Suzan Mahmoud Mohammed Faculty of science, Helwan University

Transcription

1 Europa Joural of Statstcs ad Probablty Vol.3, No., pp.4-37, Ju 015 Publshd by Europa Ctr for Rsarch Trag ad Dvlopmt UK ( ESTIMATION OF PARAMETERS OF THE MARSHALL-OLKIN WEIBULL DISTRIBUTION FROM PROGRESSIVELY CENSORED SAMPLES Mahmoud Rad Mahmoud Isttut of Statstcs, Caro Uvrsty Suza Mahmoud Mohammd Faculty of scc, Hlwa Uvrsty ABSTRACT: Ths modl ca b cosdrd as aothr usful 3-paramtr gralzato of th Wbull dstrbuto. Th stmato of paramtrs of Marshall ad Olk Wbull dstrbuto udr progrssv csorg s vstgatd, maxmum lklhood stmators of th ukow paramtrs ar obtad usg statstcal softwar (Mathmatca, MLE prforms for dffrt samplg schms wth dffrt sampl szs s obsrvd, ad th asymptotc varac covarac matrx s computd also. KEYWORDS Modfd Wbull dstrbuto, progrssv typ II csorg, Maxmum lklhood stmators, Asymptotc varac covarac matrx. INTRODUCTION Marshall ad Olk (1997 proposd a modfcato of th stadard Wbull modl through th 0, ladg to a cumulatv dstrbuto troducto of a addtoal paramtr fucto of th form ( t 1 Ft,,, 0. t 1 1 (1-1 Wth probablty dstrbuto fucto t 1 t f ( t. t 1 1 (1- Th quatl fucto s gv by 1u 1 log 1u ( 1 t F u 1 (1-3 4 ISSN (Prt, ISSN (Ol

2 Europa Joural of Statstcs ad Probablty Vol.3, No., pp.4-37, Ju 015 Publshd by Europa Ctr for Rsarch Trag ad Dvlopmt UK ( Th hazard rat s t v 1 ht (. t 1 1 (1-4 Th followg dagram provds graphs of th modfd Wbull dsty fucto for slctd valus of th paramtrs. Fg (1-1 dsty fucto of modfd Wbull dstrbuto Th followg dagram provds graphs of modfd Wbull hazard rats Fg (1- hazard rats of modfd Wbull dstrbuto 5 ISSN (Prt, ISSN (Ol

3 Europa Joural of Statstcs ad Probablty Vol.3, No., pp.4-37, Ju 015 Publshd by Europa Ctr for Rsarch Trag ad Dvlopmt UK ( Marshal ad Olk (1997 studd th hazard rat ad foud that t s crasg f α 1, β 1 ad dcrasg f 1, β 1. If β > 1 th th hazard rat s tally crasg ad vtually crasg, but thr may b o trval whr t s dcrasg.smlarly, f β < 1 th th hazard rat s tally dcrasg ad vtually dcrasg, but thr may b o trval whr t s crasg. Marshal ad Olk (1997 proposd a way of troducg a paramtr, to xpad a famly of dstrbutos. Ramsh C. Gupta t al. (010 compar th modfd dstrbuto ad th orgal dstrbuto wth rspct to som stochastc ordrgs. Also vstgatd thoroughly th mootocty of th falur rat of th rsultg dstrbuto wh th basl dstrbuto s tak as wbull. It turs out that th falur rat s crasg, dcrasg, or o-mootoc wth o or two turg pots dpdg o th paramtrs. For o-mootoc typs, th turg pots of th falur rat ar stmatd ad thr cofdc trvals ar provdd. Smulato studs ar carrd out to xam th prformac of ths trvals. MAXIMUM LIKELIHOOD ESTIMATION I cas complt data Cosdr a radom sampl cosstg of obsrvatos. Th lklhood fucto of ths sampl s L( t, t,..., t ;,, f ( t ( ( t 1 1 ( ( t 1 (,,, (1 6 t Lt 1 O takg logarthms of (1-6, (1 ( L. L log( ( t ( 1 ( t log(1 (1 (1 7 Dffrtatg (1-7 wth rspct to α, λ ad β ad quatg to zro ar: t L. L 0, (1 8 t 1 1 (1 t 6 ISSN (Prt, ISSN (Ol

