IEEE TRANSACTIONS ON RELIABILITY, VOL. 45, NO. 3,1996 SEPTEMBER 47 1

Bayes Estimation for the Pareto Failure-Model Using Gibbs Sampling

Ram C. Tiwari

Yizhou Yang

Jyoti N. Zalkikar

University of North Carolina, Charlotte

Florida State University, Tallahassee

Florida International University, Miami

Keywords - Bayes estimation, Type-1 & Type-2 censored data, RejectionlAcceptance algorithm, Conjugate prior, Improper prior, Gamma distribution, Credible interval, Generalized uniform distribution.

Summary & Conclusions - Bayes estimation of the parameters and the reliability function based on type-1 and type-2 censored samples from a Pareto failure model is considered. The analysis is extended to situations wherein the exact survival times are not available but only the number of deaths in prescribed time inter- vals is recorded. Bayes calculations can be implemented easily by means of the Gibbs Sampler.

1. INTRODUCTION .

Acronyms

PFM Pareto failure model CI symmetric credibility interval.

Notation

f ( t l8 ,m) pdf of the PFM; see (1-2) Xi i = 1,. .. ,r: first r order statistics from a sample of size

n from the PFM f (XI >

R ( x ) generic pdf, Sf m, 8 P

R ( t )

r CY,^) complete Gamma distribution with shape parameter a and scale parameter p

9( -) Indicator function, S(True) = 1, S(False) = O r (a,P) .Z (A) r ( CY,@ distribution restricted over A

implies: Bayes estimate 6 convergence in distribution XI Y - implies: conditional on Y, X is distributed as r number of failures in a censored sample

parameters of the PFM model; m > 0, 8 > 0 mean life of the PFM model (exists for 8 > 1)

[hazard rate, reliability] of PFM at mission time t h ( t ) >

V log@ )

’The singular I% plural of an acronym are always spelled the same.

s* log(xi) + ( n - r ) .log(x,): total log time on 1=1

test in a type-2 censored sample (O.O5N), (0.95N) [0.05Nth, 0.95Ph] order statistic GU (a ; b,c) generalized uniform pdf

S (sI, ..., s d ) : l x d vector

U(0 , l ) uniform distribution on (0, 1).

Other, standard notation is given in “Information for Readers & Authors” at the rear of each issue.

The PFM arises as a mixture of exponential failure distribu- tions, and its use for situations with randomly varying environ- ment is amply justified [6,7]. The mixture of exponential failure distribution has a decreasing hazard rate. Thus for a minimum life or warranty period m and a decreasing hazard rate:

g,( e ) pdf of F(a,, b,) , i=1,2

h ( t ) = 8/t , t > m > 0, (1-1)

the PFM model has pdf & S f

The mean life is finite if 8> 1:

Bayes estimation for the PFM was considered in [ 11, when m is known and 8 is unknown. When both m & 8 are unknown, a fully Bayes approach of estimation of h ( t ) , R ( t ) , and p, of the Pareto failure model wherein the range depends on one of the parameters, is developed here, using the Gibbs sampler and the rejection/acceptance algorithm. The Bayes analysis in sec- tion 2 is based on type-2 censored data. With a simple modifica- tion in the likelihood function, the analysis can be used for type-1 censored data.

1.1 Type-2 Censoring

In problems such as life-testing, the ordered observations are a common occurrence. In that case, time & cost can be saved by stopping the experiment after the r ordered observations have occurred, rather than waiting for all n failures. The pdf of the first r order statistics from a random sample of size n from a continuous pdf f(x) is:

0018-9529/96/$5.00 01996 IEEE

472 IEEE TRANSACTIONS ON RELIABILITY, VOL. 45, NO. 3, 1996 SEPTEMBER

for -03 < x1 < ... < x, < 03; r is fixed but x, is a r.v.

1.2 Type 1 Censoring

If the experiment is terminated at a fixed time, to, this is type-1 censored sampling. The r is a rev. , and has a binomial distribution with parameters p = F ( to) & n. For a given r ,

for x1 < . . . < x, < to. The form of (1-6) is similar to (1-5) with to replacing x, as the argument of R .

2. MODEL DEVELOPMENT

2.1 Type-2 Censoring'

Assumption

the degree of belief for (v,B):

v - F(ul ,b l ) , 8 - I'(a2,b2).9(0 > l) , v & 0 are

A. 1 Gamma distributions are used as priors to represent

s-independent . 1

Bayes computation for the PFM using the Gibbs Sampler is considered. A random sample of n items is drawn from the PFM (1-2) and is put on life test. The observed sample consists of, for a preassigned r , the ordered failure times, x1 < x2 < ... < xr, and ( n - r ) survivors. The likelihood reliability are:

f(SrlU,e) a exp[-O-(S,. - n . ~ ) ] , v < lOg(x1)

R ( t ) = exp[-O.(log(t) - v)], U < log@).

