Exact Moments of Record Values from Burr Distribution. with Applications

Transcription

1 Ieo Jou of Comuo d Theoec Sc ISSN -59 I. J. Com. Theo. S. No. Nov-5 Ec Mome of Recod Vue fom Bu Dbuo wh Aco M. J. S. Kh A. Shm M. I. Kh d S. Kum 3 Deme of Sc d Oeo Reech Agh Mum Uve Agh Id Deme of Mhemc Imc Uve Mdh Mdh KSA 3 Deme of Aed Sc Bbheb Bhmo Ambedk Uve Luckow Id Receved 7 Jue 5 Reved 3 Seembe 5 Acceed 8 Seembe 5 Pubhed Novembe 5 Abc: I h e he ec mome of ecod vue e obed fo wo mee Bu e II dbuo. Ec eeo fo ge d oduc mome of ecod c e o deved. The me vce d covce of he ecod c e comued fo vou vue of he he mee. Thee vue e he ued o comue he coeffce of he be e ubed emo of he oco d ce mee. The vce of hee emo e o eeed. The edco of he fuue ecod c e o dcued. Kewod: Bu e II dbuo; Recod c; Mome of ecod c; Loco d ce mee; Be e ubed emo; Be e ubed Pedco.. INTRODUCTION Eeme vue feque occu he feme d e.g. whe he comoe e coeced ee em he em bek dow whe he f comoe f d ce of e coeco he em o wokg whe he comoe f. Moeove ce of ge d ed of coecg he d fo ech obevo oe m be eeed ecodg he ecod vue eeme. The choce of kg ecod vue c be ufed m uo he d e vbe fom of he eeme.e. o hoe d e ecoded whch e hghe h evou hghe vue. Recod vue e f defed b Chde 95 mode of ucceve eeme equece of dec d deede dom vbe. Recod vue e wde ued eeme wehe codo o ecoomc d m e fe obem. Le be equece of deede d dec dbued d couou dom vbe v wh he cumuve dbuo fuco cdf F d obb de fuco df f ove he uo. Deoe he ue ecod me b u d fo u m{ k u : k u }. The ecod vue equece he defed b The df of u gve b Ahuh 995. u u f u og F f..! The o df of u d u gve b f u og F u!! E-m: hgkh@gm.com 44gc@gm.com zh.@gm.com ude@gm.com **F uho hkfu o Uve G Commo fo wdg UGC-BSR u g No. F.3 9/5BSR. h://ou.uob.edu.bh

2 8 M. J. S. Kh e..: Ec Mome of Recod Vue fom Bu whee F F. og F og F f F f A dom vbe d o hve Bu Te II dbuo f he df of of he fom f ;.3 ohewe wh he cdf gve F..4 Theefoe fo he Bu dbuo we hve f F..5 Bu e II dbuo w oduced b Bu 94. The Bu II dbuo wde ued eb moe febe eve o Webu dbuo e.g. Wgo 993; Zmme e Rodguez 977 d Tdkm 98 h wde uded he Bu dbuo. Tdkm 98 h ebhed eoh wh ome ohe dbuo. Suffce eue e vbe egdg he emo of mee of Bu dbuo ug odeed dom vbe.e. ce of ogeve ceog oe m efe o Wgo 993 A Mou d Zhee Wu 3 Som5 Wu d Yu 5 Wu e. 7 d Som e.. Be feece bed o ecod vue w codeed b Wg d Sh d Nd d Pdoouo. Aghzdeh d Abd obed he cofdece ev fo he Pmee of he Bu Te II dbuo Bed o Recod. Pw d Sz 999 S d Puhk obed he ecuece eo h fo ge d oduc mome of k ecod vue. I h e we hve obed he ec eeo fo he ge d oduc mome of Bu e II dbuo bed o ecod vue. Th e dvded o hee eco. I eco we hve obed he o d vee mome of ecod c fom Bu II dbuo. I eco 3 ug he eu obed eco we hve obed he be e ubed emo BLUE fo oco d ce mee of Bu II dbuo fo kow he mee. Fuhe be e ubed edco BLUP o ccued b ug he eu obed eco... COMPUTATION OF MOMENTS I h eco we hve deved he ec eeo fo o d vee mome of ecod c fom Bu II dbuo. Fuhe hee eu e ued o evue he me vce d covce of ecod c fom Bu II dbuo.5:.5:6 fo he f fve ecod d hee e eeed Tbe d Tbe eecve. Moeove hee vue e ued o ob he coeffce equed fo he be e ubed emo of he oco d ce mee of Bu II dbuo bed o ome obeved ecod c eco 3. The coeffce of he emo e eeed fo vou choce of obeved ecod Tbe 3 d 4 eecve. Vce d covce of he emo hve o bee comued d eeed Tbe 5. Some u eu: Hee ome eu e gve whch e ued o ob he mome of ecod c fom Bu II dbuo. k b k k b b b k b k k b whee b b he be fuco k h d og ' he k devve of d whch dgmm fuco d. he gmm fuco. h://ou.uob.edu.bh

