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1 Esimaors Values of α Values of α PRE( i) s MSE( 9)mi

2 THE EFFIIET USE OF SUPPLEMETAR IFORMATIO I FIITE POPULATIO SAMPLIG Rajes Sig Deparme of Saisics, BHU, Varaasi (U.P.), Idia Edior Florei Smaradace Deparme of Maemaics, Uiversi of ew Meico, Gallup, USA Edior 04

3 Educaio Publisig 33 esapeake Aveue olumbus, Oio 43 USA Tel. (64) oprig 04 b Educaioal Publiser ad e auors for eir papers Peer-Reviewers: Dr. A. A. Salama, Facul of Sciece, Por Said Uiversi, Egp. Said Broumi, Uiv. of Hassa II Moammedia, asablaca, Morocco. Pabira Kumar Maji, Ma Deparme, K.. Uiversi, WB, Idia. Mumaz Ali, Deparme of Maemaics, Quaid-i-Azam Uiversi, Islamabad, 44000, Pakisa. EA: ISB:

4 oes Preface: 4. Dual o raio cum produc esimaor i sraified radom samplig: 5. Epoeial raio-produc pe esimaors uder secod order approimaio i sraified radom samplig: 8 3. Two-pase samplig i esimaio of populaio mea i e presece of orespose: 8 4. A famil of media based esimaors i simple radom samplig: 4 5. Differece-pe esimaors for esimaio of mea i e presece of measureme error: 5 3

5 Preface Te purpose of wriig is book is o sugges some improved esimaors usig auiliar iformaio i samplig scemes like simple radom samplig, ssemaic samplig ad sraified radom samplig. Tis volume is a collecio of five papers, wrie b ie co-auors (lised i e order of e papers): Rajes Sig, Mukes Kumar, Maoj Kr. audar, em Kadilar, Praas Sarma, Florei Smaradace, Ail Prajapai, Hema Verma, ad Viplav Kr. Sig. I firs paper dual o raio-cum-produc esimaor is suggesed ad is properies are sudied. I secod paper a epoeial raio-produc pe esimaor i sraified radom samplig is proposed ad is properies are sudied uder secod order approimaio. I ird paper some esimaors are proposed i wo-pase samplig ad eir properies are sudied i e presece of o-respose. I four caper a famil of media based esimaor is proposed i simple radom samplig. I fif paper some differece pe esimaors are suggesed i simple radom samplig ad sraified radom samplig ad eir properies are sudied i presece of measureme error. Te auors ope a book will be elpful for e researcers ad sudes wo are workig i e field of samplig eciques. 4

6 Dual To Raio um Produc Esimaor I Sraified Radom Samplig Rajes Sig, Mukes Kumar ad Maoj K. audar Deparme of Saisics, B.H.U., Varaasi (U.P.), Idia em Kadilar Deparme of Saisics, Haceepe Uiversi, Beepe 06800, Akara, Turke orrespodig auor, rsigsa@aoo.com Absrac Trac e al.[8] ave iroduced a famil of esimaors usig Srivekaaramaa ad Trac ([6],[7]) rasformaio i simple radom samplig. I is aricle, we ave proposed a dual o raio-cum-produc esimaor i sraified radom samplig. Te epressios of e mea square error of e proposed esimaors are derived. Also, e eoreical fidigs are suppored b a umerical eample. Ke words: Auiliar iformaio, dual, raio-cum-produc esimaor, sraified radom samplig, mea square error ad efficiec.. Iroducio I plaig surves, sraified samplig as ofe proved as useful i improvig e precisio of u-sraified samplig sraegies o esimae e fiie populaio mea of e sud variable, L i. Le, ad z respecivel, be e sud ad auiliar variaes i o eac ui U (,,3, ---, ) of e populaio U. Here e size of e sraum U is, ad e size of simple radom sample i sraum U is, were,,---,l. I is sud, 5

7 uder sraified radom samplig wiou replaceme sceme, we sugges esimaors o esimae b cosiderig e esimaors i Plikusas [3] ad i Trac e al. [8]. To obai e bias ad MSE of e proposed esimaors, we use e followig oaios i e res of e aricle: were, w. Suc a, were ad are e sample ad populaio meas of e sud variable i e sraum, respecivel. Similar epressios for ad Z ca also be defied. Usig (), we ca wrie 6

8 were Te combied raio ad e combied produc esimaors are, respecivel, defied as Ad e MSE of ad o e firs degree of approimaio are, respecivel, give b (4) (5) oe a Similar epressios for ad Z ca also be defied.. lassical Esimaors 7

