Maximum Likelihood Estimation of Two Unknown. Parameter of Beta-Weibull Distribution. under Type II Censored Samples

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1 Applied Mthemticl Scieces, Vol. 6, 12, o. 48, Mximum Likelihood Estimtio of Two Ukow Prmeter of Bet-Weibull Distributio uder Type II Cesored Smples M. R. Mhmoud d R. M. Mdouh Istitute of Sttisticl Studies & Reserch Ciro Uiversity, Egypt mhmoudrid@live.com, rshmdoh@yhoo.com Abstrct I this pper, the mximumlikelihood estimtes (mles) re obtied for the two ukow prmeters of the Bet-Weibull(B-W)distributio uder type II cesored smples. Also,symptotic vrices d covrice mtrix of the estimtors regive. A itertive procedure is used to obti the estimtors umericlly usig MthCd Pckge. To study the properties of mximum likelihood estimtors simultio results re icluded for differet smple sizes. Keywords: the Bet-Weibull distributio; cesored type II;mximum likelihood estimtes; vrice covrice mtrix 1. Itroductio The Weibull distributio, med fter WllodiWeibull (1939) is oe of the best kow distributios d hs wide pplictios i diverse disciplies (see Johso et l d Murthy et l. 04). The Weibull fmily, chrcterized by shpe prmeter d scle prmeter, hs bee exteded i vrious wys to coti other distributios. Mudholkr d Srivstv (1993) itroduced expoetited versio of the Weibull model tht icluded dditiol shpe prmeter. The distributio hs closed form of probbility desity, survivl, d hzrd fuctios tht re flexible d ble to geerte wide vriety of frequetly observed hzrd shpes, icludig uimodl d bthtub. Mudholkret l. (1996) proposed other geerliztio of the Weibull model, which is ble to geerte similr types of hzrd shpes s the expoetited model; however, irregulrities my rise s the

2 2370 M. R. Mhmoud d R. M. Mdouh support of the distributio becomes depedet o theprmeter spce. Fmoyeet l. (05) itroduced further extesio of the expoetitedweibull distributio which they clled the Bet- Weibull distributio. Let G(x) be the cumultive distributio fuctio (cdf) of rdom vrible X. The cumultive distributio fuctio for geerlized clss of distributios for the rdom vrible X ws defied by Eugee et l. (02) s follows: Let x 0 I x (, b) = B(, b) 1 w 1 (1 w) b 1 dw, deotes the icomplete bet fuctio rtio (the cdf of the bet distributio with prmeters > 0 d b > 0) d B(, b) is the stdrd bet fuctio, the F(x) = I G(x) (, b) = 1 B(,b) 0 G(x) w 1 (1 w) b 1 dw, (1) Joes (04) oted tht the prmeters > 0 d b > 0 hve role of the skewess d the til weight. The distributio F well be clled the bet G distributio d the correspodig probbility desity fuctio for this geerlized clss of distributio F(x) is give by f(x) = 1 B(,b) G(x) 1 1 G(x) b 1 G (x) (2) Sigh et l. (1988) first itroduced the fmily of distributios i (2) while cosiderig pplictios i survivl lysis of lug ccer dt d they studied the cse whe G(x) is the cdf of log-logistic distributio. Eugee et l. (02) studied the cse whe G(x) is the orml cumultive distributio fuctio with prmeters μ d σ d poited out the reltioship of the fmily i (2) with the distributio of logistic order sttistics. Joes (04) studied some geerl properties of this fmily of distributios d itroduced the specil cses of skewed t d log F distributios. From (1), replcig G(x) by the cdf of Weibull distributio with prmeters c d λ, the cdf of the bet Weibull distributio is obtied s follow: Let F(x) = I 1 exp{ (λx) c }(, b) (3) for x > 0,, b, c d λ > 0.The correspodig probbility desity fuctio (pdf) d the hzrd rte fuctio ssocited with (3) re: f(x) = cλ c B(,b) xc 1 exp{ b(λx) c }[1 exp{ (λx) c }] 1 (4) Ad

