Optimal Design of Step Stress Partially Accelerated Life Test under Progressive Type-II Censored Data with Random Removal for Gompertz Distribution

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1 Aecan Jounal of Appled Matheatcs and Statstcs, 09, Vol 7, No, 37-4 Avalable onlne at Scence and Educaton Publshng DOI:069/ajas-7--6 Optal Desgn of Step Stess Patally Acceleated Lfe Test unde Pogessve Type-II Censoed Data wth Rando Reoval fo Gopetz Dstbuton Dna H Abdel Hady * Depatent of Statstcs, Matheatcs and Insuance, Faculty of Coece, Tanta Unvesty *Coespondng autho: dnaabdelhady@coecetantaedueg Receved Novebe, 08; Revsed Decebe 9, 08; Accepted Januay 0, 09 Abstact Ths pape deals wth ando eoval of pogessvely type II censoed data The eoval of the data s assued to follow a bnoal o a unfo dstbuton, and the lfe te testng s assued to follow a Gopetz dstbuton Paaetes of these dstbutons ae estated usng the Maxu Lkelhood estaton pocedue Fshe nfoaton atx s used to estate the asyptotc ean squaed eo and to constuct confdence ntevals of odel paaetes The optal patally acceleated lfete testng (PALT) s estated by nzng the Genealzed Asyptotc Vaance (GAV) Sulaton study s pefoed to Clafcaton the statstcal popetes of the paaetes A sulaton esults eveal that fo the fxed values of the paaetes, the eo and optal te decease wth nceasng saple sze n; estates of bnoal ae oe stable wth a elatvely sall eo wth nceasng saple sze; and the test desgn s obust and woks well fo bnoal eoval Keywods: patally acceleated lfe test, PALT, SS-PALT, pogessvely type II censoed data, Gopetz dstbuton optal desgn, D-optalty Cte Ths Atcle: Dna H Abdel Hady, Optal Desgn of Step Stess Patally Acceleated Lfe Test unde Pogessve Type-II Censoed Data wth Rando Reoval fo Gopetz Dstbuton Aecan Jounal of Appled Matheatcs and Statstcs, vol 7, no (09): 37-4 do: 069/ajas-7--6 Intoducton Tadtonal lfe testng ay show no falue o few falues of soe hghly elable unts because of soe sevee condtons (stesses) that ay occu n the fo of pessue, voltage, tepeatue, vbaton, load, cyclng ate, etc In such stuatons lfe testng has to be pefoed at hghe than usual condtons n ode to obtan falues as fast as possble The data collected unde stesses o sevee condtons should not be used to estate the lfe dstbuton at noal use assuptons Thee ae thee types of stess, a) step stess; b) pogessve stess, and c) constant stess; Isal, [] when testng s conducted unde stess, t s called ethe Acceleated Lfe Testng (ALT) o Patally Acceleated Lfe Testng (PALT) In Alt testng, unts ae put unde stess to each oe falue n shot te, and the atheatcal stess odel s known; whle n PALT, data can be extapolated to noal use condtons, and the statstcal odel paaetes could be estated and step-stess could be appled Nelson [] poposed the step stess PALT (SS-PALT) whee unts ae tested at noal use condtons wthn a specfed te nteval Censoed schees ase when not all lfete testng unts ae obseved When a unt does not fal wthn that te nteval, t s put unde stess, untl falue o untl the tenaton of the te nteval (censoed schee) Rao [3] and Balakshnan and Aggawala [4] have entoned that step stess saves oney, effot and te Thee ae two censong schee types Type I ases when the expeent contnues up to a pe-specfed te T and Type-II censong schee eques the expeent to contnue untl a pe-specfed