ECONOMETRIC ANALYSIS ON EFFICIENCY OF ESTIMATOR ABSTRACT
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1 ECOOMETRIC LYSIS O EFFICIECY OF ESTIMTOR M. Khohev, Lectue, Gffth Uvet, School of ccoutg d Fce, utl F. K, tt Pofeo, Mchuett Ittute of Techolog, Deptet of Mechcl Egeeg, US; cuetl t Shf Uvet, I. Houl P. Sgh, Rjeh Sgh, Pofeo of Stttc, Vk Uvet, Deptet of Mthetc d Stttc, Id. F. Sdche, octe Pofeo, Deptet of Mthetc, Uvet of ew Meco, US. BSTRCT Th ppe vetgte the effcec of ltetve to to etto ude the upe populto odel wth ucoelted eo d gdtbuted ul vble. Copo wth uul to d ubed etto e lo de. Ke wod: B, Me Sque Eo, Rto Etto Supe Populto.. ITRODUCTIO It well kow tht the to ethod of etto occupe pott plce ple uve. Whe the tud vte d the ul vte potvel (hgh) coelted, the to ethod of etto qute effectve ettg the populto e of the tud vte utlzg the foto o ul vte. Code fte populto wth ut d let d deote the vlue fo two potvel coelted vte d epectvel fo the th ut th populto,,,,. ue tht the populto e X of kow. Let d be the ple e of d epectvel bed o ple do ple of ze ( < ) ut dw wthout eplceet chee. The the clcl to etto fo Y defed b ( X ) (.) The b d e que eo (MSE) of e, up to ecod ode oet, B( ) λ ( R S S ) X (.) M( ) λ ( S R S R S ), (.3)
2 whee ( ) ( ) R X λ, Y, S ( ) ( Y ), ( -) ( - X ), d S (-) ( - Y )( - X ). It cle fo (.3) tht M ( ) wll be u whe R S S β, (.4) whee β the egeo coeffcet of o. lo fo R β, the b of (.) zeo. Tht, lot ubed fo Y. Let E ( ) β be the le of egeo of o, whee E deote vegg ove ll poble ple deg ple do plg wthout eplceet (SRSWOR).The β S S d Y β X o tht, geel, R ( X ) β (.5) It obvou fo (.4) d (.5) tht tfoto tht bg the to of populto e cloe to β wll be helpful educg the e que eo (MSE) well the b of the to etto. Th led Svekt d Tc (986) to ugget ltetve to to etto z( X ) { ( X ) } (.6) whch bed o the tfoto z, (.7) whee E( z) Z ( Y ) d utbl choe cl. I th ppe ect epeo of b d MSE of e woked out ude upe populto odel d coped wth the uul to etto.. THE SUPER POPULTIO MODEL
3 Followg Dub (959) d Ro (968) t ued tht the fte populto ude codeto telf do ple fo upe populto d the elto betwee d of the fo: β u ; (,,,) whee d β e ukow el cott; u e ucoelted do eo wth codtol (gve ) epectto E( u ) 0 g E ( u ) δ (,,.,), ο δ, ο g d e depedetl detcll dtbuted (..d.) wth coo g det G ( ) e Γ, ο,. (.) We wll wte E to deote epectto opeto wth epect to the coo dtbuto of (,,3,,) d E E c, the ove ll epectto opeto fo the odel. We deote deg b p d the deg epectto E p, fo tce, ee Chudhu d dhk (983,89) d Shh d Gupt (987). Let deote ple do ple of dtct lbel choe wthout eplceet out of,,3. The X( X ) Followg Ro d Webte (966) we wll utlze the dtbutol popete of j,,, ou ubequet devto. 3. THE BIS D ME SQURE ERROR The etto (.6) c be wtte 3
4 4 ( ) (3.) bed o ple do ple of dtct lbel choe wthout eplceet out of,,,. The b B E p ( - Y ) (3.) of h odel epectto E (B) whch wok out follow: E ( B ( ) ) E p E E c ( ) u β - - E E c ( β U ) E p E E c ( ) u β - E E c ( β X ) E p E β β X E ( ) X E ( ) ( ) ( ) ( ) ( ) { } - ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) { } [ ]
