Estimation in Step-Stress Partially Accelerated Life Tests for the Burr Type XII distribution Using Type I Censoring By. Abstract

Transcription

1 Estmto Step-Stress Prtll elerted fe Tests for the Burr Tpe XII dstrbuto Usg Tpe I Cesorg B bd-elftth,. M. _ftth@hotml.om ml S. Hss ml5_solm@hoo.om d Nssr, S. G. sdbosd@hoo.om Isttute of Sttstl Studes & Reserh Cro Uverst bstrt Some trdtol lfe tests result ver few flures b the ed of the test. So tht to obt flure qul test uts re ru t hgher th usul stress odtos. Ths stud s devoted to the estmtg flure tme dt uder step-stress prtll elerted lfe tests bsed o tpe I esorg. The lfetme dstrbuto of the test tems s ssumed to follow Burr tpe XII dstrbuto. The mxmum lelhood estmtes re obted for the dstrbuto prmeters d elerto ftor. I ddto, smptot vre d ovre mtrx of the estmtors re gve. tertve proedure s used to obt the estmtors umerll usg Mthd (). Furthermore, ofdee tervls of the estmtors re preseted. For llustrtg the preso d vrtos of mxmum lelhood estmtors smulto results re luded for dfferet smple szes. Ke words: Relblt; Step stress-prtll elerted lfe test; elerted ftor, Burr tpe XII dstrbuto; Mxmum lelhood method; Fsher formto mtrx;. Itroduto ue to otul mprovemet mufturg desg, t s more dffult to obt formto bout lfetme of produts or mterls wth hgh relblt t the tme of testg uder orml odtos. Ths mes lfetme testg uder these odtos ver ostve, te log tme. To get the formto bout the lfetme dstrbuto of these mterls, smple of these mterls s subjeted to more severe operto odtos th orml oes. These odtos re lled stresses whh m be the form of temperture, voltge, pressure, vbrto, lg rte, lod, et. Ths d of testg s lled elerted lfe test (T), where produts re put uder stresses hgher th usul to eld more flure dt short tme. The lfe dt from the hgh stresses re used to estmte the lfe dstrbuto t desg

2 odto. There re ml three T methods. The frst method s lled the ostt stress T; the seod oe s referred to s step-stress T d the thrd s the progressve stress T. the frst method s used whe the stress rems uhged, so tht f the stress s we, the test hs to lost for log tme. The other two methods, redue the testg tme d sve lot of mpower, mterl soures d moe (see Ro; (99)). The mjor ssumpto T s tht the mthemtl model reltg the lfetme of the ut d the stress s ow or be ssumed. I some ses, suh lfe-stress reltoshps re ot ow d ot be ssumed,.e T dt ot be extrpolted to use odto. So, suh ses, prtll elerted lfe tests (PT) s more sutble test to be performed for whh tested uts re subjeted to both orml d elerted odtos. ordg to Nelso (99), the stress be ppled vrous ws. Oe w to elerte flure step-stress, whh reses the stress ppled to test produt spefed dsrete sequee. Step-stress prtll elerted lfe test (SS- PT) s used to get qul formto for the lfetme of produt wth hgh relblt; spell, whe the mthemtl model relted to test odtos of me lfetme of the produt s uow d ot ssumed. The step stress sheme pples stress to test uts the w tht the stress wll be hged t prespefed tme. Geerll, test ut strts t spefed low stress. If the ut does ot fl t spefed tme, stress o t rsed d held spefed tme.. Stress s repetedl resed utl test ut fls or esorg tme s rehed. For overvew of SS-PT, there s mout of lterture o desgg SS- PT. Goel (97) osdered the estmto problem of the elerted ftor usg both mxmum lelhood d Bes methods for tems hvg expoetl d uform dstrbutos. egroot d Goel (979) estmted the prmeters of the expoetl dstrbuto d elerto ftor SS-PT usg Bes pproh, wth dfferet loss futos. lso, Bhtthr d Soejoet (989) estmted the prmeters of the Webull dstrbuto d elerto ftor usg mxmum lelhood method. B d hug (99) estmted the sle prmeter d elerto ftor for expoetl dstrbuto uder tpe I esored smple usg mxmum lelhood method. tt et l (996) osdered the mxmum lelhood method for estmtg the elerto ftor d the prmeters of Webull dstrbuto SS-PT uder tpe I esorg. bel-ghl et l (997) used Bes pproh for estmtg the prmeters of Webull dstrbuto prmeters wth ow shpe prmeter. The

