Accelerated Life Test Sampling Plans under Progressive Type II Interval Censoring with Random Removals

Transcription

1 Iteratoal Joural of Statstcs ad Probablty; Vol. 7, No. ; Jauary 8 ISSN E-ISSN Publshed by Caada Ceter of Scece ad Educato Accelerated Lfe Test Samplg Plas uder Progressve Type II Iterval Cesorg wth Radom Removals Su Keug Tse & Chag Dg Departmet of Maagemet Sceces, Cty Uversty of Hog Kog, Cha Yua Uversty of Face & Ecoomcs, Cha Correspodece: Su Keug Tse, Departmet of Maagemet Sceces, Cty Uversty of Hog Kog, Cha. Receved: September 6, 7 Accepted: October 7, 7 Ole Publshed: November 7, 7 do:.5539/jsp.v7p6 URL: Abstract Ths paper vestgates the desg of accelerated lfe test (ALT) samplg plas uder progressve Type II terval cesorg wth radom removals. For ALT samplg plas wth two over-stress levels, the optmal stress levels ad the allocato proportos to them are obtaed by mmzg the asymptotc geeralzed varace of the maxmum lkelhood estmato of model parameters. The requred sample sze ad the acceptablty costat whch satsfy gve levels of producer s rsk ad cosumer s rsk are foud. ALT samplg plas wth three over-stress levels are also cosdered uder some specfc settgs. The propertes of the derved ALT samplg plas uder dfferet parameter values are vestgated by a umercal study. Some terestg patters, whch ca provde useful sght to practtoers related areas, are foud. The true acceptace probabltes are computed usg a Mote Carlo smulato ad the results show that the accuracy of the derved ALT samplg plas s satsfactory. A umercal example s also provded for llustratve purpose. Keywords: accelerated lfe test, progressve Type II terval cesorg, radom removal, samplg pla. Itroducto The desg of relablty samplg plas uder Type II cesorg schemes has bee studed by may researchers (Fertg & Ma, 98; Hosoo, Ohta, & Kase, 98; Kocherlakota & Balakrsha, 986; Scheder, 989; Balasoorya, 995; Wu, Hug, & Tsa, 3). I practce, t s ot ucommo that some uts are removed durg the test, whch leads to progressve cesorg schemes. Balasoorya ad Saw (998), Balasoorya ad Balakrsha (), ad Balasoorya, Saw, ad Gadag () dscussed relablty samplg plas for the two-parameter expoetal, logormal ad Webull dstrbutos uder progressve Type II cesorg schemes, respectvely. The umber of removals at each falure was assumed to be pre-fxed those works. However, practce t mght be feasble to pre-determe the removal patter ad the decso of removg ay uts s based o the status of the expermet at that specfc tme, such as excessve heat or pressure, reducto of budget ad faclty, etc. Therefore, the umber of removals should be a radom outcome (Yue & Tse, 996). Tse ad Yag (3) dscussed the desg of relablty samplg plas for the Webull dstrbuto uder progressve Type II cesorg wth radom removals, where the umber of uts removed at each falure was assumed to follow a bomal dstrbuto. I recet years the feature of radom removal has bee adopted by may researchers desgg varous kds of progressve cesorg schemes, such as Ashour ad Affy (7), Wu, Che, ad Chag (7), ad S. Dey ad T. Dey (4). Uts are supposed to ru at use codto tradtoal relablty samplg plas. Whe t s desred to test the acceptace of hghly relable products, t s mpractcal to use such relablty samplg plas due to tme costrat. Wallace (985) stressed the eed for troducg ALT to relablty samplg plas. Ba, Km, ad Chu (993) studed the desg of falure cesored ALT samplg plas for logormal ad Webull dstrbutos. Hseh (994) vestgated relablty samplg plas wth ALT uder Type II cesorg for expoetal dstrbuto. The optmal desg of ALT samplg plas wth a o-costat shape parameter uder both Type I ad Type II cesorg schemes was gve by Seo, Jug, ad Km (9). Note that cotuous spectos were assumed the above works. Nevertheless, sometmes t s coveet to coduct a test wth cotuous spectos due to the hgh cost ad/or possble dager motorg the test cotuously. Uder these crcumstaces, the terval specto schemes, whch oly the umber of falures betwee two successve 6

