Bayesian Test for Lifetime Performance Index of Exponential Distribution under Symmetric Entropy Loss Function

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1 Mathmatics ttrs 08; 4(): doi: 0.648/j.ml ISSN: X (Prit); ISSN: (Oli) aysia Tst for iftim Prformac Idx of Expotial Distributio udr Symmtric Etropy oss Fuctio Guobig Fa School of Mathmatics ad Statistics, Hua Uivrsity of Fiac ad Ecoomics, Chagsha, Chia addrss: To cit this articl: Guobig Fa. aysia Tst for iftim Prformac Idx of Expotial Distributio udr Symmtric Etropy oss Fuctio. Mathmatics ttrs. Vol. 4, No., 08, pp doi: 0.648/j.ml Rcivd: March, 08; Accptd: March 7, 08; Publishd: May 4, 08 Abstract: Th task of this papr is to stimat th liftim prformac idx of Expotial distributio. A aysia tst procdur is stablishd udr symmtric tropy loss fuctio. Firstly, aysia stimatio of lif prformac idx is obtaid, th a aysia tst procdur for liftim prformac idx is proposd. Fially, a applid xampl is usd to illustrat th ffctivss of th proposd tst mthod. Kywords: iftim Prformac Idx, aysia Tst, Expotial Distributio, Symmtric Etropy oss Fuctio. Itroductio Procss capability idics (PCIs) hav b itroducd for masurig procss rproductio capability of a maufacturig idustry []. Thy ar importat i th maufacturig idustry to masur procss pottial ad prformac []. Thr ar may wll-kow PCIs hav b put forward. For xampl, th first procss capability idx CP is first itroducd by Jura i rfrc [3]. Th othr PCIs: C pk, C pm, C pmk ar proposd rspctivly [4-6]. Th study ad applicatios of PCIs hav draw grat atttio. For xampl, Abdolshah t al. [7] studid th applicatio of Cpmk i th fuzzy cotxt ad compar it with othr fuzzy PCIs. Kuriati [8] proposd a variabls rsubmittd samplig schm for o-sidd spcificatio limit basd o o-sidd PCIs. Sifi ad Nzhad [9] dvlopd a variabl samplig pla for rsubmittd lots basd o PCI ad aysia approach. To assss th products with th largr-th-bttr typ charactristics, Motgomry [0] proposd a spcial uilatral spcificatio PCI, amd as liftim prformac idx C, which is dfid as follows C µ = () σ whr is th lowr boud of th spcificatios. Th statistical ifrc of liftim prformac idx for products whos liftim distributd diffrt distributios hav b widly studid i rct yars. For xampl, t al. [] basd o typ II right csord sampl discussd th computatioal procdur of prformac assssmt of liftim idx of ormal products i th fuzzy cotxt. iu ad R [] discussd th aysia poit stimatio ad tst of liftim prformac idx wh th products liftim is distributd with xpotial distributio udr progrssivly typ II csord sampls. Wu t al. [3] dsigd accptac-samplig plas for a xpotial populatio basd o a liftim-prformac idx by miimiz th umbr of failurs rquird durig ispctio. Solima [4] discussd th aysia ad o-aysia stimators for liftim prformac idx basd o progrssiv first-failur csorig sampls arisig from th xpotiatd Frcht distributio. i [5] discussd th aysia stimatio ad tst procdur of liftim prformac idx for Ailamujia distributio udr squard rror loss fuctio. Suppos that X is th products liftim, ad it distributd with xpotial distributio whos probability dsity fuctio (pdf) ad cumulativ distributio fuctio (cdf) rspctivly: x f( x θ) = θ θ, x > 0 ()