9 Srivekaaramaa ad Trac ([6],[7]) cosidered a simple rasformaio as ( i,,,) were A is a scalar o be cose. Tis rasformaio reders e siuaio suiable for a produc meod isead of raio meod. learl is ubiased for. Usig is rasformaio, a esimaor i e sraified radom samplig is defied as Tis is a produc pe esimaor ( aleraive o combied raio pe esimaor) i sraified radom samplig. Te eac epressio for MSE of is give b (7) I some surve siuaios, iformaio o a secod auiliar variable, Z, correlaed egaivel wi e sud variable,, is readil available. Le be e kow populaio mea of Z. To esimae, Sig[4] cosidered raio-cum-produc esimaor as were Perri[] used α( ) ad z β( Z z) respecivel. Here, α ad β are cosas a make e MSE miimum. z isead of ad z, 8

10 Adapig o e sraified radom samplig, e raio cum produc esimaor usig wo auiliar variables ca be defied as Te approimae MSE of is esimaor is 3. Suggesed Esimaors Trac e al. [8] iroduced a produc esimaor usig wo auiliar variables i e simple radom samplig give b Moivaed b Trac e al. [8], we propose e followig produc esimaor for e sraified radom samplig sceme as Epressig i erms of e s, we ca wrie () as Te MSE o e firs order of approimaio, is give as ad is MSE equaio is miimised for 9

11 oe a e correspodig A is B puig e opimum value of i (), we ca obai e miimum MSE equaio for e firs proposed esimaor,. Remark 3. : Te value of is kow, bu e eac values of V0, V0 ad V00 are rarel available i pracice. However i repeaed surves or sudies based o mulipase samplig, were iformaio is gaered o several occasios i ma be possible o guess e values of V0, V0 ad V00 quie accurael. Eve oug is approac ma reduce e precisio, i ma be saisfacor provided e relaive decrease i precisio is margial, see Trac e al. [8]. as Plikusas[3] defied dual o raio cum produc esimaor i sraified radom samplig were ad g. ( ) osiderig e esimaor i (3) ad moivaed b Sig e al. [5], we propose a famil of dual o raio cum produc esimaor as 0

12 To obai e MSE of e secod proposed esimaor,, we wrie. Epressig (4) i erms of e s, we ave (5) Epadig e rig ad side of (5), o e firs order of approimaio, we ge Squarig bo sides of (6) ad e akig epecaio, we obai e MSE of e secod proposed esimaor,, o e firs order approimaio, as (7) were Tis MSE equaio is miimized for e opimum values of ad give b

13 Puig ese values of ad i MSE ( ), give i (7), we obai e miimum MSE of e secod proposed esimaor,. 4. Teoreical Efficiec omparisos I is secio, we firs compare e efficiec bewee e firs proposed esimaor,, wi e classical combied esimaor, as follows:. Te esimaor is beer a e usual esimaor if ad ol if, B B <, () were, B θ V00 V00 ad B θv0 V0 θv0. If e codiio () is saisfied, e firs proposed esimaor,, performs beer a e classical combied esimaor. We also fid e codiio uder wic e secod proposed esimaor,, performs beer a e classical combied esimaor i eor as follows:,,

14 Te esimaor 9 is beer a e usual esimaor if ad ol if, were, ad 5. umerical Eample I is secio, we use e daa se earlier used i Koucu ad Kadilar[]. Te populaio saisics are give i Table. I is daa se, e sud variable () is e umber of eacers, e firs auiliar variable () is e umber of sudes, ad e secod auiliar variable (Z) is e umber of classes i bo primar ad secodar scools for 93 disrics a 6 regios ( as : Marmara, : Agea, 3: Medierraea, 4: eral Aaolia, 5: Black Sea, 6: Eas ad Sou eas Aaolia) i Turke i 007, see Koucu ad Kadilar[]. Koucu ad Kadilar[] ave used ema allocaio for allocaig e samples o differe sraa. oe a all correlaios bewee e sud ad auiliar variables are posiive. Terefore, we decide o o use produc esimaors for is daa se for efficiec compariso. For is reaso, we appl e classical combied esimaor,, combied raio esimaor,, e raio-cum-produc esimaor,, Plikusas [3] esimaor,, ad e secod proposed esimaor,, o e daa se. For e efficiec compariso, we compue perce relaive efficiecies as 3

15 Table. Daa Saisics of Populaio

16 Table. Perce Relaive Efficiecies (PRE) of esimaors Esimaors Values of PRE( ) MSE( ) mi Table3. Te MSE values accordig o A Value of orrespodig value of A <0.8 - >V(s) (op) 863.6(op) >.40 - >V(s) MSE (mi) a e value A(opimal). 5