3 Mximum likelihood estimtio 2371 h(x) = cλc x c 1 exp{ b(λx) c }[1 exp{ (λx) c }] 1, (5) B exp{ (λx) c } (b,) respectively. Fmoyeet l. (05) studied some of the properties of extesio of the Weibull fmily which they clled the bet-weibull distributio.lee et l. (07) discussed some properties of the bet-weibull model d providedformuls for the hzrd fuctio d discussed their properties uder differet vlues for the prmeters. Cordeiroet l. (08) gve expsio for the cumultive distributio fuctio, the momets d the momet geertig fuctio of the bet-weibull distributio. They lso discussed mximum likelihood estimtio from complete smples d provided formuls for the elemets of the fisher iformtio mtrix, s well s demostrtio of its usefuless o rel dt set.whedet l. (09) ivestigted the potetil usefuless of the bet-weibull distributio for modelig cesored survivl dt from biomedicl studies. Specil Cses: The BW distributio cotis severl well-kow distributios s specil cses. For exmple, (i) The Weibull distributio is clerly specil cse whe = b = 1. Also if = 1, the BW distributio becomes the Weibull distributio with prmeters λb 1/c d c. (ii) Whe c=-k the BW reduces to the bet- type 2 Extreme vlue distributio (bet-fréchet distributio). (iii) It simplifies to the bet- Ryleigh (BR) distributio whe c = 2. (iv) If c = 1, it reduces to the bet- expoetil (BE) distributio. (v) The expoetitedweibull (EW) distributio is lso specil cse whe b =1. (vi) If c = 2 d b = 1, it gives s specil cse the geerlized Ryleigh (GR) distributio. (vii) For = 2, b=1/αd λ c = αβ, the BW distributio reduces to weighted Weibull distributio (Shhbz et l. (11)). (viii) Whe (= i) d b re itegers, the BW is the distributio of the i th order sttistic from Weibull popultio with prmeters c d λ. (ix) Whe b=1 d c=2, the BW reduces to the two-prmeter Burr type X distributio with desity fuctio f(x) = 2λ 2 1 e (λx)2 1 e (λx)2. I sectio (2), we cosider mximum likelihood estimtio (MLE) of two ukow prmeters of bet Wiebull distributio uder type II cesored smples. Sectio (3) gives the symptotic vrice covrice mtrix. The results of simultio studies will be discussed i sectio (4). Tbles d figures re displyed i the Appedix.

4 2372 M. R. Mhmoud d R. M. Mdouh 2. Mximum Likelihood Estimtio If d b re kow the problem of estimtig c d λ becomes simply estimtig the prmeters of the stdrd Weibull distributio d it will ot be discussed here. It shll be ssumed tht c d re ukow d their ML estimtes will be obtied. Suppose tht items, whose life times follow bet-weibull distributio (4) where the prmeters b d λ re kow, re put o test, the test is termited whe the rth item fils; the lifetimes of these first r filed items sy x (1), x (2),, x (r) re observed. The likelihood fuctio is give by: r cλ L(c, ) = C c x B(,b) (i) c 1 exp bλx (i) c 1 exp λx (i) c 1 r i=1 Ie λx c(b, ) (6) (r) Tkig logrithm of (4.1), the log-likelihood fuctio is ll = lc + rl c + rc l λ rl B(, b) + (c 1) lx (i) ( 1) l1 exp λx (i) c + ( r) l Ie λx (r) c(b, ) r i=1 r b λx (i) c r i=1 (7) After tkig prtil derivtives of the log-likelihood i (7) with respect to c d, d equtig the derivtives to zero, the followig equtios re obtied: ll = c r c + rl λ + r i=1 l (x (i)) b r λx (i) c i=1 lλx (i) + ( 1) exp λx (i) c λx (i) c lλx (i) r i=1 1 exp λx (i) c + ( r) c I exp λx (r) c (b,) i=1 I exp λx(r) c (b,) = 0, (8) + d ll = rψ() + rψ( + b) + r l1 exp λx (i) c i=1 + ( r) I exp λx (r) c (b,) (9) I exp λx(r) c (b,) where ψ() = lγ() which is clled digmm fuctio. Now I 1 c exp λx (r) c (b, ) = λx B(,b) (r) c lλx (r) e bλx (r) c 1 e λx (r) c 1, d

