Bayesian Shrinkage Estimator for the Scale Parameter of Exponential Distribution under Improper Prior Distribution

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1 Itratoal Joural of Statstcs ad Applcatos, (3): 35-3 DOI:.593/j.statstcs.3. Baysa Shrkag Estmator for th Scal Paramtr of Expotal Dstrbuto udr Impropr Pror Dstrbuto Abbas Najm Salma *, Rada Al Sharf Dpartmt of Mathmatcs-Ib-Al-Hatham Collg of Educato - Uvrsty of Baghdad Abstract Ths papr dals wth prlmary tst sgl stag Baysa Shrkag stmator for th scal paramtr () of a xpotal dstrbuto wh a guss valu ( ) for () avalabl from th past studs udr th mpropr pror dstrbuto ad th quadratc loss fucto. Th proposd stmators ar show to b a mor ffct tha th usual stmators wh s clos to th ss of ma squard rror (MSE). I whch th xprsso for bas ad ma squard rror of th proposd stmator ar drvd. Numrcal rsults for th bas ad MSE ar usg dffrt costats wr volvd t whch had b gv as wll as comparsos. Kywords Expotal Dstrbuto, Maxmum Lklhood Estmator, Impropr Pror Dstrbuto, Baysa Estmator, Sgl Stag Shrkag Estmator, Ma Squard Error, Rlatv Effccy. Itroducto O of th most usful ad wdly xplotd modl s th xpotal dstrbuto, Epst [7] rmarks that s th xpotal dstrbuto whch plays as mportat rol lf xprmts as th part playd by th ormal dstrbuto agrcultural xprmts. It s appld a vry wd varty of statstcal procdurs. Amog th most promt applcatos wr foud th fld of lf tstg ad rlablty thors. Th scal paramtr () s kow as ma lf tm. Th maxmum lklhood stmator for s a sampl ma whch s ubasd ad a mmum varac ubasd lar stmat. Th o paramtr xpotal dstrbuto has th followg probablty dsty fucto (p.d.f) t xp( ),t, > = (),o.w whr s a avrag or th ma lf ad t s also acts as scal paramtr, whl = / s calld th ma tm to falur (MTTF). Thompso [3] troducd th da of Shrkag th MVULE towards th pror stmat ordr to gt a bttr stmat, ad proposd a class of shrkag * Corrspodg author: abbasajm@yahoo.com (Abbas Najm Salma) Publshd ol at Copyrght Sctfc & Acadmc Publshg. All Rghts Rsrvd Τ k + ˆ ( k), whr k (costat) had stmators b kow as a shrkag wght fucto, < k <, whch s spcfd by th xprmtr advac accordg to hs blf. H compard th stmator T wth () ˆ, th trms of MSE. Aothr class of shrkag stmators wr boudd MSE ad had a bttr prformac tha th usual stmator, whch hav b dscussd [9] ad []. Bhattacharya ad Srvastava [] wr usd th atcdt pror stmat to propos a prlmary tst sgl stag shrkag stmator for as blow ψ ( )( ˆ ˆ ) +, f H : = s accptd SS = (a) ˆ ˆ ψ ( )( ) +, othrws ψ ˆ ( ) = ad ψ ˆ ( ) =. wh Also, svral authors had b studd th gral prlmary sgl stag Shrkag stmator form (a) s by tak may dffrt chocs for th shrkag wght factors ψ (ˆ ) ( =,), ψ () ˆ. For xampl, t may b tak as SS = ( ψ ))( ˆ ˆ ˆ + whr ψ ( )( ˆ ˆ ) +, f ˆ R (b) ( ), f R ψ( ) ˆ, ψ() ˆ t may b costat or a fucto of ˆ whch to rprsts o's dgr of blf

2 3 Abbas Najm Salma t al.: Baysa Shrkag Estmator for th Scal Paramtr of Expotal Dstrbuto udr Impropr Pror Dstrbuto th pror stmat, R s a sutabl rgo th paramtr spac whch may b prtst rgo. S [], [], [3], [], [] ad []. Th da of ths papr s cocr wth th dvlopmt of prlmary sgl stag shrkag stmators (a) s for stmat th scal paramtr of xpotal dstrbuto b usg th Baysa stmato tchqu udr th mpropr of pror dstrbuto ad quadratc loss fucto. Varous chocs of shrkag wght fucto had b cosdrd as wll as bg prtst rgo R for complt sampls. Th xprssos for Bas, Ma Squard Error ad Rlatv Effccy wr drvd. Numrcal rsults for Bas ad Rlatv Effccy (R.Eff.) b gv for a dffrt costat volvs th stmators. 3. Baysa Estmator Cosdr th o paramtr xpotal dstrbuto (), ad assum th followg mpropr pror dstrbuto of ; g() = (a + ) b/, >, < a <, b. s [] (3) L(t, t,,t ) = f (t ) = Ad th postror dstrbuto fucto s dfd as blow : f f f ( t, t,,t ) t xp = f (t; ) g( ) = f (t; ) g( )d = t ( a+ ) (b/ ) ( t, t,...,t ) ( t, t,...,t ) = = t ( a ) (b/ ) = + d b + t ( a + ) b + t ( a + ) d b + t ( + a) + a Γ ( + a) b+ t () (5) ()

