An Approximate E-Bayesian Estimation of Step-stress Accelerated Life Testing with Exponential Distribution

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1 Send Orders for Reprnts to The Open Cybernetcs & Systemcs Journal, 25, 9, Open Access An Approxmate E-Bayesan Estmaton of Step-stress Accelerated Lfe Testng wth Exponental Dstrbuton Lpng Hu and Xanbn Wu 2,* College of Informaton and Management, Henan Agrculture Unversty, Zhengzhou, 452, Chna; 2 Junor College, Zhejang Wanl Unversty, Nngbo, Zhejang, 35, Chna Abstract: The data analyss problem of step-stress accelerated lfe testng wth exponental dstrbuton s dscussed At the step-stress accelerated lfe testng condtons, an approxmate E-Bayesan parameter estmaton of step-stress accelerated lfe testng wth exponental dstrbuton s gven by consderng the pror dstrbutons of the hyperparameters and usng Gbbs samplng method Fnally, a smulaton example s gven, the results show that the Gbbs samplng method s smple and the convergence s better E-Bayesan parameter estmaton s more effectve than the maxmum lelhood estmaton Keywords: E-Bayesan estmaton, step-stress lfe testng, exponental dstrbuton, Gbbs samplng INTRODUCTION Wth the development of technology and the mprovement of the products qualty, hgh relablty and long-lfe products are avalable everywhere However, at normal worng condtons, the mplementaton of lfe testng can not meet the requrements of relablty evaluaton Accelerated lfe testng s a lfe testng method that s used to shorten the lfe testng cycle by ncreasng the stresses Accelerated lfe testng method can be used to assess the relablty of the products n a relatvely short perod of tme and to dentfy the reasons for product falure Accelerated lfe testng data analyss and parameter estmatons are theoretcal and practcal applcaton value Step-stress accelerated lfe testng (brefly step-stress lfe testng) s an mportant lfe testng of accelerated lfe testng In recent years, usng the gven statstcal model of stepstress lfe testng wth exponental dstrbuton, the paper [] gave the statstcal analyss method for type censorng lfe testng samples; the paper [2] gave the necessary and suffcent condton for the exstence and unqueness of the MLE of the step-stress lfe testng n type and censorng cases and got the approxmate confdence nterval of the mean lfe at the normal stress level on that bass; n type censorng case, the paper [3] got the Bayesan parameter estmaton wth constrants of step-stress lfe testng under the exponental dstrbuton; the paper [4] gave an approxmate Bayesan parameter estmaton for step-stress lfe testng under the exponental dstrbuton; the paper [5] gave a herarchcal Bayesan parameter estmaton for step-stress lfe testng under the exponental dstrbuton Although n type censorng case, the Bayesan parameter estmaton for step-stress lfe testng wth exponental 874-X/5 dstrbuton was gven n the papers [3-5], on the one hand there s no a good consderaton on the parameters n the pror dstrbutons; on the other hand the calculaton of the posteror margnal dstrbuton functon nvolves complex ntegral calculatons Based on the above consderatons, the E-Bayesan method s gven for parameter estmaton of step-stress lfe testng wth exponental dstrbuton n ths paper Pror dstrbutons of the hyperparameters n the pror dstrbuton are also gven, thus the jont posteror densty functon s got For the calculaton of the parameter estmaton n the jont posteror densty functon, Gbbs samplng s used for the teraton of parameters to be estmated Fnally, a smulaton example s gven to analog comparator the E- Bayesan estmaton and the maxmum lelhood estmaton, the results show that the E-Bayesan estmaton s more effectve than the maxmum lelhood estmaton Ths paper s organzed as follows: In Secton 2, the basc assumptons for step-stress lfe testng are stated E-Bayesan estmatons of step-stress lfe testng parameters are stated n Secton 3 The estmaton of the relablty