5 - [ ( ) {( ) ( ) }] (-) ( ) ( ) (3.3) Fo SRSWOR plg chee, the e que eo M ( ) E p ( ) Y (3.4) of h the followg foul fo odel epectto E ( M ( )) : E ( M ( )) E M ( ) whee [ ( ) ( )( )( ) ( )( ) ] (3.5) M ( ) E ( Y ) (3.6) p the MSE of ude SRSWOR chee h the odel epectto E { } ( M ( )) ( ) ( ) ( )( ) [ See, Ro(968, p.439) ] ( g )( g ) ( ) δ ( g )( g ) Futhe, we ote tht fo SRSWOR plg chee, the b B ( ) E ( Y ) p Γ Γ ( g) (3.7) (3.8) of uul to etto h the odel epectto E ( B( )) ( ) ( ) (3.9) We ote fo (3.3) d (3.9) tht f E ( B( )) ( B ( )) E 5
6 o f o f ( ) ( ) ο (3.0) Futhe we hve fo (3.5) tht E ( M ( )) E ( M ( )) < ο f o f ( ) < ο ο (3.) whch the e (3.0). Thu we tte the followg theoe: Theoe 3. : The etto le bed well oe effcet th uul to etto f ο ( ο). e. whe le betwee ο d. Theefoe, whe tecept te ( ο) the odel (.) zble, thee wll be uffcet fleblt pckg. It to be oted tht fo ο, ubed d effcet th. The zto of (3.5) wth epect to led to opt () (3.) Subttuto of (3.) (3.5) eld the u vlue of E ( M ( )). E ( ( )) ( ) δ [( g )( g ) ( ) ] ( g )( g ) M ( g) Γ Γ (3.3) whch equl to E ( M ( )) whe ο. It teetg to ote tht whe, ubed d tted t u vege MSE odel (.). 6
7 I pctce the vlue of wll hve to be eed, t the etto tge, to be ued. To e, we ue ctte dg of veu fo dt fo plot tud, o pt of the dt fo the ctul tud d judge the -tecept of the bet fttg le. Fo (3.7) d (3.3) we hve { } { ( )( )} E ( M ( )). E ( M ( )) ( )( ) ο (3.4) whch how tht oe effcet th to etto whe kow ectl. Fo ο.e ( M ( )) E ( M ( )) (3.5) Fo SRSWOR, the vce V( ) E ( Y ) p (3.6) of uul ubed etto h the odel epectto: [ ] ( V ( ) ) ( ) β { δγ( g) Γ} (3.7) E The epeo of E ( M ( )) d E ( V ( ) ) e ot e tk to cope lgebcll. Theefoe ode to fcltte the copo, deotg ( V ( ) E M ( ) 00E ( ) d E 00E ( V ( )) E ( M ( )) E, we peet below tble,,3, the vlue of the eltve effcece of wth epect to d fo few cobto of the petc vlue ude the odel (.). Vlue e gve fo 60, δ.0, 8, 0.5,.0,.5, β 0.5,.0,. 5 d g, 0.5,.0,.5,.0. The ge of, fo to be bette th fo gve 0.5,.0,. 5 e epectvel ( 0,), ( 0,), (0,3). Th clel dcte tht the ze of cee the ge of fo to be bette th cee.e. fleblt of choog cee. We hve de the followg obevto fo the tble, d 3 : () g cee both E d E decee. Whe cee E cee whle E decee. 7
8 () () (v) cee (.e. f the tecept te dept fo og potve decto) eltve effcec of wth epect to decee whle E cee. β cee E cee fo fed g whle E uffected. The u g effcec obeved ove well ove f cocde wth the vlue of. Fll, the etto to be pefeed whe the tecept te dept ubttll fo og. REFERECES Chudhu,. d dhk,.k. (983): O the effcec of Mdzuo d Se tteg eltve to evel to-tpe etto ude ptcul odel. Boetk, 70,3, Chudhu,. d dhk,.k.(989): O effcec of the to etto. Metk, 36, Dub, J. (959): ote o the pplcto of Queoulle ethod of b educto etto of to. Boetk,46, Ro, J..K. d Webte, J.T. (966): O two ethod of b educto etto of to. Boetk, 53, Ro, P.S.R.S. (968): O thee pocedue of plg fo fte populto. Boetk, 55,, Shh, D.. d Gupt, M. R. (987): effcec copo of dul to to d poduct etto. Cou. Sttt. Theo eth. 6 (3), Svekt, T. d Tc, D.S. (986) : Tfoto fte plg. Stttc, 7,4,
9 Tble : Reltve effcece of wth epect to d Γ 0.5 g β 0 E E g β 0 E E
10 Tble : Reltve effcece of wth epect to d.0 g β 0 E E g β 0 E E
11 Tble 3: Reltve effcece of wth epect to d g β 0.5 E E G β 0.5 E E
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