3 studed the estmto problem SS-PT uder both tpe I d tpe II esored dt. bdel-gh (998) osdered the estmto problem of the prmeters of Webull dstrbuto d the elerto ftor for both SS-PT d ostt-stress PT. Mxmum lelhood d Bes methods uder tpe I d tpe II esored dt re ppled ths stud. bdel-ghl et l () studed the estmto problem of the elerto ftor d the prmeters of Webull dstrbuto SS- PT usg mxmum lelhood method two tpes of dt, mel tpe I d tpe II esorg. bdel-ghl et l (b, 3) studed both the estmto d optml desg problems for the Preto dstrbuto uder SS-PT wth tpe I d tpe II esorg. bdel-gh (4) osdered the estmto problem of log-logst dstrbuto prmeters uder SS-PT. Reetl, Isml (4) used mxmum lelhood d Bes methods for estmtg the elerto ftor d the prmeters of Preto dstrbuto of the seod d. Isml (6) studed the estmto d optml desg problems for the Gompertz dstrbuto SS-PT wth tpe I esored dt. Ths rtle s to fous o the mxmum lelhood method for estmtg the elerto ftor d the prmeters of Burr tpe XII dstrbuto. Ths wor ws oduted for SS-PT uder tpe I esored smple. The performe of the obted estmtors s vestgted terms of reltve bsolute bs, me squre error d the stdrd error. Moreover, the ofdee tervls of the estmtors wll be obted. Ths rtle be orgzed s follows. I seto the Burr tpe XII dstrbuto s trodued s lfetme model d the test method s lso desrbed. Seto 3 presets pot d tervl estmtes of prmeters d elerto ftor for the Burr tpe XII uder tpe I esorg usg mxmum lelhood method. I seto 4 the pproxmte smptot vres d ovres mtrx re vestgted. Seto 5 expls the smulto studes for llustrtg the theoretl results. Fll, olusos re luded seto 6. Tbles re dspled the ppedx.. The Model d Test Model s member of the Burr (94) fml of dstrbutos, ths ludes twelve tpes of umultve dstrbuto futos wth vret of dest shpes. The two prmeter Burr tpe XII dstrbuto deoted b Burr (, ) hs dest futo of the form 3

4 f (,, ) ( ) ( t t t ), t >, > d > where, d re the shpe prmeters of the dstrbuto. The umultve dstrbuto futo s F ( ) ( ) ( t,, t ), t >, > d > The relblt futo of the Burr tpe XII dstrbuto s gve b: (.) (.) (, ) ( t ) ( ) R t, (.3) The Burr (, ) dstrbuto ws frst proposed s lfetme model b ube (97, 973). Evs d Smos (975) studed further the dstrbuto s flure model d the lso derved the mxmum lelhood estmtors s well momets of the Burr (, ) probblt dest futo. ews (98) oted tht the Webull d expoetl dstrbutos re spel lmtg ses of the prmeter vlues of the Burr (, ) dstrbuto. She proposed the use of the Burr (, ) dstrbuto s model elerted lfe test dt. I SS-PT, ll of the uts re tested frst uder orml odto, f the ut does ot fl for prespefed tme, the t rus t elerted odto utl flure. Ths mes tht f the tem hs ot fled b some prespefed tme, the test s swthed to the hgher level of stress d t s otued utl tems fls. The effet of ths swth s to multpl the remg lfetme of the tem b the verse of the elerto ftor. I ths se the swthg to the hgher stress level wll shorte the lfe of test tem. Thus the totl lfetme of test tem, deoted b Y, psses through two stges, whh re the orml d elerted odtos. The the lfetme of the ut SS-PT s gve s follows: T f T Y (.4) ( T ) f T >, where, T s the lfetme of tem t use odto, s the stress hge tme d s the elerto ftor whh s the rto of me lfe t use odto to tht t elerted odto, usull >. ssume tht the lfetme of the test tem follows Burr tpe XII dstrbuto wth shpe prmeters d dest futo of totl lfetme Y of tem s gve b:. Therefore, the probblt 4

5 f ( ) f ( ) f ( ) f f f < > (.5) ( ) ( ) ( ) where, f,, s the equvlet form to equto (.), d, f > ( ) [ ( )] { ( )} [ ] ( ),, >, > b the trsformto vrble tehque usg equtos (.) d (.4). 3. Mxmum elhood Estmto, s obted The mxmum lelhood s oe of the most mportt d wdel methods, used sttsts. The de behd mxmum lelhood prmeter estmto s to determe the prmeters tht mxmze the probblt (lelhood) of the smple dt. Furthermore, mxmum lelhood estmtors re osstet d smptotll ormll dstrbuted. I ths seto pot d tervl estmto for the prmeters d elerto ftor of Burr tpe XII dstrbuto bsed o tpe I esorg re evluted usg mxmum lelhood method. 3. Pot Estmtes I tpe I esorg the test termtes whe the esorg tme η s rehed. The observed vlues of the totl lfetme Y re..., where u d re the umbers of () <... < ( ) < ( ) < < ( ) η u u u tems fled t orml odtos d elerted odtos respetvel. et, be dtor futos, suh tht (),,..., otherwse d, < ( j ) η,,..., otherwse For smplfg () be expressed b. Se the lfetmes,..., of tems re depedet d detll dstrbuted rdom vrbles, the ther lelhood futo s gve b 5

