Bayesian Test for Lifetime Performance Index of Ailamujia Distribution Under Squared Error Loss Function

Transcription

1 Pur ad Appld Mathmatcs Joural 6; 5(6): do:.648/j.pamj.656. ISSN: (Prt); ISSN: (Ol) Baysa Tst for ftm Prformac Idx of Alamuja Dstrbuto Udr Squard Error oss Fucto apg Dpartmt of Basc Subjcts, Hua Uvrsty of Fac ad Ecoomcs, hagsha, ha Emal addrss: To ct ths artcl: apg. Baysa Tst for ftm Prformac Idx of Alamuja Dstrbuto Udr Squard Error oss Fucto. Pur ad Appld Mathmatcs Joural. Vol. 5, No. 6, 6, pp do:.648/j.pamj.656. Rcvd: Octobr 5, 6; Accptd: Octobr 4, 6; Publshd: Novmbr 7, 6 Abstract: Th am of ths papr s to stmat th lftm prformac dx of Alamuja dstrbuto. A Baysa tst procdur s dvlopd udr squard rror loss fucto. Frstly, Bays stmato of lf prformac dx s drvd, th a Baysa tst procdur for lftm prformac dx usg postror probablty rato tst mthod. Fally, a xampl s usd to llustrat th ffctvss ad fasblty of th mthod. Kywords: Bays Estmato, ftm Prformac Idx, Bays Tst, Alamuja Dstrbuto, Squard Error oss Fucto. Itroducto For maufactur dustry, procss capablty dx s a ffctv ad covt tool for qualty assssmt. May procss capablty dcs hav b put forward. For xampl, Jura [] troducd th frst procss capablty dx P, Ka [] proposd th procss capablty dx pk, whch ar th two most commoly usd dcs, Boyls [3] ad Par t al. [4] troduc two mor-advacd dcs pm ad pmk rspctvly. Thos four procss capablty dcs hav b dfd xplctly as: US S () P =, 6σ whr US ad S ar th uppr ad th lowr spcfcato lmts ad σ s th procss stadard dvato. d µ M m{ US µ, µ S} () pk = =, 3σ 3σ whr µ s th procss ma. d d () pm = =, 6 σ + ( µ T) 6 E[( X T) ] whr T s th targt valu. US µ µ S (v) pmk = m{, } 3 σ + ( µ T) 3 σ + ( µ T) Th statstcal frcs of procss capablty dcs hav draw grat attto by may authors. For xampl, Shau t al. [5] appld Baysa mthod to th stmato of pm ad pk udr th rstrcto that th procss ma quals to th mdpot of th two spcfcato lmts. Par ad Wu [6] studd th tst of pk for gral stuato wthout rstrcto o th procss ma basd o Baysa approach. h ad Hsu [7] proposd a tst for pk that s asymptotcally quvalt to th lklhood rato tst. Baral ad As [8] dvlopd mthod of gralzd cofdc trval to masur th procss capablty dx pm prsc of masurmt rrors. Mactyr[9] usg Bays approach to stmat th procss capablty dcs ad systm avalablty for th vrs Raylgh lftm dstrbuto. To assss th products wth th largr-th-bttr typ of th qualty charactrstcs, Motgomry [] proposd a spcal ulatral spcfcato procss capablty dx, amd as lftm prformac dx, µ = () σ whr s th lowr boud of th spcfcatos. Th statstcal aalyss of lftm prformac dx for products whos lftm dstrbutd varous dstrbutos hav b wdly studd. For xampl, Wu t al. [] dscussd th maxmum lklhood stmato, mmum varac

2 Pur ad Appld Mathmatcs Joural 6; 5(6): ubasd stmato lftm prformac dx of Raylgh dstrbuto product udr progrssvly typ II csord tst. t al. [] drvd maxmum lklhood stmator ad costructd a hypothss tstg procdur for lftm prformac dx basd o csord sampls, whch products lftm coms from th ormal dstrbuto but sampl data modld by fuzzy umbrs. t al. [3] studd th Baysa stmato ad Baysa tstg procdurs for lftm prformac dx udr squard rror loss fucto whch products lftm dstrbutd wth Raylgh dstrbuto. u ad R [4] studd th Baysa stmato ad Baysa tst of lftm prformac dx for xpotal product udr progrssvly typ II csord sampls. Th statstcal frcs about varous lftm dstrbutos, such as xpotal dstrbuto, Wbull dstrbuto, ormal dstrbuto ad Parto dstrbuto, tc. hav b studd a lot [5-]. Rct yars, may w dstrbutos ar proposd for varous gr applcato. Alamuja (Эрланга) dstrbuto s o of ths dstrbuto