From (2-1) and assumption A.1, the constrained

function and

(2-1)

(2-2)

(the range of S, depends on v) Bayes model (likelihood x prior) is:

f(Srlv,8).f(v)-f(8) a [exp[-8.b2 - + n.v.8

- v.bl]].(ja2+r-1.val-1

v < log(xl), b1 > n.0 , 8 > 1. (2-3)

'The likelihood function in (2-1) is a function of v, 0 for a fixed value of S,. If x , , ... ,x, are type-1 censored data (censored on the right by to ) , then the likelihood function in (2-1) has the same form except x , in the second term in the definition of S, is replaced by to. By mak- ing this change, the Bayes analysis in this section will forthrightly apply to the type-1 censored sample.

From (2-3), the full conditional pdf are:

f(vlO,Sr) a exp[-v-(bl - n . 8 ) ] . ~ ' ~ - ~ , bl > n.0 , 8 > 1,

ie, vl8, S, - I'(al, bl - n - 0 ) . 9 ( b 1 > n.8), 0 > 1;

(2-4)

0 > 1,

8 > 1. (2-5)

The conditional posterior pdf o f U & 8 are truncated Gamma pdf, with restrictions on the scale parameters.

2.2 Gibbs Sampler

Let a collection of p r.v., ul,. . . , up whose full conditional pdf denoted, generically, by f ( u , I U,, r Z s ) , s = 1,. . . , p , is available for sampling. Under mild conditions [5] , these full conditional pdf uniquely determine the joint pdf f ( ul,. . . ,up) and hence all the marginal pdff(u,), s= 1 ,..., p .

Algorithm 1 [4, 51

1. Given an arbitrary starting set of values,

U{'), U$') ,... , upo), draw

...)

to complete one iteration of the scheme. 2. k such iterations give ( u f k ) , ..., u j k ) ) . 3. N parallel replications yield N p-tuples:

Under mild conditions [7, 81:

ujk) &U, - f(u,) as k - 03.

For any function T of ul, ..., up whose s-expectation exists.

almost surely.

TIWARI ET A L BAYES ESTIMATION FOR THE PARETO FAILURE-MODEL USING GIBBS SAMPLING

~

473

The distribution of ( u l , ..., u p ) can be approximated by the empirical distribution of (U?!) ,..., U;,:)), j = 1 ,..., N. Similar- ly the marginal of U, can be approximated by the empirical distribution of U:,;), j = 1 ,. . . ,N. When a lower dimensional marginal is required, eg, f( U,), and i f f ( U, I U,, r # s) can be computed, the preferred estimate [4] is:

For any T ( u l , ..., u p ) , let

Tjk) J = T(u$) ,... , U$)), j = l , . . . , N ,

the empirical distribution of T / k ) , . . . , T$) provides an estimate off( T(u1,. . . 4,) 1.

2.3 Example3

Gibbs Sampler for the Pareto Failure Model

Given

n=20, r=7 , m=5, 0=1.2;

observations xj generated from PFM (1-2) using the Epstein-

Algorithm 1 is implemented for the conditional posterior Sobel transformation [2, 31. 4

pdf of U & 8 in (2-4) & (2-5):

vo 3 0, vj = log(xj) - v,

Wj Vj - Vj-1, j = l , ..., r.

Then yj = ( r - j + 1) .wj, for j= 1,. . . ,r, are i.i.d. exponen- tial with scale parameter 0.

Based on N = lo3 replications in algorithm 1:

A = 4.79, 6 = 1.19, fi = 30.13.

The solid lines in figures 1 - 4 are graphs off( m I S,) , f ( d IS,), f ( h ( t ) IS,.), f ( R ( t ) IS,) at t = 8 . From the Gibbs sampler, {R,(k)(t),...,RISk)(t)} is a sample from f ( R ( t ) IS,.), and {h{k’ ( t ) , . . . ,h$)N( t )} is a sample fromf(h(t) IS,.). The 90% CI are:

(h((:!O:)05N) ( t ) , h((l$)9595N) ( t ) ) .

The true values of R ( t ) , h ( t ) for various t , given in tables 1 & 2, are computed using (1-3) & (1-1).

