Some Unbiased Classes of Estimators of Finite Population Mean

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1 Itertol Jourl O Mtemtcs Ad ttstcs Iveto (IJMI) E-IN: P-IN: Volume Issue 09 etember. 04 PP-3-37 ome Ubsed lsses o Estmtors o Fte Poulto Me Prvee Kumr Msr d s Bus. Dertmet o ttstcs, Amt Uverst, Luckow, Id.. Dertmet o Aled ttstcs,bbseb Bmro Ambedkr Uverst, Luckow, Id ABTAT: I ts er, clss o bsed multvrte estmtors roosed b Abu De et l s recosdered uder te jck ke tecque o Gr d cuc d uktme et l. Te roosed clsses o estmtors re sow to be ubsed wle retg te otmum me squre error. I order to ece te rctcl utlt Abu De estmtors, te clss o estmtors utlzg te estmted otmum vlues s lso roosed. Te comrtve stud sows tt te roosed clss o jck ked estmtors d te roosed clss o estmtors bsed o te estmted otmum vlues re better t te Abu De clsses o estmtors te sese o ubsedess d rctcl utlt resectvel. KEYWOD: Ubsedess, geerlzed clss o estmtors d me squre error. I. INTODUTION W.G.ocr d.j.jesse[3,4] dscussed te use o ulr ormto to crese te recso o estmtors. Te rto estmtor mog te most commol used estmtor o te oulto me or totl o some vrble o terest o te oulto wt te el o ulr vrble we te correlto coecet betwee te two vrbles s ostve. I cse o egtve correlto, roduct estmtor s used. Tese estmtors re more ecet.e. s smller vrce t te usul estmtor o te oulto me bsed o te smle me o smle rdom smle. I ts er, mutvrte clss o estmtors gve b Abu De et l re recosdered uder te jck ke tecque o Gr d cuc d uktme et l secto. I te oulto, let deote te vrble wose oulto me Y s to be estmted b usg ormto o two ulr vrbles d.ssumg tt te oulto mes d o te ulr vrble re kow multvrte clss o estmtors o te oulto me were gve b Abu De et l(003) te ollowg () were ;,,..., re sutbl cose rel umbers.. Detos d results Assume tt te oulto mes o te ulr vrbles re kow. Let, d resectvel deote te smle mes o te vrbles, d bsed o smle rdom smle wtout relcemet (WO) o sze drw rom te oulto Dee: e Y 0, e ;=,, Y Te E e0 E e ( ) ( ) P g e

2 ome Ubsed lsses o Estmtors o Fte Poulto Me E( e ), E( e ) 0 E( e0e ), ( ) E e e j j j Were N,,, N Y Y, N Y N N Y Y N Also,, d j deote te correlto coecet betwee Y d deote te coecet o vrto o Y,. Proertes o te estmtor Bs o te Abu De clsses o estmtors resectvel., d j resectvel d Bs( ) Y j j j j () ME o te Abu De clss o estmtors Te ME o s gve b ME Y ( ) j j j j Y j j j j = ' ' b Ab b e (3) 33 P g e

3 j were A ( j ) j, ' b (,, ) e' ( e, e,..., e ), e / Te otmum vlue mmzg te ME s b e' A. Te mmum ME o te clss o estmtors s gve b ME( ) were m... Y... s te multle correlto coecet. ome Ubsed lsses o Estmtors o Fte Poulto Me II. THE POPOED JAK-KNIFE ETIMATO j Let smle rdom smle o sze =m s drw wtout relcemet rom te oulto o sze N. ts smle o sze =m s te slt u t rdom to two sub smles ec o sze m Let us dee Y e, e 0 ubsttutg tese vlues d smlg we get Bs( ) Y j j j j = B (s) (5) (4) Usg te jck - ke tecque we roose to estmte te oulto me b J 3 ' (6) B (7) 3 were ; B Bs d B Bs ' B estmtor bsed o smle o sze m 3 ' ooled estmtor bsed o sub smles o sze m ec (8) Now, E ( ) j E( ) E( ) (9) Y (.e j s ubsed estmtor o oulto me Y ) (0). Me squre error o j ME( j ) E( j Y ) E Y 34 P g e