17 6. oclusio We we eamie Table, we observe a e secod proposed esimaor,, uder opimum codiio cerail performs quie beer a all oer esimaors discussed ere. Aloug e correlaios are egaive, we also eamie e performace of e firs proposed esimaor,, accordig o e classical combied esimaor. Terefore, for various values of A ad i Table 3, e MSE values of ad are compued. From Table 3, we observe a e firs proposed esimaor,, performs beer a e esimaor,, for a wide rage of as, eve i e egaive correlaios. Refereces [] Koucu,. ad Kadilar,. Famil of Esimaors of Populaio Mea Usig Two Auiliar Variables i Sraified Radom Samplig ommu. i Sais. Teor. ad Me, 38, 009, [] Perri, P.F. Improved raio-cum-produc pe esimaors. Sais. I Tras, 007, [3] Plikusas, A. Some overview of e raio pe esimaors I: Worksop o surve samplig eor ad meodolog, Saisics Esoia, 008. [4] Sig, M. P. Raio-cum-produc meod of esimaio. Merika, 967, [5] Sig, R., Kumar, M., aua, P., Sawa,. ad Smaradace, F. A geeral famil of dual o raio-cum-produc esimaor i sample surves. Sais. I Tras- ew series. IJSA, 0, (), [6] Srivekaaramaa, T. ad Trac, D.S. A aleraive o raio meod i sample surves. A. Is. Sais. Ma.3 A, 980, -0. 6

18 [7] Srivekaaramaa, T. ad Trac, D.S. Eedig produc meod of esimaio o posiive correlaio case i surves. Ausral. J. Sais. 3, 98, [8] Trac, D.S., Sig, H.P. ad Sig, R. A aleraive o e raio-cum-produc esimaor i sample surves. Jour. of Sais. Pla. ad Ifere. 53, 996,

19 Epoeial Raio-Produc Tpe Esimaors Uder Secod Order Approimaio I Sraified Radom Samplig Rajes Sig, Praas Sarma ad Florei Smaradace Deparme of Saisics, Baaras Hidu Uiversi Varaasi-005, Idia Deparme of Maemaics, Uiversi of ew Meico, Gallup, USA orrespodig auor, rsigsa@aoo.com Absrac Sig e al. (0009) iroduced a famil of epoeial raio ad produc pe esimaors i sraified radom samplig. Uder sraified radom samplig wiou replaceme sceme, e epressios of bias ad mea square error (MSE) of Sig e al. (009) ad some oer esimaors, up o e firs- ad secod-order approimaios are derived. Also, e eoreical fidigs are suppored b a umerical eample. Kewords: Sraified Radom Samplig, populaio mea, sud variable, auiliar variable, epoeial raio pe esimaor, epoeial produc esimaor, Bias ad MSE.. ITRODUTIO I surve samplig, i is well esablised a e use of auiliar iformaio resuls i subsaial gai i efficiec over e esimaors wic do o use suc iformaio. However, i plaig surves, e sraified samplig as ofe proved eedful i improvig e precisio of esimaes over simple radom samplig. Assume a e populaio U cosis of L sraa as UU, U,,UL. Here e size of e sraum U is, ad e size of simple radom sample i sraum U is, were,,---,l. We e populaio mea of e auiliar variable,, is kow, Sig e al. (009) suggesed a combied epoeial raio-pe esimaor for esimaig e populaio mea of e sud variable ( ) : 8

20 s S ep (.) s were, i,, i i i L L s w, s w, ad w. L Te epoeial produc-pe esimaor uder sraified radom samplig is give b s S ep (.) s Followig Srivasava (967) a esimaor 3s i sraified radom samplig is defied as : α s 3 S ep (.3) s were α is a cosa suiabl cose b miimizig MSE of 3S. For α, 3S is same as coveioal epoeial raio-pe esimaor wereas for α -, i becomes coveioal epoeial produc pe esimaor. Sig e al. (008) iroduced a esimaor wic is liear combiaio of epoeial raiope ad epoeial produc-pe esimaor for esimaig e populaio mea of e sud variable ( ) i simple radom samplig. Adapig Sig e al. (008) esimaor i sraified radom samplig we propose a esimaor 4s as : s s 4 S θep ( θ)ep (.4) s s 9