3 Itratoal Joural of Statstcs ad Applcatos, (3): Thrfor, o ca obta th bays stmator of whch udr quadratc loss fucto ad rsk fucto as follow : b + t ( + a) ˆ = E( t, t,...,t ) = f ( t, t,...,t )d= d Β + a Γ ( + a) b+ t (7) + a b t b + t + (+ a ) Γ ( + a ) = Β + a + a Γ ( + a) b+ t Γ ( + a ) b+ t ˆ = d Ad by smpl calculatos, w gt b+ ˆ Β = + a t, < a <, b (). Prlmary Sgl Stag Baysa Shrkag Estmator BS Ths scto s cocr wth th poolg approach btw shrkag stmato whch had b usd a pror formato about a ukow paramtr as tal valus ad Baysa stmato wr uss a pror formato about ukow paramtr bg a pror dstrbuto for th scal paramtr () of xpotal dstrbuto wr usg spcfc shrkag wght factors as wll as prtst rgo (R) wh a pror formato about () s avalabl as tal valu ( ). Gral prlmary tst sgl stag Baysa Shrkag (PSSBS) stmator wr dfd blow whr ad S []. ψ ( )( ˆ ˆ ) +, f ˆ R B BS = ψ ˆ ˆ ˆ ( )( Β ) +, f R (9) ˆ B had b rprstd to Bays stmator for s dfd wth quato (), R whch s sutabl rgo (say prtst) ψ ( ˆ) (,), ψ () ˆ s shrkag wght fucto whch mght b a fucto of ˆ (MLE) or a costat,.. Prlmary Tst Sgl Stag Baysa Shrkag Estmator BS Usg th form (9), th proposd PSSBSE.. BS had th followg forms: ψ ( ) ˆ = ad ψ ( ) ˆ = k (costat), < k < ;, f ˆ R BS = k ( ˆ ), f ˆ Β + R ()

4 3 Abbas Najm Salma t al.: Baysa Shrkag Estmator for th Scal Paramtr of Expotal Dstrbuto udr Impropr Pror Dstrbuto ad suppos that a= ad b= - quato(). whr R s pr-tst rgo of accptac to sz for tstg th hypothss H : = agast th hypothss H A : usg th tst statstc T that Assum that, R=[a,b], a < b. ˆ ˆ ( ) = R = X /,, X/,..; () a= X /,, b= X/, () whr X /, ad X/, had b rspctvly a lowr ad a uppr (/) prctl pot of ch-squar dstrbuto wth dgr of frdom (). Also, ˆ rfr to Bays stmator, ˆ s MLE of ad was a pror formato of. B Th xprssos for Bas [] ad ma squar rror [MSE] of B(,R) = E( ) BS BS R BS wr rprstd rspctvly as follows: = [ ]f ( ˆ )d + ˆ [k( ˆ ) + ] f ( ˆ )dˆ Β R whr R s th complmt rgo of R ral spac ad f ( ˆ ) s a p.d.f. of ˆ whch has th followg form w coclud: ˆ - [ ] xp[ ˆ / ], for ˆ ( ˆ << f ) = Γ( )( /), othrws B(,R) k ( ) ( ) k J ( *, *) J ( *, *) BS = + a b a b () whr = /, Ad y J ( a*, b*) = y y dy Γ() b* a* /, /, (3) (5) a* = X, b* = X ()

