Estimation in Step-stress Partially Accelerated Life Test for Exponentiated Pareto Distribution under Progressive Censoring with Random Removal

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1 Journal of Advances n Matheatcs and Coputer Scence 5(): -6, 07; Artcle no.jamcs.3469 Prevously known as Brtsh Journal of Matheatcs & Coputer Scence ISSN: Estaton n Step-stress Partally Accelerated Lfe Test for Exponentated Pareto Dstrbuton under Progressve Censorng wth Rando Reoval Aal S. Hassan, Marwa Abd-Allah * and Hady G. A. El-Elaa Departent of Matheatcal Statstcs, Insttute of Statstcal Studes and Research, Caro Unversty Egypt. Authors contrbutons Ths work was carred out n collaboraton between all authors. Author ASH desgned the study, perfored the statstcal analyss, wrote the protocol and wrote the frst draft of the anuscrpt. Authors MAA and HGAEE anaged the analyses of the study. Author HGAEE anaged the lterature searches. All authors read and approved the fnal anuscrpt. Artcle Inforaton DOI: /JAMCS/07/3469 Edtor(s): () Jacek Dzok, Insttute of Matheatcs, Unversty of Rzeszow, Poland. () Metn Basarr, Departent of Matheatcs, Sakarya Unversty, Turkey. Revewers: () Suchandan Kayal, Natonal Insttute of Technology Rourkela, Inda. () Jong-Wuu Wu, Natonal Chay Unversty, Tawan. (3) Marwa Osan Mohaed, Zagazg Unversty, Egypt. Coplete Peer revew Hstory: Receved: 5 th June 07 Accepted: nd October 07 Revew Artcle Publshed: 6 th October 07 Abstract Accelerated lfe testng or partally accelerated lfe testng s generally used n anufacturng ndustres snce t affords sgnfcant nzaton n the cost and test te. In ths paper, a step-stress partally accelerated lfe test under progressve type-ii censorng wth rando reovals s consdered. The lfete of testng tes under use condton follow the exponentated Pareto dstrbuton and the reovals fro the test are assued to have bnoal dstrbuton. Maxu lkelhood estators for the odel paraeters and acceleraton factor are obtaned. Approxate confdence ntervals for the paraeters are fored va the noral approxaton to the asyptotc dstrbuton of axu lkelhood estators. Sulaton study s carred out to nvestgate the perforance of estators for dfferent paraeters values and saple sze. *Correspondng author: E-al: erostat336@gal.co, erostat@yahoo.co;

2 Hassan et al.; JAMCS, 5(): -6, 07; Artcle no.jamcs.3469 Keywords: Partally accelerated lfe test; axu lkelhood estaton; bnoal dstrbuton; progressvely type II censored. ACRONYMS AND NOTATIONS ALTs : Accelerated lfe tests CP : Coverage probablty EP : Exponentated Pareto MLE : Maxu lkelhood estate/estator MSE : Mean square error PALTs : Partally accelerated lfe tests SS-PALT : Step stress-partally accelerated lfe test RB : Relatve bas x : Observed lfete of unt tested under SS-PALT : Stress change te u, u : Indcator functons: u I ( x ), ( ) u I x, : Nuber of unts faled at noral use and accelerated condtons respectvely ( ) : Total nuber of unts faled under SS-PALT I(A) : State functon whch equals f the A s true and 0 f not. Introducton Recent developents n technologcal areas lead to hghly relablty products wth very long lfetes. When testng the lfe of these products under noral operatng condtons, the result gves no or very lttle falure by the end of the test. So, the accelerated lfe tests (ALTs) or partally accelerated lfe tests (PALTs) are preferred to be appled to obtan nforaton on the lfe of the products shortly and rapd. If all test tes are subjected to hgher than usual stress levels, then the test s called ALT. Whle, n PALTs tes are tested at both accelerated and use condtons. The nforaton data collected fro the test executed n the ALTs or PALTs are used to estate falure behavor of the tes at noral use condtons. Nelson [] ndcated that the stress can be appled n varous ways. The constant-stress PALTs and step-stress PALTs (SS-PALTs) are the coonly used ethods. In SS-PALTs, a test te s frst run at noral (use) condton and, f t does not fal for a specfed te, then t s run at accelerated condton untl the test ternates. In constant-stress PALTs each te s run at constant hgh stress untl ether falure occurs or the test s ternated. In any lfe tests and relablty studes, the experenter ay not always obtan coplete nforaton on falure tes for all experental unts. Data obtaned fro such experents are called censored data. The ost coon censorng schees are type-i censorng and type-ii censorng. These two censorng schees do not have the flexblty of allowng reoval of unts at ponts other than the ternal pont of the experent. Progressve censorng s a ore general censorng schee whch allows the unts to be reoved fro the test (Balakrshnan and Aggarwala []). Progressve type-ii censorng schee can be descrbed as follows. Consder an experent n whch ndependent and dentcally unts are placed on a lfe test and falure are gong to be observed. When the frst falure occurs, say at te t (), r of unts are randoly reoved fro the reanng survvng unts. The test contnues untl the th falure occurs at whch te, all the reanng survvng unts r n r r r... are all reoved fro the test. Note that, n ths schee, r r,,..., r are all prefxed. However, n soe practcal stuatons, these nubers ay occur at rando. For exaple, Yuen and Tse [3] ponted out that the nuber of patents that wthdraw fro a clncal test at each stage s rando and cannot be prefxed. Therefore, the statstcal nference on lfete dstrbutons under