4 Europa Joural of Statstcs ad Probablty Vol.3, No., pp.4-37, Ju 015 Publshd by Europa Ctr for Rsarch Trag ad Dvlopmt UK ( L. L ( 1 (1 ( t * ( t 0, (1 9 t 1 1 (1 (1 L. L (1 ( t log( t ( t log( t log( t 0, (1 10 t (1 Th maxmum lklhood stmat (MLE ( ^, ^, ^ of (,, s obtad by solvg th olar lklhood quatos (1-8, (1-9 ad (1-10. Ths quatos caot b solvd aalytcally ad popular softwar ca b usd to solv ths quatos umrcally. I cas progrssv typ two rght csord data. Lt t1, t, t b dpdt ad dtcally dstrbutd radom lftms of tms. A typ ll progrssvly rght csord sampl may b obtad th followg way: th falur tm of th frst t1,r1 ar ot obsrvd; at th tm of th (r+1st falur, dotd wth tr+1,rr+1 umbr of th rmag uts ar wthdraw from th tst radomly. At th tm of scod falur, dotd wth t, Rr+survvg tms ar rmovd at radom from th rmag tms, ad so o. At th tm of th mth falur, wh m s a prdtrmd umbr all th rmag Rr+m tms ar csord. Thrfor, a progrssvly typ-ll rght csorg schm s spcfd by tgr umbrs, m ad r1, rm-1 wth th costrats - m- r1 -.rm-1 0, m 1. Lt Th lklhood fucto to b maxmzd wh gral progrssv typ II csord sampl basd o dpdt wth F (t s L (θ ( * mr r R r 1,, 1, (1 11 r1 L c F t f t F t Whr,, t t... t, r1 r m t t! , r! r 1! * c R r R R m r 1 r 1 m 1 Th lklhood fucto gv Eq. (1.11 ca b xprssd as: L t * ;,, c 1 r t t R r mr t r 1 t t 1 t t (1 1 7 ISSN (Prt, ISSN (Ol

5 Europa Joural of Statstcs ad Probablty Vol.3, No., pp.4-37, Ju 015 Publshd by Europa Ctr for Rsarch Trag ad Dvlopmt UK ( Th log-lklhood fucto, dotd by t ;,, l l, taks th form t r 1 t ;,, costat m r log rlog 1 t r log log 1 1 t t t mr 1 R m r m r m r t log t Th thr quatos obtad by dffrtatg (1.13 wth rspct to α, λ ad β ad quatg to zro ar: r1 F ( t r1 r( mr ( t m r (1 B ( t r1 (1 B ( t r1 r 1 r1 1 1 Bt ( 1 Bt ( r1 r 1 r1 F ( t r1 (1 B ( t ( R mr (1 B ( t r1 (1 B ( t r1 0, (1 14 r 1 G ( t r 1 F ( t r 1 G ( t r 1 r( m r m r ( t m r ( m r ( 1 1 B ( t r1 (1 B ( t r1 B ( t G ( t Gt ( r 1 r 1 r Bt ( 1 Bt ( mr r1 r 1 G ( t F ( t G ( t (1 B ( t R ( 1 B ( t (1 B ( t 0, (1 15 ( r 1 C ( t r 1 F ( t r 1 C ( t r 1 r m r m r m r ( t m r 1 B ( t r1 (1 B ( t r1 B ( t C ( t log[ t ] C ( t r 1 r 1 r 1 r Bt ( 1 Bt ( mr r 1 C ( t F ( t C ( t (1 B ( t R ( 1 B ( t (1 B ( t 0, (1 16 r1 8 ISSN (Prt, ISSN (Ol

6 Europa Joural of Statstcs ad Probablty Vol.3, No., pp.4-37, Ju 015 Publshd by Europa Ctr for Rsarch Trag ad Dvlopmt UK ( Whr, r 1 r 1 r 1 r 1 t t r 1 B ( t (1 r 1 t t r 1 r 1 r 1 r 1 C ( t log[ t ]( t r 1 r 1 r 1 F ( t (1 G ( t t t r 1 1 Th maxmum lklhood stmat (MLE ˆ, ˆ, ˆ of,, s obtad by solvg th olar lklhood quatos (1-14, (1-15 ad (1-16. Ths quatos caot b solvd aalytcally ad popular softwar ca b usd to solv th quatos umrcally. Fshr formato matrx I ( l l l E E E 1 I ( l l l E E E. l l l E E E Th lmts of th sampl formato matrx, for progrssvly typ II csord wll b r( K ( t H ( t r ( (1 1 (1 B ( t (1 B ( t r 1 r 1 r 1 r 1 3 m r (1 B ( t r 1 (1 B ( t r 1 (1 B ( t r 1 (1 B ( t r 1 r1 r1 Qt ( mr m r 1 r1 (1 Bt ( r 1 r1 r1 r1 Q (t H( t r1((1 B ( t ( R 3 (1 B ( t (1 B ( t K ( t K ( t r 1((1 B ( t ( R ( R (1 B ( t (1 B ( t (1 B ( t (1 B ( t ( ISSN (Prt, ISSN (Ol