nube of falues n occu These two types do not allow fo the eoval of any unt untl lfe testng s tenated Nevetheless, ths allowance ay be desable to copose between at least one obsevaton s sought and a educed te nteval fo the expeentaton At ths stuaton, pogessve censong s needed In the pogessve Type II censong schee, the expeente selects n (d) unts fo lfe testng; and obseves the occuence of the fst falue, at te tt (), and ths s eoved fo testng When the second falue occus at te t (), one of the suvvng unts ae andoly selected and eoved fo the test, the expeent tenates when the th falue occus at te t () and the eanng suvvng unts n ae all eoved fo the Test Pogessve type-ii censong schee wth fxed nube of eovals,,, s consdeed by Cohen [5] and Cohen and Nogaad [6] In soe elablty

2 38 Aecan Jounal of Appled Matheatcs and Statstcs expeents and when the nube of eovals s not fxed, a pogessve censong wth ando eovals odel could be consdeed [7,8,9] The ando eoval s assued to follow the bnoal dstbuton, and used to estate the paaetes of the elablty odel Isal [] dscusses step-stess patally-acceleated odel The odel dffes than the pogessve type II censong schee n that the lfete of the test follows the exponental dstbuton athe than the bnoal Ths pape consdes the step-stess patally-acceleated odel, whee the eovals ae assued to follow the bnoal and the lfe te-testng follows the Gopetz dstbuton The optal stess change te s also beng detened by nzng the genealzed asyptotc vaance of the MLE paaetes In Secton, assuptons of the Gopetz dstbuton ae gven; Secton 3 pesents the assuptons of the patally acceleated odel Estaton of odel paaetes s gven n Secton 3; sulaton study esults ae gven n Secton 4 The Gopetz Dstbuton The pobablty densty functon of the Gopetz dstbuton takes the fo: x x ( ) { } f x αe exp α e, 0 < x, α, > 0 And the cuulatve dstbuton functon s: ( ) { α } F x exp e, 0 < x, α, > 0 The followng assuptons ae used: n dentcal and ndependent unts ae selected fo lfe testng The lfete of each unt has Gopetz dstbuton The followng steps ae to be followed: a) Each of the n unts s fst un unde noal use condton If t does not fal o eove fo the test by a pe-specfed teτ t s put unde acceleated condton (stess) b) At the th falue a ando nube of the suvvng unts, RR,,,, ae andoly selected and eoved fo the test c) Fnally, at the th falue the eanng suvvng unts RR nn RR ae all eoved fo the test and the test s tenated The lfete, say X, of a unt unde SS-PALT can be ewtten as T f T τ X T τ τ + f T τ Whee, the pdf of X s gven by x x ( ) { } () () f x αe exp α e,0 < x τ τ + ( x τ f ) ( x) αe (3) f ( x), x τ τ + ( x τ) > exp α e In addton, the suvval functons unde noal and acceleate use condtons espectvely s gven by And ( ) { α } S x exp e, 0 < x< τ S ( x) exp e τ α + τ, x > τ 3 Estaton of Paaetes (4) (5) Let (xx ),,,,, denote the obsevaton obtaned fo a pogessvely type-ii censoed saple wth ando eovals n a step-stess PALT Hee xx () xx () xx () Gven a pe-detened nube of eovals RR (RR,, RR ), the condtonal lkelhood functon of the obsevatons xx {(xx, ),,,, } takes the followng fo (, α,,,, ) { f( x) ( S( x) ) } f( x) ( S( x) ) L x u u R u u { } (6) Equatons () and (3) ae nseted n (6) and splfy, we get (, α,,,, ) L x u u R u x x exp{ } αe α e (7) exp{ α e } u τ ( x τ) αe + τ exp e τ α + exp τ α e 3 Paaete Estaton wth Bnoal Reovals Gven that the nube of unts eoved RR fo the test at each falue te follows a bnoal dstbuton bbbbbb(nn, pp), and RR ~bbbbbb (nn jj jj, pp) fo,, 33, nn Thus, the nube of unts eoved at each falue te follows a bnoal dstbuton such that n ( ) n P R p ( p), And fo, 33,, (, ) P R R R n j j n p ( p) (8) j j

