Bayes shrinkage estimation of reliability and the parameters of a finite range failure time model

Microelectron. Reltab., Vol. 33, No. 13, pp. 2039-2042, 1993. Printed in Great Britain.

0026-2714/9356.00 + .00 © 1993PergamonPress Ltd

BAYES S H R I N K A G E ESTIMATION OF RELIABILITY A N D THE P A R A M E T E R S OF A FINITE R A N G E F A I L U R E TIME M O D E L M. PANDEY Department of Zoology, Banaras Hindu University, Varanasi 221 005, India and V. P. SINGH Department of Statistics, Udai Pratap College, Varansi 221 002, India (Received for publication 28 May 1992)

Abstraet--A finite range failure time distribution has been proposed and studied. For estimating the two parameters of this distribution, this paper considers a prior assumption that (1 - b) is the probability that the scale parameter 0 and shape parameter p have the values 00 and P0, respectively, and that the rest of the probability mass b(0 ~
1. INTRODUCTION The theory does not prescribe a finite upper limit to the length of life (time to failure) of a piece of equipment and all failure time distributions are defined over the range (0, ~ ) . However, the life of any equipment should be finite. It has been suggested [1] that a finite range probability model for failure times defined by the probability density function (p.d.f.) should be used: f ( x ) = ( p / x ) ( x / O ) p, O<<.x<~O, p , O > O

(1)

with cumulative distribution function (c.d.f.) F(x)

~(x/O)p,

0 <~ x <~ 0

l 1,

x>O

'

(2)

which can have both increasing and decreasing failure rates. The reliability of a component which has failure time c.d.f. (2), is R ( t ) = P[x >i t] = 1 - (t/O) e.

(3)

Some other interesting properties of this distribution have been considered [2]. As an example of actual data following a finite life time distribution, as given in (1) above, it can be mentioned that the p.d.f. (1) fits data regarding the survival times before cancer deaths and remission times of patients treated for Hodgkin's disease well [3]. In fact, a test using the F-statistic to test the goodness of fit of p.d.f. (1) and exponential distribution for the above mentioned data gave a higher value of F f o r (1), indicating that the test rejects p.d.f. (1) at a lower level than it rejects the exponential

distribution. Hence p.d.f. (l) may also serve as a good model for data obtained in real life. We considered estimation of its scale parameter 0 assuming that p is known [4]. In this paper, estimation of both the parameters is considered. An unbiased estimator ~ was suggested to shrink towards a natural origin or prior value q~0 by using the shrunken estimator kq~ + (1 - k)~b0 (0 ~
2039

2040

M. PANDEYand V, P. SINGH

findings and recommendations for the choice of the constants k and b are summarized in Section 4.

In this section we first obtain the Bayes estimator of the shape parameter p, scale parameter 0 and reliability R(t) of a component with a finite range failure time distribution, and subsequently Bayesian shrinkage estimators are proposed. Let us consider that n items, with a failure time distribution given by (1), are subjected to life testing and that the test is terminated as soon as the rth item fails (r ~< n). Also let xl, x2 . . . . . x, be the observed failure times for these r items. The likelihood function for such a Type II censored sample is then given by:

(n!/(n -

r)!)[l -- (x,/O)']"-"

pC~(p) ~" C(j)x~/bT,(j,p)dp i~0

+poC2(po) ~ C(j)xf°:(l - b)Tz(j,p) /=0

if:

C,(p) ~ C(j)xP/bT3(j,p) dp

]

j=O ° '

+Cz(po) ~ C(j)xW(1-b)T4(j,p) ,

(7)

/=0

where C ( j ) = ( - I)J/(j!(n

- r - j)!)

= p,o I - I x f o - I

T l (j, p) = [V(a + s(p,j))Za/x, (~ + s(p,j))]/

T:(j,p) = 1/0~o~°°':) T3(j,p) = [F(~ + s ( p , j ) - l)l#/x, (c~ + s ( p , j ) - Ol/

(ar~fl,
1/0 4~°J)-

s(p,j) =p(r +j)

O
s(po,j) =po(r + j )

which is uniform for (0, a) and

a, r, 0 > O,

which is gamma inverted on 0, so that h(p, O)=

( fl')/(ar~ ) o-('+ 1)exp(-fl/O); a, r, 0 > O.