2 Iteratoal Joural of Statstcs ad Probablty Vol. 7, No. ; 8 spectos s recorded, would be more favorable. Studes o lfe test ad/or accelerated lfe test whch employ terval cesorg schemes are umerous. To umber some of them, Tse, Dg, ad Yag (8) vestgated the optmal desg of accelerated lfe test uder terval cesorg wth radom removals for Webull dstrbuto; Che ad Lo () compared the maxmum lkelhood estmato, momet estmato ad probablty plot estmato of parameters the geeralzed expoetal dstrbuto uder progressve Type I terval cesorg; Dg, Yag, ad Tse () dscussed the desg of optmal ALT samplg plas uder progressve Type I terval cesorg wth radom removals. Most recetly, Dg ad Tse (3) vestgated the desg of optmal ALT plas uder progressve Type II terval cesorg wth bomal removals for the Webull dstrbuto. However, as far as our kowledge goes, there s o relevat study that vestgates the desg of ALT samplg plas uder smlar expermetal settgs wth a Type II cesorg scheme. The optmal relablty samplg plas whch combe ALT, terval specto ad progressve Type II cesorg wth radom removals are developed ths paper. Ths study ca be oted as a exteso to the work of Dg ad Tse (3) alog three drectos: () the research topc s exteded from the desg of optmal ALT plas to the desg of optmal ALT relablty samplg plas, whch both the cosumer s rsk ad the producer s rsk are satsfed. I ths sese ths paper resolves a more practcal problem; () stead of mmzg the asymptotc varace of a estmated quatle of uts lfetme dstrbuto, ths paper mmzes the asymptotc geeralzed varace of the maxmum lkelhood estmato of model parameters. It eables us to compare the outcomes derved usg two dfferet crtera optmzato; () the true acceptace probabltes of the derved optmal ALT samplg plas are smulated, whch provdes us a way to evaluate the accuracy of the proposed method. The rest of ths paper s orgazed as follows: Secto descrbes the basc model of the proposed scheme. The desg of optmal ALT samplg plas uder progressve Type II terval cesorg wth radom removals s dscussed Secto 3. A umercal study s coducted Secto 4 to exame the propertes of the derved samplg plas. I Secto 5 the accuracy of the proposed ALT samplg plas s evaluated by a Mote Carlo smulato. Secto 6 provdes a umercal example. Coclusos are draw Secto 7.. Model Descrpto Cosder a ALT wth the followg settgs:. A total of detcal ad depedet uts are avalable at the begg of the test.. There are m over-stress levels,.e., s, s,..., s m. Deote s as the stress level at use codto. th 3. Suppose that uts are radomly allocated to the stress level (,,..., m). The the allocato proporto to th the stress level s gve by /. 4. A progressve Type II cesorg scheme s employed, ad the test o the (,,..., m) or more uts fal. th stress level wll be termated after c 5. Iterval spectos are coducted at tme pots t, t,, t, k ( ) ad the umber of falures x j betwee specto terval ( t, j, tj ) s recorded. It should be poted out that both the expermet tme t, k ( ) ad the umber of spectos k () are radom varables. 6. Suppose that rj (,,..., m; j,,..., k( ) ) o-faled uts are radomly removed at specto tme t j. To esure that there are at least c faled uts at the ed of the test o stress level s, rj s restrcted to be ay teger value betwee ad j c r. Further assume that r j follows a bomal dstrbuto wth probablty p, the we l l have r ~ B c j r j, p l l. For otatoal coveece, deote of uts left. The process of ths testg scheme s depcted Fgure. k ( ) k ( ), k ( ) j j j j as the umber r x r 7