21 were θ is e cosa ad suiabl cose b miimizig mea square error of e esimaor 4S. I is observed a e esimaors cosidered ere are equall efficie we erms up o firs order of approimaio are ake. Hossai e al. (006) ad Sig ad Smaradace (03) sudied some esimaors i SRSWOR uder secod order approimaio. Koucu ad Kadilar (009, 00) ), ave sudied some esimaors i sraified radom samplig uder secod order approimaio. To ave more clear picure abou e bes esimaor, i is sud we ave derived e epressios of MSE s of e esimaors cosidered i is paper up o secod order of approimaio i sraified radom samplig. 3. oaios used Le us defie, e suc a 0 s ad e s, V rs L W r s E r [( ) ( ) ] s To obai e bias ad MSE of e proposed esimaors, we use e followig oaios i e res of e aricle: were ad are e sample ad populaio meas of e sud variable i e sraum, respecivel. Similar epressios for ad Z ca also be defied. Also, we ave 0

22 were f γ, f, w. Some addiioal oaios for secod order approimaio: V rs L W r s r s E s [( ) ( ) ] r [ ] s r were, ( ) ( ) rs(), i L 3 () () V W, k k L L 3 () () V W, 30 k() 3 30() V W, 3 L 3 () 03() V 03 W, 3 3 k L k () 3() 3k 4 3() 0() 0() V W, 3

23 L k () 04() 3k 3() 4 0() V 04 W, 4 ( ) L k () () k 4 3() 0() 0() () V W, were k () ( )( ), ( )( ) k () ( )( ) 6 ( ), 3 ( )( )( 3) k 3() ( ) ( )( ). 3 ( )( )( 3) 4. Firs Order Biases ad Mea Squared Errors uder sraified radom samplig Te epressios for biases ad MSE,s of e esimaors S, S ad 3S respecivel, are : 3 Bias ( S ) V0 V 8 (4.) MSE ( S ) V0 V0 V (4.) Bias ( S ) V V0 8 (4.3) MSE ( S ) V0 V0 V 4 (4.4) Bias ( 3S ) α V0 α V0 αv 4 8 (4.5)

24 MSE ( 3S ) V0 α V0 αv 4 (4.6) V B miimizig MSE (3s), e opimum value of α is obaied as α o. B puig is V0 opimum value of α i equaio (4.5) ad (4.6), we ge e miimum value for bias ad MSE of e esimaor 3S. Te epressio for e bias ad MSE of 4s o e firs order of approimaio are give respecivel, as 3 Bias ( 4s) θ V0 V ( θ) V V0 (4.7) 8 8 MSE ( 4S ) V0 θ V0 θv (4.8) V B miimizig MSE ( 4S), e opimum value of θ is obaied as θ o. B puig is V0 opimum value of α i equaio (4.7) ad (4.8) we ge e miimum value for bias ad MSE of e esimaor 3S. We observe a for e opimum cases e biases of e esimaors 3S ad 4S are differe bu e MSE of 3S ad 4S are same. I is also observed a e MSE s of e esimaors 3S ad 4S are alwas less a e MSE s of e esimaors S ad S. Tis promped us o sud e esimaors 3S ad 4S uder secod order approimaio. 5. Secod Order Biases ad Mea Squared Errors i sraified radom samplig Epressig esimaor i s(i,,3,4) i erms of e i s (i0,), we ge s e ( e ) ep 0 e Or 3

25 e s e 0 e0e e e0e e e0e e (5.) Takig epecaios, we ge e bias of e esimaor approimaio as s up o e secod order of Bias ) V V0 V V03 V (s V (5.) Squarig equaio (5.) ad akig epecaios ad usig lemmas we ge MSE of order of approimaio as s up o secod MSE( Or, e S ) E e0 e e0e e0e e 3 8 MSE ( Ee e 4 3 e 8 5 e 4 5 e s) 0 e0e e0 e e0 e 0e 0e e Or, (5.3) MSE ( ) V V V V V V V s V Similarl we ge e biases ad MSE s of e esimaors S, 3S ad 4S up o secod order of approimaio respecivel, as (5.4) 5 5 Bias ( s ) V V0 V V3 V04 V (5.5) MSE ( ) V V V V V V V V ) (5.6) S

26 Bias α 8 α 4 α 8 α 4 α 8 α 48 α 8 α ( 3S ) V0 V V V 03 V3 α 3 4 α α α V (5.7) MSE α 4 α α V ( 3S ) V0 V0 αv V αv V α α α 3α 4 α 4 α 8 V 3α 4 7α 4 V α α 7α V (5.8) Bias ( 4S ) E( 4S ) αv α{ V0 V} α V04 48 ( α 5){ V 03 V } 3 (5.9) MSE ( 4θ ) ( 4θ ) ( 4S ) V0 θ V0 θ V θ V ( 4θ ) θ( θ 5) V 04 θv θ( 4θ ) V 03 4 ( θ 5) θ ( 4θ ) V 3 (5.0) Te opimum value of α we ge b miimizig MSE ( 3S ). Bu eoreicall e deermiaio of e opimum value of α is ver difficul, we ave calculaed e opimum value b usig umerical eciques. Similarl e opimum value of θ wic miimizes e MSE of e esimaor 4s is obaied b usig umerical eciques. 6. umerical Illusraio 5