3 Aecan Jounal of Appled Matheatcs and Statstcs 39 The lkelhood of the saple of sze n s gven as follows ( α ) ( α ) ( ) L x,,,, p L x,,,, pr P R, (9) Whee That s ( ) ( ) P ( R R, R ) P ( R R 3 3, R ) P( R R ) P( R ) P R P R, R,, R ( ) P R ( n ) ( ) n! ( )( n ) ( )( p ) ( p)! (0) The log-lkelhood functon [Eq 0] can be wtten as follows: ( α ) ( α ) ( ) l x,,,, p ln L x,,, lnp R (, α,,, p) l x lnα + ln + ln + () ( ) + α + e + x τ + ( x τ) α + e lnp ( ) α( ) + lnn( p) ( )( n ) ( )( ) + c ( n )! whee c ( n )! and u + u () To obtan the estate of the αα,, aaaaaa pp, the fst patal devatves wth espect to αα,, and pp of Equaton () ae obtaned and equated to zeo as follows: l ( + ) e α α e τ ( ) ( ) l (3) x ( ) α xe x τ( ) (4) ( ) ( ) τ α + τ + x τ e 0 l x xe τ + α τ + 0 ( ) l ( )( ) ( )( ) n 0 p p p τ Fo Equaton (6), pp s estated as follows ( )( n ) ( )( ) pˆ (5) (6) (7) Thee s no closed-fo soluton to ths syste of equatons 3, 4 and 5 usng the Newton-Raphson teaton ethod, fo oot fndng (αα,, ) s eeeeeeeeeeeeeeeeee Iteatvely as follows: ˆ j+ ˆ j G g (8) whee g s the vecto of noal equatons, and ( ) g [ g gg3] g ( + ) e α e τ ( ) g + x ( ) α + xe + x + τ ( ) ( ) ( ) τ α + τ + x τ e 3 g x τ ( x ) xe τ + α + τ ( ) and G [Equaton 7] s the atx of second devatves Whee g g g α g g g G α g3 g3 g3 g α α (9) (0)

4 40 Aecan Jounal of Appled Matheatcs and Statstcs g g ( + ) xe α ( ) ( ) τ τ x τ e + + () g g3 ( ) xe τ + τ + () α g x α ( ) + x e (3) g g3 x ( ) ( ) x xe τ + α τ τ + + τ τ (4) g3 α ( ) x e τ + τ + (5) Convegence of the Newton-Raphson algoth fo the estates of αα, and depends on the toleance lt change wth each successve teaton, to αα, and Nuecally nvetng the above G atx above, we easly obtan Fshe Infoaton atx, e FGG The appoxate ( γγ) % two sded confdence ntevals fo αα, and can be, constucted as follows: ˆ ± Z ˆ ˆ γ ˆ α, ± Zγ ˆ and ± Zγ ˆ α σ σ σ 3 Paaete Estaton wth the Unfo Reovals The odel assues that eoved unts ae ndependent and that the pobablty of each eoved unt s the sae, such that: n P( R ) n + (6) And fo,3,, P( R R, R ) n j j + The jont pobablty dstbuton of R ( R,, R ) s gven by P( R ) (7) n + whee 0 nn jj jj, 0,,, The axu lkelhood estatos can be deved dectly by axzng the equatons () and then solvng fo equatons (3), (4) and (5) 33 D-optalty Fshe's nfoaton atx s used to detene t hee oooooooooooooo vvvvvvvvvv oooo ττ n the SS-PALT type II pogessve censong schee poposed cteon s based on the detenant of Fshe's nfoaton atx The genealzed asyptotc vaance (GAV) s the ecpocal of the detenant of Fshe's nfoaton atx F [0], thus ( ˆ ˆ ) GAF ˆ, α, (8) F The D-optalty cteon s the optal value of ττ that axzes the detenant of the Fshe's nfoaton atx F and nzes the GAV Table Sulaton study esults wth Bnoal Reovals fo, α, 3, τ3 Estates 95% Confdence Inteval Coveage ττ αα CCCC αα CCCC CCCC FF BBBBBBBB αα BBBBBBBB BBBBBBBB

