348 IEEE TRANSACTIONS ON RELIABILITY, VOL. R-26, NO. 5, DECEMBER 1977

Estimation of Reliability After Corrective Action

Ram C. Dahiya R is a r.v. with mean

Key Words - Reliability estimation, Asymptotic distribution, I^-E{R} = + S2a1q1 [1 - (1 - q1)N] (1.2)Reliability growth.

Definition (1.1) of current reliability assumes that no newReader Aids - failure modes are introduced by corrective action.

Purpose: Widen state of the art Seven different estimators for the mean reliability, ,, areSpecial math needed: NoneRpesualmatsusef :ANne considered in [1 ] . EstimatingR is more practical than estimat-

ing,. Estimators of,, however, are also candidates for esti-Abstract - Asymptotic distributions of several estimators, proposed mating R. Since R is a r.v. we define s-bias and mean square

in the literature for estimating reliability after corrective action, are error (MSE) ofR as an estimator of p43= ,, R) by:derived here. Furthermore, the maximum likelihood estimators for thespecial case of equal failure probabilities are obtained. Some of theestimators appearing in the literature are shown to be not s-consistent. bias B(R ) E{R -

(1.3)1. INTRODUCTION MSE: MSE(R, q)-E{R - (1.3

Asymptotic distributions of several estimators proposed in B(R, R) = B(R, i), but MSE(R, R) f MSE(R, j)[1,3] for estimating reliability after corrective action, aredeie hee The 7 estimators of ,u [1] are:derived here.

Notation list 1= PO + 2iyiqiX, R2 =P;0/(l - Eiaiqi)

N number of trials in the sample R3 = PO + liaiqi [1 - (I - qi)N]No number of observed successesNi number of observed failures of type i / N

(^

k unknown number of failure modes R4 PO+iaiqi - liaiqi q )Nvd number of failure modes detected in the samplepo Po initial reliability, initial unreliability NiNqi unknown probability of failure of type i R5 Po + 2iaiqi + i(- 1) Y Ni) (1.4)ai known probability of removing failure mode iLi sum over i from 1 to k {

A ai if Ni> 1a* average ai; a* -iailk d

=P + Zi otherwisea*b average observed a1; a*b= I ai/dR reliability after corrective action A

A

, mean reliability after corrective action= z

denotes an estimator A

Po NoIN, qi-=Ni/N.Corcoran et al. [1] consider the problem of reliability esti-

mation in the following framework. A set ofN s-independent A aRl.and R are MLE (maximum likelihood estimators) ofRtests is performed. Each test results either in success or in 3 A

exactly one of k fixed but unspecified failure modes; thus and ,u respectively. R2 is obtained by ignoring all tests withFo+ Sgql = 1landNo + S1N1 = N. There is a known probability corrected failures and is labeled as no-test estimator in [1]. R

a1 of removing failure model (if it is observed in the set of is obtained as a modification ofR3. R6 is such that no credittests) by corrective action after the set of tests is finished, is given for correction of a failure mode that is observed only

Current reliability, after N tests and corrective action, is onlce. Detailed justification of all these estimators is given in

As observed in [1]], R 1 takes on the value unity regardless

{00, i Ni = 0 of the outcome of the tests if all a1 = 1. This is a clear draw-R=P+SYiQi Yi d :4Tifn1>0) observeoftiesmaori2whn cas

>0.alN.Smlrbeairi

IEEE TRANSACTIONS ON RELIABILITY, VOL. R-26, NO. 5, DECEMBER 1977 349

Exact s-biases of some of these estimators are [1]: R = po + apjod/k(2.3)

B(R1,g) = z1a1qP(o qi)N; 8=po + apo [I -(1 -olk)Nf]XB(R5, u) = ( )N iqi where p-0 1- p0. The expression forR in (1.1) can also be

(1.5) written asB(R6, ,u) =- aiq2(1 - q,)N-l;

R =po S- asiqi (2.4)B(R7,p)= iaiqi(l _ q,)N._ (1/N) Y-ai[I - (1 -.q,)N]. R

and this has been used to get (2.3).The exact s-biases of the other three estimators cannot be In order to derive the MLE of po and k, the joint pdf of theobtained in simple closed form. sample observations No, N1, ..., Nd, can be written as

Since all qi are small in practice, the s-bias ofR5 is thesmallest as compared to other estimators for which exact Pr{N Nl,.... Nd} = PrfNo} x Pr{N1 N2 NdlN d}s-bias expressions are given in(1.5) Furthermore [1],B(R7, p) <B(R6,,), and since B(R6, ,u) < 0, it follows that

x Pr{d'N1J. (2.5)IB(R6, 0') < IB(R7, i').

