Estimation of R= P [Y < X] for Two-parameter Burr Type XII Distribution

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1 World Acade of Scece, Egeerg ad Techolog Iteratoal Joural of Matheatcal ad Coputatoal Sceces Vol:4, No:, Estato of R P [Y < X] for Two-paraeter Burr Tpe XII Dstruto H.Paah, S.Asad Iteratoal Scece Ide, Matheatcal ad Coputatoal Sceces Vol:4, No:, waset.org/pulcato/5474 Astract I ths artcle, we cosder the estato of P[Y < X], whe stregth, X ad stress, Y are two depedet varales of Burr Tpe XII dstruto. The of the R ased o oe sple teratve procedure s otaed. Assug that the coo paraeter s kow, the au lkelhood estator, uforl u varace uased estator ad Baes estator of P[Y < X] are dscussed. The eact cofdece terval of the R s also otaed. Mote Carlo sulatos are perfored to copare the dfferet proposed ethods. Kewords Stress-Stregth odel; Mau lkelhood estator; Baes estator; Burr tpe XII dstruto. I. INTRODUCTION URR troduced twelve dfferet fors of cuulatve Bdstruto fuctos for odelg lfete data or survval data []. Out of those twelve dstrutos, Burr Tpe XII ad Burr Tpe X have receved the au atteto. Several authors cosdered dfferet aspects of these two dstrutos, see for eaple, []- [7]. The Burr Tpe XII has the followg dstruto fucto for X > : p ( ( ( ; p, ; for p >, > Therefore, the Burr Tpe XII has the dest fucto for > as: p f( ; p, p ; for p >, > ( ( I stress-stregth odel, the stress (Y ad the stregth (X are treated as rado varales ad the relalt of a copoet durg a gve perod s take to e the proalt that ts stregth eceeds the stress durg the etre terval. Due to the practcal pot of vew of relalt stress-stregth odel, the estato prole of RP(Y<X has attracted the atteto of a authors. Ahad et al. [8] ad Surles & Padgett [9] cosdered the estato of P[Y < X], where X ad Y are Burr Tpe X rado varales, Ad-Elfattah & Madouh [] cosdered the estato of P[Y < X], whe X ad Y are depedet Loa rado varales wth kow scale paraeter ad Recetl Kudu ad Gupta [] have cosdered estato of P(Y<X, whe X ad Y are depedet geeralzed epoetal Dstruto. H.Paah s facult eer of statstcs departet of Azad Uverst Laha Brach, Laha, Ira. (e-al: h.paah@ahoo.co. S.Asad s assocate professor of egeerg college of Paae Noor Uverst, Tehra, Ira. ( correspodg author to provde phoe: ; e-al: asad@kstp.r, asadfo@ahoo.co. I the preset artcle, the ferece of RP(Y<X, s studed whe X ad Y are two depedet ut ot detcall rado varales elogg to as urr tpe XII dstruto wth two paraeters. I Secto (II, the pot estato of relalt R s otaed usg au lkelhood ethod. Also we dscuss dfferet estato procedures of R f s kow Secto (III. Mote Carlo sulato results are preseted Secto (IV ad fall we draw coclusos Secto (V. II. MAXIMUM LIKELIHOOD ESTIMATOR O R Let X ad Y are two depedet Burr Tpe XII rado varales wth paraeters p, ad, respectvel. Therefore R P Y ( < X f ( f ( dd p ( ( ( p ( dd ( p Now to copute the of R, frst we ota the, s of p ad. Let X,..., X e a rado saple fro BurrXII ( p, ad Y,...,Y e a rado saple fro BurrXII (,. Therefore the log-lkelhood fucto L of p, ad for the oserved saple s L( p,, l p l ( l (3 ( l l ( p l( ( l( Dfferetatg partall wth respect to p, ad, settg the results eual to zero we get three olear euatos. L p p L ( l, (4 ( l, (5 Iteratoal Scholarl ad Scetfc Research & Iovato 4( 475 scholar.waset.org/37-689/5474

