Estimation and Optimum Constant-Stress Partially Accelerated Life Test Plans for Pareto Distribution of the Second Kind with Type-I Censoring

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1 Esmo d Opmm Cos-ress Prlly Accelered fe Tes Pls for Preo Dsrbo of he ecod Kd wh Type-I Cesorg By Abdll A. Abdel-Ghly Em H. El-Khodry Al A. Isml 3 Key Words d Phrses: Relbly; Preo dsrbo; prl ccelero; cos-sress; mmm lkelhood esmo; Fsher formo mr; opmm es pls; ype-i cesorg. Absrc Ths pper cosders he cse of Cos-ress Prlly Accelered fe Tesg CPAT whe wo sress levels re volved der ype-i cesorg. The lfemes of es ems re ssmed o follow wo-prmeer Preo lfeme dsrbo. Mmm kelhood M mehod s sed o esme he prmeers of CPAT model. Cofdece ervls for he model prmeers re cosrced. Opmm CPAT pls h deerme he bes choce of he proporo of es s lloced o ech sress re developed. ch opmm es pls mmze he Geerlzed Asympoc Vrce GAV of he M esmors of he model prmeers. For llsro mercl emples re preseed.. Irodco Uder cog qes for beer.e. more relble prodcs s more dffcl o cqre flre formo qckly for prodcs esed he sl-se codo. I order o shore he esg perod ll or some of es s my be sbeced o more severe codos h orml oes. ch Accelered fe Tesg AT or Prlly Accelered fe Tesg PAT resls shorer lves h wold be observed der orml operg codos. I AT es ems re r oly ccelered codos whle PAT hey re r boh orml d ccelered codos. As Nelso 990 dces he sress c be ppled vros wys commoly sed mehods re sep-sress d cos-sress. Uder sep-sress PAT es em s frs r se codo d f does o fl for specfed me he s r ccelered codo l flre occrs or he observo s cesored whle he cos-sress PAT rs ech em eher orml se or ccelered codo oly.e. ech s r cos-sress level l he es s ermed. Accelered es sresses volve hgher h sl emperre volge pressre lod hmdy ec. or some combo of hem. The obec of PAT s o collec more flre d lmed me who ecessrly sg hgh sresses o ll es s.. Professor of scs Deprme of scs Fcly of Ecoomcs & Polcl cece Cro Uversy Orm Gz Egyp.. Assoce Professor of scs Deprme of scs Fcly of Ecoomcs & Polcl cece Cro Uversy Orm Gz Egyp. 3. Asss Professor of scs Deprme of scs Fcly of Ecoomcs & Polcl cece Cro Uversy Orm Gz Egyp.

2 For overvew of sep-sress PAT wh ype-i cesorg B & Chg 99 dscssed boh he problems of esmo d opmlly desgg PAT for es ems hvg epoel dsrbo. For ems hvg logormlly dsrbed lves PAT pls were developed by B Chg & Ch 993. A e l. 996 cosdered oly he esmo problem of he Webll dsrbo prmeers sg he M mehod. Cocerg he cos-sress PAT here re oly hree sdes o he opmlly desgg cos-sress PAT see B & Chg 99 B Chg & Ch 993 d Isml 006. Abdel-Gh 998 cosdered oly he esmo problem cos-sress PAT for he Webll dsrbo. I hs pper he problems of boh esmo d opml desg cos-sress PAT re cosdered der Preo dsrbo of he secod kd sg ype-i cesorg. Nelso 990 poed o h he cos-sress esg hs severl dvges: frs s eser o m cos-sress level mos ess. ecod ccelered es models for cos-sress re beer developed for some merls d prodcs. Thrd d lyss for relbly esmo s well developed. Also s Yg 994 dced cos-sress ccelered lfe ess re wdely sed o sve me & moey. The pper s orgzed s follows: I eco he sed model d es mehod re descrbed. The mmm-lkelhood esmors of he model prmeers re obed eco 3. eco 4 s devoed o he dervo of he cofdece ervls for he model prmeers. I eco 5 opml pls for smple cos-sress PAT re developed. Nmercl emples re gve eco 6 o llsre he heorecl resls. Noo: T X ol mber of es ems PAT cesorg me of PAT lfeme of em se codo. The Model d Tes Mehod lfeme of em ccelered codo ccelero fcor > P P probbly h em esed oly se codo fls by probbly h em esed oly ccelered codo fls by mples mmm lkelhood esme Preo scle prmeer Preo shpe prmeer observed lfeme of em esed se codo observed lfeme of em esed ccelered codo dcor fcos: IT IX * proporo of smple s lloced o ccelered codo opmm proporo of smple s lloced o ccelered codo