ndexes wth exponental dstrbuton n Secton 4 An example s gven to llustrate the proposed procedure n Secton 5 The concluson of ths study s gven n Secton 6 2 THE BASIC ASSUMPTIONS OF STEP-STRESS LIFE TESTING Determne the normal stress level S and the accelerated stress levels S, S 2,,S, the stress levels meet S <S <S 2 < <S, n samples are taen from a number of products for step-stress lfe testng At the stress level S, the worng tme of the falure products are t t 2 r In the case of type censorng, s the pre-gven tme for stoppng the tests at the stress level S, r s the number of falure products before the tme at the stress level S ; n the case of type censorng, r s the pre-gven number of samples for stoppng the tests 25 Bentham Open t

2 73 The Open Cybernetcs & Systemcs Journal, 25, Volume 9 Hu and Wu Assumpton At the normal stress level S and the accelerated stress levels S <S 2 < <S, the lfe dstrbuton of a test unt all obey the exponental dstrbuton, ts cumulatve dstrbuton functon s: F ( t S) = exp [ ( S) t], t> ( where (S ) > s the falure rate of the products at the stress level S, ts mean lfe s (S )=/ (S ) Assumpton 2 The mean lfe of the products: s the accelerated lfe functon of the stress: ln =μ+ (S ) (2 where μ, s parameters to be estmated; (S ) s the nown functon of the stress level S Assumpton 3 Resdual lfe of the products depends only on the already cumulatve falure part and the stress level at that tme, but has nothng to do wth the cumulatve [6] 3 E-BAYESIAN ESTIMATION OF STEP-STRESS LIFE TESTING PARAMETERS 3 The Lelhood Functon of Step-stress Lfe Testng Parameters For step-stress lfe testng data wth exponental dstrbuton, the falure data t,t 2,, t r are the lfe data of the samples at the stress level S ; but the falure data s not the real lfe of the samples at the stress level S when > Therefore, the falure data need to be converted nto the real lfe data Accordng to Assumpton 3, the cumulatve falure probablty at the stress level S when the samples worng tme s t equvalent to the cumulatve falure probablty at the stress level S j when the samples worng tme s t j That s to say: F ( t) = F ( t ),,2,, S S j j Then accordng to assumpton one, we can get: exp [ ( S) t] = exp j( S j) tj So, t j =,,,2,, j t R = j Let r, =,2,,, so the total testng tme at the stress level S s: T r = tj + n, (type Censorng case) ( R) r, ( type Censorng case) T = t + ( n ) R t r j Thus accordng to relevant theorems, the lelhood functon s got: LD (, 2,, ) r exp( T ) (3) = = where D=t,, t r, t 2,, t 2r,,t 2,, t r 2 Accordng to Assumpton 2, for =/, we can see: = exp{b(s )-(S )}= where s the falure rate of the products at the normal stress level; = exp { b[ ( S) ( S) ]} = / s the acceleratng factor between the stress level S and S, ( >; S ) ( S) =, =,2, ( S) ( S) Let T = = r, T 2 = T, r = r = = So the lelhood functon (3) can be converted to: L(, ) r = T exp( T (3 Now, there are only two parameters n the lelhood functon (3 In the actual producton, people are most concerned about the falure rate at the normal stress level and the acceleraton factor 32 Defnton of E-Bayesan Estmaton Defnton [7] Wth ( ab, ) beng contnuous, E = ( a, b) ( a, b) dadb = E[ ( a, b)] D, (33) s called the expected Bayesan estmaton of (brefly E-Bayesan estmaton), where ( ab, ) s Bayesan estmaton of wth hyperparameters a and b, D s the doman of (a,b), and (a,b) s the densty functon of a and b over D By Defnton, the E-Bayesan estmaton of s the expectaton of the Bayesan estmaton of for the hyperparameters a and b The E-Bayesan estmaton of s not Bayesan estmaton or herarchcal Bayesan estmaton [8], t can be seen as a nd of modfed Herarchcal Bayesan estmaton The E-Bayesan estmaton method have wde scope potental applcatons n many felds [9,] 33 The Pror Dstrbuton of the Parameters, and the Jont Posteror Densty Functon Based on the engneerng experence, the range of the acceleraton factor: s << 2, for ths, tae the pror densty functon of as:

3 An Approxmate E-Bayesan Estmaton The Open Cybernetcs & Systemcs Journal, 25, Volume 9 73 ( ) =, << 2 (34) If the pror dstrbuton of o be ts conjugated dstrbuton Beta (a, b) wth densty functon as follows: ( ) b a ( ab, ) = Bab (, ) where < <, a>, b>, (, ) a b = ( ) d Bab t t t s the Beta functon, a>, b>, both a and b are hyperparameters In the case of modern hgh-relablty products, the possblty of the falure rate o beng larger s smaller than the possblty of the falure rate o beng smaller For ths, accordng to the paper [8], a and b should be chosen so that ( a,b) s a decreasng functon of When <a<, <b, ( a,b) s a decreasng functon of To determne the specfc value of a, b s very dffcult, for the two hyperparameters are unobservable and the nformaton obtaned n practcal applcaton s nsuffcent to determne the value of a, b Therefore, a unform dstrbuton can be respectvely defned over the range of a and b as the pror dstrbutons of hyperparameters a and b For ths, tae the pror dstrbutons of hyperparameters a, b as follows: (a)=u(,), 2 (b)=u(,c) where c s a constant Consderng that n the case of a<, the bgger b s, the thnner s the tal of the Beta densty functon But n vew of the robustness of the Bayesan estmaton, the thnner taled pror dstrbuton often leads to the worse robustness of the Bayesan estmate Accordngly, b should not be too bg, t s better to be chosen below some gven upper bound c (c > s a constant to be determned) When the parameters a, b s of ndependence, accordng to the Defnton, we can get the pror densty functon of : c a b ( ) ( ) = dadb (35) c- B(,) a b So by the pror densty functon of and : (34) and (35),as well as the lelhood functon (3, accordng to Bayesan theorem, we can get ts jont posteror densty functon, then accordng to Defnton,we can get the E- Bayesan jont posteror densty functon of (,,a,b): (,,a,b T ) B(a, b) ( ) b T exp( T ) 2 r + a where T=(n, r,, t j,,2,, r, =,2,,) (36) 34 The E-Bayesan Estmaton of the Parameters and The problem of usng Bayesan method for statstcal nference s the calculaton of the posteror margnal dstrbuton functon In many cases, t s dffcult or even mpossble to obtan the analytc expresson of the posteror margnal dstrbuton functon, sometmes we can get the analytc expresson but the results are rather complcated that s not easy for applcaton and promoton Therefore, ths paper uses Gbbs samplng approach for the teraton of parameters to be estmated, the mean of the parameters to be estmated s obtaned The bggest advantage of the approach s mplementng smple and wth the convergence of teraton 34 Gbbs Samplng MCMC (Marov Chan Monte Carlo) method s through the establshment of the Marov chan wth a stable dstrbuton to obtan the samples of p( x n ), then mae statstcal nference for the samples obtaned One of the most smple and extensve applcaton of MCMC methods s Gbbs samplng method It was proposed by S Geman and D Geman n 984 [] Tae random varables as, 2,,, assume ts full condtonal dstrbuton p( s r ) (rs), s=,2,, s avalable samplng, that s to say, when a set of values for random varable r (rs) are gven, we can generate a random sample of s The paper [2] proved that at approprate condtons, the jont dstrbuton functon p(, 2,, ) was only decded by the full condtonal dstrbuton, so all of the margnal dstrbuton functon p( s ), s=,2,, are only determned by the full condtonal dstrbuton Gbbs samplng process s as follows: the startng pont s gven () =( (), () 2, () ), () () Samplng from the full condtonal dstrbuton p( x n, () 2,, () ) ; () ( Samplng 2 from the full condtonal dstrbuton p( 2 x n, (), () 3,, () ); () () Samplng from the full condtonal dstrbuton p( x n, (), () 2,, () -, () +,, () ); () () Samplng from the full condtonal dstrbuton p( x n, (),, () - ) Repeat the above ⑴ to () steps, after t-steps teratons; we can get a Marov chan: () =( (), 2 (), () ); ( =( (, ( 2, ( ) (t) =( (t), (t) 2, (t) ) The sample whch mae Marov chan reach equlbrum can be as a sample of p( x n ) The fact can be proved that at approprate condtons, when Iteraton tmes t, then p( (t), (t) 2, (t) ) p(, 2,, )

