"Science Stays True Here" Journal of Mathematics and Statistical Science, Volume 2016, Science Signpost Publishing

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1 "Science Stys True Here" Journl of Mthemtics nd Sttisticl Science, Volume 06, Science Signpost Publishing Estimtions in Step-Prtilly Accelerted Life Tests for n Exponentil Lifetime Model Under Progressive Type-I Censoring nd Generl Entropy Loss Function Al H. Abdel-Hmid Deprtment of Mthemtics, Fculty of Science, Beni-Suef University, Beni-Suef, Egypt. Abstrct Bsed on progressively type-i censored smples, this pper discusses some estimtion methods in step-prtilly ccelerted life tests when the lifetimes of items under use condition follow the exponentil distribution. Mximum lielihood estimtions for the considered prmeters re obtined in closed forms. The observed Fisher informtion mtrix is derived to clculte confidence intervls for the considered prmeters. Byesin estimtions for the prmeters re crried out bsed on () informtive prior for the scle prmeter nd discrete prior for the ccelertion fctor, (b) both the symmetric loss (squred error loss) function nd symmetric loss (generl entropy loss) function. The resulting Byes estimtes re obtined in closed forms. The precision of the estimtes nd comprison mong them re investigted through Monte Crlo simultion study. eywords: Prtilly ccelerted life tests, Progressive type-i censoring, Exponentil distribution, Mximum lielihood nd Byesin estimtions, generl entropy loss, Simultion.. Introduction Censoring is of gret importnce in plnning durtion experiments in relibility nd lifetime studies whenever the experimenter does not observe the lifetimes of ll test units. The trditionl censoring schemes (type-i nd type-ii censoring) do not llow for units to be removed from the test t points other thn the terminl point of the experiment. This llownce will be importnt

2 76 when compromise between reduced time of experimenttion nd the observtions of t lest some extreme lifetimes re sought. Also when some of the surviving units in the experiment tht re removed erly one cn be used for some other test. These resons led us into the re of progressive censoring. Accelerted life tests (ALTs) re often used for relibility nlysis. Test units re run t higher-thn-usul stress levels to induce erly filures. A model relting life length to stress is fitted to the ccelerted filure times nd then extrpolted to estimte the filure time distribution under norml use condition. The stress loding in n ALT cn be pplied in different wys: commonly used methods re constnt stress, progressive stress nd step stress. Nelson (990, Chpter ) discussed their dvntges nd disdvntges. In ALTs the units re tested only t ccelerted conditions (see AL-Hussini nd Abdel-Hmid (006)) wheres in prtilly ALTs (PALTs) the units re tested t both ccelerted nd norml conditions. PALTs include two types, one is clled step PALTs (see Abdel-Hmid nd AL-Hussini (008)) nd the other is clled constnt PALTs (see Abdel-Hmid (009)). The step PALT (which is considered in this pper) permits the test to be chnged from norml use condition to ccelerted condition t predetermined time. Bi nd Chung (99) used the mximum lielihood (ML) method to estimte the scle prmeter nd the ccelertion fctor (the rtio of the min life t norml condition to tht t ccelerted condition) for exponentilly distributed lifetime using type-i censoring dt, nd in (993) Bi, et l obtined the sme results when the lifetime is subjected to the log-norml distribution. The novelty in this pper is to pply the step PALTs to the exponentil distribution using progressively type-i censored dt nd then estimte the prmeters under considertion using ML nd Byes methods. Severl uthors investigted inferences under progressively censored dt using different lifetime distributions, mong others, Viveros nd Blrishnn (994), Blrishnn nd Sndhu (995, 996), Blsooriy nd Blrishnn (000), Ng, et l (00, 004) nd Solimn (005, 008). Gouno, et l (004) considered -level step-stress ALT under progressive type-i censoring while Wu, et l (008) discussed the sme problem considering progressive type-i censoring with grouped dt. Blsooriy nd Low (004) discussed progressively type-i censored vrible-smpling plns for Weibull lifetime distribution under competing cuses of filures.