5 Itratoal Joural of Statstcs ad Applcatos, (3): MSE(,R) = E( ) BS BS Th Rlatv Effccy of stmator s for xampl [],[],[3]ad[3] + ( ) = k + ( ) k( ) ( ) + + ( ) ( ) k J ( a*, b*) J ( a*, b*) + J ( a*, b*) k( ) J ( a*, b*) J ( a*, b*) BS wth rspct to th classcal stmator ( ˆ ) s dfd as blow R.Eff ( / BS,R) = Μ S Ε( BS.. Prlmary Tst Sgl Stag Baysa Shrkag Estmator BS,R) (7) () By usg th form (9), whch th proposd PSSBS stmator.. ad suppos that a= ad b=. ψ ( ) ˆ = ad ψ ( ) ˆ = k (costat), < k < ; Th xprssos for Bas [] ad th Ma Squard Error [MSE] of ad, B(,R) = E( ) BS BS BS had th followg forms:, f ˆ R BS = k ( ˆ ), f ˆ Β + R (9) R Β R BS wr rprstd rspctvly as follow up: = [ ]f ( ˆ )d + ˆ [k( ˆ ) + ] f ( ˆ )dˆ {( )( j ( a*, b* ) ) k(j ( a*, b* ) j (a*, b*) ) } = + + () { MSE(,R) = k / + ( ) (k + ) k( )[ ( j ( a*, b* ) + 5. Numrcal Rsults BS a* b* a* b* a* b* + j (, ) ] k (j (, ) j (, ) j (a*,b*)] Th computatos of rlatv Effccy [] ad th Bas rato [] b usd for th stmator BS (,). Ths computatos wr prformd for =.,.5,., k =.,.,.3,.5, =.(.),, =,,,,. Som of ths computatos had b gv axd tabls. Th obsrvato mtod th tabls lad to th followg rsults:. of BS ( =, ) ar advrsly proportoal wth th small valu of ad thos of ad k. } ()

6 Abbas Najm Salma t al.: Baysa Shrkag Estmator for th Scal Paramtr of Expotal Dstrbuto udr Impropr Pror Dstrbuto. of BS ( =, ) ar maxmum howvr wh = ( = ) for all, ad k. 3. R.Eff B ( )s bttr tha R.Eff c ( ) of BS ( =, ).. Bas rato [] of BS ( =, ) ar rasoably a small wh s, othrws wll b maxmum for all ad. 5. of BS ( =, ) ar a small compard wth th small sampl sz () ad also wth th small ad k.. Effctv Itrval [th valus of that maks R.Eff. ar gratr tha o] for BS ( =, ) s [.5,.5]. 7. Th suggstd stmator BS ( =, ) ar mor ffct tha th stmators troducd by [], [9] ad [] th trms to Ma Squard Error (MSE).. Th suggstd stmator BS s mor ffct tha th stmator BS th ss of ma squard rror (MSE). Tabl (). Show Bas Rato [] ad Rlatv Effccy [R.Eff.( )] of BS wh k = (.773).939 (.59) 3.9 (.3).5 (.5). (.75).73 (.937).937. (.7).33 (.5).9 (.7) 75. (.37).9 (.7).33 (.935).9 (.77).53 (.5).979 (.59) 7. (.333).95 (.939).5 (.99).75 (.99).535 (.5) 3.93 (.7) (.335).55 (.7).5 (.93).7.5 (.5).35 (.55).9 (.95) 5.7 (.) 3. (.9).375 (.9).97 (.).9 (.5).933 (.9).33 (.77).35 (.7).39 (.935).95 Tabl (). Show Bas Rato [] ad Rlatv Effccy [R.Eff.( )] of BS wh k = (.39).59 (.59) (.93) 3.3 (.75).39 (.75).5 (.).3. (.37).39 (.5337).579 (.) 75.5 (.) 3.7 (.9). (.3). (.).33 (.537).937 (.793) 5.3 (.99).7 (.57).35 (.75).7 (.5597).595 (.573) 3. (.).37 (.97) (.).5 (.79) (.5577).539 (.57).7 (.77) 5.3 (.3).99 (.57).355 (.755). (.55).3 (.5). (.75).59 (.97) 3.3 (.57).33 (.757).57

7 Itratoal Joural of Statstcs ad Applcatos, (3): 35-3 Tabl (3). Show Bas Rato [] ad Rlatv Effccy [R.Eff.( )] of BS wh k = (.5355).55 (.5) (.573) 7.59 (.5).9 (.5) (.).33. (.57).9 (.55).5 (.335) 7.9 (.77) 3.77 (.533).9 (.77). (.5337).3555 (.5).77 (.9) 9. (.3).7 (.95). (.595).3 (.9). (.) (.) 3.97 (.7) 5.3 (.5) (.5).3.5 (.9).5799 (.7).33 (.3).7 (.39).3 (.5).5 (.599). (.).57 (.).7939 (.5).53 (.3) (.3).59 (.57).3 Tabl (). Show Bas Rato [] ad Rlatv Effccy [R.Eff.( )] of BS wh k = (.9).995 (.3) (.35).5 (.955).35 (.7).39 (.59)..59 (.335).37 (.33) 3.7 (.57).397 (.55).57 (.57).9 (.).7 (.5).939 (.39) 7.7 (.77).9 (.).57 (.3).39 (.9).355 (.5353).99 (.33) 37. (.).7 (.77).397 (.).353 (.7).5.53 (.).3 (.335).97 (.39). (.7).53 (.3).95 (.3).95 (.3).973 (.333) 7.55 (.5).5 (.75).7 (.57).95 (.)