proposd by v t al. []. Pa t al. [] studd th trval stmato ad hypothss tst of Alamuja dstrbuto basd o small sampl. og [3] studd th Bays stmato of Эрлангa dstrbuto udr typ-ii csord sampls basd o thr dffrt pror dstrbutos. [4] dscussd th mmax stmato of th paramtr ofэрлангa dstrbuto udr a o-formatv pror dstrbuto wth thr dffrt loss fuctos. Assum that Xs th products s lftm, ad t dstrbutd th Alamuja dstrbuto whos probablty dsty fucto (pdf) ad cumulatv dstrbuto fucto (cdf) rspctvly: θx f( x; θ) = 4 xθ, x, θ > () θx F( x; θ) = ( + θx), x, θ > (3) Hr, θ s th ukow paramtr. I ths papr, w study th stmato of lftm prformac dx usg Baysa approach, ad th Bays tst procdur of wll also b costructd. Scto troducs som proprts of th lftm of product wth th Alamuja dstrbuto. Morovr, th rlatoshp btw ad coformg rat s also dscussd. Furthrmor, th Baysa stmator of basd o th cojugat Gamma pror dstrbuto s also obtad udr squard rror loss fucto. A w Baysa hypothss tstg procdur s dvlopd Scto 3, ad a practcal xampl s gv Scto 4. Fally, a cocluso s gv Scto 5.. ftm Prformac Idx t X b th lftm of such a product whos lftm dstrbuto s Alamuja dstrbuto wth pdf (). It s asly to vrfy that th procss ma µ = EX = / θ ad th procss stadard dvato σ = Var( X) = / θ. Th th paramtr θ s oft calld th ma tm. Th th lftm prformac dx of Alamuja dstrbuto ca b rwrtt as follows µ / θ = = = ( θ) (4) σ / θ Th falur rat fucto r( x) s dfd by f ( x θ) 4xθ r( x; θ) = = F( x θ) + xθ Th drvatv of th falur rat fucto r( x ) wth rspct to θ s dr( x; θ) 8 xθ ( + xθ) = ( + xθ) dr( x; θ) From Eq. (6), w s that s always strctly bggr tha zro. Th th falur rat fucto r( x) s strctly crasg fucto wth rspct to th paramtr θ. Thus w ca s that th smallrθ,.. th largr th ma/θ, th smallr th falur rat ad largr th lftm prformac dx. Thrfor, th lftm prformac dx s a rasoably ad accuratly dx for assss th prformac of products. Morovr, coformg rat Pr of th product ca b dfd as a probablty of th lftm of a product X xcdg th lowr spcfcato lmt, that s P = P( X ) = 4 θ r = ( + θ) θ θx x dx (3 ), = < < It s asly to vrfy that th coformg ratp r ad lftm prformac dx hav a strctly crasg rlatoshp. Th rlatoshps of Pr ad ca b calculatd usg Matlab softwar for a lst of varous valus. For th valus whch ar ot lstd Tabl, th coformg rat Pr ca b obtad through trpolato. Tog t al. [5] potd out that th coformg rat Pr ca b calculatd by dvdg th umbr of coformg products by total umbr of products sampld, whl Motgomry 985 suggstd that th sampl sz must b larg to accuratly stmato. Howvr, a larg sampl sz s usually ot practcal cosdrg wth th prspctv of cost. Sc thr xst a o-to-o mathmatcal rlatoshp btw th coformg rat P r ad th lftm prformac dx. Th lftm prformac dx ca b a flxbl ad ffctv tool for stmatg th coformg rat P r. 3. Estmato Ths scto wll dscuss th maxmul lklhood (5) (6) (7)

3 83 apg : Baysa Tst for ftm Prformac Idx of Alamuja Dstrbuto Udr Squard Error oss Fucto stmato ad Bays stmato of lftm prformac dx of Alamuja dstrbuto 3.. Maxmum klhood Estmato t X, X,, X rprst th lftm of sampl from th Alamuja dstrbuto wth pdf (), x = ( x, x,, x ) s th obsrvato of X = ( X, X,, X ) ad t = x s th obsrvato of = = T = X. Th th lklhood fucto corrspodg to pdf () s gv by θx l( θ; x) = f ( x; θ) = 4θ x = = = θ ( 4 x ) θ = θ x = Th th log- lklhood fucto ca b obtad as follows: [ l( θ; x)] = lθ + l(4 x ) θ x = = Th maxmum lklhood stmator of θ ca b asly drvd from th log-lklhood quato dl [ ( θ; x)] =. Th maxmum lklhood stmator of θ s Whr T = X. = (8) ˆME θ = (9) T Th by varac of maxmum lklhood stmato, w ca gt th maxmum lklhood stmator of lftm prformac dx as