Optimal Progressive Group-Censoring Plans for. Weibull Distribution in Presence. of Cost Constraint

Transcription

1 It J Cotmp Mat Sccs Vol 7 0 o Optmal Progrssv Group-Csorg Plas for Wbull Dstrbuto Prsc of Cost Costrat A F Atta Dpartmt of Matmatcal Statstcs Isttut of Statstcal Stus & Rsarc Caro Uvrsty Egypt atta65@yaoocom S M Assar Dpartmt of Matmatcal Statstcs Isttut of Statstcal Stus & Rsarc Caro Uvrsty Egypt salwaassar@yaoocom Abstract Ts artcl scusss a lf tst ur progrssv typ-i group-csorg W us maxmum lloo mto to obta t pot stmator of t uow paramtr of lftm strbuto W propos tat scal paramtr s ow a fx I orr to obta a prcs stmat of sap paramtr of t Wbull strbuto o s to sg a optmal lf tst Tus ts artcl proposs a approac to trm t umbr of tst uts umbr of spctos a lgt of spcto trval of a lf tst ur a pr-trm bugt of xprmt suc tat t asymptotc varac of stmator of sap paramtr s mmum T mto wll b appl to a umrcal xampl a t sstvty aalyss wll b vstgat Kywors: Group ata; Progrssv csorg; Sstvty aalyss; Typ-I csorg; Varac optmalty Itroucto T Wbull strbuto as b rcogz as a approprat mol rlablty stus a lf tstg bcaus of ts vrsatlty fttg tm to falur

2 338 A F Atta a S M Assar strbutos of a ratr xtsv varty of complx mcasms Sc t as a varty of saps ts ma t Wbull strbuto flxbl for fttg ata a as b us as a mol for t strbuto of tm to falur of proucts Also rlablty grg t Wbull sap paramtr s oft t to t falur mcasm of t prouct I ustral lf tstg a mcal survval aalyss t s vry oft tat objct s lost or wtraw bfor falur or t objct lftm s oly ow wt a trval Hc t obta sampl s call a csor sampl Aggarwala 00 trouc typ-i trval a progrssv csorg a vlop t statstcal frc for t xpotal strbuto bas o progrssvly typ-i trval csor ata Ur progrssv typ-i trval csorg obsrvatos ar oly ow wt two coscutvly pr-scul tms a tms woul b allow to wtraw at pr-scul tm pots Progrssv csorg Progrssvly typ-i trval csor sampl s a uo of typ-i trval a progrssv csorg Suc a sampl s collct as follows: uts ar put o lf tst at tm 0 Uts ar obsrv at pr-st tms s also fx At ts tms r r r lv uts ar rmov from xprmtato rspctvly T valus r r r may b pr-spcf as prctags of t rmag lv uts sc t umbrs of uts rmag at tms ar raom varabls or postv tgrs wt t provso tat at tms T tr ar r uts avalabl for rmoval I ts cas t umbr of lv uts rmov obs at tm s r m r umbr of uts rmag Aga obs r all rmag uts at tm w xprmtato s scul to trmat Lt ot t obsrv ata wc wll b t umbr of uts ow to av fal t trvals 0 ] ] ] rspctvly T lloo fucto wll b: L x; θ [ F T - F T ] [ - F T ] wr 0 0 a r Extsv publcatos ca b fou t ltratur wc scuss t statstcal frc for progrssvly csor ata ur varous lftm strbutos Som of tm ar Balasoorya a Saw 998 Balasoorya t al 000 Balarsa a Aggarwala 000 Al Mousa a Ja 00 Wu 003 Gouo t al 004 a L t al 006 Dscusso of group ata ca b fou Cg a C 988 C a M 996 Aggarwala 00 r obs

3 Optmal progrssv group-csorg plas 339 Xog a Mg 004 Xag a Ts 005 Yag a Ts 005 a Wu t al 008 To couct a progrssv typ-i group-csor lf tst mor ffctly o as to arss t problm of trmg t umbr of tst uts t umbr of spctos a t lgt of t spcto trvals I practc t bugt of a lf xprmt s lmt I ts stuy w wll obta t optmal sttgs of a progrssv typ-i group-csor lf tst by mmzg t asymptotc varac of sap paramtr of t Wbull strbuto ur t costrat tat t total xprmtal cost os ot xc a pr-trm bugt Numrous problms appl statstcs ca b formulat as optmzato problms I practc optmzato tcus ar wly us for solvg tos problms I varous practcal fls t cso problm of obtag approprat umbr of tst uts umbr of spctos a lgt of spcto trval ur rstrct bugt of xprmt s mportat for xprmtrs spcally rlablty aalyss a ualty cotrol T rst of t papr s orgaz as follows: Scto 3 scrbs t mol a som cssary assumptos W us t maxmum lloo mto to obta t pot stmator of t sap paramtr Scto 4 proposs a procur to trm t umbr of uts to tst t umbr of spctos a t lgt of spcto trvals Scto 5 appls t propos procur to a umrcal xampl a Scto 6 stus t sstvty aalyss of t propos procur a som coclusos ar prst 3 Paramtr Estmato Suppos a progrssvly typ-i group-csor sampl s collct bgg wt a raom sampl of uts wt a Wbull lftm strbuto Lt b t umbr of uts ow to av fal t trval ] a lt r b t umbr of survvg uts bg wtraw from t tst at tm for wr 0 0 T valus of r r r ar trm by t prspcf prctags of t rmag lv uts p p p p a Tat for wr m j rj s t umbr of s r m p o-rmov survvg uts at t bgg of t t stag T w av t fact tat r r bomal m 3 wr j F T F T s t probablty tat a ut fals t tm trval F T ] for T form of t Wbull strbuto cosr r as probablty sty fucto as follows j