3 mbers of ems fled se d ccelered codos respecvely ordered flre mes se codo ordered flre mes ccelered codo. The Preo Dsrbo: As feme model The lfemes of he es ems re ssmed o follow wo-prmeer Preo dsrbo of he secod kd. The Preo dsrbo ws rodced by Preo 987 s model for he dsrbo of come. I rece yers s models severl dffere forms hve bee sded by my hors Dvs d Feldse 979 Cohe d Whe 988 Grmshw 993 mog ohers. The Preo dsrbo of he secod kd lso kow s om or Perso's Type VI dsrbo Johso e l. 994 hs bee fod o provde good model bomedcl problems sch s srvvl me followg her rspl B d Egelhrd 99. Usg he Preo dsrbo Dyer 98 sded l wge d of prodco le workers lrge dsrl frm. om 954 sed hs dsrbo he lyss of bsess flre d. The legh of wre bewee flws lso follows Preo dsrbo B d Egelhrd 99. ce he Preo dsrbo hs decresg hzrd or flre re hs ofe bee sed o model comes d srvvl mes Howlder d Hoss 00. The probbly desy fco of he Preo dsrbo of he secod kd s gve by f ; ; > 0 > 0 > 0 T The srvvl fco kes he form: R d he correspodg hzrd or seos flre re s s follows: h 3 As dced by McCe d McCe 000 he Preo dsrbo hs clssclly bee sed ecoomc sdes of come sze of ces d frms servce me qeg sysems d so o. Also hs bee sed coeco wh relbly heory d srvvl lyss Dvs & Feldse Cos-ress PAT The es procedre of he cos-sress PAT d s ssmpos re descrbed s follows: Tes Procedre I cos-sress PAT he ol smple sze of es s s dvded o wo prs sch h:. ems rdomly chose mog es ems smpled re lloced o ccelered codo d he remg re lloced o se codo. 3

4 . Ech es em s r l cesorg me d he es codo s o chged. Assmpos. The lfemes T - of ems lloced o se codo re..d. r.v.'s.. The lfemes X of ems lloced o ccelered codo re..d r.v.'s. 3. The lfemes T d X re mlly ssclly-depede. 3. Compg Mmm kelhood Prmeer-Esmes Mmm lkelhood esmors MEs of he prmeers re sed sce hey re sympoclly ormlly dsrbed d sympoclly effce my cses Grmshw 993. Also Bgghs 988 dced h he M procedre geerlly yelds effce esmors. However hese esmors do o lwys es closed form so mercl echqes re sed o compe hem. I smple cos-sress PAT he es em s r eher se codo or ccelered codo oly. A smple cos-sress es ses oly wo sresses d lloces he smple s o hem Mller & Nelso 983. ce he lfemes of he es ems follow Preo dsrbo of he secod kd he probbly desy fco of em esed se codo s gve by: f T 0 4 Whle for em esed ccelered codo he probbly desy fco s gve by: f X 0 5 where X - T The lkelhood for he lkelhood for d he ol lkelhood for ; - ; - ; ; re respecvely s follows: / 6 4

5 5 Where. /. / / 7 8 d. I s slly eser o mmze he rl logrhm of he lkelhood fco rher h he lkelhood fco self. The frs dervves of he rl logrhm of he ol lkelhood fco 8 wh respec o d re gve by: 9 0 / l l

6 6 0 The M esmes of he prmeers re he vles of d whch solve he eqos obed by leg ech of hem be zero. o from he ls eqo he M esme of s gve by: where By sbsg for o he wo eqos 9 d 0 d eqg ech of hem o zero he sysem eqos re redced o he followg wo o-ler eqos: 3 4 l l l l l l l l l l l 0

7 7 Obvosly s very dffcl o ob closed-form solo for he wo o-ler eqos 3 d 4. o erve procedres ms be sed o solve hese eqos merclly. The Newo-Rphso mehod s sed o deerme he M esmes of d. Ths oce he vles of d re deermed esme of s esly obed from. I relo o he sympoc vrce-covrce mr of he M esmors of he prmeers c be ppromed by merclly verg he Fsher-formo mr F. I s composed of he egve secod dervves of he rl logrhm of he lkelhood fco evled he M esmes. Therefore he sympoc Fsher-formo mr c be wre s follows: 5 The elemes of he bove mr F c be epressed by he followg eqos: l l l l l l l l l F l l l l