5 Aecan Jounal of Appled Matheatcs and Statstcs 4 n Table Sulaton study esults wth unfo Reovals fo, αα, 33, ττ 33 Estates 95% Confdence Inteval Coveage ττ αα CCCC αα CCCC CCCC FF BBBBBBBB αα BBBBBBBB BBBBBBBB Sulaton Study A sulaton study s pefoed to study the popetes of the estatos usng ML ethod; the study nvolves the coputaton of Mean squaed eos (MSEs), the constucton of confdence ntevals fo dffeent saple szes; and the detenaton of, the optal stess change te The followng steps wee followed: a)value of n and to be specfed b)value of the paaetes wee set as:, αα, 3, ττ 3 c) A ando saple wth sze n and censong sze wee geneated, wth ando eovals,,,,, fo the ando vaable X gven by (3) d) Geneate a goup value RR ~bbbbbbbb (nn jj jj, pp) and also RR ~uuuuuuuu (0, nn jj jj ) whee 0 nn jj jj, 0,,, and nn e) ML estates wee coputed, fo n 0, 50, 80 and 00, f) The ean squaed eo (MSE), the 95% confdence nteval of paaetes and the bas assocated wth the MLE of the paaetes, optal value of ττ, the Optal GAV of the MLEs of the odel paaetes ae obtaned nuecally fo each saple sze 5 Conclusons The (SS-PALT) unde pogessve type-ii censoed data wth bnoal and unfo ando eoval assung Gopetz dstbuton was studed The Newton-Raphson ethod s appled to obtan the optal stess-change te whch nzes the Genealzed Asyptotc Vaance (GAV) The axu Lkelhood estaton pocedue was used the estaton of odel paaetes, ean squaed eo and the optu plan fo the bnoal and unfo eovals fo dffeent saple sze wee coputed and shown Sulaton esults show that the eo and the optal te decease as saple sze ncease, when paaetes ae fxed that both the aveage value of τ and the aveage value of GAV fo type-ii pogessve censong ae gettng close to those of coplete saple wth the bgge and close faste fo bgge n Hence fo the nuecal esult we can conclude that estates of bnoal ae oe stable wth elatvely sall eo wth nceasng saple sze Theefoe, the test desgn obtaned hee s obust desgn and wok well fo bnoal eoval Refeences [] A Isal, (009) Optal Desgn of Step-Stess Lfe Test wth Pogessvely type-ii Censoed Exponental Data, Intenatonal Matheatcal Fou, 4, no 40, [] W Nelson (990) Acceleated Lfe Testng: Statstcal Models, Data Analyss and Test Plans, John Wley and sons, New Yok [3] R Rao, (99), Equvalence of the tapeed ando vaables and tapeed falue ate odels n ALT fo a class of lfe dstbuton havng the settng the clock back to zeo popety, Councaton n Statstcs Theoy and Methods, Vol, No 3, [4] N Balakshnan, and R Aggawala (000) Pogessve Censong: Theoy, Methods, and Applcatons, Bkhause, Boston [5] C Cohen (963) Pogessvely Censoed Saples n the Lfe Testng, Technoetecs, 5, [6] C Cohen and N J Nogaad (977) Pogessvely Censoed Saplng n the Thee Paaete Gaa Dstbuton, Technoetecs, 9, [7] H K Yuen and S K (996) Tse, Paaetes Estaton fo Webull dstbuted Lfete unde pogessve Censong wth ando eovals, Jounal of Statstcal Coputaton and Sulaton, 55, 57-7 [8] C Wu, S F Wu, and, H Y Chan (004) MLE and the Estated Expected Test Te fo Paeto Dstbuton unde Pogessve Censong Data, Intenatonal Jounal of Infoaton Manageent Scences, 5(3), 9-4

6 4 Aecan Jounal of Appled Matheatcs and Statstcs [9] Tse, S K, Yang, C and Yuen, H K, 000 Statstcal Analyss of Webull Dstbuted Lfete Data unde Type-II Pogessve Censong wth Bnoal Reovals Jounal of Appled Statstcs, vol 7, no 8, pp [0] S Ba, J G K and Y R Chun, (993) Desgn of Falue- Censoed Acceleated Lfe-Test Saplng Plans fo Lognoal and Webull Dstbutons, Eng Opt Vol, 97-