3 Iteratoal Joural of Statstcs ad Probablty Vol. 7, No. ; 8 Fgure. The testg scheme of a ALT uder progressve Type II terval cesorg wth radom removals (o the stress level) Suppose that the lfetme of a ut T follows a Webull dstrbuto wth probablty desty fucto (pdf) / / exp / f t t t, t. () Further assume that the scale parameter ad stress level s are related as exp s, () where ad are ukow costats ad the shape parameter does ot deped o s. Defe Y l( T ), the Y has a extreme value dstrbuto wth cumulatve dstrbuto fucto (cdf) exp exp ( ) / G y y, y, (3) where = l = + s, =/. Gve observatos ( x, r,,,..., m; j,,..., k( )), the logarthm of the lkelhood fucto ca be derved as: j l L,,, p; x, r, k( ) where y l( t ). j j j j k ( ) k ( ) m x r x j j r j j l l... l x x x, k ( ) j th k() k x l, l l!/! j j G yj G y j r j G y j c rj c j k j k ( ) ( ) r l ( ) ( ) l j p k j c k j r j p ( ) k( ) j j, (4) r! The maxmum lkelhood estmates of, ad, p (deoted by ˆ ˆ,, ˆ ad ˆp, respectvely) ca be solved from equatos l L / l L / l L / l L / p Besdes, the Fsher formato matrx of. (,,, p) s gve by l L l L l L l L l L l L I,,, p E l L l L l L I,,, (5) I p l L p where I(,, ) s the upper left 3 3 sub-matrx of I(,,, p) I ( p) E l L / p. The asymptotc covarace matrx of ˆ ˆ ˆ (,, ) s the gve by I (,, ). The detaled formulato for the etres of Eq. (5) ca be foud the Appedx of Dg ad Tse (3). Gve sample sze, use codto s ad hgh stress level ad s m, removal probablty p, predetermed umber of 8

4 Iteratoal Joural of Statstcs ad Probablty Vol. 7, No. ; 8 falures c ad spectos tmes ( tj,,,..., m; j,,..., k( )) o each stress, the stress levels ( s,,,..., m ) ad the allocato proportos (,,,..., m ) to these stress levels are selected such a way that the geeralzed asymptotc varace of ˆ, ˆ, ad ˆ, whch s gve by I (,, ), s mmzed. 3. Desg of ALT Samplg Plas Suppose that a sample of sze s radomly draw from the lot ad the test s coducted at the accelerated settgs descrbed Secto. Assume that the lfetme of a ut T follows a Webull dstrbuto F(, ), where the relatoshp betwee the scale parameter ad the stress s s gve by Eq. () ad the shape parameter does ot deped o s. Suppose that a ut wth lfetme less tha s cosdered to be ocoformg. Defe Y l( T), the Y follows a extreme value dstrbuto G(, ) ad the lower specfcato lmt for the log lfetme s gve by l( ). Defe d, where s the locato parameter of G (.) at use codto ad d s the acceptablty costat. Sce the stresses ca be stadardzed such that s, s ad s (,,..., m ), t follows from Eq. (3) that m s. By the varace prcple of the maxmum lkelhood method, the MLE of s the gve ˆ ˆ d ˆ ˆ d ˆ. To judge whether a lot should be accepted or ot, ˆ s compared wth the lower by specfcato lmt. If ˆ, the lot s accepted; otherwse, t s rejected. Defe the ocoformg fracto of the lot by p f, whch s calculated as p P( Y ) exp exp ( ) /. (6) f The sample sze ad the acceptablty costat d are determed such that lots wth ocoformg fracto pf p are accepted wth a probablty of at least ad lots wth pf, where ad are the gve levels of producer s ad cosumer s rsks, respectvely. It follows from ˆ ˆ ˆ d Var ˆ Var( ˆ ) d Cov( ˆ, ˆ ) d Var( ˆ ). that p are rejected wth a probablty of at least Sce ˆ ˆ / U ( d) / Var( ) s parameter-free ad asymptotcally stadard ormal, the operatg characterstc (OC) curve s gve by O( p ) ( ˆ f P ) l l( p f) d / Var( ˆ ) where (.) s the cdf of the stadard ormal dstrbuto., (7) The sample sze ad the acceptablty costat d are determed such that the OC curve goes through two pots ( p, ) ad (, ). Ths mples It follows that p p d Var ˆ l l /, l( l( p )) d / Var( ˆ ). (8) d u l l p u l l p / u u, Var ˆ l l p d / u, (9) where u z () z. The acceptablty costat d s calculated drectly from the frst part of Eq. (9), whle the requred sample sze ca be obtaed by a search method from the secod part (the detaled algorthm s provded Secto 4.). 4. Numercal Study 4.. ALT Samplg Plas wth Two Over-stress Levels The propertes of the derved ALT samplg plas uder dfferet parameter values are evaluated by a umercal study ths secto. The followg settgs are made:. Two over-stress levels s, s are employed,.e., m. 9