Iz(s)= ( I / r s )

f:

e x p ( _ z ) z ~- l dz.

The Bayes estimator T B of 0 is obtained as

TB = E(O IX)

Thus

~bh(p, 0), i f p #P0, 0 # 00 g(p, 0) = ((1 - b), i f p =P0, 0 = 00.

= f,~ffOL(X/p,O)g(p,O)dpdO/

(5)

The joint posterior o f p and 0, via the Bayes theorem, will be: ~(P,

(6)

(4)

Following Ref. [5] consider a prior g(p, O) o f p and 0, which places a weight (1 - b) on the prior or guess values P0 and 00 of the parameters p and 0, and distributes the rest of probability mass b over some suitable interval of p and 0 according to the joint density h (p, 0). Further it is assumed that p and 0 are independent with the marginals

h2(0 ) = (fl~)/(Fot) 0-t,+ 1)exp(-fl/O);

P.=

c2(po)

O<~xt<~x2<~'"<~x,<~O, p,O>O.

f

fo' L(X/p, O)g(p, 0) dp dO.

c , ( p ) = p" [-I x V 1

r x p ' O -rp H xpi- 1. i=l

hl(p)=l/a;

r

This, on simplification, reduces to:

2. BAYESIAN SHRINKAGE ESTIMATORS OF p, 0 AND R(t)

L(X /P, O) =

f

O) = L(X/p, O)g(p, 0)/

fff[L(X/p,O)g(p,O)dpdO.,

(8)

After simplification

/

fff[L(X/p,O)g(p,O)dpdO., To obtain the Bayesian shrinkage estimator of p, 0 and R(t), we first find Bayes estimators by considering a squared error loss function. The Bayes estimator PB of p for the above prior g(p,O) is given by

PB = E(p IX)

TB=

C~(p) ~. C(j)xP/bT3(j,p)dp ]=0

+

C,(po) ~', C(j)x~°(1-b)r4(j,p) 1=0

if:

C~(p) ~ C(j)x~JbT3(j,p)dp j=O

+ C2(po)n-r ~. C(j)xW(1 - b)T4(j, p) 1 .

(9)

j=O

= f~; f[pL(X/p,O)g(p,O)dpdO/

The reliability function R(t) can easily be obtained without explicit knowledge of the posterior of R

A failure time model because R(t) is an increasing function of 0. The Bayes estimator/~a(t) of R(t) is given by

R B ( t ) = [ f f f : R(t)L(x/O,p)g(p,O)dp dO]/ I f f f : L(x/O,p)g(p,O)dpdO] Rs(t) = 1 -

I[;

(10)

r

tPC~(p) ~ C(j)xP/bTs(j,p)dp j=O

+tP°C2(po) ~ C(j)x~e(1 - b)T6(j,p) C~(p) Y~ C(j)xfbT3(j,p) dp

2041

and E3 = MSE(RML(t))/MSE(RBs (t)), respectively. It is not feasible to determine El, E 2 and E 3 by analytical methods so Monte Carlo methods have been used to determine empirical efficiencies. These are based on 500 simulated samples from a finite range failure time distribution of size n = 10, censored at r = 5. Prior values of the parameters were chosen as 00 = 30.00 and P0 = 0.70. Efficiency computations are carried out for 0/00= 0.25, 1.00, 2.00, 4.00, p/po=0.50, 1.00, 2.00, k =0.25, 0.50, 0.75, b = 0.25, 0.50, 0.75, ~ = 12.00, fl = 3.00, t = 0.40 and a = 2.00.

]=0

4. NUMERICAL FINDINGS AND CONCLUSION

+C:(Po) ~ C(j)XPr°J(1 - b)T4(zp ) , j=O

(ll)

where

Ts(j,p) - [F(a + p(r,j) + p)I~/x, (~ + s(p,j) +p)]/ ( a r ~t3 ,¢pJ ~+ , )

T6(j,p) = l/0ficp°J~+p°. Thus the ordinary Bayes estimators of p, 0 and are given by (7), (9) and (ll). The re.I.e, of 0 given by Ref. [1] is

R(t)

TM,=

(12)

x~,~.