3 77 The exponentil distribution is useful to describe filure times of units which subject to wer out. Pl, et l (006, p.5) indicted tht filure times of electric bulbs, pplinces, btteries, trnsistors, etc cn be modeled by exponentil distribution. Therefore, this distribution is frequently discussed in relibility pplictions. The rest of the pper is orgnized s follows: In Section, description of the model nd discussion of progressive type-i censoring scheme re presented. Closed forms of the mximum lielihood estimtes (MLEs) of the prmeters under considertion re derived in Section 3. Bsed on the squred error loss (SEL) nd generl entropy loss (GEL) functions, Byesin estimtion of the prmeters is obtined in Section 4. A simultion study nd n illustrtive exmple re presented in Section 5. Finlly, some concluding remrs re given in Section 6.. Model Description nd Progressive Type-I Censoring According to step PALT ech of n test units under considertion is first run t norml condition nd if it does not fil by stress chnge time τ, then the test is chnged to ccelerted condition nd held until ll units fil. Suppose tht Y is the totl lifetime of unit under norml nd ccelerted conditions. Thus Y T, 0 < T τ, = τ + ( T τ) / β, T > τ, (.) where T is the lifetime of unit t norml condition, τ is the stress chnge time nd β ( > ) is the ccelertion fctor. Suppose lso tht the rndom vrible T hs exponentil distribution with scle prmeter θ ( > 0). Thus the cumultive distribution function (CDF) of T is given by Ft ( ) = exp( t/ θ ), t> 0. (.). Progressive Type-I Censoring Scheme The progressive type-i censoring is pplied to step PALT s follows: The n test units re initilly plced on norml stress condition nd run until time τ ( 0) >, t which point the number of filed units n re counted nd R surviving units re removed from the test; strting from time τ the reminder

4 78 n n R surviving units re run until time τ t which point the number of filed units n re counted nd R surviving units re removed from the test. The test continues in this mnner until time τ t which point the reminder n n R surviving units re then plced on ccelerted condition nd run until time τ + t which point the number of filures n + re counted nd R + surviving units re removed from the test. The test continues in this mnner under ccelerted condition until time τ t which R = n Σ n Σ R surviving units re removed, thereby terminte the test. The i= i i= i removed units re often used in other experiments in the sme or t different fcilities. The censoring times τ,, τ,, τ re fixed in dvnce. The experimenter my notice the following iid observtions: y y τ y y y y τ n n ( ) ( ) n y y τ y y τ y y τ. n ( + ) ( + ) n+ + n (.3) The PDF of unit under step PALT my be written s follows. y f( y) = exp, 0 < y τ, θ θ g( y) = β τ + β( y τ) f( y) = exp, y > τ. θ θ (.4) The survivl function (SF) nd hzrd rte function (HRF) of the rndom vrible Y re given, respectively, by S( y) = exp( y/ θ), y τ, S( y) = S( y) = exp{ [ τ + β( y τ)] / θ}, y > τ. (.5) H( y) = / θ, y τ, H( y) = H( y) = β / θ, y > τ. (.6)

5 79 3. Mximum Lielihood Estimtion Bsed on the censoring times ( τ,, τ,, τ ) nd progressively type-i censored smple given in (.3), the lielihood function cn be written in the form ni Ri Ri L( θβ, ;y)_ f( yij )[ F( τi )] f( yij )[ F( τi )]. ni i= j= i= + j= (3.) Bsed on Equtions (.4) nd (3.), the log-lielihood function ( θβ, ; y) = log L( θβ, ; y) is given by ni ( θβ, ;y)_ Nlogθ nr i iτi y + ij N ( τ θlog β) + θ i j = = ni + nr i i ( τ + βτ ( i τ )) + β ( yij τ ), i= + j= (3.) where y ij is the j th observed filure in the i th pre-specified time intervl ] τ, i τi], τ = 0 0, i= i i= + i is the totl number of observed filures before(fter) N = n ( N = n ) N = N+ N. The lielihood equtions re then given by τ nd = 0 = Nτ Nθ + nrτ + y + τ nr + β( Q + W), n i i i i ij i i θ i= j= i= + (3.3) N Q+ W β β θ = 0 =, (3.4) where Q= nr i i( τi τ) nd i= + n i i= +. j= W = ( y τ ) ij The ML estimtes (MLEs) ˆ θ nd ˆβ of θ nd β cn be obtined by solving Equtions (3.3) nd (3.4) with respect to θ nd β. Thus