3 Hassan et al.; JAMCS, 5(): -6, 07; Artcle no.jamcs.3469 progressve type II censorng wth rando reovals has been studed by varous authors, for nstance; Wu et al. [4], Yan et al. [5], Tse et al. [6], and Dey and Dey [7]. Specfcally, PALTs were studed under type-i and type-ii censorng schees by several authors; for exaple, see Goel [8] DeGroot and Goel [9], Bhattacharyya and Soejoet [0], Ba and Chung [], Abdel- Ghan [], Abd-Elfattah et al. [3], Aly and Isal [4], Hassan and Thobety [5]. Works for PALTs have been studed under progressve censorng, for nstance; Isal and Sarhan [6], Srvastava and Mttal [7], and Mohe EL-Dn et al. [8]. The Pareto dstrbuton s the ost popular odel for analyzng skewed data. The Pareto dstrbuton was orgnally proposed to odel the unequal dstrbuton of wealth snce t was observed the way that a larger porton of the wealth of any socety s owned by a saller percentage of the people. Ever snce, t plays an portant role n analyzng a wde range of real-world stuatons, not only n the feld of econocs. Exaples of approxately Pareto dstrbuted phenoena ay be found n szes of sand partcles and clusters of Bose-Ensten condensate close to absolute zero. Exponentated Pareto dstrbuton has been receved the greatest attenton fro theoretcal and appled statstcans prarly due to ts use n relablty and lfe testng studes snce t has decreasng and upsde-down bathtub shaped falure rates. It s consdered to be useful for odelng and analyzng the lfe te data n edcal and bologcal scences, engneerng, etc. A new two-paraeter dstrbuton, called the exponentated Pareto (EP) dstrbuton has been ntroduced by Gupta et al. [9]. The EP dstrbuton can be defned by rasng the dstrbuton functon of a Pareto dstrbuton to a postve power. The probablty densty functon of EP dstrbuton, wth two shape paraeters and, s gven by: ( ) The survval functon of the EP dstrbuton s as follows In ths paper, the progressvely type-ii censored saplng wth bnoal reovals data s used to obtan the pont and approxate confdence nterval estates of the EP paraeters n SS-PALTs. The paper can be arranged as follows. A descrpton of the odel, test procedure, and ts assuptons are presented n Secton. In Secton 3, the axu lkelhood estates of the odel paraeters are provded n Secton 3. Approxate confdence nterval estates of the odel paraeters are derved n Secton 4. Sulaton study s perfored to llustrate theoretcal results n Secton 5. Soe concludng rearks are suarzed at last secton. Model and Test Procedure f ( t ) t t,,, t 0 () The followng assuptons of SS-PALT are consdered: S ( t ) t. (). n dentcal and ndependent unts are put on the lfe under used condton and the lfete of each testng unt has the EP dstrbuton.. The test s ternated at the th falure, where s prefxed n. 3. Each of the n unts s frst run under noral use condton. If t does not fal or reove fro the test by a pre-specfed te, t s put under accelerated condton. r,,,..., are randoly 4. At the th falure a rando nuber of the survvng unts, selected and reoved fro the test. Fnally, at the th falure the reanng survvng unts R n R are all reoved fro the test and the test s ternated. 3