5 Iteratoal Joural of Statstcs ad Probablty Vol. 7, No. ; 8. The spectos o each stress level are equally spaced,.e., t, tj t, j l, (,,..., m; j,,..., k( )), th where l s the specto legth o the stress level. Defe MT exp( s) ( / ) as the mea of uts lfetme dstrbuto o the same stress ad l / MT as the proporto of the specto legth to the correspodg mea., whch s proportoal to the specto legth l, s used ths umercal study sce t s more coveet to use a relatve value tha a absolute oe. The case of s cosdered. th 3. Defe the cesorg fracto o the stress level as fc c / (,). The cases of both fc fc ad fc fc are cosdered sce uts are much easer to fal o the hgh stress level tha o the low oe. Wthout loss of geeralty, set s, sm. I practce, t s ofte dffcult for a expermeter to gve pror estmates of parameters ad. O the cotrary, based o the expermeters expereces ad/or the formato collected from prelmary or smlar studes, the estmato of the probablty that a ut falls to a certa terval s much easer. Defe P u = P (a ut s lfetme T falls to (, ) at use codto) ad P h = P (a ut s lfetme T falls to (, ) o hgh stress level), the we have P Pu Ph l l, u l l l l. () I order to obta a optmal ALT samplg pla uder progressve Type II terval cesorg wth radom removals, the values of, d, s ad have to be determed. The acceptablty costat d depeds o ( p, ) ad ( p, ) oly, ad t ca be calculated from Eq. (9) drectly. The determato of the other three parameters requres the combato of a grd search method ad the Mote Carlo smulato. For the sake of smplcty, let deote. The,, s ad are calculated usg the followg algorthm: l l p d / u () () * *. Set a tal value for. Cosder the smallest sample sze ad set. Fd the optmal ( s, ) whch mmzes I (,, ) usg a grd search method over ut square (,) (,). Calculate the correspodg value * * of Var( ˆ ) at ( s, ).. Set () (), fd 3. Repeat step utl for ( s, ) based o sample sze * * Var( ) ad for (), ˆ () ad compute ˆ Var( ). Defe ( ), ˆ ( ) ( l) ( u) * * 4. Set ( ) /. Fd ( s, ) ad calculate Var( ˆ ). If ( u) ( ). 5. Repeat step 4 utl Var( ˆ ) approxmately holds or. ( u) ( l) Var( ) accordgly. ad ( l) ( ) Var( ˆ ), set ( l) ( ). ( u) ( ) ; otherwse, set For gve parameter values, specfcally ( p, ) =(.4,.95); ( p, ) =(.84,.); P u =.; Ph. ;.5,, ; ( fc, f c) =(.5,.5), (.8,.8), (.5,.7), (.5,.9); p =,.5,.,.3 ad =.,.5,.,.3, the optmal ALT samplg plas (, d, s, ) are determed usg the algorthm descrbed above. A cosstet patter * * emerged based o the results of these combatos. The requred sample sze decreases as the cesorg fractos f c, f crease. I order to provde a better sght o the effect of p (the probablty of radom removal) ad (whch s c proportoal to the specto legth), some cases are selected for llustrato ad the correspodg results are depcted Fgure. 3