8 l 0 l Coseqely he mmm lkelhood esmors of d hve sympoc vrcecovrce mr defed by verg he Fsher formo mr F s dced before. 4. Cofdece Iervls for he Model Prmeers As dced by Vder Wel d Meeker 990 he mos commo mehod o se cofdece bods for he prmeers s o se he lrge-smple sympoc orml dsrbo of he M esmors. To cosrc cofdece ervl for poplo prmeer λ; ssme h λ λ y y d U λ U λ y y re fcos of he smple d y y sch h P λ λ λ U λ γ where he ervl λ U λ s clled wo-sded γ 00 % cofdece ervl for λ. λ d U λ re he lower d pper cofdece lms for λ respecvely. The rdom lms λ d U λ eclose λ wh probbly γ. Asympoclly he mmm lkelhood esmors der ppropre reglry codos re cosse d ormlly dsrbed. Therefore he wo-sded pprome γ 00 % cofdece lms for poplo prmeer λ c be cosrced sch h: P - z λ λ σ λ z γ 3 where z s he 00-γ/ h sdrd orml percele. Therefore he wo-sded pprome γ 00 % cofdece lms for d re gve respecvely s follows: 8

9 zσ zσ zσ U zσ U zσ U zσ Opmm mple Cos-ress Tes Pls Mos of he es pls re eqlly-spced es sresses.e. he sme mber of es s re lloced o ech sress. ch es pls re slly effce for esmg he me lfe desg sress Yg 994. I hs seco es pls ssclly opmm re developed o decde he opml smple-proporo lloced o ech sress. Therefore o deerme he opml smpleproporo * lloced o ccelered codo s chose sch h he Geerlzed Asympoc Vrce GAV of he M esmors of he model prmeers s mmzed. The GAV of he M esmors of he model prmeers s opmly crero s commoly sed d defed below s he recprocl of he deerm of he Fsher-formo mr F B Km & Ch 993. Th s GAV F The Newo-Rphso mehod s ppled o merclly deerme he bes choce of he smpleproporo lloced o ccelered codo whch mmzes he GAV s defed before. Accordgly he correspodg epeced opml mbers of ems fled se d ccelered codos c be obed respecvely s follows: P d P where P d P. 6. mlo des The m obecve of hs seco s o mke mercl vesgo for llsrg he heorecl resls of boh esmo d opml desg problems. everl d ses re geered from Preo dsrbo for dffere combos of he re prmeer vles of d d for smple szes d 500 sg 500 replcos for ech smple sze. The re prmeer vles sed hs smlo sdy re d Comper progrms re prepred d he Newo-Rphso mehod s sed for he prccl pplco of he M esmors of d. Therefore he derved oler logrhmc lkelhood eqos 3 d 4 re solved ervely. Oce he vles of d re deermed esme of he shpe prmeer s esly obed from eqo. For dffere smple szes d re vles of he prmeers 9

10 he M esmes esmed sympoc vrces d cofdece ervls for prmeer-esmes re repored Tbles d 3. Whle Tbles d 4 he opml smple-proporo * lloced o ccelered codo he epeced opml mbers of ems fled se d ccelered codos d he opml GAV of he M esmors of he model prmeers re clded. Resls of smlo sdes provde sgh o he smplg behvor of he esmors. The mercl resls dce h he M esmes pprome he re vles of he prmeers s he smple sze creses. Also s show from he mercl resls he sympoc vrces of he esmors decrese s he smple sze s geg o be lrger. The eqos 4 re sed o cosrc he pprome cofdece lms for he hree prmeers d wh resls show Tbles d 3. These Tbles prese -sded pprome cofdece bods bsed o 95% cofdece degree for he prmeers. As see from he resls he ervls of he prmeers pper o be rrow s he smple sze creses. Also opmm es pls re developed merclly. I c be observed from he mercl resls v * preseed Tbles d 4 h he opmm es pls do o lloce he sme mber of es s o ech sress. I prcce he opmm es pls re mpor for mprovg precso prmeer esmo d hs mprovg he qly of he ferece. o hese opmm pls re more sefl d more effce for esmg he lfe dsrbo desg sress. Also Tbles d 4 clde he epeced mbers of ems fled se d ccelered codos represeed by * d * respecvely for ech smple sze. Flly hese Tbles lso prese he opml GAV of he M esmors of he model prmeers whch s obed merclly wh * plce of for dffere szed smples. As dced from he resls he opml GAV decreses s he smple sze creses. Tble The M esmes esmed sympoc vrces of he M esmors d cofdece bods of he prmeers se respecvely gve 0.5 d 50 for dffere szed smples der ype-i cesorg cos-sress PAT Prmeer Esme Vrce ower bod Upper bod