3 I. J. Com. Theo. S. No. 7-5 Nov-5 9 h://ou.uob.edu.bh Pudkov e Vo. I! og d Pudkov e Vo. I. og d Theoem.: Sge mome of ecod c fom Bu II dbuo gve b! E u. whee. Poof: We hve d f F E u og! d og! Se o ge d E u og! Now he eu oved vew of u eu. Ug u eu ecuve we c ob he mome fo vue of. Theoem.: The Poduc mome of ecod vue fom Bu II dbuo!! E u u. d!.3 whee d.

4 M. J. S. Kh e..: Ec Mome of Recod Vue fom Bu h://ou.uob.edu.bh Poof: We hve d I og!.4 whee d I og og! Se o ge og! I d Now ug he Mcu ee eo we hve! Hece ug he eu we ge I! Pug he vue of I.4 d eg we ge og!! d Now ug he u eu he Theoem oved. Poceedg he m w d edg bom.3 c be ebhed. Remk.: Theoem. educe o Theoem. f we u d.3. Remk.: The eu obed.. d.3 c be uzed fo obg he mome of Lom dbuo d og ogc dbuo he e he ec ce of Bu II dbuo.

5 I. J. Com. Theo. S. No. 7-5 Nov-5 Tbe : Me of ecod c fom Bu dbuo Tbe : Covce of ecod c fom Bu dbuo Aco of he mome The ec d ec eeo fo ge mome of ecod c gve. ow u o evue he me of ecod c. Tbe ee he me of u 5 fo Bu II dbuo. Fo he comuo of vce d covce he oduc mome vce d covce e comued. Fo he vue of wee comued f d he wee fed b ug he mme of he vce covce m. Tbe ovde he vce d covce of ecod c fo.5:.5:6. MATLAB h bee ued fo comuo of he mome be d ogmm fuco e vbe hee. Aume Y Y o be fe equece of d.v. wh df g ; 3. h://ou.uob.edu.bh

6 M. J. S. Kh e..: Ec Mome of Recod Vue fom Bu Le Yu Yu Yu be he f obeved ecod vue fom he bove equece. The whee Yu u he veco of obeved ecod c fom ouo wh he dd Bu II dbuo df d cdf gve.3 d.4 eecve. The we c we he be e ubed emo BLUE of d Aod e ˆ Y u Yu Yu 3. ˆ by u byu byu 3.3 Hee ' d b ' e he ee of he m C V A V wh A ' ' whee he me of he f ecod vue d V he vee of he covce m V. Vce d covce of hee emo e gve b V ˆ d V ˆ d d Cov ˆ ˆ d 3.4 whee d d D V A. d d The ece coeffce 3. d 3.3 equed fo he BLUE of d bed o of ecod c fom he Bu II dbuo e comued Tbe 3 d Tbe 4 eecve. The vce d covce of he BLUE e eeed Tbe 5. Hee coeffce e comued fo Bu II dbuo.5:.5:6. Sm we c ob BLUE fo ohe choce of he mee. Tbe 3: Coeffce fo he BLUE of fo h://ou.uob.edu.bh

7 I. J. Com. Theo. S. No. 7-5 Nov-5 3 Tbe 4: Coeffce fo he BLUE of fo Tbe 5: Vce d covce of he BLUE of d em of Eme. Le u code he ce whee he comoe hve fue me whch foow Bu II dbuo wh 34. Suoe h we obeve he foowg mued obeved fue me Theefoe we obeve he ecod c fom he obeved d foow: Hee he fo he ecoded d wh 3 3 d ; d V e obed fom Tbe d Tbe eecve. The coeffce 3. d 3.3 e eeed Tbe 3 d Tbe 4 eecve. h://ou.uob.edu.bh