7 Europa Joural of Statstcs ad Probablty Vol.3, No., pp.4-37, Ju 015 Publshd by Europa Ctr for Rsarch Trag ad Dvlopmt UK ( r 1 G ( t r 1 L ( t r 1 G ( t r 1 K ( t r 1 G ( t r 1 S ( t r 1 G ( t r 1 r ( 3 (1 B ( t r 1 (1 B ( t r 1 (1 B ( t r 1 (1 B ( t r 1 r 1 1 (1 Bt ( r 1 r 1 K ( t r 1 r 1 G ( t r 1 a( t r1 G ( t r1 r( ( (1 B ( t r1 (1 B ( t r1 (1 B ( t r1 (1 B ( t r1 r 1 (1 (1 Bt ( r 1 G ( t L ( t G ( t ( mr r1 (1 B ( t (1 B ( t K ( t (1 B ( t ( G ( t R mr (1 B ( t (1 B ( t ( r1 K ( t (1 ( G ( t R (1 B ( t (1 B ( t G ( t L ( t G ( t K ( t G ( t S ( t r 1 G ( t r 1 ((1 B ( t R ( 3 (1 Bt ( (1 B ( t (1 B ( t (1 B ( t (1 18 r 1 C ( t r 1 L ( t r 1 C ( t r 1 K ( t r 1 C ( t r 1 S ( t r 1 C ( t r 1 r ( 3 (1 B ( t r 1 (1 B ( t r 1 (1 B ( t r 1 (1 B ( t r 1 r 1 1 (1 Bt ( r 1 r 1 K ( t r 1 r 1 C ( t r 1 a( t r1 C ( t r1 r ( ( (1 B ( t r1 (1 B ( t r1 (1 B ( t r1 (1 B ( t r1 r 1 (1 (1 Bt ( r 1 C ( t L( t C ( t ( mr r1 (1 B ( t (1 B ( t 30 ISSN (Prt, ISSN (Ol

8 Europa Joural of Statstcs ad Probablty Vol.3, No., pp.4-37, Ju 015 Publshd by Europa Ctr for Rsarch Trag ad Dvlopmt UK ( K ( t (1 B ( t ( C ( t R mr (1 B ( t (1 B ( t 1 K ( t (1 ( C ( t R (1 B ( t (1 B ( t C ( t L ( t C ( t K ( t C ( t S ( t C ( t ((1 B ( t R ( 3 (1 B ( t (1 B ( t (1 B ( t (1 Bt ( m r ( m r ( 1 r 1 G ( t r 1 a( t r 1 G ( t r 1 r ( (1 B ( t r1 (1 B ( t r1 r 1 (1 (1 Bt ( r 1 ( 1 N ( t r 1 a( t r 1 ( 1 N ( t r 1 r 1 w ( t r 1 3 a( t r 1 w ( t r 1 d ( t r 1 w ( t r 1 r ( 3 (1 B ( t r 1 (1 B ( t r 1 (1 B ( t r 1 (1 B ( t r1 (1 B ( t r1 r 1 1 (1 Bt ( (1 B ( t ( 1 N ( t (1 B ( t w ( t f ( t N ( t ( 1 ( ( m r m r N t r 1 r 1 (1 B ( t (1 B ( t (1 B ( t r 1 r 1 (1 19 G ( t a( t G ( t (1 R G ( t ( (1 B ( t (1 B ( t ( 1 N ( t a( t ( 1 N ( t w ( t 3 a( t w ( t d ( t w ( t (1 B ( t R ( 3 (1 B ( t (1 B ( t (1 B ( t (1 B ( t (1 B ( t ( ISSN (Prt, ISSN (Ol

