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)
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
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 ( ) (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 ( ) ( ) ( ) ( ) ( ) { } - ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) { } [ ]
- [ ( ) {( ) ( ) }] (-) ( ) ( ) (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
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
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
() () (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, 689-693. Chudhu,. d dhk,.k.(989): O effcec of the to etto. Metk, 36, 55-59. Dub, J. (959): ote o the pplcto of Queoulle ethod of b educto etto of to. Boetk,46,477-480. Ro, J..K. d Webte, J.T. (966): O two ethod of b educto etto of to. Boetk, 53, 57-577. Ro, P.S.R.S. (968): O thee pocedue of plg fo fte populto. Boetk, 55,,438-44. Shh, D.. d Gupt, M. R. (987): effcec copo of dul to to d poduct etto. Cou. Sttt. Theo eth. 6 (3), 693-703. Svekt, T. d Tc, D.S. (986) : Tfoto fte plg. Stttc, 7,4,597-608. 8
Tble : Reltve effcece of wth epect to d Γ 0.5 g β 0 E E 0.30 0.60 0.90 0.30 0.60 0.90 0.5 9.86 93.3 9.40 0.34 0.54 00.57.0 48.6 483.6 478.09 0.34 0.54 00.57.5 964.3 966.7 956.98 0.34 0.54 00.57 0.5 3.67 3.77 3.30 00.49 00.56 00. 0.5.0 37.8 37.99 37.6 00.49 00.56 00..5 43.08 43.36 4.93 00.49 00.56 00. 0.5.06.08 0.95 0.7 00.9 07.0.0 48.08 48. 47.93 0.7 00.9 07.5 09.78 09.83 09.57 0.7 00.9 07 0.5 03.99 04.00 03.96 06 07 03.5.0 6.64 6.65 6.60 06 07 03.5 37.7 37.7 37.66 06 07 03 0.5 0.3 0.3 0. 0 0 0.0.0 06.43 06.43 06.4 0 0 0.5 3.43 3.43 3.4 0 0 0 0.5 g β 0 E E 0.30 0.60 0.90 0.30 0.60 0.90 0.5 96.58 96.96 95. 03.33 0.5 00.56.0 49.46 49.39 487.77 03.33 0.5 00.56.5 98.9 984.39 975.53 03.33 0.5 00.56 0.5 34.37 34.46 34.46 00.48 00.55 0 0.5.0 40.86 4.0 4 00.48 00.55 0.5 48.35 48.63 47.0 00.48 00.55 0 0.5.76.79.65 00.7 00.9 07.0.0 49.0 49.05 48.87 00.7 00.9 07.5.0.6 0.90 00.7 00.9 07 0.5 04.00 04.00 03.96 06 07 0.5.0 6.64 6.65 6.60 06 07 0.5 37.7 37.73 37.67 06 07 0 0.5 0.60 0.60 0.58 0 0 0.0.0 05.77 05.77 05.76 0 0 0.5.73.73.73 0 0 0 9
Tble : Reltve effcece of wth epect to d.0 g β 0 E E 0.50.0.50.90 0.50.0.50.90 0.5 90.3 93.36 90.3 83.8 04.73 06.4 04.73 0.6.0 475.78 483.40 475.78 459.55 04.73 06.4 04.73 0.6.5 95.55 966.79 95.55 99.0 04.73 06.4 04.73 0.6 0.5 3.03 3.80 3.03 30.34 0.73 0.3 0.73 00.43 0.5.0 36.67 38.05 36.67 33.65 0.73 0.3 0.73 00.43.5 4.07 43.46 4.07 405.8 0.73 0.3 0.73 00.43 0.5 0.87.09 0.87 0.36 00.6 00.8 00.6 00.5.0.0 47.8 48. 47.8 47.5 00.6 00.8 00.6 00.5.5 09.4 09.84 09.4 08.46 00.6 00.8 00.6 00.5 0.5 03.93 04.00 03.93 03.77 00. 