4 340 A F Atta a S M Assar xp ; > > > x x x x f wr s t scal paramtr a s t sap paramtr a t cumulatv strbuto fucto s : 0 xp > x x x F 3 I t cas of s ow a s uow t lloo fucto ur a progrssvly typ-i group-csor sampl wt bomal rmoval bcoms m L 33 Lt T m L 34 Sc F F F 35 xp So tat T artm of t lloo fucto 34 wll b m L 36 T frst partal rvatv for s obta as follows m L 37 wr T sco partal rvatv for s obta as follows

5 Optmal progrssv group-csorg plas 34 [ ] [ ] m L 38 wr a xp I t spcal cas wr t spcto trvals ar of ual lgt say t w ca f tat ] [ { } ] [ ] [ a { } { } ] [ ] [ ] [ ] [ T Fsr s formato s as follows j j j p I 39 wr xp

6 34 A F Atta a S M Assar I t spcal cas wr t spcto trvals ar of ual lgt say a t prctags of rmovals ar t sam p p T asymptotc varac of ˆ ca b fou as follows [ [ ] a ] [ By substtutg 39 w av [ I ] ] [ ] 30 Var ˆ 3 Β Β Β ψ [ ψ Β ψ Β ] ˆ Var Β Β ψ Β ψ Β p T ψ j j wr Β Β ψ [ ] a ψ [ ] T w ca b obta t asymptotc varac umrcally 4 Plag of Lf Tst for Wbull Dstrbuto Ts scto scuss a lf tst ur progrssv typ-i group csorg plas for Wbull strbuto I orr to obta a prcs stmat of sap paramtr o s to sg a optmal lf tst W us a approac wc propos by Wu a Huag 00 to trm t umbr of tst uts umbr of spctos a lgt of spcto trval of a lf tst ur a pr-trm bugt of xprmt suc tat t asymptotc varac of stmator of sap paramtr s mmum W assum tat t lgts of spcto trvals ar all ual for smplcty of scusso T u-lgt assumpto s also covt for practtors Lt ot t umbr of uts o tst b t umbr of spctos a b t lgt of spcto trval Obvously t cso varabls affct bot t cost of xprmt a t prcso of stmatg ma lftm Lt TC ot t total cost of couctg a progrssv group csorg xprmt It volvs four parts: T cost of stallg all tst uts t bgg of a lf xprmt say C a T cost of tst uts s C s wr C s ots t cost of o tst ut

7 Optmal progrssv group-csorg plas T cost of spcto s C I wr C I ots t cost of o spcto 4 T cost of opratg a xprmt s C o wr C o s t oprato cost t tm trval btw two spctos Trfor t total cost of xprmt s TC C a C s C I C o I orr to obta a prcs stmator of t sap paramtr of lftm strbuto a typcal cso problm ca b formulat as follows: mmz Var ˆ subjct to Ca CS CI C0 Cr 4 Ν a > 0 wr C r ots t bugt for t xprmt a N s t st of postv tgrs Sc t cso varabls a ar tgr t cso varabl s ral a t objctv fucto a costrat ar bot olar fuctos of a t olar mx tgr programmg ca b us to f t optmal soluto A xcllt rvw of olar mx tgr programmg ca b fou Grossma 00 5 Illustratv Exampl To llustrat t us of our mto to obta varac optmalty t followg xampl s scuss W apply t propos mtos as Wu a Huag 00 to a sampl wc w grat from t Wbull strbuto wt scal paramtr a sap paramtr by usg Matca 3 pacag T sampl sz 5 umbr of spctos 4 a lgt of spcto trval T pr-trm prctags of rmovals ar p p p3 p T ata ar prst Tabl Tabl Progrssvly typ-i group-csor sampl from t Wbull strbuto r From 36 w obta t maxmum lloo stmat of to b ˆ W us ts valu t sg of our w xprmt Assum tat t prctags of rmovals ar p p p3 0 5 a p 4 Suppos furtr tat t valus of cost paramtrs ar as follows: Ca $800 Cs $85 pr ut CI $40 Co $8 pr 0 a Cr $6000 Cosr t objctv fucto to b t asymptotc varac of t sap paramtr Tus t optmal sg problm s:

8 344 A F Atta a S M Assar Mmz Var ˆ subjct to By usg t algortm propos Wu a Huag 00 W ca obta t optmal sg as follows: * 5 * 4 a * 9375 Tat s t optmal umbr of tst uts s 5 t optmal umbr of spctos s 4 a t optmal lgt of spcto trval s 9375 Tus t trmato tm of t xprmt s 775 T valu of Varˆ s From ts obta optmal sg w ca f t followg rsult W t prctags of rmovals ar all st to b 05 a t pr-trm bugt of xprmt s $6000 o s mor umbr of tst uts logr lgt of spcto trval ta tos t orgal sg orr to gt t mmum asymptotc varac of t sap paramtr 6 Sstvty aalyss & Cocluso T sstvty stuy of t optmal soluto to cag t valus of t ffrt paramtrs s a mportat ssu to t plag of lf tst Ts paramtrs cosst of two parts: t prctags of rmovals p ; a t paramtrs of xprmtal cost Ca Cs CI Co a Cr W wll vstgat t ffcts of t prctags of rmovals a t cost paramtrs o t optmal soluto Sctos 6 a 6 rspctvly T sstvty stuy s bas o t last Exampl Scto 5 6 T ffct of prctags of rmovals I ts subscto w assum tat t pr-spcf prctags of rmovals from t rmag lv uts at ac stag ar all ual Tat s p p p p a p Ur fx cost paramtrs Ca Cs CI Co Cr us abov xampl t optmal solutos of a for ffrt cocs of prctag of rmovals p ar prst Tabl Tabl Optmal valus of a ur varous combatos of a p p K From Tabl w ca f tat:

9 Optmal progrssv group-csorg plas 345 T valu of crass as p crass T lgt of spcto trval s a crasg fucto of p 3 T valu of Varˆ crass as p crass 6 T ffct of cost paramtrs Cags cost paramtrs ca also affct t optmal solutos Lt us cosr t valu of strbuto paramtr a t cost paramtrs Ca Cs CI Co Cr Usg t sam valu of strbuto paramtr t fluc of cost paramtrs o a s vstgat Tabl 3 Optmal valus for ffrt costs valus ur a C a C s C I C 0 C r

10 346 A F Atta a S M Assar Tabl 4 Optmal valus for ffrt costs valus ur a C a C s C I C 0 C r N Tabl 5 Optmal valus for ffrt costs valus ur a C a C s C I C 0 C r

11 Optmal progrssv group-csorg plas From Tabls 3 4 a 5 w f t followg rsults: A gr valu of Cr las to a gr valu of T umbr of tst uts s a crasg fucto of Ca 3 T umbr of tst uts s sstv to cags CI a Co 4 A gr valu of Cs causs a lowr valu of 5 T lgt of spcto trval a t trmato tm ar sstv to t cags ts valus of cost paramtrs w p s small Rfrcs [] C-T L SJS Wu a N Balarsa Ifrc for -gamma strbuto bas o progrssvly typ-ii csor ata Commu Stat Tory Mtos [] C Xog a J Mg Aalyss of group a csor ata from stp-strss lf tst IEEE Tras Rlab

12 348 A F Atta a S M Assar [3] C Yag a S-K Ts Plag acclrat lf tsts ur progrssv typ I trval csorg wt raom rmovals Commu Stat Smul Comput [4] E Gouo A S a N Balarsa Optmal stp-strss tst ur progrssv typ-i csorg IEEE Tras Rlab [5] IE Grossma Rvw of olar mx-tgr a sjuctv programmg tcus Optm Eg [6] K F Cg a C H C Estmato of t Wbull paramtrs wt group ata Commu Stat Tory Mtos [7] L Xag a S-K Ts Maxmum lloo stmato survval stus ur progrssv trval csorg wt raom rmovals J Boparm Stat [8] MAM Al Mousa a ZF Ja Baysa prcto for progrssvly csor ata from t Burr mol Stat Paprs [9] N Balarsa a R Aggarwala Progrssv csorg tory mtos a applcatos Bräusr Bosto 000 [0] R Aggarwala Progrssv Itrval Csorg: Som matmatcal rsults wt applcatos to frc Commu Stat Tory Mtos [] S-J Wu Estmato for t two-paramtr Parto strbuto ur progrssv csorg wt uform rmovals J Stat Comput Smul [] S-J Wu a S-R Huag Optmal progrssv group-csorg plas for xpotal strbuto prsc of cost costrat Stat Paprs [3] S-J Wu Y-P L a S-T C Optmal stp-strss tst ur typ I progrssv group-csorg wt raom rmovals J Stat Pla Ifrc [4] U Balasoorya a SLC Saw Rlablty samplg plas for t twoparamtr xpotal strbuto ur progrssv csorg J Appl Stat

13 Optmal progrssv group-csorg plas 349 [5] U Balasoorya SLC Saw a V Gaag Progrssvly csor rlablty samplg plas for t Wbull strbuto Tcomtrcs [6] Z C a J M Cofc trval for t ma of t xpotal strbuto bas o group ata IEEE Tras Rlab Rcv: Jauary 0