5 Mximum likelihood estimtio 2373 I exp λx (r) c (b, ) = 1 e λx (r) c B(,b) 0 tb 1 (1 t) 1 l(1 t) dt {ψ() ψ( + b)}. B(, b)i exp λx(r) c (b, ) (10) For computtiol purposes we rewrite the R.H.S of (10) s 1 1 B(, b) k + 1 k=0 k=0 0 e λx (r) c t b+k (1 t) 1 dt {ψ() ψ( + b)}. B(, b)i exp λx(r) c (b, ) 1 1 = I B(, b) k + 1 exp λx(r) c (b + k + 1, ) {ψ() ψ( + b)}. B(, b)i exp λx (r) c (b, ), wherel(1 u) = b)}. k=0 uk+1 k+1, u 1, u 1 d B(, b) = B(, b){ψ() ψ( + The L.H.S. of (10) c be obtied by usig the reltio betwee the icomplete bet fuctio d the Guss hypergeometric fuctio s follows: I exp λx (r) c (b, ) = I 1 exp λx (r) c (, b) = l1 exp λx (r) c + ψ() ψ( + b)i 1 exp λx(r) c (, b) + 1 exp λx (r) c 2 3F B(, b) 2,, 1 b; + 1, + 1; 1 exp λx (r) c = l1 exp λx (r) c + ψ() ψ( + b)i 1 exp λx(r) c (, b) Γ()Γ( + b) + 1 exp λx Γ(b) (r) c 3F 2,, 1 b; + 1, + 1; 1 exp λx (r) c. where F p q 1,., p ; b 1,., b q ; z = hypergeometric fuctio (see referece [16]). p F q 1,., p ;b 1,.,b q ;z Γ(b 1 ).Γb q is regulrized The equtios (8) d (9) do ot hve lyticl solutio d they must be solved umericl. For this purpose, the populr umericl softwre pckge MthCd ws used.

6 2374 M. R. Mhmoud d R. M. Mdouh Usig ivrice property of mximum likelihood estimtio (see Mood et l 1974), we c obti the MLE of the hzrd rte fuctio of B-W distributio. From (5), replcig the prmeters c d by their MLEs c d i.e, h(x) = c λc xc 1 exp b(λx) c 1 1 exp (λx) c B exp (λx) c (b, ). 3. The Asymptotic Vrice Covrice Mtrix The symptotic vrice-covrice mtrix for the estimtorsc d c be obtied by ivertig the iformtio mtrix with the elemets tht re egtive of the expected vlues of the secod order derivtive of logrithms of the likelihood fuctios. The secod prtil derivtives of the log- likelihood fuctio re obtied i the followig: 2 ll c 2 = r c 2 b r λx (i) c i=1 1) λx (i) c lλx (i) 2 e λx i r r) i=1 lλx (i) 2 + ( 1 e λx (i) c c 1 λx(i) c λx (i) c 2 lλx (i) e λx (i) c 1 e λx (i) c I exp λx(r) c (b,). 2 c 2I exp λx (r) c (b,) c I exp λx (r) c (b,) 2, I exp λx(r) c (b,) where 2 c 2 I exp λx (r) c ( (b, ) = 1 lλx B(,b) (r) 2 λx (r) c e bλx (r) c 1 e λx (r) c 2 ( 1)λx (r) c e λx (r) c + 1 e λx (r) c (1 bλx (r) c ). 2 ll = c exp λx (r) c λx (i) c lλx (i) r i=1 1 exp λx (i) c r) + ( I exp λx(r) c 2 (b,). c I exp λx (r) c (b,) c I exp λx (r) c (b,). I exp λx (r) c (b,) 2, I exp λx(r) c (b,) 2 c I exp λx (r) c (b, ) 1 = [ψ() ψ( + b) B(, b) l 1 e λx (r) c ]λx (r) c lλx (r) e bλx (r) c 1 e λx (r) c 1.