6 ( ) ( ) ( ) { } ( ) [ ] { } ( ) [ ] ( ) { } ( ) [ ] { },, ; η (3.) where, d. It s usull eser to mxmze the turl logrthm of the lelhood futo rther th the lelhood futo tself. Therefore, the logrthm of the lelhood futo s (3.) ( ) ( ) [ ] [ ] ( ) [ ] l l l l l l l l l where, { }] [, ( )] [ η,, u, u d u Mxmum lelhood estmtors of, d re solutos to the sstem of equtos obted b lettg the frst prtl dervtves of the totl log lelhood be zero wth respet to, d, respetvel. Therefore, the sstem of equtos s s follows: ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ), l η (3.3) ( )( ) ( ) ( ) ( ) ( ) ( ) l l l l l l (3.4) d[ ( ) ( ) ( ) ( l l l l ). (3.5) From equto (3.5) the mxmum lelhood estmte of s expressed b 6

7 o, (3.6) where, ( ) ( ) ( ) ( ) l l l Cosequetl, b substtutg for to equtos (3.3) d (3.4), the sstem equtos re redued to two oler equtos s follows: l l 3 (3.7) d ( ) ( ) 5 4 (3.8) where, ( ) ( ) ( ) l l,, ( )( ) ( ) 3 l ( ) ( )( ) 4 d ( ) ( ) ( )( ) 5 η. Se the losed form soluto to oler equtos (3.7) d (3.8) re ver hrd to obt. tertve proedure s ppled to solve these equtos umerll usg Mthd () sttstl pge. Newto-Rphso method s ppled for smulteousl solvg the oler equtos to obt d. Therefore s lulted esl from equto (3.6). 3. Itervl Estmtes If d ),..., ( θ θ ),..., ( U U θ θ re futo of the smple dt the ofdee tervl for populto prmeter,..., θ s gve b γ θ θ θ ] [ U p (3.9) where, d re the lower d upper ofdee lmts whh elose θ θ U θ wth probbltγ. The tervl s lled two sded ], [ θ θ U % γ ofdee tervl for θ. For lrge smple sze, the mxmum lelhood estmtes, uder pproprte regulrt odtos, re osstet d smptotll ormll dstrbuted. 7

8 Therefore, the two sded pproxmte γ % ofdee lmts for the mxmum lelhood estmte θ of populto prmeter θ be ostruted, suh tht θ θ p [ z z] γ (3.) σ ( ) θ where, ( γ ) z s the [ ] stdrd orml peretle. Therefore, the two sded pproxmte γ % ofdee lmts for populto prmeter θ be obted, suh tht p[ θ zσ ( ) θ θ θ zσ ( θ )] γ (3.) The, the two sded pproxmte ofdee lmts for, d wll be ostruted usg equto (3.) wth ofdee levels 95 % d 99 %. 4. smptot Vres d Covre s of Estmtes The smptot vres of mxmum lelhood estmtes re gve b the elemets of the verse of the Fsher formto mtrx I ( θ ) E{ l θ θ } Ufortutel, the ext mthemtl expressos for the bove expetto re ver dffult to obt. Therefore, the observed Fsher formto mtrx s gve b I ( θ ) { l θ θ } j j, whh s obted b droppg the expetto o operto E [see Cohe (965)]. The pproxmte (observed) smptot vre ovre mtrx F for the mxmum lelhood estmtes be wrtte s follows F [ ( θ )],, j,, 3 I j d ( ) (,,) j θ (4.) [ The seod prtl dervtves of the mxmum lelhood futo re gve s the followg: l ( ) ( ) ( ) [ ] ( ) ( )( )( )( ) ( ) ( )( ) ( ) [ ], (4.) ( ) ( η )( )( η )( ) ( ) ( η )( ) ( ) j. 8

9 ( )( ) ( )( )( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ], l l l l l η (4.3) [[ ( ) ( )( ) ( ) ( ) ( )( ), l η (4.4) [ ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( )( ) { } ( ) ( ) ( ) ( ) ( ) { }, l l l l l l l l l l (4.5) ( ) ( ) ( ) ( ) ( )( ) ( ), l l l l (4.6) [[ d l. (4.7) Cosequetl, the mxmum lelhood estmtors of, d hve smptot vre ovre mtrx defed b vertg the Fsher formto mtrx d b substtutg for F, for d for. 5. Smulto Studes Smulto studes hve bee performed usg Mthd () for llustrtg the theoretl results of estmto problem. The performe of the resultg estmtors of the elerto ftor d two shpe prmeters hs bee osdered terms of ther bsolute reltve bs (RBs), me squre error (MSE), d reltve error (RE). Furthermore, the smptot vre d ovre mtrx d twosded ofdee tervls of the elerto ftor d two shpe prmeters re obted. The Smulto proedures were desrbed below: 9