27 For e oe aural populaio daa, we sall calculae e bias ad e mea square error of e esimaor ad compare Bias ad MSE for e firs ad secod order of approimaio. Daa Se- To illusrae e performace of above esimaors, we ave cosidered e aural daa give i Sig ad audar (986, p.6). Te daa were colleced i a pilo surve for esimaig e ee of culivaio ad producio of fres fruis i ree disrics of Uar- Prades i e ear Table 6.: Bias ad MSE of esimaors Esimaor Bias MSE Firs order Secod order Firs order Secod order s s 3s s OLUSIO I e Table 6. e bias ad MSE of e esimaors S, S, 3S ad 4S are wrie uder firs order ad secod order of approimaio. Te esimaor S is epoeial produc-pe esimaor ad i is cosidered i case of egaive correlaio. So e bias ad mea squared error for is esimaor is more a e oer esimaors cosidered ere. For e classical epoeial raio-pe esimaor, i is observed a e biases ad e mea squared errors icreased for secod order. Te esimaor 3S ad 4S ave e same mea squared error for e firs order bu e mea squared error of 3S is less a 4S for e secod order. So, o 6

28 e basis of e give daa se we coclude a e esimaor 3S is bes followed b e esimaor 4S amog e esimaors cosidered ere. REFEREES Koucu,. ad Kadilar,. (009) : Famil of esimaors of populaio mea usig wo auiliar variables i sraified radom samplig. ommu. i Sais. Teor. ad Me, 38, Koucu,. ad Kadilar,. (00) : O e famil of esimaors of populaio mea i sraified radom samplig. Pak. Jour. Sa., 6(), Sig, D. ad udar, F.S. (986): Teor ad aalsis of sample surve desigs. Wile Easer Limied, ew Deli. Sig, R., aua, P. ad Sawa,.(008): O liear combiaio of Raio-produc pe epoeial esimaor for esimaig fiie populaio mea. Saisics i Trasiio,9(),05-5. Sig, R., Kumar, M., audar, M. K., Kadilar,. (009) : Improved Epoeial Esimaor i Sraified Radom Samplig. Pak. J. Sa. Oper. Res. 5(), pp Sig, R. ad Smaradace, F. (03): O improveme i esimaig populaio parameer(s) usig auiliar iformaio. Educaioal Publisig & Joural of Maer Regulari (Beijig) pg

29 TWO-PHASE SAMPLIG I ESTIMATIO OF POPULATIO MEA I THE PRESEE OF O-RESPOSE Maoj Kr. audar, Ail Prajapai, Rajes Sig ad Florei Smaradace Deparme of Saisics, Baaras Hidu Uiversi, Varaasi-005 Deparme of Maemaics, Uiversi of ew Meico, Gallup, USA orrespodig auor, rsigsa@aoo.com Absrac Te prese paper preses e deail discussio o esimaio of populaio mea i simple radom samplig i e presece of o-respose. Moivaed b Gupa ad Sabbir (008), we ave suggesed e class of esimaors of populaio mea usig a auiliar variable uder o-respose. A eoreical sud is carried ou usig wo-pase samplig sceme we e populaio mea of auiliar variable is o kow. A empirical sud as also bee doe i e suppor of eoreical resuls. Kewords: Two-pase samplig, class of esimaors, opimum esimaor, o-respose, umerical illusraios.. Iroducio Te auiliar iformaio is geerall used o improve e efficiec of e esimaors. ocra (940) proposed e raio esimaor for esimaig e populaio mea weever sud variable is posiivel correlaed wi auiliar variable. orar o e siuaio of raio esimaor, if e sud ad auiliar variables are egaivel correlaed, Mur (964) suggesed e produc esimaor o esimae e populaio mea. Hase e al. (953) proposed e differece esimaor wic was subsequel modified o provide e liear regressio esimaor for e populaio mea or oal. Moa (967) suggesed a esimaor b combiig e raio ad regressio meods for esimaig e populaio parameers. I order o esimae e populaio mea or populaio oal of e sud caracer uilizig auiliar iformaio, several oer auors icludig Srivasava ( 97), Redd (974), Ra ad Saai (980), Srivekaaramaa (980), Srivasava ad Jajj (98) 8