3The number of significant figures is not intended to imply any ac- curacy in the estimates, but to illustrate the arithmetic.

Table 1. Posterior Mean and 90% CI of R ( t )

t True R ( t ) R ( t ) R ( t ) - R ( t ) 90% CI

7 0.6678 0.6033 0.0645 (0.4306, 0.9249) 8 0.5689 0.5141 0.0548 (0.3670, 0.7730) 9 0.4939 0.4464 0.0475 (0.3200, 0.6750)

10 0.4352 0.3935 0.0417 (0.2821, 0.5883) 15 0.2675 0.2420 0.0255 (0.1701, 0.3585) 20 0.1894 0.1715 0.0179 (0.1186, 0.2555) 30 0.1164 0.1055 0.0109 (0.0690, 0.1611)

Table 2. Posterior Mean and 90% CI of h ( t )

t True h ( t ) h^(t) h ( t ) -h^( t ) 90% CI

7 0.1714 0.1711 0.0003 (0.1462, 0.2071) 8 0.15 0.1498 0.0002 (0.1280, 0.1812) 9 0.1333 0.1331 0.0002 (0.1137, 0.1611)

10 0.12 0.1198 0.0002 (0.1024, 0.1450) 15 0.08 0.0599 0.0001 (0.0512, 0.0725) 20 0.06 0.0599 0.0001 (0.0512, 0.0725) 30 0.04 0.0399 0.0001 (0.0341, 0.0483)

2.4 Example

Alternative Priors for 8, rn Assumptions

A.l 8 & m are s-independent. A.2 8 - r ( a 2 , b2).9(0 > 1). A.3 m - f ( m ) , improper prior:

f ( m ) cc constant, 0 < m < w. 4

Under assumptions A.l - A.3, the posterior pdf of (e, m ) and full conditional pdf are:

f(O,mlS,.) 0: exp[-8.(b2 + S,. -

1 < e < M, < iog(xl);

Olv, S, - r(a2 + r, b2 + S,. - n.v).9(v < log(xl)),

e > 1; (2-6)

ml6, S, - GU(n.8; 0 , ~ ~ ) . (2-7)

The Gibbs Sampler is implemented using the conditional posterior pdf (2-6) & (2-7).

Given

m=5, 0=1.2, n=20, r = 7 ;

a2 =75, b2 =48, N = lo3 replications.

Then,

A = 4.80, 6 = 1.19, C; = 30.67.

4

474 IEEE TRANSACTIONS ON RELIABILITY, VOL. 45, NO. 3,1996 SEPTEMBER

g . x x 0 N 0 -

The dotted lines in figures 1 - 4 give estimated pdf. Tables 3 & 4 give the true values, posterior means, and 90% CI. The Bayes estimates of m, 8, p are close to their true values when v has the informative prior r(70,70). For the two choices of priors in section 2.1 (assumption A.l) and in section 2.4 (assumptions A.l - A.3), the CI for h ( t ) are not as different as the CI for R ( t ) .

'\ I I I I

I I

4 6 8 10

- - r(70,70), o - r(75,48).g(o > 1) ......... v = constant, 0 - r(75,48).9(0 > 1) -_____ based on Death Records with Y - r(70,70), 0 - r(75,48). S(6'

Figure 1. Estimated pdf{m)S,}

> 1)

1 .o 1.2 1.4 1.6 1.8

~ - r(70,70), o - r(75,48).~(o > 1) ......... v = constant, 0 - r(75,48).9(0 > 1) ______ based on Death Records with v - r(70,70), 0 - F(75,48) .g(0

Figure 2. Estimated pdf{O(S,) > 1)

3 . BAYES ESTIMATION BASED ON DEATH RECORDS

3.1 Theory

In some clinical studies, the experimenter can decide to terminate the experiment after a preassigned period of time T.

0 m 9 '\

0 ,

0 1000 2000 3000

~ - r(70,70), 0 - r(75,48).9(o > 1) v = constant, 0 - I'(75,48).9(0 > 1)

> 1)

......... _--___ based on Death Records with v - I'(70,70), 0 - r(75,48).9(0

Figure 3. Estimated pdf{h(t)lS,} at t =8

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

~ v - ry70,70), o - r(75,48).9(0 > 1) ......... v = constant, 8 - I'(75,48).9(0 > 1) _____- based on Death Records with v - r(70,70), 0 - r(75,48) .g(O

Figure 4. Estimated pdf{R(t)lS,} at t = 8

> 1)