6 Iteratoal Joural of Statstcs ad Probablty Vol. 7, No. ; 8 The followg patters are observed: Fgure. Two over-stress levels ALT samplg plas uder progressve Type II terval cesorg wth radom removals a. For the cases of.5, creases as creases for all values of p. For the cases of, whe p =,.5 ad., creases as creases; whe p =.3, frst decreases ad the creases as creases. For the cases of, whe p = ad.5, creases as creases; whe p =. ad.3, frst decreases ad the creases as creases. Ths patter ca be terpreted ths way: Larger meas wder specto tervals, from whch the collected formato o uts lfetme s less accurate ad thus more uts are requred to judge whether to accept the lot or ot. However, whe p, a larger also mples that uts are less lkely to be removed at the early stage of the test. Cosequetly, more formato o the lfetme dstrbuto s collected ad the requred sample sze s decreased. Takg these two kds of effect to cosderato, shorter specto terval does t always yeld smaller requred sample sze for ALT samplg plas uder progressve Type II terval cesorg wth radom removals. b. For the cases of.5 ad, decreases as p creases for all values of. For the cases of, whe =. ad.3, decreases as p creases; whe =. ad.5, frst decreases ad the creases as p creases. Ths patter s caused by the two-sded effects of the removal probablty p. Geerally speakg, a test s lkely to be prologed as p creases. Thus more formato o the lfetme dstrbuto ca be observed ad the requred sample sze s decreased. Nevertheless, whe the specto tervals are too small, a o-zero removal probablty p also causes more uts beg removed at the early stage of the test. I ths case, less data ca be collected ad thus s creased. I cocluso, except for several cases ( ad =./.5), the removal probablty p s helpful reducg the requred sample sze. 4.. ALT Samplg Plas wth Three Over-stress Levels ALT plas wth three over-stress levels are useful practce sce they ca provde a way to check the assumed straght-le relatoshp betwee dstrbuto parameter ad stress level s by addg a mddle stress. The desg of three over-stress levels ALT samplg plas uder progressve Type II terval cesorg wth radom removals s dscussed ths secto. They are developed uder the followg settgs:. Three over-stress levels, s, s ad s 3 are employed. I partcular, set s, s3 ad s ( s s3 ) /.. The allocato proportos to three over-stress levels (,, 3) are set to be (/ 3, / 3, / 3) ad (.5,.3,.). 3. Three settgs of cesorg fractos are cosdered, amely, ( fc, fc, f c3) equals (.5,.5,.5), (.8,.8,.8) ad (.5,.7,.9). 4. The proporto of the specto legth to the correspodg mea, that s, s set to be equal o three over-stress levels,.e., 3. A umercal study s coducted to determe ALT samplg plas wth three over-stress levels uder equally spaced specto tmes. For gve parameters, the optmal low stress level s s foud by a grd search method over terval (, 3

7 Iteratoal Joural of Statstcs ad Probablty Vol. 7, No. ; 8 ). The step sze s.. For dfferet parameter values, specfcally, ( p, ) =(.4,.95); ( p, ) =(.84,.); P u =.; Ph.; =.5,, ; p =,.5,.,.3 ad =.,.5,.,.3, the optmal low stress level s ad the requred sample sze are calculated. The effects of p ad o the requred sample sze are depcted Fgure 3. We ote that: Fgure 3. Three over-stress levels ALT samplg plas uder progressve Type II terval cesorg wth radom removals a. For the cases of.5 ad, creases as creases. For the cases of, whe p = ad.5, creases as creases; whe p =. ad.3, frst decreases ad the creases as creases. b. For the cases of.5, decreases as p creases. For the cases of, whe =.5,. ad.3, decreases as p creases; whe., frst decreases ad the creases as p creases. For the cases of, whe =. ad.3, decreases as p creases; whe =. ad.5, frst decreases ad the creases as p creases. Note that these patters are smlar to those observed the two over-stress levels case. 5. Accuracy of Large Sample Approxmato Sce the proposed ALT samplg plas are derved based o asymptotc theory, there s a eed to evaluate the fte sample behavor of them. The accuracy of the derved ALT samplg plas s assessed by a smulato study. The OC curve s set to go through two pots, that s, ( p, ) =(.4,.95) ad ( p, ) =(.84,.). For each combato of parameters, the ocoformg fracto of a lot uder gve acceptace probablty (99% ad 95%) o the pre-defed OC curve s computed, ad the the true acceptace probablty of a lot wth that correspodg ocoformg fracto s calculated by a Mote Carlo smulato wth rus. The results for ALT samplg plas wth two over-stress levels ad three over-stress levels are preseted Table ad Table, respectvely. Actually, several dfferet values of (.5,, ) are cosdered ths umercal study. Sce they show smlar patters, oly parts of the results of are provded for smplcty. We ote from Table ad Table that the smulated acceptace probabltes are close to ther omal values most cases. Ths dcates that the optmal ALT samplg plas derved based o asymptotc approxmato have satsfactory accuracy. 6. A Numercal Example Suppose that there s a agreemet betwee a cosumer ad a producer to determe the acceptablty of a lot. I partcular, f the ocoformg fracto of a lot s smaller tha.4, the the lot should be accepted wth a probablty of at least.95; whle f the ocoformg fracto of a lot s larger tha.84, the t should be rejected wth a probablty of at least.9. Assume that a ALT relablty samplg pla wth two over-stress levels s used to determe the acceptablty 3