follows: ˆ ME ˆ = ( θme) = ( ) () T W ca also asly show that T s a radom varabl dstrbutd wth th Gamma dstrbuto Γ (, θ), whch has th followg probablty dsty fucto: 3.. Bays Estmato θ ft( t; θ) = t, t >, θ > Γ( ) () I ths scto, w shall dscuss th Bays stmato of th lftm prformac dx of Alamuja dstrbuto wth pdf () usg Baysa approach. Squard rror loss fucto s o of th most mportat loss fuctos Baysa statstcal aalyss, ad th formula of t s ( ˆ θ, θ) ( ˆ θ θ) = () Assum that th cojugat pror dstrbuto of θ s th Gamma pror dstrbuto Γ (, ), wth pdf π θ θ θ Γ( ) θ ( ;, ) =,, >, > (3) Whr, > ar two pror hypr paramtrs. ombg th lklhood fucto (8) wth th pror probablty dsty fucto Eq. (3), th postror pdf of θ ca b drvd usg Bays Thorm as follows That s h( θ x) l( θ; x) π( θ) θ θ θ Γ ( ) θ + ( + t) θ (4) θ X ~ Γ ( +, + T). (5) Th udr th squard rror loss fucto (), Bays stmator of θ s th postror ma,.. ˆ + θb = E[ θ x] = + T Furthr th Bays stmator of s ˆ = E( X) = E( ( θ) X) B = [ E( θ X)] + = [ ] + T 4. Bays Tst of ftm Prformac Idx (6) (7) Ths scto wll costruct a Baysa tstg procdur to assss whthr th lftm prformac dx adhrs to th rqurd lvl. Assum that th rqurd valu of lftm prformac s largr tha th targt valu. c. Frst, w stablsh th followg hypothss: H : c H : > c. (8) Th th w proposd Baysa tstg procdur of s as follows: Stp. Dtrm th lowr lftm lmt ad sampl sz. Stp. alculat th Baysa stmator

4 Pur ad Appld Mathmatcs Joural 6; 5(6): Whr T = X. = ˆ + B = [ ], (9) + T Stp 3. alculat th postror probablty odds rato BF P( H X) P( H X) P( H X) P( H X) = =, () + whr P( H X) = c π( θ X), π( θ x) = θ Γ ( + ) ad t = x s th obsrvato of T = X. = = + ( + t) θ Stp 4. Th dcso ruls ar provdd as follows: If ˆB > cadbf <, w rjct to th ull hypothssh, th t s cocludd that th lftm prformac dx or coformg rat of th products mts th rqurd lvl; If ˆB < cad BF >, w accpt th ull hypothssh, th t s cocludd that th lftm prformac dx or coformg rat of th products dos ot mt th rqurd lvl. 5. Numrcal Exampl To llustrat th practcablty ad fasblty of th proposd tstg mthod, a Mot arlo smulato s usd to grat a sampl of Alamuja dstrbuto wth θ =. ad =. Th data st s: , , 3.74, 7.84, 3.98, , 7.84, 4.67, 6.86,.54, 7.64, 4.86, 4.333,.9649, , 4.6, 4.3, , , Now w gv th stps of th proposd Baysa tstg procdur about as follows: Stp. alculat t = x = ad hr w = assum th lowr lftm lmt =4.78. To dal wth th lftm prformacs, th coformg rat P r s rqurd to xcd.8. Accordg to Eq. (7), th valu of s rqurd to xcd Thus, th targt valu of prformac dx s st at c=.8375, th w stablsh th followg tstg hypothss H :.8375 H : > Stp. Udr squard loss fucto, w gt th Baysa stmat ˆ B =.74; Stp 3. Suppos that th pror paramtr valus =. ad =., th P( H X) = π( θ X). c Thrfor, th postror odds rato BF P( H X) P( H X) = >. Stp 4. Obvously, ˆ =.74 < c =.8375 adbf >, B th w ca accpt th ull hypothss H : c. That s, w coclud that th lftm prformac dx dos ot mt th rqurd lvl. 6. oclusos Procss capablty dcs ar wll ffctv tools to assss th prformac ad pottalty of thr procss ad wdly mployd by maufacturs. ftm prformac dx s a thlargr-th-bttr dx whch s spcally usful for o-ormal dstrbutos. Ths papr studd th Baysa stmato ad Baysa tst of lf prformac dx udr squard rror loss fucto. Th w proposd Baysa tstg mthod s asr tha othr classcal approachs ad th tst procss s asy to oprat by usg ordary programmg softwar as Matlab, Excl, tc. Ths tstg mthod ca b smlar usd to othr lf dstrbutos. Th tstg procdur ca provd rfrc for th trprs grs to assss whthr th tru lftm prformac of products mts th rqurmts. Ackowldgmt Ths study s partally supportd by Natural Scc Foudato of