8 Abbas Najm Salma t al.: Baysa Shrkag Estmator for th Scal Paramtr of Expotal Dstrbuto udr Impropr Pror Dstrbuto Tabl (5). Show Bas Rato [] ad Rlatv Effccy [R.Eff.( )] of BS wh k = (.7).95 (.3).37 (.5).9 (.37). (.).3 (.7).. (.9).3 (.39) 5.97 (.5).377 (.3595).7355 (.5).9 (.37). (.7).995 (.).75 (.).575 (.3).35 (.). (.39).557 (.).5 (.33).7 (.3).379 (.339).7 (.5).55 (.35).5.7 (.55).99 (.75).997 (.37).5599 (.375).97 (.955).97 (.393).33 (.33).793 (.) (.357).7 (.37).55 (.975).55 (.3) Tabl (). Show Bas Rato [] ad Rlatv Effccy [R.Eff.( )] of BS wh k = (.35).933 (.553) (.77).97 (.577).79 (.3). (.73)..79 (.37).3 (.557) (.77).3 (.).35 (.53).93 (.3).93 (.9).77 (.573) 3.3 (.353).9 (.).97 (.).9977 (.5).5 (.77).77 (.57) 5.75 (.539). (.33).937 (.).7 (.57).5. (.5).59 (.579).79 (.9).3 (.3). (.9).5 (.55). (.).777 (.777).5 (.5). (.).337 (.337).957 (.957) REFERENCES [] Al-Joboor, A.N., (), "Prlmary Tst Sgl Stag Shruk Estmator for Th Paramtrs of Smpl Lar Rgrsso Modl", Ib Al-Hatham J. for Pur ad Appld Sc., Vol. 3(3), pp [] Al-Joboor, A.N., (), O Sgfcac Tst Estmator for th Shap Paramtr of Gralzd Raylgh Dstrbuto.AL-Qadsya J.for computr ad mathmatcs Sccs, Vol.3,No.,PP [3] Al-Hmyar,Z.A., Khurshd,A ad Al-Joboor, A. N.,(9), "O Thompso Typ Estmators for th Ma of Normal Dstrbuto", REVISTA INVESTIGACIÓN OPERACIONAL J. Vol. 3, No., pp.9-. [] Al-Joboor, A.N. ad Mohammad, M.A., (), "Prlmary Tst Baysa Shruk Estmators for th ma of Normal Dstrbuto wth Kow Varac", Dyala, Jour., Vol. 3, pp.99-. [5] Al-Joboor,A.N. ad Salma, M.D., ()," O Doubl Stag Shrkag- Baysa Estmator for th Scal Paramtr of Expotal Dstrbuto", Ib Al-Hatham J. for Pur ad Appld Sc., Vol. 5(), pp

9 Itratoal Joural of Statstcs ad Applcatos, (3): [] Bhattacharya, S.K. ad Srvastava, V. K., (97), "A prlmary Tst Procdur Lf Tstg", J. Amr. Statst. Assoc., Vol. 9, pp [7] Epst, B. ad Sobl, M., (95), "Som Thorms to Lf Tstg from a Expotal Dstrbuto", Aals of Mathmatcal Statstcs, Vol.5, pp [] Kalaf,B.A., (7), "A Effct Shrkag Estmators for th Ma of Normal Dstrbuto", M.Sc. Thss, Baghdad Uvrsty, Collg of Educato (Ib- Al- Hatham), Baghdad, Iraq. [9] Mhta,J.S. ad Srvasa, R., (97), "Estmato of th Ma by Shrkag to a Pot", J. Amr. Statst. Assoc., Vol., pp.-9. [] Pady, B.N. ad Srvastava, R., (95), "O Shrkag Estmato of Expotal Scal Paramtr", IEEE Tras, Rl., Vol. R.3, pp. -. [] Salma,M.D.,()," Estmat th Paramtr ad Rlablty Fucto of Expotal Dstrbuto". M.Sc. Thss, Baghdad Uvrsty, Educato Collg (Ib AL-Hatham). [] Abdulrahma,S.T.()," Estmat Th Ma of Normal Dstrbuto Va Prlmary Tst Shrkag Tchqu", Ib Al-Hatham J.for Pur ad Appld Sc., Vol. 5(), pp [3] Thompso,J.R., (9), "Som Shrkag Tchqus for Estmatg th Ma", J. Amr. Statst. Assoc, Vol.(3), pp.3-.