6 80 n i ˆ θ = τ ni ( Ri ) nr i iτi y ij, N i= + i= i= + j= ˆ ˆ Nθ β =, Q+ W (3.5) where Q nd W re s given bove. 3. Approximte Confidence Intervl The locl Fisher informtion mtrix, F, for MLEs ( ˆ θβ, ˆ ) is the symmetric mtrix of negtive second prtil derivtives of ( θβ, ) with respect to θ nd β. The inverse of F is the locl estimte of the symptotic vrince-covrince mtrix of ˆ θ nd ˆβ, tht is F ˆ ˆ θ θβ vr( ˆ θ) cov( ˆ θβ, ˆ) = =, ˆ ˆ cov( ˆ θβ, ˆ) vr( ˆ β) β θ β (3.6) where the cret ˆ indictes tht the derivtive is clculted t ( ˆ θβ, ˆ). The elements of this mtrix cn be esily obtined. Following the generl symptotic theory of MLEs, the smpling distribution of ˆ β β vr( ˆ β ) ˆ θ θ vr( ˆ θ ) cn be pproximted by stndrd norml distribution which is useful in constructing confidence intervls (CIs) for the unnown prmeters. A two sided 00( α)% norml pproximtion CIs for the two prmeters θ nd β cn then be constructed s nd ˆ θ ± z vr( ˆ θ) nd ˆ β ± z vr( ˆ β), α/ α/

7 8 where z α / is the vlue of stndrd norml rndom vrible leving n re α / to the right nd both of vr( ˆ θ ) nd vr( ˆ β ) cn be obtined from (3.6). The vlue z α / should be replced by t α / (the vlue of t -distribution leving n re α / to the right) if n < Byesin Estimtion The Byesin pproch plys n importnt rule in nlyzing filure dt nd other time-to-event. It hs been proposed s n lterntive procedure to trditionl sttisticl perspective. Due to the complicted computtions rising from generl Byesin procedure (see, for exmple, Abdel-Hmid (008)), it is preferred to consider n lterntive procedure which my be regrded s n pproximtion to more generl procedure. In this pper, we suppose tht β is restricted to finite number of vlues β,, βq with respective prior probbilities p,, pq such tht q = p =, i.e. P( β = β ) = p, see Solimn (008). The use of discrete distribution with equl probbilities for the scle prmeter β resulted in closed form expression for the posterior distribution. This csed n element of uncertinly, which is sometimes desirble in some cses. Furthermore, suppose tht, conditionl upon β = β, θ hs inverted gmm ( c, d ) with density d d πθ β= β = θ, θ>,, >, c c ( ) exp 0 ( c d 0) Γ( c ) θ (4.) where c nd d re chosen so s to reflect prior beliefs on θ given tht β = β. Bsed on (.4), (3.) nd (4.), our ctul opinion bout θ is summrized by the conditionl posterior distribution of θ given β = β which is given by Byes theorem s [ θ] ( N+ c+ ) N+ c θ Ψ exp Ψ / π ( θ β = β; y) =, Γ ( N + c ) (4.) where