4 Hassan et al.; JAMCS, 5(): -6, 07; Artcle no.jamcs Suppose that an ndvdual unt beng reoved fro the test s ndependent of the others but wth the sae reoval probablty p. Then, the nuber of unts reoved at each falure te follows a bnoal dstrbuton. That s, R ~ bno ( n, p) and for,3,...,, R ~ bno( n r, p) and r nr r... r 6. The lfete, say X, of a unt under SS-PALT can be rewrtten as X T f T T f T (3) where, T s the lfete of an te at noral condton, and s the acceleraton factor. Thus, fro the assuptons, the probablty densty functon (3) of a total lfete of test te takes the followng for 3 Paraeters Estaton In ths secton, the axu lkelhood estators of the odel paraeters;, and are obtaned n step-stress PALT based on the progressvely type II censored data wth bnoal reoval. We assue that the nuber of unts reoved fro the test at each falure te follows a bnoal dstrbuton. x r u u,,..., Let(,,, ),, be the observed values of lfete X obtaned fro a progressvely type-ii censored saple under a step-stress PALT. Hence, for the progressve censorng wth the pre deterned nuber of the reovals R ( R r, R r,... R ), r the condtonal lkelhood functon of the observatons x {( x, r, u, u ),,,..., } can be defned as follows where, ( ) f ( x ) x x, 0 x f ( x ) (4) ( ) f ( x ) ( x ( x ), x Insertng the probablty densty functons functons n (5), then we have r u r u f ( x), f ( x ) defned n (3) and ther correspondng survval L ( x,,,, u, u R r) f ( x ) S ( x ) f ( x ) S ( x ), (5) S ( x ) x, S ( x ) ( x ). 4

5 Hassan et al.; JAMCS, 5(): -6, 07; Artcle no.jamcs.3469 r ( ) L ( x,,,, u, u R r) x x x u ( ) r ( x ( x ) ( x ).(6) The nuber of unts reoved at each falure te assued to follow a bnoal dstrbuton wth the followng probablty ass functon u n P R r p p r n r ( ) ( ). r Whle, for,3,..., P R r R r R r p p r n r j (,..., ) n r j r ( ). Moreover, suppose that follows Where That s, R s ndependent of X for all. Hence the lkelhood functon can be expressed as L( x,,,, p) L ( x,,,, u, u R r) P( R r), (7) P( R r) P ( R r, R r,..., R r ) P( R r R r,..., R r ) P( R r R r,..., R r )... P( R r R r ) P( R r ). 3 3 ( n )! P R r p p ( n r )! r r ( )( n ) ( ) r ( ) ( ), The natural logarth of the condtonal lkelhood functon, denoted by, ln L, can be obtaned fro (6) as follows D H r H H lnl ln ln ln ( ) ln( D ) r ln[ ( D ) ] ( ) ln ( ) ln( ) ln( ( ) ) ( ) ln, 5

6 Hassan et al.; JAMCS, 5(): -6, 07; Artcle no.jamcs.3469 where, D and ( ) x H x u u Snce P ( R r) does not nvolve the paraeters;, and, so the axu lkelhood estators, ˆ, ˆ, and ˆ ln L drectly. Hence, the axu lkelhood estates, can be obtaned by axzng (MLEs) of, and can be obtaned by solvng the followng non-lnear equatons ln( ) ln( ), [( ) ] (( ) ) ln L r ln( D ) r ln( H ) D H D H ln L ln D r ( D ) D ln D ( ) ln D ( D ) [ ( D ) ] ln H r ( H ) H ln H H ( H ) ( ( H ) ) ( ) ln, ln L H ( x ) r ( H ) H ( x ) ( ) ( H ) ( H ) ( x ) ( ). H The axu lkelhood estators of the odel paraeters are deterned by settng ln L ln L ln L,, to be zero. These equatons cannot be solved analytcally and statstcal software can be used to solve the nuercally va teratve technque. Slarly, snce L( x,,,, u, u R r ) does not nvolve the bnoal paraeter p, the MLE of p can be derved by axzng (6) drectly. Hence the MLE of p s obtaned by solvng the followng equaton r ( )( n ) ( ) r ln L p p p. Hence, ˆ. p ( )( n ) ( ) r r 6