10 Step : rdom smples of szes (5) 4 d 5 were geerted from Burr tpe XII dstrbuto. The geerto of Burr tpe XII dstrbuto s ver smple, fu hs uform (,) rdom umber, the Y ( ) ( ) [( U ) ] follows Burr tpe XII dstrbuto. The true prmeters seleted vlues re (,.5, ) d (.5,,.5) Step : Choosg the esorg tme t the orml odto to be 3 d esorg tme of PT to beη 6. Step 3: For eh smple d for the two sets of prmeters, the elerto ftor d the prmeters of dstrbuto were estmted SS-PT uder tpe I esored smple. Step 4. Newto Rphso method ws used for solvg the two oler lelhood for d gve equtos (3.7) d (3.8), respetvel. The estmte of the shpe prmeter ws esl obted from equto (3.6) Step 5: The RBs, MSE, d RE of the estmtors for elerto ftor d two shpe prmeters for ll smple szes d for two sets of prmeters were tbulted. Step 6: The smptot vre d ovre mtrx of the estmtors for dfferet smple szes were obted. Step 7: The two sded ofdee lmt wth ofdee level γ. 95 d γ. 99 of the elerto ftor d the two shpe prmeters were ostruted. Smulto results re summrzed Tbles, d 3. Tble gves the RBs, MSE, d RE of the estmtors. The smptot vres d ovre mtrx of the estmtors re dspled Tble. Whle Tble 3 presets the pproxmted two-sded ofdee lmts t 95 % d 99% for the prmeters d elerto ftor. From these tbles, the followg observtos be mde o the performe of SS-PT prmeter estmto of Burr tpe XII lfetme dstrbuto:. For the frst set of prmeters (.5,,.5), the mxmum lelhood estmtors hve good sttstl propertes th the seod set of prmeters (,.5, ) for ll smple szes (see Tble).. s the elerto ftor reses the estmtes hve smller MSE, d RE. s the smple sze reses the RBses d MSEs of the estmted prmeters derese. Ths dtes tht the mxmum lelhood estmtes provde

11 smptotll ormll dstrbuted d osstet estmtors for the prmeters d elerto ftor 3. The smptot vres of the estmtors re deresg whe the smple sze resg (see Tble). 4. The tervl of the estmtors dereses whe the smple sze s resg. lso, the tervl of the estmtor t γ. 95 s smller th the tervl of estmtor t γ.99 (see Tble3). 6. Coluso For produts hvg hgh relblt, the test of produt lfe uder orml use ofte requres log perod of tme. So T or PT s used to fltte estmtg the relblt of the ut short perod of tme. I T test tems re ru ol t elerted odtos, whle PT the re ru t both orml d elerted odtos. Oe w to elerte flure s the SS- PT whh reses the stresses ppled to test produt spefed dsrete sequee. The lfetme of the test tems s ssumed to follow the Burr tpe XII dstrbuto. Uder tpe I esorg, the test ut s frst ru t orml use odto, d f t does ot fl for spefed tme, the t s ru t elerted odto utl esorg tmeη s rehed. The mxmum lelhood method s used for estmtg the elerto ftor d prmeters of Burr tpe XII dstrbuto uder tpe I esorg. Performe of step-stress testg pls d model ssumptos re usull evluted b the propertes of the mxmum lelhood estmtes of model prmeters. I ths stud, the frst set of prmeters hve good sttstl propertes th the seod set of prmeters for ll smple szes Mxmum lelhood estmtors re osstet d smptotll ormll dstrbuted. s the smple sze reses the smptot vre d ovre of estmtors dereses. Regrdg the tervl of estmtors, t be oted tht the tervl of the estmtors t γ. 99 s greter th the orrespodg t γ. 95. lso, s the smple sze reses the tervl of the estmtors dereses for the two ofdee level.

12 ppedx Tble: The RBs, MSE d RE of the Prmeters (,,,, η) Uder Tpe I Cesorg Prmeters (,.5,,3,6) (.5,,.5,3,6) (,,,, η) RBs MSE RE RBs MSE RE

13 Tble: smptot Vres d Covres of Estmtes Uder Tpe Cesorg (,.5,,3,6) (.5,,.5,3,6) ĉ ĉ

14 Tble3: Cofdee Bouds of the Estmtes t Cofdee level t.95 d.99 Prmeters (,.5,,3,6) (.5,,.5,3,6) (,,,, η) Stdrd ower Upper Stdrd ower Upper Boud devto Boud Boud devto Boud

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