30 ad Sig ad Kumar (008, 0) ave proposed esimaors wic lead improvemes over usual per ui esimaor. I is observed a e o-respose is a commo problem i a pe of surve. Hase ad Hurwiz (946) were e firs o corac e problem of o-respose wile coducig mail surves. Te suggesed a ecique, kow as sub-samplig of orespodes, o deal wi e problem of o-respose ad is adjusmes. I fac e developed a ubiased esimaor for populaio mea i e presece of o-respose b dividig e populaio io wo groups, viz. respose group ad o-respose group. To avoid bias due o o-respose, e suggesed for akig a sub-sample of e o-respodig uis. Le us cosider a populaio cosiss of uis ad a sample of size is seleced from e populaio usig simple radom samplig wiou replaceme (SRSWOR) sceme. Le us assume a ad be e sud ad auiliar variables wi respecive populaio meas ad. Le us cosider e siuaio i wic sud variable is subjeced o orespose ad auiliar variable is free from e o-respose. I is observed a ere are respode ad o-respode uis i e sample of uis for e sud variable. Usig e ecique of sub samplig of o-respodes suggesed b Hase ad Hurwiz (946), we selec a sub-sample of o-respode uis from uis suc a k,k ad collec e iformaio o sub-sample b persoal ierview meod. Te usual sample mea, raio ad regressio esimaors for esimaig e populaio mea uder o-respose are respecivel represeed b (.) 9

31 R (.) lr ( ) b (.3) were ad are e meas based o respode ad o-respode uis respecivel. is e sample mea esimaor of populaio mea, based o sample of size ad b is e sample regressio coefficie of o. Te variace ad mea square errors (MSE) of e above esimaors, are respecivel give b R ad lr were ( ) ( k ) S WS V MSE MSE S ad (.4) ( ) ( ) ( k ) R ρ WS (.5) ( ) ( ) ( k ) lr ρ WS (.6) S are respecivel e mea squares of ad i e populaio. ( S ) ad ( S ) are e coefficies of variaio of ad respecivel. S ad W are respecivel e mea square ad o-respose rae of e o-respose group i e populaio for e sud variable. ρ is e populaio correlaio coefficie bewee ad. We e iformaio o populaio mea of auiliar variable is o available, oe ca use e wo-pase samplig sceme i obaiig e improved esimaor raer a e previous oes. ema (938) was e firs wo gave cocep of wo-pase samplig i esimaig e populaio parameers. Two-pase samplig is cos effecive as well as easier. Tis samplig sceme is used o obai e iformaio abou auiliar variable ceapl from 30

32 a bigger sample a firs pase ad relaivel small sample a e secod sage. Sukame (96) used wo-pase samplig sceme o propose a geeral raio-pe esimaor. Rao (973) used wo-pase samplig o sraificaio, o-respose problems ad ivesigaive comparisos. ocra (977) supplied some basic iformaio for wo-pase samplig. Saoo e al. (993) provided regressio approac i esimaio b usig wo auiliar variables for wo-pase samplig. I e sequece of improvig e efficiec of e esimaors, Sig ad Upadaa (995) suggesed a geeralized esimaor o esimae populaio mea usig wo auiliar variables i wo-pase samplig. I esimaig e populaio mea, if is ukow, firs, we obai e esimae of i usig wo-pase samplig sceme ad e esimae. Uder wo-pase samplig sceme, firs we selec a larger sample of ' uis from e populaio of size wi e elp of SRSWOR sceme. Secodl, we selec a small sample of size from ' uis. Le us agai assume a e siuaio i wic e o-respose is observed o sud variable ol ad auiliar variable is free from e o-respose. Te usual raio ad regressio esimaors of populaio mea uder wo-pase samplig i e presece of o-respose are respecivel give b ' R (.7) ' ad lr b( ) were (.8) ' is e mea based o ' uis for e auiliar variable. Te MSE s of e esimaors followig epressios R ad lr are respecivel represeed b e MSE ( ) ( ) ( k ) R ρ WS ' ' (.9) 3

33 ad MSE ( ) ( ) ( k ) lr ρ WS ' ' (.0) I e prese paper, we ave discussed e sud of o-respose of a geeral class of esimaors usig a auiliar variable. We ave suggesed e class of esimaors i wopase samplig we e populaio mea of auiliar variable is ukow. Te opimum proper of e class is also discussed ad i is compared o raio ad regressio esimaors uder o-respose. Te eoreical sud is also suppored wi e umerical illusraios.. Suggesed lass of Esimaors Le us assume a e o-respose is observed o e sud variable ad auiliar variable provides complee respose o e uis. Moivaed b Gupa ad Sabbir (008), we sugges a class of esimaors of populaio mea uder o-respose as [ ( )] η λ α α η λ (.) were α ad α are e cosas ad wose values are o be deermied. λ ad η( 0) eier cosas or fucios of e kow parameers. are I order o obai e bias ad MSE of, we use e large sample approimaio. Le us assume a ( e ), ( ) suc a ( e ) E( e ) 0 E e, V ( ) ( ) ( k ) e S E W, E V ( ) ( ) e 3

