Tolerance Interval for Exponentiated Exponential Distribution Based on Grouped Data

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1 Itratoal Rfrd Joural of Egrg ad Scc (IRJES) ISSN (Ol) X, (Prt) Volum, Issu 10 (Octobr 013), PP Tolrac Itrval for Expotatd Expotal Dstrbuto Basd o Groupd Data C. S. Kaad 1, D. T. Shr 1 Prcpal, Aadba Raora Arts,Commrc ad Scc Collg, Vabhavwad, Sdhudurga , Ida. Had, Dpartmt of Statstcs, Shvaj vrsty, Kolhapur , Ida. Abstract:- ppr -xpctato ad -cott -lvl tolrac trval for xpotatd xpotal dstrbuto basd o groupd data ar cosdrd. Th adquacy of proposd tolrac trvals ar valuatd usg smulato study ad through ral lf data du to Lawlss (198). Kywords : Expotatd Expotal Dstrbuto, ppr xpctato ad cott lvl tolrac trval, Maxmum llhood stmator, groupd data. I. INTRODCTION Gupta t al. (1998) troducd th Expotatd Expotal Dstrbuto (EE) as a gralzato of th stadard xpotal dstrbuto. Gupta ad Kudu (001) Studd ths dstrbuto ad obsrvd that t ca b usd qut ffctvly aalyzg may lf tm data partcularly plac of two paramtr gamma ad Wbull dstrbutos. I thr srs of paprs o EE dstrbuto ( also amd as Gralzd xpotal dstrbuto), thy dscussd dffrt stmato procdurs, tstg of hypothss, costructo of cofdc trval, stmatg th strss ad strgth paramtr ad closss of gralzd xpotal wth Wbull, gamma ad logormal dstrbuto. Shr t al. (005) obtad -xpctato ad -cott -lvl tolrac trval for ths dstrbuto basd o ugroupd data. May tms a lf tstg problm, t s ot possbl to rcord xact tm of falur of a compots du to svral rasos. Hc t s mor coomcal to obsrv o of falurs of compots prdfd tm trvals whch form groupd data. Th ma am of ths papr s to costruct xpctato ad cott lvl tolrac trval for EE dstrbuto basd o groupd data. Lt b a statstc basd o data obsrvd from a dstrbuto wth dsty fucto f(x, θ ) whr θ rprsts a vctor of uow paramtrs th th trval (-, ) s a -xpctato tolrac trval (TI) f E f (x,θ)dx β ad trval (-, ) s uppr cott lvl TI f P f (x,θ)dx β γ C s calld th covrag of TI (-, ) f for vry θ (1.1) for gv, є (0,1). (1.) C f (x,θ)dx. O sdd TI s wh f(x,) s a two paramtr EE dstrbuto basd o maxmum llhood stmators for ugroupd data wr dscussd by Shr t al. (005). Ths papr xtds th procdur for sttg TI basd o groupd data. I scto w provd uppr -xpctato TI alog wth ts approxmat xpctd covrag. -cott -lvl TI basd o MLE s of ar dscussd scto 3. W study th prformac of both ths TI usg smulato tchqu scto 4. I scto 5 w llustrat th practcal applcatos of th procdur by applyg t to ral lf data st. II. -EXPECTATION TOLERANCE INTERVAL Gupta ad Kudu (001) dfd EE dstrbuto th followg way. Lt Y b a two paramtr EE radom varabl wth dstrbuto fucto 6 Pag