8 8 φ ψ d Ψ = + +, ni φ = ( yij + Riτi ), i= j= ni ψ = τ + β( yij τ) + Ri( τ + β( τi τ)) i= + j= (4.3) β is On pplying the discrete version of Byes theorem, the mrginl posterior probbility distribution of 0 = P( = y) = q π β β 0 p πθ ( β= β) L( θβ, ) dθ = p πθ ( β= β) L( θβ, ) dθ β pd Γ ( N+ c) Ψ =, Γ( c ) Σ p ( d /Γ( c )) Γ ( N + c ) Ψ N c N c q N c N c = β (4.4) where Ψ is s defined in (4.3). 4. Estimtion Bsed on Squred Error Loss Function Under squred error loss (SEL) function, the Byes estimtor for θ is θ SE = θπ π ( θ β = β ) dθ 0 = q = q Ψ = π, N + c (4.5) nd the Byes estimtor for β is given by SE q β = βπ. (4.6) = Similrly, the Byes estimtors for the SF nd HRF t some y respectively, by = y under SEL re given,

9 83 S SE N+ c q Ψ π ε ε = Ψ + ( y ) ( τ + β( y τ)) ( y ) =, (4.7) H N + c =, q ε SE( y ) β π = Ψ (4.8) where y τ ε =,, (4.9) 0, τ < y < τ. 4. Asymmetric Loss Function The loss function L( ϑϑ, ) provides mesure of the finncil consequences rising from wrong estimte ϑ of the unnown quntity ϑ. Due to its good mthemticl properties, not its pplicbility to represent true loss structure, most of the Byesin inference procedures hve been employed using symmetric SEL function (Tribus nd Szonyi (989) nd Leon, et l (99)), L( ϑϑ ) ζ( ϑ ϑ) (4.0), =, where ζ is constnt. Although the qudrtic loss function in (4.0) is resonble choice for mny estimtion problems, there re severl situtions where it is not pproprite. For exmple, during the estimtion of the verge relible woring life of the components of spce shuttle or n ircrft, over-estimtion is usully more serious thn under-estimtion of the sme mgnitude, mińs nd Porosińsi (009). So tht loss function should represent the consequences of different errors which my rise from over- nd under-estimtions. To overcome this problem, symmetric loss functions hve been introduced such s liner-exponentil (LINEX) loss function, introduced by Vrin (975), nd generl entropy loss function (GEL), introduced by Clbri nd Pulcini (996). Despite the populrity nd flexibility of the LINEX loss function to del with estimtion of the loction prmeter, it seems to be not pproprite for estimtion of the scle prmeter nd other quntities, Bsu nd Ebrhimi (99), Prsin nd Snjri Fripour (993) nd Srivstv nd Tnn (007).

10 84 A suitble lterntive to LINEX loss is the GEL given by: ν ϑ ϑ L( ϑϑ, ) νlog, (4.) ϑ ϑ whose minimum occurs t ϑ = ϑ. This loss is generliztion of the entropy loss proposed by severl uthors where ν =, Dey, et l (987) nd Dey nd Liu (99). For positive vlues of ν, positive error ( ϑ ϑ > 0 ) cuses more serious error thn negtive error. The Byesin estimtor ϑ GE of ϑ under GEL is given by ν / ( E [ ]) ϑ ϑ ν ϑ GE =, (4.) ν provided existence nd finiteness of Eϑ [ ϑ ]. It is cler tht when ν =, Byesin estimtor (4.) coincides with the Byesin estimtor under the SEL function, wheres when ν = this estimtor coincides with the Byesin estimtor of the SF under the weighted SEL function,. L( ϑϑ, ) = ( ϑ ϑ) / ϑ 4.3 Estimtion Bsed on Generl Entropy Loss Function It is shown bove tht under GEL (4.) the Byesin estimtion of the unnown prmeter/function cn be clculted from Eqution (4.). Therefore, if in Eqution (4.) ϑ = θ, then the Byesin estimtor of the scle prmeter θ of Eqution (.4) under GEL is given by θ GE = E θ ν / ν / ν q ν θ = π ( ) 0 π θ β = β dθ. = (4.3) Using Equtions (4.) nd (4.4), Eqution (4.3) becomes q ν = ( N c) Γ + Ψ / ν Γ ( N + c + ν ) θ GE = π. (4.4)