2 Mathmatics ttrs 08; 4(): 0-4 x F( x; θ) = θ θ, x > 0 (3) Hr, θ > 0 is th ukow paramtr. Th aim of this papr is to study th aysia statistical ifrc of liftim prformac idx C. Th aysia stimatio ad th aysia tst procdur of C will b drivd. Th orgaizatio of this articl is as follows. Sctio will rcall som proprtis of th liftim prformac idx of product with th xpotial distributio. Th rlatioship btw C ad coformig rat will b also discussd. Furthrmor, th aysia stimator of C basd o th cojugat Gamma prior distributio is also obtaid udr symmtric tropy loss fuctio i Sctio 3. A w aysia hypothsis tstig procdur of liftim prformac idx is dvlopd i Sctio 4, ad a applicatio xampl is giv i Sctio 5. Fially, coclusios ar giv i Sctio 6.. iftim Prformac Idx Suppos that X is th liftim of such a product whos liftim distributio is xpotial distributio with pdf (). Th th procss ma µ = EX = / θ ad th procss stadard dviatio σ = Var (X) = / θ. Th th paramtr θ is oft calld th ma tim. Th th liftim prformac idx C of xpotial product ca b rwritt as follows C µ / θ = = = θ (4) σ / θ Th failur rat fuctio r( x) ca b drivd as f(x θ) θxp( θx) r (x) = = = θ (5) F(x θ) xp( θx) From (4) ad (5), w ca s that th largr th ma / θ is, th smallr th failur rat ad largr th liftim prformac idx C ar. Thrfor, th liftim prformac idx C ca rasoably ad accuratly rprst th product prformac of w products. Morovr, if th liftim X of a product xcds th lowr spcificatio limit, th th product is dfid as a coformig rat, ad ca b dfid as Pr P(X ) xp( x)dx = = θ θ θ C = =, < C < (6) Accordig to th quatio (6), w ca s thr xists a strictly icrasig rlatioship btw coformig rat P r ad th liftim prformac idx C. Tabl lists valus of C ad th corrspodig coformig rat P r. Wh th valus of C which ar ot listd i Tabl, th coformig rat P r ca b drivd through itrpolatio. Tabl. Th liftim prformac idx C vrsus th coformig rat P r Th coformig rat ca b calculatd by dividig th umbr of coformig products by total umbr of products sampld. To accuratly stimatig rat, Motgomry [0] suggstd that th sampl siz must b ough larg. Howvr, a larg umbr of sampls ar usually ot practical from th poit of cost. I additio, a larg sampl is also ot practical du to tim limitatio or othr rstrictios such as matrial rsourcs, mchaical or xprimtal difficultis ad so o. Sic a o-to-o mappig xists btw th coformig rat P r ad th liftim prformac idx C. Thrfor, liftim prformac idx ca b ot oly a flxibl ad ffctiv tool to valuat th products prformac, but also b a ffctiv tool to stimat th coformig rat P r. 3. Estimatio 3.. Maximum iklihood Estimatio t X, X,, X b a liftim of sampl from th xpotial distributio with pdf (), x = ( x, x,, x ) is th obsrvatio of X = ( X, X,, X ) ad t = xi is th obsrvatio of statistic T = Xi. Th th liklihood fuctio corrspodig to pdf () is giv by (x; θ) = θt f(x θ) = θxp( θxi) = θ (7) Th maximum liklihood stimator of θ ca b asily dl [ ( θ; x)] drivd from th log-liklihood quatio = 0 as dθ follows: ˆM θ = (8) T W ca also asily prov that T is a radom variabl distributd with Gamma distributio Γ(, θ) with th

3 Guobig Fa: aysia Tst for iftim Prformac Idx of Expotial Distributio udr Symmtric Etropy oss Fuctio followig probability dsity fuctio: 3.. ays Estimatio θ θt ft( t; θ) = t, t > 0, θ > 0 Γ( ) This sctio will discuss th aysiaia stimatio of th liftim prformac idx of xpotial distributio with pdf () udr symmtric tropy loss fuctio (Zhao [6] ad i [7]) (9) ˆ θ θ ( ˆ, θ θ) = + (0) θ ˆ θ W ar itrstd i stimatig θ udr th symmtric tropy loss fuctio (0). mma t X, X,, X b a liftim of sampl from th xpotial distributio with pdf (), x = ( x, x,, x ) is th obsrvatio of X = ( X, X,, X ), Th udr th symmtric tropy loss fuctio (0), th uiqu aysia stimator of θ, say θˆ, is giv by ˆ E( θ X) θ = E( θ X) / () Proof. Obviously, th symmtric loss i () is strictly covx, th th uiqu aysia stimator is obtaid from th rlatio de[ ( ˆ θ, θ) X] = 0 dθ which rducs to (). Assum that th prior distributio of th paramtr θ is th Gamma prior distributio Γ (, β), with pdf β π θ β θ β θ Γ( ) βθ ( ;, ) =,, > 0, > 0 () Th combiig th liklihood fuctio (7) with th prior pdf (), th postrior pdf of θ ca b drivd usig ays Thorm as follows That is h( θ x) l( θ; x) π( θ) θ θ β θ Γ ( ) θt βθ + ( β + t) θ (3) θ X ~ Γ ( +, β + T). (4) Th udr th symmtric tropy loss fuctio (0), th aysia stimator of θ ca b drivd as follows + / ˆ E( θ X) β + T θ = = E( θ X) β + T + = ( β + T) ( + )( + ) Furthr th aysia stimator of C is Cˆ = ˆ θ β + T = ( + )( + ) / (5) (6) 4. aysia Tst of iftim Prformac Idx This sctio will propos a aysia tstig procdur to assss whthr th liftim prformac idx adhrs to th rquird lvl. Assum that th rquird idx valu of liftim prformac is largr tha c, whr c is th targt valu. First, w costruct th followig hypothsis: H : 0 C c H : C > c. (7) t Y= θβ ( + T) X, th for giv Sigificac lvl γ, w ca asily show that θ X ~ Γ ( +, β + T). t b th γ χ γ(( + )) prctil of χ (( + )). Th That is P( θβ ( + T) χ (( + )) X) = γ (8) γ χ γ(( + )) P( θ X) = γ ( β+ T) χ γ(( + )) P( θ X) = γ ( β+ T) (9) (0) Th th lowr cofidc limit of liftim prformac idx C with lvl γ ca b obtaid, as follows: χ (( )) ( ˆ γ + = C) ( + ) () Th th w proposd aysia tstig procdur of C is as follows: (i) For giv sampl siz, dtrmi th lowr liftim spcificatio limit. (ii) Udr symmtric tropy loss fuctio, calculat th aysia stimator