Hua Provc (No. 5JJ33 ad No. 6JJ4) ad Foudato of Hua Educatoal ommtt (No.58). Th author also gratfully ackowldgs th hlpful commts ad suggstos of th rvwrs, whch hav mprovd th prstato. Rfrcs [] Jura J. M., Grya F. M., Bgham R. S. J., 974. Qualty otrol Hadbook. Nw York: McGraw-Hll. [] Ka V. E., 986. Procss capablty dcs. Joural of Qualty Tchology, 8(): 4-5. [3] ha. K., hg S. W., Sprg F. A., 988. A w masur of procss capablty: pm. Joural of Qualty Tchology, (3): [4] Par W.., Kotz S., Johso N.., 99. Dstrbutoal ad frtal proprts of procss capablty dcs. Joural of Qualty Tchology, 4(4): [5] Shau J. H., hag. T., Hug H. N., 999. A Baysa procdur for procss capablty assssmt. Qualty & Rlablty Egrg, 5(5): [6] Par W.., Wu. W., 5. Procss capablty assssmt for dx pk, basd o Baysa approach. Mtrka, 6(): -34. [7] h S. M., Hsu Y. S., 6. Uformly most powrful tst for procss capablty dx. Qualty Tchology & Quattatv Maagmt, ():

5 85 apg : Baysa Tst for ftm Prformac Idx of Alamuja Dstrbuto Udr Squard Error oss Fucto [8] Baral A. K., As M. Z., 5. Assssmt of pm th prsc of masurmt rrors. Joural of Statstcal Thory & Applcatos, 4(): 3-7. [9] Mactyr A., 5. O procss capablty ad systm avalablty aalyss of th vrs Raylgh dstrbuto. Paksta Joural of Statstcs & Oprato Rsarch, (): [] Motgomry, D.., 985. Itroducto to Statstcal Qualty otrol, Nw York: Joh Wly & Sos,. [] Wu.., h.., h Y. J., 3. Dcso procdur of lftm prformac assssmt of Raylgh products udr progrssvly Typ II rght csord sampls. Itratoal Joural of Iformato & Maagmt Sccs, 4(3): [] W.., Hog. W., Wu J. W., 5. omputatoal procdur of prformac assssmt of lftm dx of ormal products wth fuzzy data udr th typ II rght csord samplg pla. Joural of Itllgt & Fuzzy Systms, 8(4): [3] W.., Wu J. W., Hog M.., t al.,. Assssg th lftm prformac dx of Raylgh products basd o th Baysa stmato udr progrssv typ II rght csord sampls. Joural of omputatoal & Appld Mathmatcs, 35(6): [4] u M. F., R H. P., 3. Baysa tst procdur of lftm prformac dx for xpotal dstrbuto udr progrssv typ-ii csorg. Itratoal Joural of Appld Mathmatcs & Statstcs, 3(): [5] Shafay A. R., 6. Exact frc for a smpl stp-strss modl wth gralzd Typ-I hybrd csord data from th xpotal dstrbuto. ommucato Statstcs- Smulato ad omputato, 45(): 8-6. [6] Bradar B. S., Satosha. D., 4. Estmato of th ma of th xpotal dstrbuto usg maxmum rakd st samplg wth uqual sampls. Op Joural of Statstcs, 4(4): [7] Alomar A. I., 4. Baysa stmato of th ma of xpotal dstrbuto usg movg xtrms rakd st samplg. Statstcal Paprs, 44(3): [8] Pg X., Ya Z., 4. Estmato ad applcato for a w xtdd Wbull dstrbuto. Rlablty Egrg & Systm Safty, 4, (): [9] Horrac W.., 5. Momts of th trucatd ormal dstrbuto. Joural of Productvty Aalyss, 43(): [] Guar K., Krst V.,. Estmato of th locato ad scal paramtrs of a Parto dstrbuto by lar fuctos of ordr statstcs. Joural of th Amrca Statstcal Assocato, 68(68): 8-7. [] v H. Q., Gao. H., h..,. Эрланга dstrbuto ad ts applcato supportablty data aalyss. Joural of Acadmy of Armord Forc Egrg, 6(3): [] Pa G. T., Wag B. H., h.., Huag Y. B., Dag M. T., 9. Th rsarch of trval stmato ad hypothtcal tst of small sampl of Эрланга dstrbuto. Applcato of Statstcs ad Maagmt, 8(3): [3] og B., 5. Baysa stmato of paramtr o Эрлангa dstrbuto udr dffrt pror dstrbuto. Mathmatcs Practc & Thory, (4): [4]. P., 6. Mmax stmato of th paramtr of ЭРланга dstrbuto udr dffrt loss fuctos, Scc Joural of Appld Mathmatcs ad Statstcs. 4(5): [5] Tog,. I., h, K. T. ad h, H. T.,. Statstcal tstg for assssg th prformac of lftm dx of lctroc compots wth xpotal dstrbuto, Itratoal Joural of Qualty & Rlablty Maagmt, 9(7): 8-84.