7 Hassan et al.; JAMCS, 5(): -6, 07; Artcle no.jamcs Approxate Confdence Interval The ost coon ethod to set confdence bounds for the paraeters s to use the asyptotc noral dstrbuton of the MLEs (see VanderWel and Meeker [0]). The asyptotc varances and covarance atrx of the MLE of the paraeters can be approxated by nuercally nvertng the asyptotc Fshernforaton atrx F. It s coposed of the negatve second and xed dervatves of the natural logarth of the lkelhood functon evaluated at the MLE. The asyptotc Fsher nforaton atrx F can be wrtten as follows ln L ln L ln L f f f 3 ln L ln L ln L F f ˆ ˆ ˆ f f 3 (,, ) f 3 f 3 f 33 ln L ln L ln L The second and xed partal dervatves of the natural logarth of lkelhood functon wth respect to, and are obtaned as follows ln( ) ( ) L ln( ) ( ) r D D r H H [( D ) ] [( H ) ] ln, ln L ln D r ( D ) D ln D ( ) ln D ( D ) [ ( D ) ] ln H r ( H ) H ln H H ( H ) ( ( H ) ) ( ) ln, lnl lnd lnh r lnd r ln( D )( D ) D lnd ( D ) ( H ) ( D )[( D ) ] [( D ) ] r ln ln( )( ) H r H H H lnh, ( H )[( H ) ] [( H ) ] ln L H ( x ) r H ( x ) r ( H ) ln( H ) H ( H ) (( H ) )( H ) (( H ) ), ln L ( x ) H ln H ( x ) ( x ) ( )[( ) ( ) ln ( ) ln ( ) ] ( ) ( H ) H ( H ) H r x H H H H H H H H ( ( H ) ) ( ) r ( H ) H ln H ( x ) ( ( H ) ) 7

8 Hassan et al.; JAMCS, 5(): -6, 07; Artcle no.jamcs.3469 ln L ( H )[ ( ) H ( x ) ] H ( x ) ( ) ( ) ( H ) ( ) r ( x ) [( ) ( H ) H ( )( H ) H ] [ ( H ) ] r ( ) ( ) ( ) H H ( x ) ( x ) ( ). [ ( H ) ] ( H ) ln L (ln D ) D r(ln D ) [( )( D ). D ( D ). D ] ( ) ( D ) [ ( D ) ] ( ) r (ln D ) D ( D ) r (ln H ) [( )( H ). H ( H ). H [ ( D ) ] [ ( H ) ] ( ) r (ln ) ( ) ln H H H H H ( ), [ ( H ) ] ( H ) In relaton to the asyptotc varance-covarance atrx of the axu lkelhood estators of the paraeters, t can be approxated by nuercally nvertng the above Fsher's nforaton atrx F. Thus, the approxate 00( )% and two sded confdence ntervals for, and can be, respectvely, easly obtaned by ˆ z, ˆ z, ˆ z ˆ ˆ ˆ,, where, z 00( ) s the axu lkelhood estates. 5 Sulaton Study th standard noral percentle and (.) s the standard devaton for the In ths secton, a sulaton study s perfored to exane the perforance of the paraeter estates and the nfluence of the avalable nforaton fro the progressvely censored data on these estates. The perforance of estators has been consdered n ters of ther relatve bas (RB), ean square error (MSE) and coverage probablty (CP). For dfferent choces of,, p and R, =,,..., ; the results are concluded n Tables, and 3. A sulaton study s perfored accordng to the followng steps:. The value of n and s specfed.. Three selected set of paraeters are consdered as; set I (.5,., 0.5) set II (.5, 3,.5) and set III (.5, 3, 0.5). Three levels of p are selected as 0.4, 0.6 and 0.8. Assung that the value of equals 7 throughout all experents. 8

9 Hassan et al.; JAMCS, 5(): -6, 07; Artcle no.jamcs Generate a rando saple wth sze n and censorng sze wth rando reovals R, =,,...,, fro the rando varable X gven by (4). 4. Generate a group value rando nuberr ~ bno ( n r, p) and r n r r.... r 5. The MLEs of the odel paraeters are coputed for saple szes n= 50, 75, 00, and Copute the values of relatve bases, ean square errors and the coverage rate of the 95% confdence nterval of MLE of the paraeters nuercally for each saple sze. 7. The above steps are repeated 000 tes for dfferent values of n,, 8. The sulaton results are lsted n Tables to 3 and represented through soe fgures.. Fro the results shown n Tables - 3, one can observe that the relatve bases and the MSEs of the paraeter estates decrease as the saple sze ncreases.. For set I, the MSEs of ˆ have the sallest values at p=0.8 for dfferent and n, whle the MSEs of ˆ and ˆ have the t sallest values at p=0.6 for dfferent and n (see for exaple Fgs. and ).. For set III, the MSEs of ˆ have the sallest values at p=0.6 for dfferent and n, whle the MSE of ˆ has the sallest values at p=0.4 for dfferent and n (see for exaple Fgs. 3 and 4). and ˆ ˆ and ˆ v. For set II, the MSEs of ˆ, have the sallest values at p=0.4 for dfferent and n, (see for exaple Fg. 5). v. As seen fro the Tables -3, the coverage probabltes are close to the nonal confdence level 0.95% for dfferent n, and selected set of paraeters. j j R =,,...,, and p p=0.4 p=0.6 p=0.8 =0 =50 =70 =90 =0 =50 =00 =40 n=00 n=50 Fg.. MSE of ˆ for set I for dfferent values of p at saple szes n= =00 and 50 Fg.. MSE of ˆ for set I for dfferent values of p at saple szes n=000 and 50 9

