ESTIMASI DAN RELIABILITAS PADA DISTRIBUSI WEIBULL DENGAN METODE BAYES

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1 LEMMA VOL I NO., NOV 24 ESTIMASI DAN RELIABILITAS PADA DISTRIBUSI WEIBULL DENGAN METODE BAYES Adev Mur Adel Progrm Stud Peddk Mtemtk, Uversts Mhutr Muhmmd Ym, Solok devmur@gml.com Abstrk. Peelt bertuju utuk meetuk estms d relblts deg metode Byes. Metode yg dguk dlh deg meglss teor yg relev deg mslh berdsrk stud ltertur. Hsl dr eelt dlh betuk dr rmeter estms utuk, d estms relblts d dstrbus Webull deg metode Byes utuk sstem deg wktu keggl X, X2,,X deg dstrbus osteror f(,)=/ dlh = ( x ) ( x ), = + ( x ) ( x ), R(t) = ( + t ) ( x ) Kt kuc: rmeter estms, relblts, dstrbus Webull, metode Byes. A. PENDAHULUAN Persg dlm bdg dustr semkest st, membut r roduse berush meujukk kults roduk hsl dustry. Sehgg sewktu rosuse memsrk roduky, hk kosume meggk bhw hk roduse dber forms mege dy throduk tersebut. Utuk megukur dy th d kedl dr sutu roduk hsl dustry, erluk sutu uj ytu uj hdu. Au tuju uj hdu meurut Soejoet (995:) dlh ) megdetfks model sttsk yg sesu bg dstrbus th hdu tu roses keggl, ytu sutu roses yg megkbtk tdk berfugsy ut deg wjr, 2) megestms rmeter-rmeter yg tdk dkethu dr model dtrbus dt d t jug dlkuk sutu uj hotess, 3) meghtug bts kofdes relblts dr komoe th hdu. Utuk megestms l dr sutu rmeter meurut Romeu (23:) membedky ts du, ytu metode klsk d metode Byes. Pd metode klsk rmeter meruk besr yg tet, sedgkd metode Byes rmeter dg sebg eubh ck d memuy sutu dstrbus yg dsebut dstrbus ror yg t mejelsk sutu dstrbus bersyrt eubh ck kotu yg dsebut dstrbus osteror. PRODI PEND. MATEMATIKA STKIP PGRI SUMBAR

2 LEMMA VOL I NO., NOV 24 Sel utuk megestms sutu rmtr, tuju uj hdu dlh utuk megethu relblts sutu sstem. Relblts sutu system ddefsk sebg elug bhw sstem k bekerj lg sedkt utuk sutu erode wktu tertetu t kerusk. Jk ssumsk sstem memlk dstrbus th hdu mk estms relblts t dlkuk mellu fugs dstrbusy. Utuk megestms Relblts meurut Romeu (23:5) metode Byes lebh efse dguk kre t meghslk forms yg lebh byk tetg estms rmeter d relbltsy. Oleh kre tu, utuk mecr relblts t dguk slh stu dr model dstrbus, ytu dstrbus Webull, dstrbus Eksoesl, dstrbus Ekstrm d l sebgy. Meurut Dudewcz (998:7) dstrbus Webull serg dguk dlm model dstrbus uj hdu, kre t memodelk lju keggl dlm berbg ked d t meghslk sebuh edekt yg bk utuk hukum elug dr beber eubh ck sert serg sesu dlm berbg bdg, seert bdg dustr, bdg keseht d l sebgy. B. METODE PENELITIAN Metode yg dguk dlm eelt dlh lss teor-teor yg relev deg ermslh yg dbhs berldskd kj keustk, seert teor dstrbus elug, rt d vrs, metode Byes, estms ttk d Relblts. Au lgkh-lgkh kerj yg dlkuk dlh: ) Meetuk betuk dstrbus ror d dstrbus Webull, 2) Meetuk betuk dr dstrbus osteror deg mesubstusk dstrbus ror yg telh dtetukd dstrbus Webull, 3) meetuk betuk estms dr rmeter d d dstrbus Webull deg metode Byes, 4) Meetuk betuk estms Relblts d dstrbus Webull deg metode Byes. C. HASIL DAN PEMBAHASAN Metode Byes Metode Byes t dguk utuk meetuk dstrbus bersyrt eubh ck kotu. Deg metode Byes, dstrbus bersyrt eubh ck kotu yg dsebut dstrbus osteror t dbetuk deg fugs kemugk deg forms yg l yg telh tersed sebelumy (forms wl) yg dytk deg dstrbus ror. PRODI PEND. MATEMATIKA STKIP PGRI SUMBAR 2