The number of survivors (0 5 s I n ) out of n patients includ- ed in the study is recorded after expiration period T, and the exact survival times for the ( n - s) deaths are not available. Thus only the triplet (n , s, T ) is recorded. This section treats a more general situation wherein the record of d such

s, I n,. The likelihood function is: s-independent triplets (ni,si, T , ) is available, i = 1,2,. .. ,d, 0 5

TIWARI ET A L BAYES ESTIMATION FOR THE PARETO FAILURE-MODEL USING GIBBS SAMPLING 475

Table 3. Posterior Mean and 90% CI of R ( t ) using Improper Prior for m

t True R ( t ) & t ) R ( t ) -&( t ) 90% CI

7 0.6678 0.6361 0.0317 (0.5669, 0.6923) 8 0.5689 0.5423 0.0266 (0.4739, 0.6017) 9 0.4939 0.4711 0.0228 (0.4037, 0,5320)

10 0.4352 0.4154 0.0198 (0.3490, 0.4766) 15 0.2675 0.2560 0.0115 (0.2004, 0.3126) 20 0.1894 0.1815 0.0079 (0.1350, 0.2317) 30 0.1164 0.1118 0.0046 (0.0775, 0.1524)

The pdf{z} isf(x) [8]. To apply algorithm 2, use discrete ap- proximations of the conditional posterior pdf (3-2) & (3-3): Given a large value of M , generate vj & ej, j = 1,. .. ,M, from pdf gl, g2 and define:

f( ej I S, v ) /g2 Cej) , < min {Tkl ,

M 1 s k s d Table 4. Posterior Mean and 90% CI of h ( t ) using Improper f(OjIv,S) =

Prior for m f(e,I ~,v)/g2(0i) i = l

t True h ( t ) h^(t) h ( t ) -h*(t) 90% CI 8; > 1. (3-5)

~I

7 0.1714 0.1693 0.0021 (0.1386, 0.2002) 8 0.15 0.1482 0.0018 (0.1222, 0.1752) 9 0.1333 0.1317 0.0016 (0.1086, 0.1557) Eq (3-4) & (3-5) are then approximated by the truncated

10 0.12 0.1186 0.0014 (0.0977, 0.1401) Gamma pdf, 15 0.08 0.0790 0.0010 (0.0652, 0.0934) 20 0.06 0.0593 0.0007 (0.0489, 0.0701) 30 0.04 0.0395 0.0005 (0.0326,0.0467) r (b1,&) . I ( v < min {log(T) I),

l s i s d

r(ri2,62) .z(e > 11, d

9 [I -exp(-o.(log(T) - V) ) ] f l i - s l , by choosing hi & 6, so that the support of these pdf cover the supports of (3-4) & (3-5). Generate vi and Oi using these truncated Gamma pdf as g and the pdf (3-4), (3-5) as f

i = l

(3-1) in algorithm 2. v < min {log(T)} l s i s d

Under assumption A. 1 , the full conditional pdf of v & 0 are: 3.2 ~~~~~l~

f(vl8,S) 0~ f(Sle,v).f(v).f(e), v < min {log(T,)}, Given l < i < d

e > 1 , (3-2) 8=1.2, m=5, d=10;

ni=20, i=1,2 ,..., d.

e > 1. (3-3) Table 5 gives triplets (ni ,si ,T), where:

The rejection/acceptance algorithm [9: p 991 is used for sampl- ing, as follows: To generate random observations from a distribution with pdff(x) , consider an integrable function g ( x ) such that g(x) 2 f(x). Define a pdf,

T, were generated from PFM (1-2),

St were generated from binm(* ; ( m / T ) ’, 4) .

Table 5. Simulated Triplets from PFM

Algorithm 2

4

n, 20 20 20 20 20 20 20 20 20 20 s, 2 2 3 2 3 2 2 1 4 1 T, 10 11 9 12 10 11 10 12 9 11

1. Generate x from q(x). 2. Generate U from U(0,l). 3. If U 5 f(x)/g(x) then set z = x; else goto 1.

The dashed lines in figures 1 - 4 give the estimated pdf. Tables 6 & 7 give the true values, the posterior means,

4 and the 90% CI.