00.8 00. 05.5.0 6.57 6.65 6.57 6.39 00. 00.8 00. 05.5 37.63 37.73 37.63 37.4 00. 00.8 00. 05 0.5 0. 0.3 0. 0.5 00.67 09 07 0.0.0 06.4 06.43 06.4 06.3 00.67 09 07 0.5 3.4 3.43 3.4 3.35 00.67 09 07 0.0 g β 0 E E 0.50.0.50.90 0.50.0.50.90 0.5 94.0 97.08 94.0 87.47 04.67 06.33 04.67 0.4.0 485.03 49.70 485.03 468.68 04.67 06.33 04.67 0.4.5 976 985.40 976 937.36 04.67 06.33 04.67 0.4 0.5 33.73 34.49 33.73 3.05 0.70 0.8 0.70 08 0.5.0 39.7 4.08 39.7 36.7 0.70 0.8 0.70 08.5 46.35 48.73 46.35 4.3 0.70 0.8 0.70 08 0.5.07.08.07.08 00.60 00.80 00.60 00.5.0.0 48.77 49.06 48.77 48. 00.60 00.80 00.60 00.5.5 0.75.7 0.75 09.8 00.60 00.80 00.60 00.5 0.5 03.94 04.0 03.94 03.78 0 00.7 0 05.5.0 6.57 6.65 6.57 6.40 0 00.7 0 05.5 37.64 37.73 37.64 37.4 0 00.7 0 05 0.5 0.58 0.60 0.58 0.5 07 09 07 0.0.0 05.75 05.77 05.75 05.70 07 09 07 0.5.7.73.7.65 07 09 07 0 0
Tble 3: Reltve effcece of wth epect to d g β 0.5 E E 0.60.0.80.40.90 0.60.0.80.40.90 0.5 83.8 9.5 9.5 83.8 7.79 08.77 3.76 3.76 08.77 0.65.0 459.55 480.6 480.6 459.55 49.47 08.77 3.76 3.76 08.77 0.65.5 99.0 96.5 96.5 99.0 858.94 08.77 3.76 3.76 08.77 0.65 0.5 30.34 3.5 3.5 30.34 7.0 03.9 05.0 05.0 03.9 00.64 0.5.0 33.64 37.55 37.55 33.65 7.67 03.9 05.0 05.0 03.9 00.64.5 405.8 4.60 4.60 405.8 395.44 03.9 05.0 05.0 03.9 00.64 0.5 0.36.0.0 0.36 09.34 0.7 0.77 0.77 0.7 00.3.0.0 47.5 48.0 48.0 47.5 47.79 0.7 0.77 0.77 0.7 00.3.5 08.46 09.69 09.69 08.46 06.53 0.7 0.77 0.77 0.7 00.3 0.5 03.77 03.98 03.98 03.77 03.44 00.40 00.60 00.60 00.40 08.5.0 6.39 6.6 6.6 6.39 6.0 00.40 00.60 00.60 00.40 08.5 37.4 37.69 37.69 37.4 39.68 00.40 00.60 00.60 00.40 08 0.5 0.5 0. 0. 0.5 4 00.3 0 0 00.3 03.0.0 06.35 06.4 06.4 06.35 06.4 00.3 0 0 00.3 03.5 3.35 3.4 3.4 3.35 3.3 00.3 0 0 00.3 03 G β 0.5 E E 0.60.0.80.40.90 0.60.0.80.40.90 0.5 87.47 96.97 95.97 87.47 75.33 08.67 3.59 3.59 08.67 0.63.0 468.68 489.9 489.9 468.68 438.34 08.67 3.59 3.59 08.67 0.63.5 937.36 979.83 979.83 937.36 876.67 08.67 3.59 3.59 08.67 0.63 0.5 3.05 34. 34. 3.05 8.73 03.3 04.9 04.9 03.3 00.63 0.5.0 36.70 40.58 40.58 36.70 30.76 03.3 04.9 04.9 03.3 00.63.5 4.3 47.87 47.87 4.3 400.80 03.3 04.9 04.9 03.3 00.63 0.5.08.7.7.08 8 0.4 0.7 0.7 0.4 00.3.0.0 48. 48.96 48.96 48. 46.77 0.4 0.7 0.7 0.4 00.3.5 09.8.0.0 09.8 07.9 0.4 0.7 0.7 0.4 00.3 0.5 03.78 03.98 03.98 03.78 03.46 00.39 00.58 00.58 00.39 08.5.0 6.40 6.6 6.6 6.40 6.40 00.39 00.58 00.58 00.39 08.5 37.43 37.70 37.70 37.43 37.00 00.39 00.58 00.58 00.39 08 0.5 0.53 0.59 0.59 0.53 0.4 00.3 00.9 00.9 03 03.0.0 05.70 05.77 05.77 05.70 05.59 00.3 00.9 00.9 03 03.5.65.7.7.65.54 00.3 00.9 00.9 03 03