The m.l.e, values o f p and PML = --(2n -

R(t) are

r)/ ~ (log X i - log)(,), (13) /

i~l

RML(t) = 1 -- (t/O) ~.

(14)

The Bayesian shrinkage estimators of the shape parameter p scale parameter 0 and reliability R(t) are, therefore, given by

Pas = kpML + (1 -- k)pa,

(15)

Ts s = kTM, + (1 - k)Ta

(16)

kRML(t ) + (1 -- k)Rs(t),

(17)

respectively, where 0 ~
The Bayes shrinkage estimators PBs, Tas and

RBs(t) are compared with m.l.e, values of p, 0 and R(t) on the basis of their efficiencies. The efficiencies of Pns, TBs and RBs(t ) with respect to corresponding m.l.e, values /'ME, TML and RML(t ) are defined as E~ = MSE(PML)/MSE(PBs),

E: = MSE( TML)/MSE( Tss)

(i) For p 1.00 for all values of b and k, the proposed estimators PBs and TBs are better than PML and TML. In this case TBs is better than TML when 0/00 < 0.25 for all values of b and k except for k = 0.25. For P >-2.00 for all values of b and k. (iii) For p >P0, PBs is more efficient than PML for all values of b and k and 0/00, except 0/0o = 2.00. TBs is more efficient than TML if 0.25 ~<0/0 o <~2.00 for all values of b and k except k =0.25. For p >P0, Rss(t) is more efficient than RML(t) if 0/00= 1.00 for all values of b and k. However, in this case, Res is better than RML(t), if0/0o = 4.00, for k = 0.50. On the basis of the above findings it may be concluded that:

and Ras(t) =

The three tables for the efficiencies EL, E 2 and E 3 have been obtained for three values of P/Po where b and k are equal. The tables are not presented here because of space constraints. Our findings are as follows.

(i) For p _1.00 we may choose any value of b and k except k = 0.75. (ii) For p =Po, if we hope 0/0o>12.00, we may choose any values of b and k for the estimation of p and R(t). (iii) For p >P0, if we expect 0/0 o = 1.00 we may choose any values of b and k for estimating p, 0 and R(t). However, for 0 < 00, k = 0.50 may be chosen. Thus, looking overall at the results it is concluded that the Bayes shrinkage estimators may be usefully used instead of the m.l.e, with a proper choice of constants.

2042

M. PANDEYand V. P. SINGH

Acknowledgements--The authors wish to thank Dr S. K. Upadhyay, Mr Borhan Uddin, Mrs J. Ferdos and Mr Rajesh Singh (comprising the research group in the Statistics Department of the B.H.U.) for their useful suggestions to debug the computer program and in typing this paper.

5. 6.

REFERENCES 7. 1. S. M. Ali and A. Islam, Parameter estimation in a failure time distribution, J. I A P Q R 2, 35-37 (1977). 2. C. D. Lai and S. P. Mukherjee, A note on a finite range distribution of failure times, Microelectron. Reliab. 26, 183-189 (1986). 3. J. Alan Gross and A. Virginia Clark, Survival Distributions Reliability Applications in the Biomedical Science. Wiley, New York (1975). 4. M. Pandey and V. P. Singh, Bayesian shrinkage estimation of reliability from a censored sample from a

8. 9. 10.

finite range failure time model, Microelectron Reliab. 29, 955-958 (1989). H. H. Lemmar, From ordinary to Bayesian shrinkage estimators, S. Aft. statist. J. 15, 57-72 (1981). M. Pandey and S. K. Upadhyaya, Bayesian shrinkage estimation of reliability from censored sample with exponential failure model, S. Aft. statist, J. 19, 21--23 (1985). M. Pandey and S. K. Upadhyay, Bayes shrinkage estimators of the Weibull parameters, IEEE. Trans. Reliab. R34, 491-494 (1985). M. Pandey, A Bayesian approach to shrinkage estimation of Weibull scale parameter, S. Afr. Statist. J. 22, 1-13 (1988). S. K. Sinha and B. K. Kale, Life Testing and Reliability Estimation. Wiley Eastern Limited, Delhi (1980). J. R. Thompson, Some shrinkage techniques for estimating the mean, J. Am. statist. Assoc. 63, 113 122 (1968).