3 LEMMA VOL I NO., NOV 24 Asumsk wktu keggl deg X dlh dstrbus Webull degrmeter (,) deg rmeter betuk yg mecrk sutu vrs d dlh rmeter skl yg mecrk rt dr dstrbus Webull deg fugs dt elugy ( Sh. 98:3) Dstrbus ror f(x;, ) = x e x x>, >, > () Jk x, x2,, x dlh l dr eubh ck dr fugs dt elug (f) dstrbus Webull deg g() d h() dsebut deg dstrbus ror. Kre l d bersft o formtf mk dlm meetuk dstrbus betuk dr dstrbus ror g() dguk tur Jeffreys d h() lh dstrbus uform. Jk rmeter tdk dkethu mk dstrbus ror g() I/2() (Robbert, 994:4). Pedekt l dr tur Jeffrey medekt kr kudrt dr forms Fsher. Seljuty utuk meetuk l dr forms Fsher I() d dstrbus Welbull, substusk teorem:i() = E ( (Robert, 994: 3) I() = E (l x ) x 2 = E + x 22 l f(x, ) 2 I() = 4 {Ex2 2E + 2 } Deg Ex 2 = 2 Γ(3) = 2 2, berdsrk teorem Γ() = ( )! (Wlole, 995: 27), mk eroleh: E = Γ() =, sehgg I() = 4 { } = 2 mk dstrbus ror bg deg megguk tur Jeffrey ytu: g() = 2 = (2) Seljuty dtetuk dstrbus ror bg ytu h()) yg lh dr dstrbus Uform, deg fugs dt elugy f(x) = b eroleh: h() = ( Freud, 999:28), sehgg, < < deg hrg kosts, oleh kre tu dmbl =, mk eroleh dstrbus rory h()= (3) Seljuty dstrbus ror g() d h() dsubstusk utuk memeroleh dstrbus osteror bg rmeter d. PRODI PEND. MATEMATIKA STKIP PGRI SUMBAR 3

4 LEMMA VOL I NO., NOV 24 Dstrbus Posteror Jk x,x2,,x dlh l dr eubh ck deg f dstrbus Webull, deg rmeter, mk fugs kemugky berdsrk defs: L(, X) = f(x, x 2,, x, ) 998: 42) L(, X) = ( ) ex ( x ) (Dudewcz, (4) Deg mesubstusk (2), (3), (4) ke ersm: π(, x,, x ) = f(x,,x,) g().h() f(x,,x,) g().h()d Ω (Soejoet, 988:44) π(. X) = mslk π(, X) = ( ) x x ex( ) ( ) x x ex( ) d x = λ, mk Deg K = mslk + λ ex( ) + λ ex( ) d x = m Mk K = λ K = λ Γ() m + λ ex ( (m ) = Γ(), K = + λ ex ( )(5) ) d, ex ( m ) d(m ) m λ ( x,k = ) K (6) Substuskersm (6) ke Persm (5), sehgg eroleh dstrbus osteror bg (,), ytu: π(, X) = λ ( ) Γ() + λ ( x ) (7) Dstrbus Posteror Mrgl Dstrbus Posteror Mrgl bg Berdsrk defs dstrbus Posteror mrgl bg, (Box&To,992:67) PRODI PEND. MATEMATIKA STKIP PGRI SUMBAR 4

5 LEMMA VOL I NO., NOV 24 π( X) = π(, X) π( X) = π(, X) = q λ ( Γ() + ) λ ( x ) π( X) = Γ() + λ ( x ) Jd dstrbus osteror mrgl bg dlh: λ ex ( x ) π( X) = + λ ex ( ) Γ() λ ( x ) (8) Dstrbus Posteror Mrgl bg Berdsrk defs dstrbus Posteror mrgl bg, (Box&To,992:67) π( X) = π(, X) d = λ ex ( Γ() + ) λ ( x ) d π( X) = K λ + ex ( x π( X) = K λ ( ) + ex ( m ) d ) d, mslk = m = K λ (m ) m ex (m )d(m ) Fktor yg d dlm kurug sku d ersm d ts meruk fugs gmm deg (m ) = y, = α, sehgg eroleh: π( X) = K λ Γ() ( x ) deg K = Jd dstrbus osteror mrgl bg dlh: Γ() λ ( x ) π( X) = λ ( x ) λ ( x ) (9) PRODI PEND. MATEMATIKA STKIP PGRI SUMBAR 5

6 LEMMA VOL I NO., NOV 24 Estms bg Prmeter d deg Metode Byes Jk rmeter d dr dstrbus Webull tdk dkethu, mk estms d t dlkuk mellu estms ttk deg metode Byes. Utuk memeroleh betuk estms rmeter d substuskersm (8) d (9) ke defs: = E( X) = π( X) d Ω = E( X) = Ω π( X) d (Sh, 98: 23) Estms bg deg Metode Byes = E( X) = π( X)d = + λ ex ( Γ() λ ( x ) ) d = Γ() λ ( x ) ( λ ex ( x ) d) = K ( λ ex ( x ) d), mslk = m = K λ ( ( ) ex ( m ) d ) = K λ ( (m ) ( ) m ex( m ) d(m )) Fktor yg d dlm kurug sku d ersm d ts meruk fugs gmm deg (m ) = y d bl ( ) = α, mk: eroleh: = K λ ( Γ( ) = Γ( ) Γ() λ ( x ) m ) = K Γ( ) λ ( x λ ( x ), sehgg ). Jd estms bg deg metode Byes: = ( x ) ( x ) () PRODI PEND. MATEMATIKA STKIP PGRI SUMBAR 6

