Comparing Different Estimators for Parameters of Kumaraswamy Distribution

Transcription

1 Compaig Diffee Esimaos fo Paamees of Kumaaswamy Disibuio ا.م.د نذير عباس ابراهيم الشمري جامعة النهرين/بغداد-العراق أ.م.د نشات جاسم محمد الجامعة التقنية الوسطى/بغداد- العراق Absac: This pape deals wih compaig diffee mehods o esimae scale paamee ( ) ad shape paamee ( ) of Kumaaswamy Disibuio, hese esimaos ae mome esimaos,maximum likelihood,pobabiliy weighed mome, mehod of maximum likelihood of odeed obsevaio ad L-mome esimaos. he compaiso is doe hough simulaio usig diffee sample size ad diffee se of values of (, ) ad he compaig he esuls usig saisical measues mea squae eo (MSE) keywods:kumaaswamy Disibuio, mome mehod,maximum likelihood mehod,pobabiliy weighed mome mehod,l-mome mehod. Ioducio: This disibuio is oe of he coiuous pobabiliy disibuios which have may applicaios,such as heigh of idividuals scoes obaied i a es which lies bewee (-) ad ca be asfomed o ieval (,).May eseaches,nadaajah discussed his disibuio as a esul of bea family,also Fle Che sudied he disibuio ad show is applicaio i hydology daa daily seam flow. also Gupa ad Nadaajah (24),wie abou he esimaio of is paamees.i (23) Tabassum Sidhu ad Navid Feoze ad Muhammed Aslam focus o o-bayesia esimaio fo he shape paamees of he Kumaaswamy disibuio. Hee we coiue he wok abou his disibuio o compae diffee esimaos fo he wo paamees by usig fou mehods such as :mome mehod,maximum likelihood mehod, pobabiliy weighed momes,maximum likelihood of odeed obsevaio, he compaiso is doe hough simulaio The pobabiliy desiy fucio of he disibuio is: f T ( ) ( ),,, () 399

2 Ad he cumulaive disibuio fucio is: F ( ) ( ) T.. (2) Ad he eliabiliy fucio is: R( ) ( ) (3) The hazad fucio is: H( ) (4) Which is ime depede. - Mehod of mome This mehod based o piciple of equaig populaio momes by hei coespodig sample momes. he populaio momes is: E( ) ( ) d (5) Le z / ( z) d dz dz ( z) / Afe some algebic opeaios,the.. (6) / ) ) E( ) (7) / ) h The fomula of populaio mome abou oigi. whe, / ) ) E ( ) / ) Fis Sample mome is i i The he fis equaio is: / ) ). (8) / ) Ad whe 2 2 / ) ) E ( 2 ) 2 / ) 4

3 Secod Sample mome 2 i i he 2 / ) ) E ( 2 ) = 2 / ) 2 i i (9) Fom hese wo equaio we obai a momes esimaos ( mom, mom ) by solvig i umeically hough Newo-Raphso Mehod. 2- Mehod of Maximum Likelihood The mehod of maximum likelihood selecs he se of values of he model paamees ha maximizes he likelihood fucio. I ideed maximizes he pobabiliy of he obseved daa ude he esulig disibuio. Maximum-likelihood esimaio gives a uified appoach o esimaio, which is well-defied i he case of he omal disibuio ad may ohe poblems. So he likelihood of he disibuio is : L i ( i )... () i i The logaihm of likelihood fucio is: LogL log( ) log( ) ( ) log( i ) ( ) log( ) () i To obai he maximum likelihood esimaos we deive equaio (),wih espec o ad,he log L log L i log( i ) i i log( i ) ( ) i i ( i ) i.. (2) log( ) i (3) Which ca be solved umeically o obai a maximum likelihood esimaos ( mle, mle ) by solvig i umeically hough Newo-Raphso Mehod. 3- Mehod of Pobabiliy Weighed Momes This mehod depeds o equaig weighed momes k wih populaio momes obaied, whee 4