7 Mximum likelihood estimtio ll 2 = rψ () + rψ ( + b) + ( r) d 2 2 I exp λx (r) c (b, ) 2 2I (b,) exp λx (r) c I exp λx (r) c (b,) I exp λx (r) c (b,) 2 Iexp λx (r) c (b,) = ψ () ψ ( + b) l 1 e λx (r) c ψ() + ψ( + b) 2 I (, b) + 1 e λx (r) c 2, 2 1 e λx (r) c l 1 e λx (r) c ψ() + ψ( + b) 2 B(, b) 2 1 e λx (r) 3 B(, b) c 3F2,, 1 b; + 1, + 1; 1 e λx (r) c 4F3,,, 1 b; + 1, + 1, + 1; 1 e λx (r) c. Cohe (1965) suggested the pproximte vrice covrice mtrix my be obtied by replcig expected vlues by their MLEs. 4. Simultio Studies A simultio is coducted to study properties of the MLE of the bet-weibull distributio. The prmeter sets for which the BW hzrd rte fuctio is bthtub, icresig d decresig re simulted. For ech simulted smple the bsolute reltive bises (ARbis), me squre errors (mse), reltive me squre errors (Rmse) d reltive root me squre errors (Rrmse) re computed. We geerte 10,000 rdom smples from B-W distributio with differet smple sizes (, 30, 50, 100 d 150) d with differet cses of the ucesored percetge r % (70%, 80%, 90% d 100%). The smples re geerted from BW distributio usig the followig trsformtio: X i = 1 λ ( l (1 B i) 1/c, i = 1,2,, whereb 1,. B re rdom umbers from bet distributio i itervl (0, 1). Simultio results re summrized i Tbles 1, 2, 3 d 4. Tbles 1, 2 d 3 give thearbis, mse, Rmse d Rrmse of the estimtors. The symptotic vrices d covrice mtrix of the estimtors re displyed i Tble 4. The followig re some observtios from the simultio study 1. For the prmeter set of bthtub hzrd fuctio (Tbles 1, 4) : The ARbis of the mlec decreses whe the smple size icreses d whe the percetge of ucesored observtio

8 2376 M. R. Mhmoud d R. M. Mdouh r % icreses. Also, the ARbis ofthe mle decreses whe the smple size icreses d whe the percetge of ucesored observtio r % icreses except for (100) %. The mse, Rmse d Rrmsof the mlesc d decresewhe the smple size icreses d whe the percetge of ucesored observtio r % icreses. The symptotic vrice of c decreses whe the smple size icreses d the percetge of ucesored observtio r % icreses. Also, The symptotic vrice of decreses whe the smple size icreses d whe the percetge of ucesored observtio r % icreses except for smll smple sizes (= d =30). 2. For the prmeter set of icresig hzrd fuctio (Tbles 2, 4) : The ARbis of the mlec decreses whe the smple size icreses d whe the percetge of ucesored observtio r % icreses. Also, the ARbis ofthe mle decreses whe the smple size icreses but i most cses it icreses whe the percetge of ucesored observtio r % icreses. The mse, Rmse d Rrmseof the mlesc d decresewhe the smple size icreses d they lso decrese whe the percetge of ucesored observtio r % icreses. The symptotic vrice of c decreses whe the smple size icreses d the percetge of ucesored observtio r % icreses. Also, The symptotic vrice of decreses whe the smple size icreses but whe the percetge of ucesored observtio r % decreses we hve three cses: it icreses for smple sizes (=, =30), costt for =50 d decreses slowly for =100 d = For the prmeter set of decresig hzrd fuctio (Tbles 3, 4): Similr results s icresig hzrd fuctio re observed.

9 Mximum likelihood estimtio For the prmeter sets for which the BW hzrd rte fuctio is bthtub, icresig d decresig, Tbles (5, 6 d 7) d figures (1, 2 d 3) illustrte the MLE of hzrd fuctio t smple size =50 d ucesored percetge 80%. I geerl, the results for the prmeter set of bthtub hzrd fuctio re better th the results for the other two cses. Tble 1: ARbis, mse, Rmse d Rrmse of prmeters c = 1.5 d = 0.5 d with kow prmeters λ=0.5 d b=0.5 (bthtub hzrd fuctio) r % Prmeter ARbis mse Rmse Rrmse c % c c % c c % c c % c

10 2378 M. R. Mhmoud d R. M. Mdouh Tble 2: ARbis, mse, Rmse d Rrmse of prmeters c = 1.5 d = 1.5 d with kow prmeters λ=0.5 d b=0.5 (icresig hzrd fuctio) r % Prmeter ARbis mse Rmse Rrmse c % c c % c c % c c % c

11 Mximum likelihood estimtio 2379 Tble 3: ARbis, mse, Rmse d Rrmse of prmeters c = 0.5 d = 1.5 d with kow prmeters λ=0.5 d b=0.5 (decresig hzrd fuctio) r % Prmeter ARbis mse Rmse Rrmse c % c c % c c % c c % c ,