34 ov ad ( ) (, ) E e e ge. ρ Puig e values of ad form e above assumpios i e equaio (.), we ( α ) α ( e τe τe e τ e ) α ( e τe ) (.) O akig epecaio of e equaio (.), e bias of o e firs order of approimaio is give b B ( ) E( ) ( α ) α( τ τρ ) [ α τ ] (.3) Squarig bo e sides of e equaio (.) ad akig epecaio, we ca obai e MSE of o e firs order of approimaio as MSE [ α ( ) ( α ) α ( τ τρ ) ( ρ )] α τ α ( k ) α WS (.4) I e sequece of obaiig e bes esimaor wii e suggesed class wi respec o α ad α, we obai e opimum values of α ad wi respec o α ad α ad equaig e derivaives o zero, we ave α. O differeiaig MSE ( ) MSE α ( ) ( α ) [ α( τ τρ ) α ( ρ τ )] ( ) k α WS 0 (.5) MSE α ( ) [ α α ( ρ τ )] 0 (.6) Solvig e equaios (.4) ad (.5), we ge 33

35 α ( op) (.7) ( ) ( k ) S ρ W ad ( op) α ( op) ( ρ τ ) α (.8) Subsiuig e values of α ( op) ad ( op) α from equaios (.7) ad (.8) io e equaio (.4), e MSE of is give b e followig epressio. MSE MSE ( ) ( lr ) mi (.9) ( ) ( k ) S ρ W 3. Suggesed lass i Two-Pase Samplig I is geerall see a e populaio mea of auiliar variable, is o kow. I is siuaio, we ma use e wo-pase samplig sceme o fid ou e esimae of. Usig wo-pase samplig, we ow sugges a class of esimaors of populaio mea i e presece of o-respose we is ukow, as ' ' [ ( )] η λ α α η λ (3.) 3. Bias ad MSE of B applig e large sample approimaio, we ca obai e bias ad mea square error of. Le us assume a ' ( e ), ( ) ad ( e ) e suc a ( e ) E( e ) E( e ) 0 E,

36 35 ( ) ( ) S W k e E, ( ) e E, ( ) ' 3 e E, ( ) e e E ρ, ( ) ' 3 e e E ρ ad ( ) ' 3 e e E. Uder e above assumpio, e equaio (3.) gives ( ) ( ) e e e e e e e e e e τ τ τ τ τ τ α α ( ) e e e e e e e e τ τ τ τ α (3.) Takig epecaio of bo e sides of equaio (3.), we ge e bias of up o e firs order of approimaio as ( ) ( ) ( ) [ ] ' B α ρ τ α α (3.3) Te MSE of up o e firs order of approimaio ca be obaied b e followig epressio ( ) ( ) ( ) E MSE α ( ) ( ) τρ τ α ' S W k ( ) [ ] ' ρ τ α α α (3.4) 3. Opimum Values of α ad α O differeiaig ( ) MSE wi respec o α ad α ad equaig e derivaives o zero, we ge e ormal equaios

37 MSE α ( ) ( ) ( ) ( k ) α α ( τ ρ ) 0 ' τ τρ S W ' α (3.5) MSE( ) ad [ α α( τ ρ )] 0 α ' (3.6) From equaios (3.5) ad (3.6), we ge e opimum values of α ad α as α ( op) (3.7) ( k ) S ' ρ W ad ( op) α ( op) ( ρ τ ) α (3.8) O subsiuig e opimum values of α ad α, e equaio (3.4) provides miimum MSE of MSE MSE ( ) ( lr ) mi 4. Empirical Sud (3.9) ( k ) S ' ρ W I e suppor of eoreical resuls, some umerical illusraios are give below: 4. I is secio, we ave illusraed e relaive efficiec of e esimaors R, lr ad ( op) wi respec o. For is purpose, we ave cosidered e daa used b Kadilar ad igi (006). Te deails of e populaio are give below: 00, 50, 500, 5, 5,, ρ k.5, 4 S S 5 36

38 Table. Perceage Relaive Efficiec (PRE) wi respec o W Esimaor R lr ( op) Te prese secio preses e relaive efficiec of e esimaors, wi respec o. Tere are wo daa ses wic ave bee cosidered o ad ( op) illusrae e eoreical resuls. R lr Daa Se : Te populaio cosidered b Srivasava (993) is used o give e umerical ierpreaio of e prese sud. Te populaio of seve villages i a Tesil of Idia alog wi eir culivaed area (i acres) i 98 is cosidered. Te culivaed area (i acres) is ake as sud variable ad e populaio is assumed o be auiliar variable. Te populaio parameers are give below: 70, ' 40, 5, 98. 9, , S , S 406.3, S 44., ρ , k. 5 37