7 LEMMA VOL I NO., NOV 24 Estms bg deg Metode Byes = E( X) = π( X) = ( x ) ( x ) Jd estms bg deg metode Byes dlh: = + ( x ) ( x ) () Relblts Relblts sutu sstem dlh elug bhw sstem k bekerj sesu deg fugsy t kerusk, lg sedkt utuk sutu erode tertetu. Meurut Sh, 98:),Mslk X dlh wktu hdu dr sutu sstem. Relblts sstem d wktu t ddefsk R(t) = P(X t) = F(t), dm F(t) dlh fugs dstrbus dr wktu keggl X. Asumsk dstrbus wktu keggl dlh dstrbus Webull degrmeter d,mk fugs dstrbusy: t F(t) = x ex ( x ) dx = ex ( t ), Relblts d wktu t dlh: R(t) = ( ex ( t )) = ex ( t ) (2) Mslk wktu keggl X=X, X2,, X berdstrbus Webull degrmeter d yg memlk dstrbus ror g(), h(), mk estms Relblts d dstrbus Webull deg metode Byes megguk defs R(t) = E(R(t) X) (Sh, 98: 28) R(t) λ ( = R(t)π(, X)d = ex ( t ) x ) Γ() + λ ( x ) d R(t) = Mslk ( K λ ex ( ( + t ) ) + d x + t ) = w, mk, R(t) = K λ ( ) + ex ( w ) d PRODI PEND. MATEMATIKA STKIP PGRI SUMBAR 7

8 LEMMA VOL I NO., NOV 24 R(t) = K λ (w ) ex( w ) d (w ) w Fktor yg d dlm kurug sku d ersm d ts meruk fugs gmm deg (w ) = Z d bl = α, mk berdsrk teorem jk blg sl mk Γ() = ( )! (Wlole, 995: 27), eroleh: R(t) = K λ Γ() w, Deg K = Γ() λ ( x ), sehgg eroleh: R(t) = Γ() Γ() λ ( x ) λ o ( x +t ) Jd estms Byes utuk relblts dlh: R(t) = λ ( x +t ) o λ ( x ) (3) Dm R(t) dlh setms Byes utk relblts sutu sstem d wktu t, yg dsebut jug tksrelug deg meggukedekt metode Byes d sutu sstem yg jug bekerj sesu deg fugsy t meglm kerusk, lg sedkt d wktu t. D. KESIMPULAN DAN SARAN Berdsrk temueelt dembhs yg telh dlkuk, dlm eelt t dsmulk bhw: Dstrbus ror gbug dr rmeter d d dstrbus Webull t drumusk: f(,)=g() h() = = Betuk estms d d dstrbus Webull, yg erolh mellu eggu dstrbus ror deg metode Byes t dtetuk deg rumus: = ( x ) ( x ) = + ( x ) ( x ) Betuk estms Relblts d dstrbus Webull yg terdr dr system deg wktu keggl X, X2,,X yg dtetuk mellu metode Byes dlh: PRODI PEND. MATEMATIKA STKIP PGRI SUMBAR 8

9 LEMMA VOL I NO., NOV 24 R(t) = ( + t ) ( x ) Au sr yg dberk sehubug degeelt dlh gr eelt berkuty t megembgkeelt utuk model dstrbus ly d megguk metode l utuk meetuk estms ttk d relbltsy tu meglksy d ermslh yt dlm kehdu sehr-hr, seert dustry bol lmu. Relblts (kedl) t dtgktk deg ketet dlm memlh dstrbus rory, kre semk besr dstrbus ror k semk besr ul relbltsy(kedl dr sutu roduk k berth lm). DAFTAR PUSTAKA. Box&To Byes Iferece Sttstcl Alyss. Jho Wley & Sos, Ic, Cd. 2. Dudewcz, E.J. d Mshr, S.N Sttstk Mtemtk Moder. ITB, Bdug. 3. Freud, J.E d Wlole, R.E Mthemtcl Sttstcs. Pretce Hll, New Jersey. 4. Robert, Crst P The Byes Choce. Srger-Verlg New York, Ic, New York. 5. Romeu, J.L.23. Use of Byes Techque or Relblty. Jourl of RAC START, volume, Number 8, htt://rc.loscece.com 6. Sh, S.K d Kle, B.K. 98. Lfe Testg d Relblty Estmto. Wle Ester lmted, New Delh. 7. Wlole, R.E d Myer, Rymod H Ilmu Pelug d Sttstk utuk Isyur d Ilmu. Eds ke-4. ITB, Bdug. 8. Soejoet, Zzw Als Dt Uj Hdu. UGM. Yogykrt. 9. Soejoet, Zzw d Soebr Iferes Byes. Uversts Terbuk, Derteme Peddk d Kebudy. PRODI PEND. MATEMATIKA STKIP PGRI SUMBAR 9