11 85 Similrly put ϑ = β, then the Byesin estimtor of the ccelertion fctor β under GEL is given by GE q = ν / ν β = β π. (4.5) Similrly, the Byesin estimtor of the SF nd HRF t some y respectively, by = y under GEL re given, S GE N+ c / ν q Ψ π ε ε = ν ( y ) ( τ β( y τ)) Ψ + ( y ) =, (4.6) H GE ν q Γ ( N + c ν ) Ψ ε = ( N c) β / ν ( y ) = π, Γ + (4.7) where ε is s defined in (4.9). The vlue of y hs been ten to equl 0.3 in the simultion study. 5. Simultion Study Due to the complicted expressions of the estimtors, n nlyticl comprison of these estimtors is impossible. Therefore, Monte Crlo simultion study is crried out in order to clculte the MLEs, Byes estimtes (BEs), men squred errors (MSEs), reltive bsolute bises (RABs) nd 90% pproximte CIs of the model prmeters, bsed on r = 000 Monte Crlo simultions. The simultion study is performed ccording to the following steps. Generte rndom smple of size n from uniform(0, ) distribution nd obtin the order sttistics ( U : n,, Unn : ).. For given vlue of the prmeter θ nd vlue of stress chnge time τ, find n such tht

12 86 U τ exp < U. θ n : n n +: n 3. For given vlues of ccelertion fctor β nd censoring time τ, find n such tht U τ + βτ ( τ) exp < U. θ n: n n n + : n n 4. From steps nd 3, the ordered observtions re clculted s follows y < < y τ < y < < y τ, : n n : n n +: n n + n: n y in : θlog( Uin : ), i n, = τ ( θlog( Uin : ) + τ) / β, n + i n + n. 5. The observtions yin :, i =,, n + n represent type-i censored smple generted from the exponentil distribution under PALT. 6. For given vlues of nd, pply the progressive type-i censoring scheme to the observtions generted in step 4 to obtin the observtions given in (.3), where i= n = n + R i i nd n = n i i + Ri. = + 7. Bsed on the progressively type-i censored smple given is step 5, clculte ˆ θ nd ˆβ ccording to Equtions (3.5). The MSE ( ˆ θ ) nd RAB( ˆ θ ) bsed on r different smples re clculted s follows MSE( θ) = ( θ θ) nd RAB( θ) = θ θ. r r ˆ ˆ ˆ ˆ ω ω r ω= rθ ω= 8. Similrly, the MSE( ˆ β ) nd RAB ( ˆ β ) cn be clculted s in step The BEs under SEL (GEL) of θ, β, SF nd HRF with their MSEs nd RABs cn be computed similrly from Equtions (4.5)-(4.8) ((4.4)-(4.7)). Suppose tht the progressive censoring is designed with three censoring times. The first two of them re occurred t use stress condition nd the third one is occurred t ccelerted stress condition. At the

13 87 second censoring time, the stress is chnged from use to ccelerted condition. The experiment termintes t the time point which corresponds to the third pre-specified censoring time. It hs been ten into ccount tht the clcultions re performed on type-i censoring, R = R = 0, nd progressive type-i censoring, R 0, R 0, for the se of comprison. Bsed on smples of sizes 5, 50 nd 00 subject to progressive type-i censoring with two different censoring schemes (C.S.), Tble shows the MLEs with their MSEs nd RABs. It shows lso the lower nd upper bounds of 90% CIs for the unnown prmeters in ddition to their lengths nd coverge probbilities (COVPs). Tble shows the BEs with their MSEs nd RABs of the model prmeters in ddition to the SF nd HRF bsed on SEL nd GEL functions with ν =,, 33. The MLEs nd BEs shown in Tbles nd re the verge estimtes over 000 different smples. The popultion prmeter vlues used in the simultion study re θ = nd β =.. The censoring time vlues re τ = 0., τ = nd τ 3 =6.0. The prmeter β hs ssigned discrete distribution with ten vlues.05(0.0).3 with equl probbilities p = 0., =,, 0. The MLE of the SF nd HRF t some y > 0 cn be computed by using the invrince property of MLEs. The following two points hve been ten into ccount in the simultion procedure: The IMSL subroutines for pseudo-rndom number genertion hve been used. It hs been numericlly shown tht the vector of prmeters in the considered popultion stisfying the log-lielihood Equtions (3.3)-(3.4) ctully mximizes log-lielihood function (3.). This is done by pplying Theorem (7-9) on p. 5 of Apostol (960).