10 Hassan et al.; JAMCS, 5(): -6, 07; Artcle no.jamcs.3469 Table. Relatve bas, ean square error and coverage probablty for set I (.5,., 0.5) P=0.4 P=0.6 P=0.8 RB MSE CP RB MSE CP RB MSE CP n=50 =0 α β θ n=50 =30 α β θ n=50 =40 α β θ n=75 =0 α β θ n=75 =50 α β θ n=75 =70 α β θ n=00 =0 α β θ n=00 =50 α β θ n=00 =70 α β θ n=00 =90 α β θ

11 Hassan et al.; JAMCS, 5(): -6, 07; Artcle no.jamcs.3469 P=0.4 P=0.6 P=0.8 RB MSE CP RB MSE CP RB MSE CP n=50 =0 α β θ n=50 =50 α β θ n=50 =00 α β θ n=50 =40 α β θ Table. Relatve bas, ean square error and coverage probablty for set II (.5, 3,.5) P=0.4 P=0.6 P=0.8 RB MSE CP RB MSE CP RB MSE CP n=50 =0 α β θ n=50 =30 α β θ n=50 =40 α β θ n=75 =0 α β θ n=75 =50 α β θ

12 Hassan et al.; JAMCS, 5(): -6, 07; Artcle no.jamcs.3469 P=0.4 P=0.6 P=0.8 RB MSE CP RB MSE CP RB MSE CP n=75 =70 α β θ n=00 =0 α β θ n=00 =50 α β θ n=00 =70 α β θ n=00 =90 α β θ n=50 =0 α β θ n=50 =50 α β θ n=50 =00 α β θ n=50 =40 α β θ

13 Hassan et al.; JAMCS, 5(): -6, 07; Artcle no.jamcs.3469 Table 3. Relatve bas, ean square error and coverage probablty for set III (.5, 3, 0.5) P=0.4 P=0.6 P=0.8 RB MSE CP RB MSE CP RB MSE CP n=50 =0 α β θ n=50 =30 α β θ n=50 =40 α β θ n=75 =0 α β θ n=75 =50 α β θ n=75 =70 α β θ n=00 =0 α β θ n=00 =50 α β θ n=00 =70 α β θ n=00 =90 α β θ

14 Hassan et al.; JAMCS, 5(): -6, 07; Artcle no.jamcs.3469 P=0.4 P=0.6 P=0.8 RB MSE CP RB MSE CP RB MSE CP n=50 =0 α β θ n=50 =50 α β θ n=50 =00 α β θ n=50 =40 α β θ MSE(β) p= p= p=0.8 0 MSE(θ) p=0.4 p=0.6 p=0.8 =0 =40 =50 =0 =70 =0 =00 =0 =40 =50 =0 =70 =0 =00 n=50 n=75 n=00 n=50 Fg. 3. MSE of ˆ for set III for dfferent values of p for dfferent saple szes n=50 n=75 n=00 n=50 Fg. 4. MSE of ˆ for set III for dfferent values of p for dfferent saple szes 4