9 Europa Joural of Statstcs ad Probablty Vol.3, No., pp.4-37, Ju 015 Publshd by Europa Ctr for Rsarch Trag ad Dvlopmt UK ( m r r r 1 G ( t r 1 a( t r 1 G ( t r 1 r 1 C ( t r 1 a( t r 1 C ( t r 1 ( ( (1 B ( t r 1 (1 B ( t r 1 (1 B ( t r 1 (1 B ( t r 1 r 1 (1 (1 Bt ( r (1 Bt ( r 1 r 1 r 1 r 1 r 1 r 1 r 1 (1 Bt ( r 1 (1 B ( t r1 (1 B ( t r1 r 1 g ( t a( t r ( g ( t b( t B ( t g ( t B ( t b( t B ( t ( t f ( t ( t ( mr r1 (1 B ( t (1 B ( t (1 B ( t (1 B ( t G ( t a( t G ( t (1 B ( t R C ( t ( mr (1 B ( t (1 B ( t ( r1 G ( t a( t G ( t (1 R C ( t ( (1 Bt ( (1 Bt ( 1 g ( t a( t g ( t b( t a( t b( t ((1 B ( t R ( (1 B ( t (1 B ( t (1 B ( t (1 B ( t ( t 3 a( t ( t d ( t ( t (1 1 (1 ( (1 ( (1 ( 3 B t B t B t 3 ISSN (Prt, ISSN (Ol

10 Europa Joural of Statstcs ad Probablty Vol.3, No., pp.4-37, Ju 015 Publshd by Europa Ctr for Rsarch Trag ad Dvlopmt UK ( r 1 C ( t r 1 a( t r 1 C ( t r 1 r ( m r (1 B ( t r1 (1 B ( t r1 r 1 (1 (1 Bt ( r 1 r 1 c( t r 1 a( t r 1 c( t r 1 r 1 h( t r 1 3 a( t r 1 h( t r 1 d ( t r1 h( t r1 r ( (1 ( B t r 1 (1 B ( t r 1 (1 B ( t r 1 (1 B ( t r 1 3 (1 Bt ( r 1 r 1 (1 (1 Bt ( r 1 B ( t B ( t h( t f ( t h( t ( ( m r m r ct r 1 r 1 (1 B ( t (1 B ( t (1 B ( t C ( t a( t C ( t (1 B ( t RC ( t ( mr (1 B ( t (1 B ( t ( r1 C ( t a( t C ( t (1 RC ( t ( (1 B ( t (1 B ( t c( t a( t c( t h( t 3 a( t h( t d ( t h( t R (1 B ( t ( 3 (1 B ( t (1 B ( t (1 B ( t (1 B ( t (1 B ( t (1 Whr, Q ( t r 1 r 1 r 1 r 1 r 1 r 1 H ( t t K ( t L ( t t 3 t r 1 t r 1 t r 1 r 1 r 1 N ( t t ( t 33 ISSN (Prt, ISSN (Ol

11 Europa Joural of Statstcs ad Probablty Vol.3, No., pp.4-37, Ju 015 Publshd by Europa Ctr for Rsarch Trag ad Dvlopmt UK ( S ( t (1 r 1 a( t (1 r 1 3 t t t t r 1 r 1 b ( t t log[ t ]( t 1 r 1 r 1 r 1 r 1 c( t log[ t ] ( t r 1 r 1 r 1 d ( t (1 r 1 3 r 1 ( t t log[ t ]( t 1 r 1 r 1 r 1 r 1 f ( t (1 r 1 g ( t t ( t r1 r1 1 r 1 h( t log[ t ]( t r 1 r 1 r 1 w ( t t ( t r1 r1 For 0, th maxmum lklhood stmators ( stmators, ad ˆ, ˆ, ˆ varac-covarac matrx ^ ^ ^,, of (,,,ar cosstt s asymptotcally to ormal wth ma vctor 0 ad 1 I. Asymptotc varac covarac matrx l l l V Cov, Cov, va r,, Cov, V Cov, l l l Cov, Cov, V l l l Numrcal xprmts ad data aalyss,,,,,,,,,,,,,,,,,, I ths scto, w prst som rsults from gratg progrssv csord typ II data from xtdd Wbull dstrbuto by usg Mathmatca to obsrv how th MLEs prform for dffrt samplg schms ad for dffrt sampl szs, Th asymptotc varac covarac matrx s computd also. W hav tak =50, 100 ad 150, m=0, 40 ad 60. Dffrt samplg schms. Dffrt valus of th paramtrs ar takα=1.8,.0, ad 3.0, λ=0.5, 1.5, ad 3, β=0.5, 1.5 ad3 I ach cas, w hav calculatd th MLEs. W rplcat th procss 1000 tms ad comput th avrag bass ad stadard dvatos of th dffrt stmats. Th schm (, m, r,. 34 ISSN (Prt, ISSN (Ol