8 Iteratoal Joural of Statstcs ad Probablty Vol. 7, No. ; 8 of the lot. The probabltes for a ut to fal at use codto ad hgh stress level are estmated to be. ad., respectvely. A progressve Type II terval cesorg scheme s employed, ad the cesorg fractos o two stress levels are.8. The proportos of the specto legth to the correspodg dstrbuto mea o both stresses are set to be.. Besdes, based o pror formato, t s assumed that uts lfetmes are Webull dstrbuted wth shape parameter ad a ut s lkely to be removed at each specto wth probablty.. The problem s to determe the umber of uts used ths ALT samplg pla ad to determe the low stress level ad the allocato proportos to two stresses so that () both the cosumer s rsk ad the producer s rsk ca be satsfed ad () the maxmum amout of formato o uts lfetme dstrbuto ca be collected. The optmal ALT samplg pla s obtaed usg the proposed method. The requred sample sze s 7, wth 7 ad uts allocated to the low ad hgh stress levels, respectvely. The low stress level should be settled at. multpled by the actual hgh stress. Besdes, the acceptablty costat whch s requred to make the decso s Cocluso The desg of ALT samplg plas uder progressve Type II terval cesorg wth radom removals was dscussed ths paper. For ALT samplg plas wth two over-stress levels, the optmal stress levels ad the correspodg allocato proportos, whch mmze the geeralzed asymptotc varace of the MLE of model parameters, were foud. The sample sze ad the acceptablty costat requred to judge the acceptablty of the lot were calculated. The propertes of the derved ALT samplg plas were examed by a umercal study. It s show that geerally the removal probablty s helpful reducg the requred sample sze. More mportatly, whe there exsts radom removal, short specto terval does t always yeld small requred sample sze, whch s dfferet from the case of o radom removal. These terestg patters would provde useful sghts to expermeter desgg smlar ALT samplg plas. The accuracy of the proposed samplg plas was evaluated by a Mote Carlo smulato. The results show the smulated acceptace probabltes are close to ther omal values most cases, whch dcates that the performace of the derved ALT samplg plas s satsfactory. Table. Smulated acceptace probabltes for two over-stress levels ALT samplg plas uder progressve Type II terval cesorg wth radom removals ( m ; ; p.4 ;.95 ; p.84 ;. ) fc fc.5 fc fc.8 Selected pots o Smulated Selected pots o Smulated OC curve probablty OC curve probablty 99% 95% 99% 95% 99% 95% 99% 95% p p p p , f.7 fc.5, fc.9 Smulated Selected pots o probablty OC curve fc c Selected pots o OC curve Smulated probablty 33

9 Iteratoal Joural of Statstcs ad Probablty Vol. 7, No. ; 8 99% 95% 99% 95% 99% 95% 99% 95% p p p p

10 Iteratoal Joural of Statstcs ad Probablty Vol. 7, No. ; 8 Table. Smulated acceptace probabltes for three over-stress levels ALT samplg plas uder progressve Type II terval cesorg wth radom removals ( m 3 ; ; p.4 ;.95 ; p.84 ;. ) Case I. : : / 3, / 3, / 3 3 c c c3.5 f f f fc fc fc3.8 fc.5, fc.7, fc3.9 Selected Smulated Selected Smulated Selected Smulated pots probabltes pots probabltes pots probabltes 99% 95% 99% 95% 99% 95% 99% 95% 99% 95% 99% 95% p p p p

11 Iteratoal Joural of Statstcs ad Probablty Vol. 7, No. ; 8 Table.(Cot d) Smulated acceptace probabltes for three over-stress levels ALT samplg plas uder progressve Type II terval cesorg wth radom removals ( m 3 ; ; p.4 ;.95 ; p.84 ;. ) : :.5,.3,. Case II. c c c3.5 3 f f f f f f 3.8 fc.5, fc.7, fc3.9 c c c Selected pots Smulated probabltes Selected pots Smulated probabltes Selected pots Smulated probabltes 99% 95% 99% 95% 99% 95% 99% 95% 99% 95% 99% 95% p p p p