2 Tolrac Itrval for Expotatd Expotal Dstrbuto Basd o Groupd Data λ y ) F(y,λ,) (1,,y>0, (.1) Thrfor corrspodg probablty dsty fucto s gv by λy 1 λy f(y,λ,) λ(1 ).,y>0 (.) Suppos X j, j=1,,., b th umbr of obsrvatos th trval (t j-1, t j ] wth t 0 =0 ad t = such that x. Thus w hav groupd data from th udrlyg dstrbuto. 1 Th log llhood fucto for th groupd data ca b wrtt as L(λ,/ x) C x l F(t) F(t1), whr c s dpdt of ad ad F(.) s dfd (.1). 1 Thrfor to obta MLE of ad, w dffrtat L(,) partally w.r.t. ad ad quatg t to zro. Ths gvs L 11 x λt λt 1 l(1 ) x l(1 ) 0 λt 1 λt λt 1 1 λt 1 1 λt λt 1 λt 1 λt 1 1 L t x (1 ) t 1x (1 ) 0 λ λt λt 1 λt λt 1 11 (1 ) (1 ) 1 (1 ) (1 ) It s clar that o closd form soluto s possbl. W ca us Nwto-Raphso mthod to solv abov two quatos. Th th prctl of (.) s gv by X (θ )= -1 l(1-1/ ). Sc θ (λ,) s uow, w rplac t by ts MLE θ (λ, ) ad w propos approxmat uppr xpctato TI as 1 1/ ˆ I1(x) 0,λˆ l(1 β ). Expctd covrag of I 1 (X) s gv by followg thorm. Thorm 1: A xpctd covrag of I 1 (X) s gv by E[ F{X ( θ); θ}] β A(θ)σ B( θ)σ C( θ)σ, β 1 1 1G whr xλ t Gλ t A θ G t Gλ t G t 0.5G λλ t Gx t -1 Bθ Gt loggt 1 0.5logG t, 1G λ t C θ G t log G t Gt λt G(t) 1, G x t G(t)/x, G λ t G(t)/, t G xλ G(t)/x, (.3), G G xλ x t t. 7 Pag

3 t Tolrac Itrval for Expotatd Expotal Dstrbuto Basd o Groupd Data λ ad rspctvly ad 1 = G λλ G(t)/ whl 1, ar asymptotc varacs of Cov( λ, ) s a asymptotc covarac of λ,. Proof : Follows from th rsult of Atwood (1984). W omt th sam for brvty. Th prformac of I 1 (X) s studd usg smulato xprmts ad has b rportd th Scto 4. I th followg w obta -cott -lvl TI for th dstrbuto (.1). III. X -CONTENT -LEVEL TOLERANCE INTERVAL Lt I (X) = ( 0, β ) b a uppr -cott -lvl TI for th dstrbuto havg cdf (.1). Th factor > 0 s to b dtrmd such that I (X) s a -cott -lvl TI for (0, 1) ad (0, 1)... PFδ Xβ;λ, β γ Xβ l 1- β 1/ λδ. Ths mpls P γ Assumg cosstcy ad asymptotc ormalty of th MLE s asymptotcally ormal wth ma Xβ (θ) ad varac varac covarac matrx of θ (λ, ) ad σ (θ) xβ(θ) H, λ, whr (3.1) λ ad, w hav Xβ(θ) σ xβ(θ). (θ) s ' HH wth as a If Z s a stadard ormal varat th w ca wrt from (3.1) 1/ l(1 - β ) PZ xβ( θ) 1 γ, λδ σ(θ) 1/ l(1- β ) Suppos Z 1- γ xβ( θ), whr Z 1- s th 100(1-) th prctl of th λδ σ(θ) 1 λ Z 1 γ σ(θ) stadard ormal varat th w hav δ 1-. l(1 - β 1/ ) Not that dpds o both th paramtrs ad. Rplacg ad by thr rspctv MLEs a asymptotc uppr -cott -lvl TI I (X) s 1 λˆ Z 1 γ σ(θ) l1 β 0, 1 1/ ˆ λˆ l 1 β Th prformac of I (X) s studd usg smulato xprmts ad has b rportd th Scto 5. IV. SIMLATION STDY Th approxmat valu of th actual covrag of th proposd xpctato TI s gv by (.3). Th prformac of th sad TI s llustratd by smulato tchqu by rplacg ad by ts MLE s λˆ ad ˆ. I th smulato study of I 1 (X), w grat (=10, 5, 50, 75, 100) obsrvatos from EE dstrbuto wth =, = ad group to fv qual spacd trvals as 0-0.5, , , ad abov.0. MLE s of ad ar otad ad ar usd to comput uppr - xpctato TI of I 1 (X) for =.90,.95,.975 ad.99. Rpatg th abov procdur 10,000 tms ad stmat xpctd covrag rportd Tabl 4.1. Sam 1/ ˆ (3.) 8 Pag