12 Europa Joural of Statstcs ad Probablty Vol.3, No., pp.4-37, Ju 015 Publshd by Europa Ctr for Rsarch Trag ad Dvlopmt UK ( Schm[1] =50, m=0, r=0, R1= R= R3=... R19=1 R0=11. Schm [] =100, m=40, r=0, R1= R= R3= R39=1 R40= Schm [3] =150, m=60, r=0, R1= R= R3= R59 =1 R60=31. Schm[4] =00, m=80, r=0, R1= R= R3= R79=1, R80=41. Tabl 1: Estmato of paramtrs wh α=3, λ=1.5, β=0.5. Schm va r,, (50, 0, (100,40, Tabl Estmato of paramtrs usg Schm[3] α λ β va r,, ISSN (Prt, ISSN (Ol

13 Europa Joural of Statstcs ad Probablty Vol.3, No., pp.4-37, Ju 015 Publshd by Europa Ctr for Rsarch Trag ad Dvlopmt UK ( Tabl 3 Estmato of paramtrs usg Schm[4] α λ β va r,, Baysa Estmato Hr w cosdr th bays stmato of th ukow paramtrs wh both ar ukow. Hr w assum gamma pror for α, λ ad β b1 f ( f f 1 3 ( ( d1 k 1 a c m If t1, t, t ar th radom sampl of sz th th lklhood fucto s gv by Lt (,,, = ( ( t 1 1 ( t 1 1 t (1 (1 (1 Th jot postror dsty fucto of α,λ ad β ca b wrtt as L( t,,, f ( f ( f ( 1 3 (,, ( L( t,,, f ( f ( f ( d ( d ( d ( 1 3 Thrfor, th bays stmator of ay fucto of α, λ ad β say g (,, E,, t (g(,, = L ( t,,, f ( f ( f ( d ( d ( d ( 1 3 L ( t,,, f ( f ( f ( d ( d ( d ( 1 3 (1 4 It s ot possbl to comput (1-4 aalytcally ths cas thrfor w us MCMC mthods to fd th bays stmator of α, λ ad β usg α=3, λ=1ad β= 36 ISSN (Prt, ISSN (Ol

14 Europa Joural of Statstcs ad Probablty Vol.3, No., pp.4-37, Ju 015 Publshd by Europa Ctr for Rsarch Trag ad Dvlopmt UK ( Numrcal xprmts ad data aalyss ma SD mc rror.5% mda 97.5% ˆ ˆ ˆ CONCLUSION W ca say whl th Nwto-Raphso mthod dd ot covrg may cass or covrgd to strag uaccptabl rsults, th bscto mthod, whl slow, gav good rsults udr dffrt progrssv samplg schms. As a rul of thumb, bttr rsults ar obtad wh th total umbr of obsrvatos s at last four tms as much as th umbr of paramtrs. Also, dffrt samplg schms gav comparabl rsults wth th sampl szs,, cosdrd. REFERENCE Balakrsha N., 007 Progrssv csorg mthodology: a apprasal. TEST 16: Balakrsha. N. ad Aggarwala, R., 000. Progrssv csorg, thory, mthods ad applcatos. Brkhausr, Bosto. Balakrsha N. ad Aggarwala R., 000. Progrssv csorg, thory, mthods ad applcatos. Bosto, Brkhausr. Balakrsha N. ad Kaa N., 001. Pot ad trval stmato for paramtrs of th logstc dstrbuto basd o progrssvly typ-ii csord data. I: Hadbook of Statstcs. Vol.0. North Hollad: Amstrdam. Coh A.C., Progrssvly csord sampls lf tstg. Tchomtrcs 5: Gupta R.D. ad Kudu D., Gralzd xpotal dstrbuto. Aust N Z J Stat 41: Gupta R.D. ad Kudu D., 001. Gralzd xpotal dstrbuto, dffrt mthods of stmatos. J Stat Comput Smul 69: Gupta R.D. ad Kudu D., 006, Comparso of th Fshr formato matrcs of th Wbull ad GE dstrbutos. J Stat Pla Ifrc 136: Gupta R.D. ad Kudu D., 007. Gralzd xpotal dstrbuto; xstg mthods ad som rct dvlopmts. J Stat Pla Ifrc 137: Gupta R. C., Lv S. ad Pg C., 010. Estmatg turg pots of th falur rat of th xtdd Wbull dstrbuto. Computatoal Statstcs ad Data Aalyss 54: Marshall, A.W. ad Olk, I., A w mthod for addg a paramtr to a famly of dstrbutos wth applcato to th xpotal ad Wbull famls. Bomtrka 84 (3: ISSN (Prt, ISSN (Ol