Olsen [3] has also investigated the problem of estimating,and has suggested a new estimator. For investigating the small The pdf{N1, N2, , Nd N d} is multinomial with equal classsample properties of these estimators, a simulation study was probability 1Id and with n N - No as the total number ofcarried out; some of the conclusions are as follows. R2 is a observations. Obtaining Pr{dlNo} is nothing but classicalpooriedout;estma ofrheiabiity ion

proviesgolodest.imtsw 'occupancy problem' with k cells and n balls and can bepoor estimator of reliabflity;R3 provides good estimates whenotie tm[,p12.S,telklho ucinithe sample size is large; and R4 is conservative. Furthermore, ,

R5, R6, Ra (suggested by Olsen [3] and defined here in,In

ktSection 2) all performed well for a variety of configurations. L(po, k) = binm(NO; po, N)(n!H Ni_ )(1 /dk) (k d!

This paper extends the work of [1,3] as follows. The MLEof , and R in the special case of equal failure probabilities are Zd1 (- 1)'(d- i)fobtained. Furthermore, the asymptotic distributions of all the i=0 i !(d - i)! (2.6)estimators are obtained. This provides interval estimation of uas well as R. It is proved that R2 is not a s-consistent estimatorof ,, which clarifies its poor behavior observed in the simula- where binm(.; p, N) is the binomial pmf.tion study of Olsen [3] . From (2.6) the MLE ofpo is

2. SPECIAL CASE OF EQUAL FAILURE PROBABILITIES Ao =NoQ/N. (2.7)AND EQUAL CORRECTION PROBABILITIES

Since k is an integer, we need integer maximization of (2.6) inOlsen [3] considers the special case when qi = q and ai = a, order to obtain MLE of k. Let t be some real number satisfy-

i = 1, ..., k, and provides a new estimator of reliability, given by ing

Ra =PO +apO d/k (2.1) L(po, k) = L(po, k - 1), (2.8)

where po 1 - p0, d is the number of failure modes detectedin the sample, and k is the solution of then the MLE of k is

k[l - (1 -lk)N-No] -d=0. (2.2) k=[k] (2.9)

The estimator Ra is labeled as adjusted successes estimator in where [x] denotes greatest integer < x. Now (2.8), on making[3] and is based on a heuristic approach. Eq. (2.2) is the use of (2.6), becomesresult of conditional moment estimation of k in [1], obtainedby equating dandcE{dNn }. k[1-(1- 1/k)r] _d=, (2.10)

Olsen [3] also considers the presence of an uncorrectablefailure mode in the system being tested. Here only correctable which is the same as (2.2) and is involved in moment estima-failure modes are assumed in the system and the results of [3] tion of k in [3] . It is obvious that 1l.h.s. of (2.10) is negativeare modified accordingly. However, all the results obtained for 0 <k 6 d and is a strictly increasing function of k. Hencehere can be modified for the system with uncorrectable failure there is a unique solution, k > d, of (2.10). A table for k formodes by just changing the definition of symbols. n . 30 is given in [3] and similar tables can easily be obtained

Simplified expressions for R and ,u in this special case are for higher values of n.

350 DAHIYA: ESTIMATION OF RELIABILITY AFTER CORRECTIVE ACTION

The MLE ofR and p, denoted by Rb and RC respectively, feature of their work. The asymptotic distribution of all theseare estimators is required to obtain interval estimation of reliability

for large samples. The asymptotic distribution of six of theseRb + apod/k estimators is obtained here and is given in the following theorem.

(2.12) Theorem ]: \N(R-,u)i N(O,u2), fori 1,3, 4-7 as

R=P + aPo [ - (1 - polk)N] N °oo. Where

Ra is heuristic and uses moment estimation of k, is very 2-Po + z2a2qi - Po +similar to the MLE Rb, the only difference being that k is usedas an estimator of k in Ran and ignores the fact that k is an This theorem is proved in the Appendix.integer. Now we show that R2 is not a s-consistent estimator of ,.