4 Tolrac Itrval for Expotatd Expotal Dstrbuto Basd o Groupd Data procdur s rpatd by groupg obsrvatos to fv uqual spacd trvals as 0-0.4, , , ad abov.. Th xpctd covrag rportd Tabl 4.. W obsrv from Tabl 4.1 ad 4. that for small sampl sz TI udrstmats xpctd covrag whl as sampl sz crass covrag covrgs to th dsrd lvl. It vrfs th cosstcy proprty of th proposd trval. Tabl 4.1: Expctd covrag of - xpctato TI (qually spacd) Tabl 4.: Expctd covrag of - xpctato TI (uqually spacd) ppr -cott -lvl TI gv (3.) rqur asymptotc varac covarac matrx of th MLE s λˆ ad ˆ whch volv complcatd tgrals to b solvd umrcally. Ths problm s rsolvd by usg bootstrap tchqu. I th followg scto w llustrat a applcato of th procdur by applyg t to ral lf data st. V. APPLICATION Lawlss data st (198, pag 8) s bst fttd to EE dstrbuto tha two paramtr Wbull ad Gamma as rportd by Gupta ad Kudu (001).Th data ar rgardg aros tst o durac of dp groov ball bargs. Th data ar th class of umbr of mllo rsolutos bfor falur for 3 ball bargs th lf tst wth qually spacd trvals as Class trval : No of ball bargs : For qually spacd data MLE s of ad ar ad rspctvly ad uppr xpctato tolrac lmts (say 1 (X)) for varous s ar : (X) : If th abov data ar wth uqually spacd trvals as, Class trval : No of ball bargs : MLE s of ad ar ad rspctvly ad uppr xpctato tolrac lmts (say (X)) for varous s ar : (X) : sg bootstrap tchqu w grat 5000 radom sampls (wth rplacmt) ach of sz 3 from th orgal data ad group thm wth trvals qually spacd. For ach of th bootstrap sampl w comput MLE s of ad. Basd o such 5000 MLE s w obta varac covarac matrx of MLE s ad s usd to propos asymptotc uppr cott lvl tolrac lmt (TL) wth combatos of =.90,.95,.975,.99 ad =.90,.95 ad tabulatd Tabl 5.1. W also grat 5000 radom bootstrap sampls (wth rplacmt) ach of sz 3 from th orgal data ad group thm wth trvals uqually spacd. By rpatg th sam procdur as abov,w propos asymptotc uppr -cott -lvl tolrac lmt ad rportd Tabl 5.. Tabl 5.1: ppr -Cott -lvl TL (qually spacd) =.90 =.95 ( θ ) (X) (X) 9 Pag

5 Tolrac Itrval for Expotatd Expotal Dstrbuto Basd o Groupd Data Tabl 5.: ppr -Cott -lvl TL (uqually spacd) =.90 =.95 ( θ ) (X) (X) It s obsrvd that -cott -lvl TI for qually spacd trvals has a wdr lgth as agast th uqually spacd trvals. Th smulato study dcats that for small sampl sz TI udrstmats xpctd covrag whl as sampl sz crass covrag covrgs to dsrd lvl. Th prformac of both th proposd TI s ar satsfactory ad ca b usd practc. Th mthod dvlopd hr ca b xtdd for dstrbutos blogs to xpotatd scal famly of dstrbutos such as xpotatd Wbull, Expotatd gamma or Expotatd Raylgh dstrbuto. REFERENCES [1]. Atwood C. L.(1984) : Approxmat tolrac trvals basd o maxmum llhood stmats, J. Amr. Stat. Assoc. 79, []. Gupta R.C. ad Gupta P.L. ad Gupta R.D.(1998): Modlg falur tm data by Lhma altratvs, Commucatos Statstcs, Thory ad Mthods 7, [3]. Gupta R. D. ad Kudu D.(001): Expotatd Expotal Famly: A altratv to Gamma ad Wbull Dstrbutos, Bomtrcal Joural,43, 1, [4]. Lawlss J.F. (198) : Statstcal Modls ad Mthods for lf tm data, Joh Wly ad Sos, Nw Yor, pp. 8. [5]. Kumbhar R. R. ad Shr D. T. (005): Tolrac Itrvals for Expotatd Scal Famly of dstrbutos, Joural of Appld Statstcs, 3, 10, Pag