12 2380 M. R. Mhmoud d R. M. Mdouh Tble 4: Asymptotic vrice covrice mtrix of estimtors uder type II cesored smples r % c c c (1.5, 0.5, 0.5, 0.5) (1.5, 1.5, 0.5, 0.5) (0.5, 1.5, 0.5, 0.5) 70% % % % % % % % % % % % % % % % % % % %

13 Mximum likelihood estimtio 2381 Tble 5: Mximum Likelihood Estimtio of Hzrd Fuctio whe c d re ukow t =50 d ucesored percetge 80% (bthtub hzrd fuctio) x h(x) x h(x) x h(x) x h(x) h( x) x Figuer1: Hzrd fuctio (bthtub)

14 2382 M. R. Mhmoud d R. M. Mdouh Tble 6:Mximum Likelihood Estimtio of Hzrd Fuctio whe c d re ukow t =50 d ucesored percetge 80% (icresig hzrd fuctio) x h(x) x h(x) x h(x) x h(x) h( x) x Figuer2: Hzrd fuctio (icresig)

15 Mximum likelihood estimtio 2383 Tble 7: Mximum Likelihood Estimtio of Hzrd Fuctio whe c d re ukow t =50 d ucesored percetge 80% (decresig hzrd fuctio) x h(x) x h(x) x h(x) x h(x) h( x) x Figuer3: Hzrd fuctio (decresig) Refereces [1]C. Cohe, Mximum Likelihood Estimtio i the Weibull Distributio Bsed o Complete d o Cesored Smples, Techometrics, 7(4) (1965), [2] G. M. Cordeiro, A. B. Sims d B. D. Stošić, Closed Form Expressios for Momets of the Bet Weibull Distributio, Ais d Acdemi Brsileir de Ciêcis, 83(2) (08),

16 2384 M. R. Mhmoud d R. M. Mdouh [3] N. Eugee, C. Lee d F. Fmoye, Bet-Norml Distributio d its Applictios, Commu. Sttist. - Theory d Methods, 31(02), [4] F.Fmoye, C.Lee d O.Olumolde,The Bet-Weibull distributio, Jourl of Sttisticl Theory d Applictios, 4(05), [5] N. L. Johso, S. Kotz d N. Blkrish, Cotiuous Uivrite Distributios, Volume 2 (secod editio), Joh Wiley d Sos, New York, [6] M. C. Joes, Fmilies of Distributios Arisig from Distributios of Order Sttistics, Test, 13(1) (04): [7] C. Lee, F. Fmoye d O. Olumolde, Bet-Weibull Distributio: Some Properties d Applictios to Cesored Dt, Jourl of Moder Applied Sttisticl Methods, 6(07), [8] A. M. Mood, F. A. Grybill d D. C. Boes, Itroductio to the Theory of Sttistics, Third Editio, McGrw-Hill, Ic., [9] G. S. Mudholkr d D. K. Srivstv, The ExpoetitedWeibull Fmily for Alyzig Bthtub Filure-Rte Dt, IEEE Trsctios o Relibility, 42(1993), [10] G. S. Mudholkr d D. K. Srivstv d G. D. Kolli, A Geerliztio of the Weibull Distributio with Applictio to the Alysis of Survivl Dt, Jourl of the Americ Sttisticl Associtio, 91(1996), [11] P. D. N. Murthy, M. Xie d R. Jig, Weibull Models, Joh Wiley d Sos, New York, 04. [12] S. Shhbz, M. Q. Shhbz d N. S. Butt, A Clss of Weighted Weibull Distributio, Pkist Jourl of Sttistics d Opertio Reserch, VI(1) (11), [13] K. P. Sigh, C. M. Lee d E. O. George, O Geerlized Log-Logistic Model for Cesored Survivl Dt, Biometricl Jourl, 30(1988), [14] A. S. Whed, T. M. Luog d J. H. Jeog, A ew Geerliztio of Weibull Distributio with Applictio to Brest Ccer Dt Set, Sttistics i Medicie, 28(09), [15] W. Weibull, Sttisticl Theory of the Stregth of Mterils, IgeioorVeteskpsAkdemiesHdligr, 151(1939), [16] lst visited t