4 732 The Open Cybernetcs & Systemcs Journal, 25, Volume 9 Hu and Wu Hence, an estmaton of the margnal dstrbuton we need can be got [3]: fˆ( s ) f ( s r, r s ) = The convergence of the teraton samplng s determned by t 342 The Gbbs Samplng Process of the Parameters,, a and b By the E-Bayesan jont posteror densty functon (36), we can see that the full condtonal posteror probablty densty functon of s: (,,, ) r a ( ) b abt + exp( T The full condtonal posteror probablty densty functon of s: ( ), abt,, T exp( T The full condtonal posteror probablty densty functon of a s: ( a,,, ) a b T / B( a, b) The full condtonal posteror probablty densty functon of b s: ( b,,, ) ( ) b a T / B( a, b) The random number for the full condtonal posteror dstrbuton of,, a and b can be produced by selected samplng method Accordng to the steps of Gbbs samplng method, the startng pont s gven as ( (), (), a (), b () ), then the t tme teraton s dvded nto the followng four steps: (t) () Samplng from the full condtonal posteror dstrbuton ( T, (t-), a (t-), b (t-) ) ( Samplng (t) from the full condtonal posteror dstrbuton ( T, (t),a (t-), b (t-) ) (3) Samplng a (t) from the full condtonal posteror dstrbuton (a T, (t), (t), b (t-) ) (4) Samplng b (t) from the full condtonal posteror dstrbuton (b T, (t), (t), a (t) ) Then ( (t), (t), a (t), b (t), t=,2, n,n+,,n ) s a Gbbs teraton sample of parameters (,, a, b), where n s the dscarded sample sze before Gbbs teratons have got a steady state, N>n s the overall sample sze We only care about the falure rate at the normal stress level: and the acceleraton factor, so the E-Bayesan parameter estmaton of and are respectvely: N ˆ = N N n, ˆ = t n N n t = n + = + 4 THE ESTIMATION OF THE RELIABILITY IN- DEXES WITH EXPONENTIAL DISTRIBUTION Our ultmate goal s to get the estmaton of the product mean lfe at the normal stress level:, then accordng to the assumpton two and the accelerated lfe equaton (, the parameters μ and n the accelerated lfe equaton are estmated Thereby, the accelerated lfe model s establshed Accordng to the Assumpton 2, the mean lfe s (S )=/ (S ) So the frst relablty ndex whch s the mean lfe s got: ˆ = /ˆ Usng the E-Bayesan parameter estmaton of the falure rate ˆ, the second relablty ndex whch s relablty s obtaned: R(t)=exp(-t)=exp(- ˆ t) Then accordng to the accelerated lfe equaton (), we have ln ˆ =μ +(S ), usng the least square method to get a dstrbuton curve wth varous ponts (μ, ), the approxmaton of the parameters μ and can be estmated : μ= ˆμ, = ˆ Thereby, we can establsh the accelerated lfe model: ln = ˆμ + ˆ (S ) 5 A SIMULATION EXAMPLE Now there are a number of electronc products, ther lfe obeys the exponental dstrbuton, 4 samples are taen from the products for the four-steps step-stress accelerated lfe testng The normal stress level s S =28V(volt), tae accelerated stress levels as S =38V, S 2 =4V, S 3 =44V, S 4 =47V, the censorng tme of each step s =h(hour), 2 =6h, 3 =25h, 3 =25h Accelerated lfe equaton ln=68-6lns s Pre-gven, where the parameters μ=68, = - 6 Usng Monte-Carlo method to obtan a set of lfetme data: Frst usng the method n the paper [2] parameters μ, n the accelerated lfe equaton are estmated, the MLE of μ, are respectvely: ˆμ =62955, ˆ =42498 Then from the accelerated lfe equaton, we can get the estmaton of the falure rate: ˆ =33 6 and the estmaton of the acceleraton factor: ˆ =77647 Next, E-Bayesan method s used to estmate the above parameters Tae the unform dstrbuton over (,5) as the pror dstrbuton of, tae Gbbs samplng teraton tmes N =55, the ntal value are gven as =, =, a=, b=2 The fact can be found that the parameters bascally reach a steady state after the 5 teraton steps Therefore, the E-Bayesan estmaton of the parameters and are obtaned from the mean of 5 samples after n = 5 steps, then we can get the estmaton of the falure Rate: ˆ =437 7 and the estmaton of the accelera-