8 4 M. J. S. Kh e..: Ec Mome of Recod Vue fom Bu Theefoe he BLUE of d ˆ e comued o be d ˆ.34. The coeodg vce d covce of ˆ d ˆ ee Tbe 5 e comued o be V ˆ d V ˆ d Cov ˆ ˆ Le u code he ue ouo me E Y Now uoe * Yu Yu Yu3 /3 he me of he obeved ecod. We woud hve *.8767 d S. E. *.995. The BLUE of ˆ ˆ ˆ.945. The dd eo of ˆ comued o be S. E.ˆ.4. Theefoe he BLUE efom bee h he me of obeved ecod he ee of dd eo. I he coe of edco of he fuue ecod obevo uoe we obeve o he f ecoded obevo Y Yu Yu Yu d he go o edc Y u whee. Whe F beog o oco d ce mee fm he mo we-kow edco he be e ubed edco BLUP ee fo eme he coe of ode c Kmk d Neo 975 of Y u gve b ˆ ˆ ˆ Yu w' V ˆ. ˆ whee he me of he f ecod vue d w ' he veco of he covce bewee he h fuue ecod c d he f ecoded obevo. The me que edco eo MSPE of Y u foud o be ee fo eme Rqb 996 MSPE Yu E' V E E' w whee E'. V A V w' V I A V A V. I ou d eu we hve obeved hee ecod c. Tbe 3 4 d 5 e ued o comue he BLUP of he fuue ecod c Y u4 d Y u5 bed o he f hee obeved ecod c. The vue e comued o be Yu d Y u d d he coeodg MSPE e gve b MSPE Y u MSPE Y u Ackowedgeme: The uho e hkfu o he umou Refeee d Chef Edo Ieo Jou of Comuo d Theoec Sc fo he fufu comme d uggeo. Refeece M. Ahuh Recod Sc Nov Scece Pubhe New Yok 995. M. A. M. A Mou d Z. F. Jhee Sc feece fo he Bu mode bed o ogeve ceoed d Com. Mh. A. vo B.C. Aod N. Bkh d H.N. Ng Recod We New Yok A. Aghzdeh d M. Abd Cofdece ev fo he mee of he Bu Te II dbuo bed o Recod I. Jou. S. d Eco. vo I.W. Bu Cumuve fequec fuco A. Mh. S. vo K. N. Chde The dbuo d fequec of ecod vue J. R. S. Soc. vo. B h://ou.uob.edu.bh

9 I. J. Com. Theo. S. No. 7-5 Nov K.S. Kmk d P.I. Neo Be Le Ubed Pedco of Ode Sc Loco d Sce Fme J. Ame. S. Aoc. vo M. Nd d A.S. Pdoouo Be fo he Bu Te II dbuo bed o ecod vue Sc h 9 P. Pw d D. Sz Recuece eo fo ge d oduc mome of k ecod vue fom Peo Geezed Peo d Bu dbuo Comm. S. Theo Mehod vo A. P. Pudkov Y. A. Bchkov d O. I. Mchev Ieg d ee Godo d Bech Scece Pubhe vo M.Z. Rqb Le edco of fuue ode c Commu. S.- Theo Meh. vo R. N. Rodguez A gude o he Bu e II dbuo Bomek vo J. S d N. Puhk Recuece eo fo mome of ecod vue fom Bu dbuo J. S. Re. vo A. A. Som Emo of mee of fe fom ogeve ceoed d ug Bu-II mode IEEE T. Reb. Vo A. A. Som A. H. Abd Eh N. A. Abou-Eheggg d A. A. Modheh Be Ifeece d Pedco of Bu Te II Dbuo fo Pogeve F Fue Ceoed Smg Ie. Ifo. Mge. vo P.R. Tdkm A ook he Bu d eed dbuo Ieo S. Rev. vo L. Wg d Y. Sh Emc Be feece fo he Bu Mode bed o ecod A. Mh. Sc. vo D. R. Wgo Mmum kehood mehod fo fg he Bu e-ii dbuo o mu ogeve ceoed fe e d Mek vo S. J. Wu Emo fo he wo-mee Peo dbuo ude ogeve ceog wh ufom emov J. S. Comu. Sm. vo S. J. Wu Y. J. Che d C. T. Chg Sc feece bed o ogeve ceoed me wh dom emov fom he Bu e II dbuo S. Comu. Sm. Vo J. W. Wu d H. Y. Yu Sc feece bou he he mee of he Bu e II dbuo ude he fueceoed mg A. Mh. Comu. Vo. 63 No W. J. Zmme J. B. Ke d F. K. Wg The Bu II dbuo eb J. Qu. Techo. vo h://ou.uob.edu.bh