4 k i k ( pi ). (4) i Whee p i ca be esimaed by ay o-paameic esimao, he commo oe is ( i.35) p i. (5) By equaig he heoeical mome by coespodig weighed momes ' M ),we ge: ( k / ) ) / ) 2 / ) ) 2 / ) = i = i k i ( pi ).. (6) i ( p 2 i )... (7) These equaio's Which ca be solved umeically o obai a weighed mome esimaos ( pwm, pwm ) by solvig i umeically hough Newo- Raphso Mehod. 4- Mehod of L-Mome The esimaio of paamees by his mehod deped o equaig populaio L-momes by coespodig sample momes.i.e Le populaio L-mome defied by: F ( ) f ( ) d. (8) Ad coespodig sample l-mome is: b C i C i ( i)..(9) Now fo he p.d.f give i equaio () ad c.d.f give i equaio (2) we mus solve he iegal ( ) ( ) d ( ) ( ) d Le: z ( ) / / z Afe usig above asfomaio we fid i i ( ) Ci Bea, (2) i Fom his fomula Bea (/, ) Bea (/,/ ) (2) Ad 42

5 2 Bea (/,2/ ) 2Bea (/,/ ) Bea (/, ). (22) Equaig wih sample momes obaied fom i b C ( i).. (23) C i These equaio's Which ca be solved umeically o obai a L-mome esimaos ( Lmom, Lmom ) by solvig i umeically hough Newo- Raphso Mehod. Resuls I his simulaio sudy, we have chose,5,25,5, o epese small, modeae ad lage sample size, seveal values of paamee.5,,.5 ad.5,,. 5.The umbe of eplicaio used was (L=).The simulaio pogam was wie by usig malab-r2b pogam. Afe esimaig he paamees he mea squae eo (MSE) was calculaed o compae he mehods of esimaio, Whee: ^ 2 ( ) ^ Mse( ) l.. (24) ^ 2 ( ) ^ Mse( ) l (25) The esuls of he simulaio sudy ae summaized ad abulaed i able() ad able(2) of he fou esimaos fo all sample size ad (, ) values especively. I each ow of able() ad able(2) we have fou values of esimaos ha is he mome mehod, maximum likelihood mehod,pobabiliy weighed mome mehod,l-mome mehod. he bes mehod is he L- mome ha gives smalles value of (MSE). So we ecommeded o use i i esimaio of he paamees of he disibuio. Table : MSE fo diffee esimaos of fo Kummaasumay disibuio sample size () Mom mle Pwm L-mom Bes L-mom L-mom L-mom L-mom L-mom L-mom L-mom 43

6 L-mom L-mom L-mom L-mom L-mom L-mom Mle Mle Table 2: MSE fo diffee esimaos of fo Kummaasumay disibuio sample size () mom mle Pwm L-mom Bes L-mom L-mom L-mom L-mom L-mom L-mom L-mom L-mom L-mom L-mom L-mom L-mom L-mom L-mom L-mom Refeces - Al-Ahai F.M.(2),"Paamee Esimaio fo he Double Paeo Disibuio"Joual of Mahemaics ad Saisics,7(4): Sidhu T.N., Feoze N. ad Aslam M.,(23)" Bayesia Aalysis of he Kumaaswamy Disibuio ude Failue Cesoig Samplig Scheme ".Ieaioal Joual of Advaced Sciece ad Techology Vol P. Kumaaswamy, (98) A Geealized Pobabiliy Desiy Fucio fo Double-Bouded Radom Pocesses, Joual of Hydology, vol. 46, pp S. Nadaajah,(28) O he Disibuio of Kumaaswamy, J Hydol, vol. 348, pp M. C. Joes,(29) Kumaaswamy s Disibuio: A Bea Type Disibuio wih Some Tacabiliy Advaages, Sa Mehodol, vol. 6, pp

7 6-M.Gag,(29) O Geealized Ode Saisics fom Kumaaswamy Disibuio, Tamsui Oxfod J Mah Sci., vol. 25, o. 2, pp