11 Tble The resls of opml desg of he lfe es for dffere szed smples der ype-i cesorg cos-sress PAT * * * Opml GAV Tble 3 The M esmes esmed sympoc vrces of he M esmors d cofdece bods of he prmeers se respecvely gve 0.5 d 50 for dffere szed smples der ype-i cesorg cos-sress PAT Prmeer Esme Vrce ower bod Upper bod Tble 4 The resls of opml desg of he lfe es for dffere szed smples der ype-i cesorg cos-sress PAT * * * Opml GAV

12 Refereces Abdel-Gh M. M. 998 "Ivesgo of some feme Models der Prlly Accelered fe Tess" Ph. D. Thess Deprme of scs Fcly of Ecoomcs & Polcl cece Cro Uversy Egyp. A A.F. Abdel-Ghly A.A. d Abdel-Gh M.M. 996 The Esmo Problem of Prlly Accelered fe Tess for he Webll Dsrbo by Mmm kelhood Mehod wh Cesored D Proceedgs of he 3 s Al coferece of sscs comper sceces d opero reserch IR Cro Uversy pp B D.. d Chg. W. 99 " Opml Desg of Prlly Accelered fe Tess for he Epoel Dsrbo der Type-I Cesorg" IEEE Tr. o relbly vol. 4 3 pp B D.. d Chg. W. Ch Y. R. 993 "Opml Desg of Prlly Accelered fe Tess for he ogorml Dsrbo Uder Type -I Cesorg" Relbly Eg. & sys. fey 40 pp B D.. Km J. G. d Ch Y. R. 993 Desg of Flre-Cesored Accelered fe-tes mplg Pls for ogorml d Webll Dsrbos Eg. Op. Vol. PP 97-. B. J. d Egelhrd M. 99 "Irodco o Probbly d Mhemcl scs" d Edo PWKENT Pblshg Compy Boso Mosschse. Bgghs M.M. 988 A Noe o Covergece Problems Nmercl Techqes for Accelered fe Tesg Alyss IEEE Tr. o Relbly Cohe A. C. d Whe B. J. 988 "Prmeer Esmo Relbly d fe p Models" Mrcel Dekker Ic. New York. Dvs H.T. d Feldse M The Geerlzed Preo w s Model for Progressvely Cesored rvvl D Bomerc Dyer D. 98 "The rcrl Probbly for he rog Preo w" C. J. s Grmshw.W. 993 Compg Mmm kelhood Esmes for he Geerlzed Preo Dsrbo Techomercs Isml A. A. 006 "Opmm Cos-ress Prlly Accelered fe Tes Pls wh Type-II Cesorg: The cse of Webll Flre Dsrbo" Ier Jly # 6. Johso N.. Koz. d Blkrsh. N. 994 "Coos Uvre Dsrbos" Vol. d Edo. Wley-Ierscece Pblco New York. om K "Bsess flres: Aoher emple of he lyss of flre d" J. Amer. s. Assoc McCe E.D. d McCe Esmo of he Preo hpe Prmeer Commco scs mlo d Compo Vol. 9 No

13 Mller R. d Nelso W. 983 Opmm mple ep-ress Pls for Accelered fe Tesg IEEE Trs. o Rel. Vol. R-3 No. Aprl Nelso W. 990 " Accelered fe Tesg: scl Models D Alyss d Tes Pls" Joh Wley d sos New York. PreoV. 987 Cors d'ecoome polqe. Roge e Ce Prs. Vder Wel.A. d Meeker W.Q. 990 Accrcy of Approme Cofdece Bods sg Cesored Webll Regresso D from Accelered fe Tess IEEE Tr. o Relbly 393 Ags Yg G.B. 994 Opmm Cos-ress Accelered fe-tes Pls IEEE Tr. o Relbly 434 December