15 Hassan et al.; JAMCS, 5(): -6, 07; Artcle no.jamcs =0 =50 =70 =90 =0 =50 =00 =40 p=0.4 p=0.6 p=0.8 n=00 n=50 Fg. 5. MSE of ˆ for set II for dfferent values of p at saple szes n=00 and 50 6 Concluson In ths paper, a SS-PALT s presented under the progressve type II censored data wth bnoal reovals. The lfe tes of the testng tes are assued to follow exponentated Pareto dstrbuton. The MLEs of the consdered paraeters are obtaned and studed ther perforance through ther relatve bases and MSEs. Also, approxate confdence ntervals for the paraeters are constructed. Generally, based on the above analyss, one can say that a large saple sze wth a large censorng saple gves a better estate n the sense of havng saller relatve bas and MSE. Also the values of bnoal paraeter p have nfluence on the accuracy of the paraeter estates. Furtherore, the coverage probabltes are close to the nonal confdence level 0.95% for dfferent censorng schees, saples szes and set of paraeters. Copetng Interests Authors have declared that no copetng nterests exst. References [] Nelson W. Accelerated lfe testng: Statstcal odels, data analyss and test plans. John Wley and Sons, New York; 990. [] Balakrshnan N, Aggarwala R. Progressve censorng: Theory, ethods, and applcatons. Brkhauser, Boston; 000. [3] Yuen HK, Tse SK. Paraeters estaton for Webull dstrbuted lfete under progressve censorng wth rando reovals. Journall of Statstcal Coputaton and Sulaton. 996;55:57-7. [4] Wu SJ, Chen YJ, Chang CT. Statstcal nference based on progressvely censored saples wth rando reovals fro the Burr type XII dstrbuton. Journal of Statstcal Coputaton & Sulaton. 007;77:9-7. [5] Yan W, Sh Y, Song B, Zhaoyong H. Statstcal analyss of generalzed exponental dstrbuton under progressve censorng wth bnoal reovals. Journal of Syste Engneerng and Electroncs. 0; (4):

16 Hassan et al.; JAMCS, 5(): -6, 07; Artcle no.jamcs.3469 [6] Tse SK, Yang C, Yuen HK. Statstcal analyss of Webull dstrbuted lfete data under type-ii progressve censorng wth bnoal reovals. Journal of Appled Statstcs. 000;7: [7] Dey S, Dey T. Statstcal Inference for the Raylegh dstrbuton under progressvely Type-II censorng wth bnoal reoval. Appled Matheatcal Modellng. 04;38(3): [8] Goel PK. Soe estaton probles n the study of tapered rando varables, Techncal report no. 50, Departent of statstcs, Carnege-Mellon Unversty, Pttspurgh, Pennsylvana; 97. [9] DeGroot MH and Goel Pk. Bayesan and optal desgn n partally accelerated lfe testng. Naval Research Logstcs Quarterly. 979;6():3-35. [0] Bhattacharyya GK, Soejoet Z. A tapered falure rate odel for steps tress accelerated lfe test. Councaton n Statstcs -Theory & Methods. 989;8(5): [] Ba S, Chung SW. Optal desgn of partally accelerated lfe tests for the exponental dstrbuton under type-i censorng. IEEE Transactons on Relablty. 99;4(3): [] Abdel-Ghan MM. The estaton proble of the log-logstc paraeters n step partally accelerated lfe tests usng type-i censored data, The Natonal Revew of Socal Scences. 004;4():-9. [3] Abd-Elfattah AM, Hassan AS and Nassr SG. Estaton n step-stress partally accelerated lfe tests for the Burr Type XII dstrbuton usng type I censorng. Statstcal Methodology. 008;5(6): [4] Aly HM, Isal AA. Optu sple te-step stress plans for partally accelerated lfe testng wth censorng. Far East Journal of Theoretcal Statstcs. 008;4(): [5] Hassan AS, Thobety AK. Optal desgn of falure step stress partally accelerated lfe tests wth Type II censored nverted Webull data. Internatonal Journal of Engneerng Research and Applcatons. 0;(3): [6] Isal A, Sarhan AM. Optal desgn of step-stress lfe test wth progressvely type-ii censored exponental data. Internatonal Matheatcal Foru. 009;4(40): [7] Srvastava PW, Mttal N. Optu step- stress partally accelerated lfe tests for truncated logstc dstrbuton wth censorng. Appled Matheatcal Modellng. 00;34: [8] Mohe EL-Dn, MM, Abu-Youssef SE, Al NSA, Abd El-Rahee AM. Estaton n step-stress accelerated lfe test for power generalzed Webull wth progressve censorng. Advances n Statstcs. 05;-3. Avalable: [9] Gupta RC, Gupta RD, Gupta PL. Modelng falure te data by Lehan alternatves. Councaton of Statstcs Theory & Method. 998;7(4): [0] VanderWel SA, Meeker WQ. Accuracy of approxate confdence bounds usng censored Webull regresson data fro accelerated lfe test. IEEE Transactons on Relablty. 990;39(3): Hassan et al.; Ths s an Open Access artcle dstrbuted under the ters of the Creatve Coons Attrbuton Lcense ( whch perts unrestrcted use, dstrbuton, and reproducton n any edu, provded the orgnal work s properly cted. Peer-revew hstory: The peer revew hstory for ths paper can be accessed here (Please copy paste the total lnk n your browser address bar) 6