for the future added random number Y ofunits that will fail by time t 0, i.e., in the interval.t; t 0/. If Z is the number of sample units that remainunfailed at time t 0, then (X, Y , Z) have a trinomialdistribution with corresponding probabilities (p, q , r),where X C Y C Z D N and p C q C r D 1.

Weibull

The probabilities p, q and r are given by thelife distribution that the N sample units come from.Here we use a Weibull life distribution with unknown

Received 9 May 1998Copyright 2000 John Wiley & Sons, Ltd. Revised 23 June 1999

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24 W. NELSON

scale parameter value and given (known) shapeparameter value . Thus

p D 1 expT.t=/ Uq D expT.t=/ U expT.t 0=/ U (1)r D expT.t 0=/ U

3. PREDICTION AND LIMITSExample

For illustration, consider a steam generator withN D 20;000 tubes, of which X D 8 have failed by theinspection at age t D 3:0 years. We seek a predictionand prediction limits for the future additional numberY of tubes that will need plugging by a futureinspection at age t 0 D 10:0 years. Here we use aWeibull shape value D 3:3.

Prediction

Based on the observed number X, the maximumlikelihood (ML) estimate of is

D t=flnT1 .X=N/Ug1= (2)The corresponding prediction for Y is

Y D Nq (3)where the estimate q of q is obtained by substituting into (1) for q , yielding

q D T1 .X=N/U T1 .X=N/U.t 0=t/ (4)For the steam generator example,

q D T1.8=20;000/U T1.8=20;000/U.10:0=3:0/3:3D 0:0206396

and the prediction is Y D 20;000.0:0206396/D 413additional tubes failing.

Limits

When the Weibull probabilities p and q above aresmall, then approximately

p=q 1=T.t 0=t/ 1U (5)which does not depend on the unknown Weibullscale parameter . Nelson [3] gives two-sided 100C%confidence limits for the multinomial odds ratio p=q ,namely,

TX=.YC1/UF.a0I 2X; 2YC2/ p=q 1=T.t 0=t/ 1U (6) T.XC1/=Y UF.1a00I 2XC2; 2Y /

where F.aIm;n/ is the 100a% point of the Fdistribution with m degrees of freedom in thenumerator and n in the denominator, and 1a0a00 DC. The lowest Y value that satisfies the left inequalityis a one-sided upper 100.1a0/% prediction limitYU for Y . The highest Y value that satisfies theright inequality is a one-sided lower 100.1a00/%prediction limit YL for Y . Together, .YL; YU / are two-sided C D .1a0a00/% prediction limits for Y .

Example

Such 90% prediction limits for the steam generatorexample are YL D 205 and YU D 756, quite wide.

Simpler limits

Solving (6) is laborious. Simpler approximate limitsfor typically large Y result from noting that, as Y goesto infinity:

the low percentile F.a0I 2X; 2YC2/ rises mono-tonically to F.a0I 2X;1/ D 2.a0I 2X/=2X, achi-square percentile; the high percentile F.1a00I 2XC2; 2Y / de-

scends monotonically to F.1a00I 2XC2;1/ D2.1a00I 2XC2/=.2XC2/.

Substituted into the left and right sides of (6),these limiting values yield conservative two-sidedapproximate 100C% prediction limits

yL D T.t 0=t/ 1U 0:52.a0I 2X/;yU D T.t 0=t/ 1U 0:52.1a00I 2XC2/

(7)

These limits are easy to calculate. They are suitable ifp and q are small and Y is large.

Example

Such simpler 90% prediction limits for the steamgenerator example, where T.10:0=3:0/3:3 1U D52:150, are

yL D 52:150 0:52.0:05I 28/ 1D 52:150 0:5 7:962 1 D 207.or 100 207=20;000 D 1:0%/

yU D 52:150 0:52.10:05I 28C2/D 52:150 0:5 28:87 D 753.or 100 753=20;000 D 3:8%/

Here p D 8=20;000 D 0:0004 and q D 0:0206 aresmall, and Y D 413 is large. Thus the simpler limitsare suitably close to those above.