4 ome Ubsed lsses o Estmtors o Fte Poulto Me ( Y ) ( Y ) E = E Y E Y Y Y ( ) ( ) ( )( ) ( ) () ME( ) E( Y ) = Y A () werea= Y j j j j Y j j j j Te ME o te roosed clss o jck-ked estmtors s gve b ME( ) b' Ab b' e wc c g be mmzed so tt te mmum ME s ME( ) Y = ME( ) J* m... m (3) were A ( ), b' (,, ), e' ( e, e,..., e ), e / j j j Te otmum vlue mmzg te ME s b e' A. Te mmum ME o te clss o estmtors s gve b... ME( ) m Y... were s te multle correlto coecet. (4) III. OMPAION OF THE ETIMATO I ts secto, we comre te roosed estmtors wt some kow estmtor. Te comrso wll be terms o te bs d te me squre error u to te order o -. We cosder te ollowg kow estmtors.. Te me. It s ubsed estmtor d ts vrce s gve b Vr( )=. Te rto estmtor Y 35 P g e

5 ome Ubsed lsses o Estmtors o Fte Poulto Me Were m be cose s or. Te bs d te me squre error o ts estmtor re resectvel gve b B( ) Y ( ) ME Y ( ) ( ) 3. Product estmtor Y Te bs d te me squre error o ts estmtor or were m be cose s re resectvel gve b B( ) Y P 4. Te estmtor were m be cose s or. 5. I = =, te te estmtor reduces to 6. I = =- te te estmtor () reduces to Te roosed jck-ke estmtors bove estmtors. j re ubsed d me squre error less or equl s comred wt te IV. NUMEIAL ILLUTATION Te comrso mog tese estmtors s gve b usg rel dt set. Te dt or ts llustrto s bee tke rom [], Dstrct Hdbook o Algr, Id. Te oulto tt we lke to stud cots 33 vllges. We cosder te vrbles Y,, were Y s te umber o cultvtors, s te re o vllges d s te umber o ouseold vllge. Trougout ts stud, two tes o clcultos re used. Te rst te deeds o te oulto dt, d te secod te deeds o smulto stud o 30,000 reeted smles wtout relcemet o szes 80 rom te dt. We comute te bs d te ME or ll estmtors. Numercl comutto Te ollowg vlues were obted usg te wole dt set: Y =093., =8.57, =43.3 =0.765, =0.7684, =0.766 =0.973, =0.86, = P g e

6 ome Ubsed lsses o Estmtors o Fte Poulto Me Usg te bove results we clculted te ME, bs, d te ecec or ll te estmtors secto-4. I te Tble, we reset te ME, ecec, d te bs ll or te estmtors gve ecto 4. As we c see tt our suggested estmtors d domte ll oter estmtors wt ecec 4.5. Te re sueror over ll oter estmtors d re ubsed. j Tble: Bs d ME o estmtors uder stud bsed o oulto dt Estmtor Aulr vrble ME Ecec Bs oe * * w,, P P j, wj M, , EFEENE [] W.A.Abu-De et l.,ome estmtors o te oulto me usg ulr ormto, Aled Mtemtcs d omutto 39 (003) [] Jck ke tecque o Gr d cuc d uktme et l. [3] W.G.ocr, mlg Tecques, trd ed. Jo Wle &sos, New York, 977. [4]..Jesse, ttstcl urve Tecques, Jo Wle &sos, New York, 978. [5] D.g,F..udr, Teor d Alss o smle urve Desg, New Age Publcto, New Del, Id 986. [6].K.rvstv, A estmtor usg ulr ormto smle surves, lcutt ttstcl Assocto Bullet 6(967) P g e