5 An Approxmate E-Bayesan Estmaton The Open Cybernetcs & Systemcs Journal, 25, Volume Table Step-stress lfe testng data of the electronc products t h t 2 t 22 t 23 t 24 t h 42676h 24553h h h t 3 t 32 t 33 t 34 t 35 t h 6732h 27626h 7957h 24247h h t 4 t 42 t 43 t h 3632h 273h 5968h ton factor: ˆ =29275, furthermore, we can get the estmaton of parameters μ and are respectvely ˆμ =67783, ˆ =5928 The results show that E-Bayesan parameter estmatons are very close to the true values; however the parameter estmatons whch are obtaned by the MLE method have a larger dfference wth the true values CONCLUSION Usng the E-Bayesan method, the parameters estmaton and the relablty ndexes estmaton for step-stress accelerated lfe testng wth exponental dstrbuton are gven The values of the hyperparameters n the pror dstrbuton are avoded Usng Gbbs samplng for the calculaton of the posteror margnal dstrbutons and parameter estmaton, the calculaton wth hgh-dmensonal ntegrals s avoded Fnally, the fact can be seen that E-Bayesan parameter estmaton s more effectve and practcal than the maxmum lelhood estmaton from the smulaton example CONFLICT OF INTEREST The authors confrm that ths artcle content has no conflct of nterest ACKNOWLEDGEMENTS Ths wor was fnancally supported by the Natural Scence Foundaton of Nngbo Grant (No 23A6276) and the Scentfc Research Project from Educaton of Zhejang Provnce (No Y243585) and the thn-tan Entrepreneurshp project of Nngbo(3B766) REFERENCES [] J S Lu, Monte Carlo strateges n scentfc computng, Sprnger-Verlag, New Yor, 2 [2] H L Fe and X X Zhang, Maxmum lelhood estmate of stepstress accelerated lfe testng under the exponental dstrbuton, Chnese Mathematcal Applcaton, vol 7, no 3, pp , 24 [3] C X Zhong and S S Mao, Bayesan approach to an accelerated lfe test under the exponental dstrbuton case, Chnese Appled Mathematcs, vol 8, no 4, pp , 993 [4] Z H Zhang, Bayes analyss of step-stress accelerated lfe testng under exponental dstrbutons, Chnese Appled Mathematcs, vol 2, no 2, pp 75-8, 997 [5] D Wu and Y C Tang, Bayesan estmaton of step-stress accelerated lfe testng of exponental dstrbuton under CE model, Chnese Applcaton of Statstcs and Management, vol 27, no 3, pp , 28 [6] W Nelson, Accelerated lfe testng-step-stress models and data analyss, IEEE Trans Relablty, vol 29, no 2, pp 3-8, 98 [7] M Han, E-Bayesan estmaton of falure probablty and ts applcaton, Mathematcal and Computer Modelng, vol 45, pp , 27 [8] M Han, The structure of herarchcal pror dstrbuton and t s applcatons, Chnese Operatons Research and Management Scence, vol 6, no 3, pp 3-4, 997 [9] G L Ca, L L Wu and X F Tang, E-Bayes relablty analyss of double hyperparameters zero falure data, Journal of Jangsu Unversty (Natural Scence Edton), vol 3, no 6, pp 2 [] G L Ca and W Q Xu, Applcaton of E-Bayes method n stoc forecast, Proceedngs of The 4th Internatonal Conference on Informaton and Computng Scence, vol, pp 54-56, 2 [] Y C Tang and H L Fe, Bayesan analyss for Webull dstrbuton accelerated lfe testng based on Gbbs samplng, Chnese Mathematcal Statstcs and Appled Probablty, vol 3, no, pp 8-88, 998 [2] J L Besag, Spatal nteracton and the statstcal analyss of lattce systems, Journal of the Royal Statstcal Socety Seres B (Methodologcal), vol 36, no 2, pp , 974 [3] G Stuart and G Donald, Stochastc Relaxaton, Gbbs dstrbutons and the Bayesan restoraton of mages, Journal of Appled Statstcs, vol 2, no 6, pp 25-62, 993 Receved: September 6, 24 Revsed: December 23, 24 Accepted: December 3, 24 Hu and Wu; Lcensee Bentham Open Ths s an open access artcle lcensed under the terms of the Creatve Commons Attrbuton Non-Commercal Lcense ( lcenses/by-nc/3/) whch permts unrestrcted, non-commercal use, dstrbuton and reproducton n any medum, provded the wor s properly cted