2 World Acade of Scece, Egeerg ad Techolog Iteratoal Joural of Matheatcal ad Coputatoal Sceces Vol:4, No:, Iteratoal Scece Ide, Matheatcal ad Coputatoal Sceces Vol:4, No:, waset.org/pulcato/5474 L l l ( p ( l l ro (4, (5 ad (6, we ota pˆ (7 l ( ˆ (8 ( l ad ˆ ca e otaed as the soluto of the o-lear euato h ( ( l( l l l l l l (6 l Coseuetl, ˆ ca e otaed solvg the olear euato ( u ( Where u ( ( ( ( l( l l l l l l (9 l Sce ˆ s a fed pot soluto of the o-lear euato (, therefore, t ca e otaed usg a sple teratve schee as follows: u ( ( ( ( Where ( s the - th terate of ˆ.Oce we ota ˆ, pˆ ad ˆ ca e otaed fro (7 ad (8 respectvel. Therefore, the of R ecoes ˆ R ( pˆ ˆ III. ESTIMATION O R I IS KNOWN I ths secto, we cosder the estato of R whe s kow. Wthout loss of geeralt, we ca assue that. Therefore, ths secto t s assued that X,...,X s a rado saple fro BurrXII ( p, ad Y,...,Y s a rado saple fro BurrXII (, ad ased o the saples we wat to estate R. rst, we cosder the of R ad ts dstrutoal propertes. A. of R Based o the aove saples, t s clear that, the of R ael Rˆ s gve ˆ Rˆ ˆ pˆ (3 Where pˆ (4 l ˆ (5 l Therefore, ( l( Rˆ (6 l( l( We cosdered u p V Therefore, Rˆ or d U U cv R Rˆ R Rˆ l( l( d d ~ χ ~ χ p (7 Iteratoal Scholarl ad Scetfc Research & Iovato 4( 476 scholar.waset.org/37-689/5474

3 World Acade of Scece, Egeerg ad Techolog Iteratoal Joural of Matheatcal ad Coputatoal Sceces Vol:4, No:, Iteratoal Scece Ide, Matheatcal ad Coputatoal Sceces Vol:4, No:, waset.org/pulcato/5474 p Here d dcates euvalet dstruto ad c. The rado varales U ad V are depedet ad follow χ dstruto, wth ad degrees of freedo respectvel. Moreover, has a dstruto wth ad degrees of freedo. Therefore, the ( % cofdece terval of R ca e otaed as: Rˆ Where, ad th : ˆ,:, R,,,,, (8 are the lower ad upper percetle pots of a dstruto wth ad degrees of freedo. B. of R I ths susecto we ota the of R.Whe the coo paraeter s kow, l (, l( s a otl suffcet statstc for (p,. Therefore usg the results of Tog [],[3] t follows that ~ ( ( (!! T R f T T ( (!! T (9 or ~ R ( ( (!! T f T T ( (!! T Where T l ( ad T l ( C. Baes Estato of R ( I ths susecto, we ota the Baes estato of R uder the assuptos that the paraeters p ad are rado varales for oth the populatos. It s assued that p ad have depedet gaa prors wth the PD's: β p π ( p p e β p > ( β ( e β > π ( respectvel. Here, β,, β >. Therefore, p ad follow Gaa (, β ad Gaa (, β respectvel. The posteror PD's of p ad are as follows: π ( p ~ Γ, β l( (3 ( ( π ~ Γ, β l (4 Assue that p ad are depedet, usg (3 ad (4, the ot posteror dest of p ad gve the data as ( T ( T π ( p,, (5 p ep pt T { } β l( ad T l( T β Applg the trasforatos techue of rado varales, let r p ad u p < r <, u > The ( T ( T π ( u, r, u ( r r ep { u [( r T rt ]} tegrate out u ( T ( T π ( r, r ( r [( r T rt ] Usg euato (6, Baes estator of R, sa Rˆ suared error loss fucto s Rˆ E ( R, r π ( r, dr r ( r [( r T rt ] (6, uder ˆ R (7 ( T ( T dr The coputato of the Rˆ s coplcated as t ca see fro euato (7.so, we wll use the MATHCAD progra to evaluate the value of Rˆ. Iteratoal Scholarl ad Scetfc Research & Iovato 4( 477 scholar.waset.org/37-689/5474

