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1 Available olie a h://scik.og J. Mah. Comu. Sci. 2 (22), No. 4, ISSN: UNBIASED ESTIMATION IN BURR DISTRIBUTION YASHBIR SINGH * Deame of Saisics, School of Mahemaics, Saisics ad Comuaioal Scieces, Ceal Uivesiy of Rajasha, Kishagah (Ajme) 3582, Idia Absac. The aim of he ese sudy is o discuss he uifomly miimum vaiace ubiased esimae (UMVUE) of he obabiliy desiy fucio (df) of he Bu disibuio. The UMVUE of he cumulaive disibuio fucio (cdf), h quaile ad h mome of he Bu disibuio ae also obaied as he coesodig fucioal of his desiy esimao. Keywods: Bu disibuio, desiy fucio, UMVUE, momes, cumulaive disibuio fucio, comlee sufficie saisic. 2 AMS Subjec Classificaio: 62F. Ioducio Le X,, X be a adom samle fom a disibuio wih df f( x, ), (aamee sace), x S (samle sace). A imoa objecive of he saisical heoy of aameic esimaio is o esimae f( x, ). To esimae f( x, ),we esimae (aamee) by a suiable esimao U ad he elace by U i f( x, ). Thus f ( x, U) is aameic esimae of he ue desiy f( x, ). Howeve i may siuaios, we ca obai he desiy esimao bee ha f ( x, U ). Fo examle, if ubiasedess is desied, he f ( x, U) is o a ubiased esimao of f( x, ) i.e. E[ f ( x, U ) ] f( x, ). Fuhe if we wa o ge a good esimao fo a class of aameic fucio, he i is bee o obai a good esimao of he udelyig desiy fucio * Coesodig auho addes: sigh.yshbi@gmail.com (Y. Sigh) Received Jauay 28, 22 83
2 UNBIASED ESTIMATION IN BURR DISTRIBUTION 83 ad he obai he esimaos fo hese aameic fucio as he coesodig fucioal of he desiy esimao. To obai good esimao of he desiy fucio we ca coside a ubiased esimao havig miimum vaiace. Tha is, we ca obai he uifomly miimum vaiace ubiased esimao of he desiy fucio f( x, ). I his ae we obai he UMVUE of he df of Bu disibuio. A adom vaiable X is said o have a Bu disibuio if is df is give by x, x,, f( x,, ) ( x ), ohewise () The Bu disibuio is a imoa disibuio i saisical heoy ad his disibuio fids may imoa alicaios i egieeig, suvival aalysis, idusy ad acuaial sciece. The UMVUE of he df of saisical disibuios have bee he subjec of ivesigaio by may auhos. Asabasdi [] obaied he UMUVUE of he df of he Paeo disibuio. Dixi ad Jabbai Nooghabi [2] obaied he UMVUE of he df of Paeo disibuio wih he esece of oulies. Sigh [3] has discussed he UMVUE of he df of seveal uivaiae obabiliy disibuios. The ogaizaio of his ae is as follows. I secio 2, we discuss he mehod of obaiig he UMVUE of he df of a saisical disibuio. I secio 3, we obaied he UMVUE of he df of Bu disibuio. I secio 4, we deived he UMVUE of cdf, h quaile ad h mome of he Bu disibuio as he coesodig fucioal of he ue desiy esimao. Fially a he ed of his ae a coclusio is give. 2. Pelimiaies Le X,, X be a adom samle fom a disibuio wih df f( x, ),, x S ad T be comlee sufficie saisic fo he family { f( x, ), }. The codiioal desiy of X give T=, deoed by g(x ), is ubiased fo f( x, ) sice E[g(x )] g( x ) h(, ) d,
3 832 YASHBIR SINGH * k( x,, ) d f( x, ) whee k( x,, ) ad h (, ) deoe esecively he joi df of X ad T ad magial df of T. Sice g(x ) is a fucio of comlee sufficie saisic which is ubiased fo f( x, ) hece g(x ) is he UMVUE of f( x, ). 3. Mai Resuls 3. UMVUE Le X,,X be a adom samle fom f( x, ) i (). We shall obai he UMVUE of f( x, ) whe is kow. If is kow, he T = log( x ) is comlee sufficie saisic fo o fo he family { f( x, ), }. To obai g(x ), he UMVUE of f( x, ), we oceed as follows. Noe ha by defiiio, k( x,, ) g( x ) (2) h (, ) = ( x) f ( x, ) h (, ) (3) whee ( x) is he codiioal df of T give X a x. The saisic T has gamma disibuio wih aamee ad. To obai ( x), we oe ha T = i log( x ) i = x log( ) log( x ) i2 i T = log( x ) W (4) whee
4 UNBIASED ESTIMATION IN BURR DISTRIBUTION 833 W = i2 log( x ) i The codiioal desiy of T give X =x is he ucodiioal desiy of W evaluaed a log(+x ). The saisic W has gamma disibuio wih aamee (-) ad. Thus fom (3) g(x ) comes ou as below g(x ) = = x ( x ) e [ log( x )] 2 e ( log( x )) 2 x log( x ) ( x ) if log( x ), ohewise (5) 3.2 UMVUE of cdf, h quaile ad h mome I his secio we obaied he UMVUE of cdf, h quaile ad h mome of he Bu disibuio as he coesodig fucioal of desiy esimao g(x ) give i (5). The cdf of he Bu disibuio is give by: x if x Fx ( ), ohewise The UMVUE of F(x) is give by u Fˆ ( u ) g ( x ) u ˆ ( ) x log( x ) F ( u ) ( x ) 2
5 834 YASHBIR SINGH * if u ˆ log( x ) ( ) ( F u if u e ) if u ( e ) (6) The UMVUE ˆ of he h quaile of he Bu disibuio is give by Fˆ( ˆ ) ˆ g( x ) ( ˆ ) e / [( ) ] (7) Fuhe, he UMVUE ˆ of he h mome ˆ E[ X T( X ) ] of he Bu disibuio is give by / ( e ) x g( x ) / ( e ) 2 ( ) x log( x ) x ( x ) Le I = / ( e ) 2 ( ) x log( x ) x log( x ) O iegaio by as we ge he followig ecuece elaio I I I, 2
6 UNBIASED ESTIMATION IN BURR DISTRIBUTION 835 Thus ˆ I I, 2 Coclusio I his ae we obaied he UMVUE of he df of Bu disibuio. This UMVUE ca be used o chaaceize he Bu disibuio ad o obai a goodess of fi es fo Bu disibuio. REFERNCES [] B.R. Asabadi, Esimaio i Paeo disibuio, Meika, 37 (99), [2] U.J. Dixi, M. Jabbai Nooghabi, Efficie esimaio i he Paeo disibuio wih he esece of oulies, Saisical Mehodology, 8 (2), [3] J. Sigh, Miimum vaiace ubiased esimaio of obabiliy desiies, Ausalia Joual of Saisics, 22 (98),
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