R2 is one of the estimators considered in [31 and was found to3. SPECIAL CASE OF EQUAL FAILURE PROBABILITIES be poor. Our observation explains this behavior of R2. Since

Po -+po andqioPqii= 1, .,k,wehaveLet the failure probabilities be equal but the probabilities of

p

correction of the different failure modes be unequal. This R2 +pol/(l - zzaiqi) (4.1)special case is more realistic as compared to the one consideredin Section 2. Expressions for R and ,u, under this assumption, Also po1(l - 2iaiqi) - p -* 0 i.f.f. all ai = 1. Hence R2 is, inare general, not a s-consistent estimator of,u. A A.

The asymptotic distribution of estimators Rb, Rc Rd, ReR =po + bpOd/k can be obtained similarly. We state the result without proof

(3.1) in Theorem 2.

ipO +PO[1-l(1-fiO/k)N] a 7heorem 2: The asymptotic distribution of VN(Rb - ,u)and \/N_7(RA - p) is N(O, a2), and the asymptotic distribution

The MLE for p0 and k are still the same as given earlier in of VV(Rd - u)and (Re 2) is N(O, u) whereSection 2, and hence the MLE ofR is

R =A + (3.2) 2 ((1 -a)2popo, a2 (1 -a*)2popo.In order to investigate the behavior of the estimators (con-Usually, at's will have to be obtained from the engineers sieeinSc2ad3)wtrspttohevltonfeql

wharinovdi .eoigte eet rmtesse sidered in Seec 2 and 3) with respect to the violation of equalwho are involved in removing the defects from the system falr prbblt suninw oeta n. . ~~~~~failure probability assumytion, we note that Rk andbeing tested. Since ai cannot be specified for an unobserved A p A b

failure mode, the MLE of p cannot be obtained by replacing RC -*PO + aPo, Rd and Re - POp + a*pO, and p -po + Ziaiqi,MLE of po and k in (3.1). One possible estimator of p, on where, is as defined in (1.2). Hence it follows that themaking use of a* as an estimator of a*, is estimators of Sec. 2 and 3 are not s-consistent estimators ofp

defined in (1.2). This implies that the estimators of Sec. 2Re = Po + Plo [1 - (1 - po0/k)N ] a0* (3.3) and 3 are very sensitive to the violation of equal failure prob-

ability assumption. In case of ai = 1, i = 1, 2, ..., k, it is obviousOne interesting property ofRd and Re is that specifying that R .-o 1, where R could be any of the estimators considered

average probability of correcting the d observed failures is in this paper. So the estimators of Sec. 2 and 3 are consistentsufficient to compute these estimators. This can be useful in this special case. The assertion in [3] that the estimator Rabecause assessing the average probability of correction is is not very sensitive to the violation of equal failure probabilitycertainly easier then assessing the individual probabilities of condition is, indeed, masked by the fact that ai = 1 is assumedcorrection. in the simulation study.

Since estimating R is more practically important as com-

4. ASYMPTOTIC RESULTS pared to estimating p, the following Lemma is given for obtain-ing interval estimation ofR for large N.

Notation List Lemma: The asymptotic distribution of \/§N(R - R) is thesame as that of vKAf(R - p), for any estimatorR considered

convergence in distribution here.p convergence in probability This Lemma is proved in the Appendix.N(p, a2) s-normal r.v. with mean pu and variance a2- equality sign for asymptotic distribution ACKNOWLEDGMENT

In order to obtain interval estimation of reliability after This research was carried out at the University of Delhi dur-corrective action, by using the estimators in [1], one needs to ing my visit as an Indo-American Fellow for Advanced Researchknow the probability distribution of the estimators. The in India. I thank the Indo-U.S. Subcommission on Educationauthors in [1 ] acknowledge that this is the most incomplete and Culture for providing the fellowship.

IEEE TRANSACTIONS ON RELIABILITY, VOL. R-26, NO. 5, DECEMBER 1977 351

APPENDIX Finally, v/ (R6 - = WAR1 -p) follows from the factp aR6 rA-

that Zi ai, and N(R7 - ,) _yN(R 1 -i,) is obvious fromProofof Theorem I the fact that yi/N P 0. Q.E.D.