4 World Acade of Scece, Egeerg ad Techolog Iteratoal Joural of Matheatcal ad Coputatoal Sceces Vol:4, No:, Iteratoal Scece Ide, Matheatcal ad Coputatoal Sceces Vol:4, No:, waset.org/pulcato/5474 IV. SIMULATION STUDY I ths secto we preset results of soe uercal eperets to copare the perforace of the dfferet estators proposed the secto (III. We perfor etesve Mote Carlo sulatos to copare the perforace of the dfferet estators, al wth respect to ther ases ad ea suared errors. we cosder the case whe the coo paraeter s kow. I ths case we cosder the followg sall saple sze; (, (,,(,,(,3,(,,(,,(,3, (3,, (3,, (3, 3 ad we take p ad 5, 8 respectvel. Wthout loss of geeralt, we take. All the results are ased o replcatos. We ota the estates of R usg the ad. We also copute the Baes estate of R as suggested susecto C wth the followg cofguratos of (,(3,3 ad., 5,, β 5,, 5, β 5,, 5. We report the average estates ad average MSE, s of the, s ad, s ased o replcatos Tale ad Baes estator ased o replcatos s reported Tale. ro tale we ca ote that the ea suare error decreasg creasg saple sze wth saple sze s costat ad the ea suare error creasg decreasg saple sze wth saple sze s costat ad also t decreasg creasg the oth of the. The chages ea suare error of Rˆ due to chage p ad ca e gored ad t s oserved Tale the ea suare error of Rˆ creasg creasg that s costat. V. CONCLUSION I ths paper we copare dfferet ethods of estatg R P(Y < X whe Y ad X oth follow Burr Tpe XII dstruto wth paraeters (p, ad (,, respectvel. Whe the paraeter s ukow, t s oserved that the, s of the three ukow paraeters ca e otaed solvg oe o-lear euato. We cosder oe sple teratve procedure to copute the, s of the ukow paraeters ad tur to copute the of R. Whe the paraeter s kow, we ota au lkelhood estator ad uforl u varace uased estator. We also ota Baes estator uder suared error loss fucto. It s oserved that the ad are ute slar ature, although ased o ea suared errors, the perforace of the, s are argall etter tha the rest. [3] I.G. Surles, ad W.J. Padgett, Iferece for relalt ad stressstregth for a scaled Burr Tpe X dstruto, Lfete Data Aalss vol. 7, [] 87-,. [4] M.Z. Raa, ad D. Kudu, Coparso of Dfferet Estators of P[Y < X] for a Scaled Burr Tpe X Dstruto, Coucato Statstcs-Coputatos ad Sulatos vol. 34(, , 5. [5] Q. Shao, Notes o au lkelhood estato for the threeparaeter Burr XII dstruto, Coputatoal Statstcs & Data Aalss vol. 45, , 4. [6] D. Moore, ad A.S. Papadopoulos, The Burr tpe XII dstruto as a falure odel uder varous loss fuctos, Mcroelectrocs Relalt 4, 7-,. [7] R.N. Rodrguez, gude to Burr Tpe XII dstrutos, Boetrka vol. 64,9-34, 977. [8] K.E. Ahad, M.E. akhr, ad Z.. Jahee, Eprcal Baes estato of P(Y < X ad characterzato of Burr-tpe X odel, Joural of Statstcal Plag ad Iferece vol. 64, 97-38, 997. [9] J.G. Surles, ad W.J. Padgett, Soe propertes of a scaled Burr tpe X dstruto, Joural of Statstcal Plag ad Iferece vol. 8, Issue, 7-8, 5. [] A.M. Ad-Elfattah, ad R.M. Madouh, Estato of Pr{Y<X} Loa case, The 39th Aual Coferece o Statstcs, Coputer Scece ad Operato Research, ISSR, Caro Uverst, Egpt, part, 56-66, 4. [] D. Kudu, ad R.D. Gupta, Estato of P(Y<X for Geeralzed Epoetal Dstrutos, Metrka 6(3, 9-38, 5. [] H. Tog, A ote o the estato of P(Y < X the epoetal case, Techoetrcs vol. 6, 65, 974. (Errara, vol. 7, 395, 975. [3] H. Tog, O the estato of P(Y < X for epoetal fales, IEE Trasactos o Relalt vol. 6, 54-56, 977. REERENCES [] I.W. Burr, Cuulatve freuec dstruto, Aals of Matheatcal Statstcs vol. 3, 5-3, 94. [] H. Paah, ad S. Asad, Burr Tpe XII Dstruto: Dfferet Method of Estatos, The Teth Islac Coutres Coferece o Statstcal Sceces The Aerca Uverst, Egpt, 9. Iteratoal Scholarl ad Scetfc Research & Iovato 4( 478 scholar.waset.org/37-689/5474

5 World Acade of Scece, Egeerg ad Techolog Iteratoal Joural of Matheatcal ad Coputatoal Sceces Vol:4, No:, (, TABLE I BIASES AND MEAN SQUARED ERRORS O THE S AND S 5 8 Methods (, -.75( (.64.7( (.398 (, -.( ( (.8 -.7(.8 Iteratoal Scece Ide, Matheatcal ad Coputatoal Sceces Vol:4, No:, waset.org/pulcato/5474 (,3 (, (, (,3 (3, (3, (3, ( (.68.5(.9.4( (.5 -.3(.8 -.5(.95.8( (.7.46(.7.98(..9(.3.4(.94.(.93 I each cell the frst, secod rows represet the average ases ad ea suared errors of the -.55( ( (.8.4(.8 -.8( ( (..6(.8.53( (.93.7(.3.48(.5 -.3( (., s,, s. Iteratoal Scholarl ad Scetfc Research & Iovato 4( 479 scholar.waset.org/37-689/5474

6 World Acade of Scece, Egeerg ad Techolog Iteratoal Joural of Matheatcal ad Coputatoal Sceces Vol:4, No:, Iteratoal Scece Ide, Matheatcal ad Coputatoal Sceces Vol:4, No:, waset.org/pulcato/5474 (, (3,3 TABLE II BIASES AND MSES O THE BAYES ESTIMATORS O R β β Rˆ MSE E E E E E E E E E E E E E E E E E E-5 Iteratoal Scholarl ad Scetfc Research & Iovato 4( 48 scholar.waset.org/37-689/5474

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