Copyright 2000 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 2000; 16: 2326

WEIBULL PREDICTION 25

LR limits

The preceding prediction limits are approximateand suitable for small p and q . The followingapproximate likelihood ratio limits are suitable forlarger p and q and do not use the approximation (5).These limits require that X and Y be large and closeto normally distributed. The unconstrained samplelikelihood for observed (x, y, z) is

L.x; y; z/ D MpxqyrzD Mpxqy.1 p q/Nxy (8)

whereM is the multinomial coefficient. The maximumunconstrained sample likelihood is

L.x; y/ D M.x=N/x.y=N/y T1 .x=N/ .y=N/UNxy (9)

Under the constrained model where (p, q , r) are givenby Weibull probabilities (1), the constrained samplelikelihood is

K.x; yIA/ D Mpxqy.1 p q/Nxy (10)where p and q are functions (1) of the unknownWeibull parameter . The maximum likelihoodestimate 0 for maximizes (10) and is a functionof .x; y/. The function 0.x; y/ cannot be foundexplicitly but must be found numerically. Themaximum of the constrained sample likelihood (10)is K.x; yI 0.x; y//. The log likelihood ratio teststatistic for the model with Weibull probabilities (1)is

Q.x; y/ D 2flnTK.x; yI 0.x; y//UlnTL.x; y/Ug (11)

When the Weibull model and chosen value arecorrect, the asymptotic distribution of Q.x; y/ isapproximately 2 with one degree of freedom. Thus,with approximate probability P%,

Q.x; y/ 2.P I 1/ (12)the P th percentile of the 2 distribution with onedegree of freedom. The two values .Y 0L; Y 0U / that mostnearly satisfy this inequality are approximateP% two-sided prediction limits for Y . Calculation of theselimits is laborious and requires a computer program.

Example

Such 90% prediction limits for the steam generatorexample are Y 0L D 216 and Y 0U D 700. Here the upperlimit is somewhat lower than the previous limits. Thisshould be expected, since X D 8 is small and the Xdistribution is far from normal.

4. DISCUSSION

This section discusses certain features of theprediction limits.

Coverage probabilities

The exact coverage probabilities of the threeapproximate prediction intervals above have beenstudied by Nordman and Meeker [4], using simulation.

Known shape

In practice the Weibull shape parameter is notknown. Instead it is estimated subjectively or fromrelevant data. Thus its value is uncertain. This uncertainty may contribute greater uncertainty to theprediction error .Y Y / than does the randomstatistical variability of X. Thus useful future workwould model the uncertainty in and incorporatedinto suitable prediction limits. Thus the actualuncertainty in such a prediction is (possibly much)greater than given by the limits above.

Other uncertainties

Such a prediction is subject to other sourcesof uncertainty, which result in greater predictionuncertainty than given by the limits above. Forexample, if the life distribution is not Weibull, thereis model error from using a wrong distribution. Thismight result, even if the distribution is Weibull, whenthe operating conditions of the sample units are notstable over time or if failures are due to a mix ofcauses.

ACKNOWLEDGEMENTS

The author gratefully acknowledges that this articlebenefited much from valuable input from a referee andProfessor Bill Meeker.

REFERENCES

1. W. Nelson, Applied Life Data Analysis, Wiley, New York, 1982.2. G. J. Hahn and W. Q. Meeker, Statistical Intervals: a Guide for

Practitioners, Wiley, New York, 1991.3. W. Nelson, Statistical methods for the ratio of two multinomial

proportions, Amer. Statistician, 26, 2227 (1972).4. D. J. Nordman and W. Q. Meeker, Weibull prediction intervals

for a future number of failures, submitted, available fromProf. Wm. Meeker, Statistics Dept., Iowa State Univ., Ames,Iowa 50011, U.S.A.

Copyright 2000 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 2000; 16: 2326

26 W. NELSON

Authors biography:

Wayne Nelson is a leading expert on reliability dataanalysis. Formerly with the General Electric Companyfor 23 years, he now consults and teaches engineeringapplications of statistics for many companies, professionalsocieties and universities. For his contributions to reliabilitydata analysis and accelerated testing, he was elected a Fellow

of the Institute of Electrical and Electronics Engineers,the American Society for Quality control (ASQC) and theAmerican Statistical Association (ASA). He has authoredtwo well-known books, Accelerated Testing and AppliedLife Data Analysis. Among his 100+ publications, he hasreceived the Brumbaugh, Wilcoxon and Youden Prizes ofthe ASQC and eight Outstanding Presentation Awards fromthe ASA.

Copyright 2000 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 2000; 16: 2326

1 INTRODUCTIONMotivationPredictionOverview

2 MODELTrinomialWeibull

3 PREDICTION AND LIMITSExamplePredictionLimitsExampleSimpler limitsExampleLR limitsExample

4 DISCUSSIONCoverage probabilitiesKnown shapeOther uncertainties