476 IEEE TRANSACTIONS ON RELIABILITY, VOL. 45, NO. 3, 1996 SEPTEMBER

Table 6. Posterior Mean and 90% CI of R ( t ) [Based on death records]

t True R ( t ) R ( t ) R ( t ) - R ( t ) 90% CI

10 0.4352 0.4135 0.0217 (0.2690, 0.6465) 11 0.3882 0.3667 0.0215 (0.2344, 0.5760) 12 0.3497 0.3286 0.021 1 (0.2080, 0.5201) 13 0.3070 0.2706 0.0201 (0.1671, 0.4362) 14 0.2676 0.2481 0.0195 (0.1513, 0.4051) 15 0.1894 0.1726 0.0168 (0.0996, 0.4051)

Table 7. Posterior Mean and 90% CI of h ( t ) [Based on death records]

t True h ( t ) h*(t) h ( r ) -h*(t) 90% CI

10 0.1200 0.1251 -0.0051 (0.1001, 0.1536) 11 0.1091 0.1137 -0.0046 (0.0910, 0.1397) 12 0.1000 0.1042 -0.0042 (0.0834, 0.1280) 13 0.0923 0.0962 -0.0039 (0.0710, 0.1182) 14 0.0857 0.0893 -0.0036 (0.0715, 0.1097) 15 0.0800 0.0834 -0.0034 (0.0667, 0.1024) 20 0.0600 0.0625 -0.0025 (0.0500, 0.0768)

REFERENCES

[I] B.C. Arnold, S.J. Press, “Bayesian inference for Pareto Population”, J. Econometrics, vol 21, 1983, pp 287-306.

[2] B. Epstein, M. Sobel, “Life testing”, J . Amer. Statistical Assoc, vol

[3] B. Epstein, M. Sobel, “Some theorems relevant to life testing from an exponential distribution”, Annals of Mathematical Statistics, vol25, 1954,

48, 1953, pp 486-502.

pp 373-380.

[4] A.E. Gelfand, A.F.M. Smith, “Sampling based approaches to calculating marginal densities”, J . Amer. Statistical Assoc, vol85, 1990, pp 398-409.

[5] S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images”, IEEE Trans. Pattern Analysis and Machine Intelligence, vol 6, 1984, pp 721-741. S. Kumar, R.C. Tiwari, “Bayes estimation of reliability under a ran- dom environment governed by a Dirichlet prior”, IEEE Trans. Reliability, vol 38, 1989 Jun, pp 218-223. D.V. Lindley, N.D. Singpurwalla, “Multivariate distributions for the life lengths of components of a system sharing a common environment”, J. Applied Probability, vol 23, 1986, pp 418-431.

[8] B. Ripley, Stochastic Simulation, 1987; John Wiley & Sons. [9] M.A. Tanner, Tools for Statistical Inference, Lecture Notes in Statistics,

vol67, 1992 (J. Berger, S. Fierberg, J. Gani, et al, Eds) Springer-Verlag.

[6]

[7]

AUTHORS

Dr. Ram C. Tiwari; Dept. of Mathematics; Univ. of North Carolina; Charlotte, North Carolina 28223 USA. Internet (e-mail): fmaOOrct@unccvm,uncc.edu

Ram C. Tiwari is a Professor & Chair’n of the Mathematics Depart- ment at the University of North Carolina at Charlotte. For biography, see IEEE Trans. Reliability, vol 41, 1992 Dec, p 607.

Yizhou Yang; Dept. of Statistics; Florida State University; Tallahassee, Florida, 32306 USA.

Yizhou Yang (born 1968) is a PhD student at the Florida State Univer- sity. He received the BS (1991) in Applied Mathematics from Beijing Institute of Technology, and MS (1994) in Applied Statistics from the University of North Carolina at Charlotte. His research interests are in reliability and survival analysis.

Dr. Jyoti N. Zalkikar; Dept. of Statistics; Florida International Univ; Univer- sity Park; Miami, Florida 33199 USA. Internet (e-mail): zalkikar@servms.fiu.edu

Jyoti N. Zalkikar is an Associate Professor at the Florida International University. For biography, see IEEE Trans. Reliability, vol41, 1992 Dec, p 607.

Manuscript received 1996 June 30

Publisher Item Identifier S 0018-9529(96)07361-7 4 T R b

CORRECTION I996 June Issue CORRECTION 1996 June Issue CORRECTION I996 June Issue CORRECTION

Correction to: Plastic Packaging Is Highly Reliable [I]

1. page 185, column 2, line -19: Change “important” to “noteworthy”.

2. page 186, figure 3 (title): Change “table 4” to “table 1”.

3. page 189, column 1, line -3 (last par., first sentence): Change “lo5” to “lo4”.

REFERENCE [I] N. Sinnadurai, “Plastic packaging is highly reliable”, IEEE Tram.

4TRb Reli(dnlity, vol 45, 1996 Jun, p p 184-193.