12 Iteratoal Joural of Statstcs ad Probablty Vol. 7, No. ; 8 Ackowledgemet Dg was supported by the Natoal Natural Scece Foudato of Cha uder Grat umber 698. Refereces Ashour, S. K., & Affy, W. M. (7). Statstcal aalyss of expoetated Webull famly uder type I progressve terval cesorg wth radom removals. Joural of Appled Sceces Research, 3(), Ba, D. S., Km, J. G., & Chu, Y. R. (993). Desg of falure-cesored accelerated lfe-test samplg plas for logormal ad Webull dstrbutos. Egeerg optmzato, (3), Balasoorya, U. (995). Falure cesored relablty samplg plas for the expoetal dstrbuto. Joural of Statstcal Computato ad Smulato, 5(4), Balasoorya, U., & Balakrsha, N. (). Relablty samplg plas for logormal dstrbuto, based o progressvely-cesored samples. IEEE Trasactos o Relablty, 49(), Balasoorya, U., & Saw, S. L. (998). Relablty samplg plas for the two-parameter expoetal dstrbuto uder progressve cesorg. Joural of Appled Statstcs, 5(5), Balasoorya, U., Saw, S. L., & Gadag, V. (). Progressvely cesored relablty samplg plas for the Webull dstrbuto. Techometrcs, 4(), Che, D. G., & Lo, Y. L. (). Parameter estmatos for geeralzed expoetal dstrbuto uder progressve type-i terval cesorg. Computatoal Statstcs & Data Aalyss, 54(6), Dey, S., & Dey, T. (4). Statstcal Iferece for the Raylegh dstrbuto uder progressvely Type-II cesorg wth bomal removal. Appled Mathematcal Modellg, 38(3), Dg, C., & Tse, S. K. (3). Desg of accelerated lfe test plas uder progressve Type II terval cesorg wth radom removals. Joural of Statstcal Computato ad Smulato, 83(7), Dg, C., Yag, C., & Tse, S. K. (). Accelerated lfe test samplg plas for the Webull dstrbuto uder type I progressve terval cesorg wth radom removals. Joural of Statstcal Computato ad Smulato, 8(8), Fertg, K. W., & Ma, N. R. (98). Lfe-test samplg plas for two-parameter Webull populatos. Techometrcs, (), Hosoo, Y., Ohta, H., & Kase, S. (98). Desg of sgle samplg plas for doubly expoetal characterstcs. Froters Qualty Cotrol, 94-. Hseh, H. K. (994). Accelerated lfe test samplg plas for expoetal dstrbutos. Commucatos Statstcs-Smulato ad Computato, 3(), Kocherlakota, S., & Balakrsha, N. (986). Oe ad two sded samplg plas based o the expoetal dstrbuto. Naval Research Logstcs Quarterly, 33(3), Scheder, H. (989). Falure-cesored varables-samplg plas for logormal ad Webull dstrbutos. Techometrcs, 3(), Seo, J. H., Jug, M., & Km, C. M. (9). Desg of accelerated lfe test samplg plas wth a ocostat shape parameter. Europea Joural of Operatoal Research, 97(), Tse, S. K., Dg, C., & Yag, C. (8). Optmal accelerated lfe tests uder terval cesorg wth radom removals: the case of Webull falure dstrbuto. Statstcs, 4(5), Tse, S. K., & Yag, C. (3). Relablty samplg plas for the Webull dstrbuto uder Type II progressve cesorg wth bomal removals. Joural of Appled Statstcs, 3(6), Wallace, W. E. (985). Preset practces ad future plas for MIL STD 78. Naval Research Logstcs Quarterly, 3(), Wu, S. J., Che, Y. J., & Chag, C. T. (7). Statstcal ferece based o progressvely cesored samples wth radom removals from the Burr type XII dstrbuto. Joural of Statstcal Computato ad Smulato, 77(),

13 Iteratoal Joural of Statstcs ad Probablty Vol. 7, No. ; 8 Wu, J. W., Hug, W. L., & Tsa, C. H. (3). Estmato of the parameters of the Gompertz dstrbuto uder the frst falure-cesored samplg pla. Statstcs, 37(6), Yue, H. K., & Tse, S. K. (996). Parameters estmato for Webull dstrbuted lfetmes uder progressve cesorg wth radom removals. Joural of Statstcal Computato ad Smulato, 55(-), Copyrghts Copyrght for ths artcle s retaed by the author(s), wth frst publcato rghts grated to the joural. Ths s a ope-access artcle dstrbuted uder the terms ad codtos of the Creatve Commos Attrbuto lcese ( 38