14 88 Tble. MLEs (CIs) of θ nd β with their MSEs nd RABs (lengths nd estimted coverge probbilities (in %) ) for different smple sizes nd censoring schemes. C.S. ˆ θ MSE( ˆ θ ) RAB( ˆ θ ) CI(θ ) LCI(θ ) COVP(θ ) ˆβ MSE( ˆβ ) RAB( ˆβ ) CI( β ) LCI( β ) COVP( β ) 5 R =R = ( ,.679) (-.88,4.506) R =R = ( ,.3340) (-.064,4.8559) R =R = (-0.58,.9007) ( ,3.495) R =R = (0.349,.7754) (0.047,3.4840) R =R = (0.463,.969) (0.90,.797) R =R = (0.5893,.64) (0.9684,.455) C.S. Censoring Schemes & LCI length of CI.

15 89 Tble. BEs of θ, β, SF nd HRF with their MSEs nd RABs under SEL nd GEL functions for different smple sizes nd censoring schemes. n C.S. E.M. θ MSE(θ ) RAB(θ ) S MSE( S ) RAB( S ) β MSE( β ) RAB( β ) H MSE( H ) RAB( H ) 5 R =R = Byes(SE) GE ( ν = 3) GE ( ν = ) GE ( ν = 3) R =R =0 Byes(SE) GE ( ν = 3) GE ( ν = ) GE ( ν = 3) R =R = Byes(SE) GE ( ν = 3) GE ( ν = ) GE ( ν = 3)

16 90 R =R =0 Byes(SE) GE ( ν = 3) GE ( ν = ) GE ( ν = 3) R =R = Byes(SE) GE ( ν = 3) GE ( ν = ) GE ( ν = 3) R =R =0 Byes(SE) GE ( ν = 3) GE ( ν = ) GE ( ν = 3) C.S. Censoring Schemes. & E.M. Estimtion Method.

17 9 6. Conclusion Censoring is common phenomenon in mny life nd ftigue tests. It hs been shown by Viveros nd Blrishnn (994) tht the inference is prcticl when the smple dt re subjected to progressively censored experimentl scheme. It hs been discussed in this pper some results on sttisticl inference when the dt re gthered ccording to step PALTs nd collected under progressive type-i censoring scheme. We hve obtined MLEs nd BEs, in closed forms, for the two unnown prmeters s well s the SF nd HRF considering n exponentil life model. The results re obtined under both symmetric nd symmetric loss functions. A simultion study hs been conducted to exmine the performnce of the MLEs s well s the BEs under different smple sizes. From the simultion results, listed in Tble, we observe the following:. For fixed vlues of n, the MSEs nd RABs of the MLEs tht correspond to R = R = (progressive type-i censoring) re greter thn those correspond to R = R = 0 (type-i censoring).. The LCI(θ ) nd LCI( β ) (the COVP(θ ) nd COVP( β )) tht correspond to R = R = re greter (less) thn those correspond to R = R = For fixed vlues of R nd R, the MSEs, RABs nd LCIs (COVPs) decrese (increse) s n increses. From the simultion results, listed in Tble, we observe the following: 4. For fixed vlues of smple size n nd censoring schemes, the MSEs nd RABs of the BEs of θβ, nd SF ( HRF ) increse (decrese) s ν increses. 5. For fixed vlues of n the MSEs nd RABs of the estimtes under progressive type-i censoring R R = = re greter thn those under type-i censoring R = R = For fixed vlues of censoring scheme, ll the MSEs nd RABs decrese s n increses. 7. Prcticlly, the negtive vlues in lower bounds of the CIs should be ten equl zero since the two prmeters re positive. It cn be observed from Tble nd Tble tht the BEs re better thn MLEs for smll smple sizes

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