In order to prove this theorem, we first obtain the asympto- Proof ofLemma:tic distribution of N(R 1 - ,) and then prove that

In order to prove the Lemma, it is sufficient to prove that\/N(R- ,u) N_(R 1 -,), i = 3, 4, ... 7. \N(R - i) 00. For proving this, V\7(R - ,u) can be written

asNow]R1, defined in (1.4), is the same as

AN(R - ,) = <N j(yi - ai)qi -vN iaiqi(l - q,)N.R1=NoIN+YiajNi1N_

It is obvious that N(1 - q,)N -+ 0, asN - oo. Furthermore,Since (No, N1, ..., Nk) have a multinomial distribution, which (Yi - ai) 4 0 which can be proved by computing the meantends to multivariate s-normal, obtaining the asymptotic and variance of v§N(yi - ai) and showing that they both tenddistribution ofN(R 1 - g) is straightforward and is the same to zero asN -+ oo. Hence it follows that y§IN(R - ,u) -P 0. Q.E.D.as given in Theorem 1. Furthermore, it is also true thatVar{V§R1N} = a2, for all N. REFERENCES

In order to establish the asymptotic distribution of R3, wehave [1] W.J. Corcoran, H. Weingarten, P.W. Zehna, "Estimating relia-

bility after corrective action", Journal ofManagement Science,(R3-1-)=aN[Po+iasQi{l~(l )}1~ \N 1 - vol 10, 1964 Jul, pp 786-95.

-v/N =R 3 - [2] W. Feller, An Introduction to Probability Theory and Its

Application, Volume I, Third Edition, John Wiley and Sons:But 1 -(1 - q)N -+1, asN-+oo(cf. [4, p. 371]). Hence, on New York 1968.making use of the result in [4, p. 319], it can be shown that [3] D.E. Olsen, "Estimating reliability growth", IEEE Trans

Reliability, vol R-26, 1977 Apr, pp 50-53.

-(R3-11) - vaN(o + zEaiq1 - M) = VAkR1 -12) [4] C.R. Rao, Linear Statistical Inference and its Applications, JohnNIN 3 ~~~~~~~~~~~~~~~Wileyand Sons: New York 1965.

which gives the asymptotic distributions of R3.Proof of VNN(?4 - i') _ (R 1 - p) follows from the fact

that [N/(N- 1)]N Aead1_ N P 0, asN-o.te, and ( - )Ram C. Dahiya; Dept. of Math. & Statistics; University of Massachusetts;For proving NIN(R5 -,) a V§N(R 1 - ,), it is sufficient to Amherst, MA 01003 USA.

show that I/(N ) = h(N.) XP 0. It is easy to show thatMr. Ram C. Dahiya is an Associate Professor of Statistics in the Depart-

qN[1 - (pi/q)N+l ] N[ (q/pi)N ] ment of Mathematics and Statistics, University of Massachusetts. HeE{h(N=)}-_________ _ =pz[* was born at Bidhlan (India) and received his MA in Mathematical

1 - (piIqi) (1 - (qj/pj)) Statistics at University of Delhi in 1964 and his PhD in Statistics atUniversity of Wisconsin in 1970. His earlier publications have appearedin Biometrika; Biometrics; Journal of Royal Statistical Society;

Using first or second part of this equality, depending on Journal of American Statistical Society; IEEE Transactions on Relia-whether pi/qi < 1 or > 1, it follows that E{h(Nj)} -+ 0. Since bility; Sankhya; and Technometrics.h(Ni) is a non-negative r.v., E{h(Ni)} -o 0, implies thath(Np)P 0 Manuscript received 1977 February 11; revised 1977 June 7, and 1977

h(NZ) ~ 0. September 2. nn

Man uscri pts Rece ived For information, write to the author at the address listed; do NOT write to the Editor.

"On the effect of sampling errors in an empirical Bayes esti- "2-unit warm-standby redundant system with delay and onemation procedure", Chris P. Tsokos; Dept of Mathematics; repair facility", Dr. P.K. Kapur; IMSOR; Bygning:349; 2800University of South Florida; Tampa, FL 33620 USA. Lyngby DENMARK.

"Inspection intervals for items with constant hazard rate failing "An efficient method for reliability evaluation of a generalin an obvious manner", David J. Sherwin; Dept of Chemical network", Dr. K.K. Aggarwal; Electronics & CommunicationEngrg; Univ of Technology; Loughborough Leicestershire LE11 Engineering Dept; Regional Engineering College; Kurukshetra3TU ENGLAND. 132 119 INDIA.