4 Mathmatics ttrs 08; 4(): Whr T ˆ θ = ( β + T) ( + )( + ) = Xi. (iii) Calculat th 00 ( γ )% o-sid cofidc itrval [, ) for liftim prformac idx C, whr is dfid as quatio (). (iv) Th dcisio ruls of statistical tst ar providd as follows: If th prformac idx valu c [, ), w rjct to th ull hypothsis H0 ad coclud that th liftim prformac idx of product mts th rquird lvl. If th prformac idx valu c [, ),, w accpt th ull hypothsis H 0 ad coclud that th liftim prformac idx dos ot mt th rquird lvl. 5. Discussio To illustrat th fasibility ad practicability of th proposd aysia tstig mthod, a practical xampl proposd i rfrc Nlso [8] (98, P. 05. Tabl.) is adoptd. Thr is a lif tstig xprimt i which spcims of a typ of lctrical isulatig fluid wr subjct to a costat voltag strss. Th data st is rportd i Tabl. Tabl. if tstig xprimt data. Data St ( = 9 ) alakrisha t al. [9] aalyzd ths =9 obsrvatios, ad thy provd that th data st has a xpotial distributio with pdf as () by usig last squars mthod ad goodss of fit tst. Gail ad Gastwirth [0] also provd th data st ca b modld by xpotial modl basd o Gii statistics. I th follow, w will giv th dtail stps of th proposd tstig procdur about C as: (i) Dtrmi th lowr liftim limit =.04. That is to say if th liftim of a lctrical isulatig fluid xcds hours, th th lctrical isulatig fluid is a coformig product. I ordr to satisfy th rgardig opratioal prformac. Th coformig rat P r is rquird to xcd 80 prct. Rfrrig to Tabl, th valu of C is rquird to xcd Thus, th prformac idx targt valu is c=0.80. (ii) Establish tstig hypothsis as blow H : C 0.80 H : C > (iii) Spcifid a sigificac lvl γ = (iv) Calculat th 00 (- γ )% o-sid cofidc itrval [, ) for C, whr, χ (( )) ( ˆ γ + = C) ( + ) Hr w suppos th prior distributio of θ is th oiformativ prior, i.. = β = 0. Th th 95% osidd cofidc itrval for C is [, ) = [0.8983, ). (v) caus of th prformac idx valu c = 0.80 [, ), w rjct th ull hypothsis H : C. Thus, w ca coclud that th liftim prformac idx of products mts th rquird lvl. 6. Coclusios Procss capability idics ar widly mployd i maufactur idustry to assss th prformac ad pottial of thir procss. iftim prformac idx is o of th most importat PCIs. This papr studis th aysia stimatio ad aysia tst of lif prformac idx wh th liftim of products is distributd with xpotial distributio. Th proposd tstig mthod is asy to complt with th hlp of som softwar, such as xcl, matlab. Th w tst procdur ca provid rfrc for th trpris girs to assss whthr th tru prformac of products mts th rquirmts. Ackowldgmts This study is partially supportd by Social Scic Foudatio of Hua Provic (No. 5YA065). Th author is gratful to th rviwrs for a vry carful radig of th mauscript ad th suggstios. Rfrcs [] Shu M H, Wu H C. Maufacturig procss prformac valuatio for fuzzy data basd o loss-basd capability idx [J]. Soft Computig, 0, 6 (): [] Ch C C, ai C M, Ni H Y. Masurig procss capability idx Cpm with fuzzy data [J]. Quality & Quatity, 00, 44 (3): [3] Jura J M, Grya F M, igham R S J. Quality Cotrol Hadbook. Nw York: McGraw-Hill., 974. [4] Eslamipoor R, Hossii-Nasab H. A modifid procss capability idx usig loss fuctio cocpt [J]. Quality & Rliability Egirig Itratioal, 06, 3 (): [5] Ch K S, Huag C F, Chag T C. A mathmatical programmig modl for costructig th cofidc itrval of procss capability idx Cpm i valuatig procss prformac: a xampl of fiv-way pip [J]. Joural of th Chis Istitut of Egirs, 07, 40 ():6-33. [6] Gu X, Ma Y, iu J, t al. Robust paramtr dsig for multivariat quality charactristics basd o procss capability idx with idividual obsrvatios [J]. Systms Egirig & Elctroics, 07, 39 ():

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