39 Table : Perceage Relaive Efficiec wi respec o W Esimaor R lr ( op) Daa Se : ow, we ave used aoer populaio cosidered b Kare ad Sia (004). Te daa are based o e psical grow of upper-socio-ecoomic group of 95 scool cildre of Varaasi disric uder a IMR sud, Deparme of Paediarics, Baaras Hidu Uiversi, Idia durig Te deails are give below: 95, ' 70, 35, , , S , S ,.355, ρ , k. 5. S 38

40 Table 3: Perceage Relaive Efficiec wi respec o W Esimaor R lr ( op) oclusio Te sud of a geeral class of esimaors of populaio mea uder o-respose as bee preseed. We ave also suggesed a class of esimaors of populaio mea i e presece of o-respose usig wo-pase samplig we populaio mea of auiliar variable is o kow. Te opimum proper of e suggesed class as bee discussed. We ave compared e opimum esimaor wi some eisig esimaors roug umerical sud. Te Tables, ad 3 represe e perceage relaive efficiec of e opimum esimaor of suggesed class, liear regressio esimaor ad raio esimaor wi respec o sample mea esimaor. I e above ables, we ave observed a e perceage relaive efficiec of e opimum esimaor is iger a e liear regressio ad raio esimaors. I is also observed a e perceage relaive efficiec decreases wi icrease i orespose. Refereces. ocra, W. G. (940) : Te esimaio of e ields of cereal eperimes b 39

41 samplig for e raio of grai i oal produce, Joural of Te Agriculural Scieces, 30, ocra, W. G. (977) : Samplig Teciques, 3rd ed., Jo Wile ad Sos, ew ork. 3. Gupa, S. ad Sabbir, J. (008) : O improveme i esimaig e populaio mea i simple radom samplig, Joural of Applied Saisics, 35(5), Hase, M. H. ad Hurwiz, W.. (946) : Te problem of o-respose i sample surves, Joural of Te America Saisical Associaio, 4, Hase, M. H., Hurwiz, W.. ad Madow,, W. G. (953): Sample Surve Meods ad Teor, Volume I, Jo Wile ad Sos, Ic., ew ork. 6. Kadilar,. ad igi, H. (006) : ew raio esimaors usig correlaio coefficie, Iersa 4,. 7. Kare, B. B. ad Sia, R. R. (004) : Esimaio of fiie populaio raio usig wo pase samplig sceme i e presece of o-respose, Aligar Joural of Saisics 4, Moa, S. (967) : ombiaio of regressio ad raio esimaes, Joural of e Idia Saisical Associaio 5, Mur, M.. (964) : Produc meod of esimaio, Saka, 6A, ema, J. (938). oribuio o e eor of samplig uma populaios, Joural of America Saisical Associaio, 33, Rao, J..K. (973) : O double samplig for sraificaio ad aalic surves, Biomerika, 60, Ra, S. K. ad Saai, A. (980) : Efficie families of raio ad produc-pe esimaors, Biomerika, 67, Redd, V.. (974) : O a rasformed raio meod of esimaio, Saka, 36, Saoo, J., Saoo, L.., Moa, S. (993) : A regressio approac o esimaio i wo pase samplig usig wo auiliar variables. urr. Sci. 65(),

42 5. Sig, G.. ad Upadaa, L.. (995) : A class of modified cai-pe esimaors usig wo auiliar variables i wo-pase samplig, Mero, Vol. LIII, o. 3-4, Sig, R., Kumar, M. ad Smaradace, F. (008): Almos Ubiased Esimaor for Esimaig Populaio Mea Usig Kow Value of Some Populaio Parameer(s). Pak. J. Sa. Oper. Res., 4() pp Sig, R. ad Kumar, M. (0): A oe o rasformaios o auiliar variable i surve samplig. MASA, 6:, 7-9. doi 0.333/MASA Srivasava, S. (993) Some problems o e esimaio of populaio mea usig auiliar caracer i presece of o-respose i sample surves. P. D. Tesis, Baaras Hidu Uiversi, Varaasi, Idia. 9. Srivasava, S.K. (97) : A geeralized esimaor for e mea of a fiie populaio usig muli-auiliar iformaio, Joural of Te America Saisical Associaio, 66 (334), Srivasava, S.K. ad Jajj, H. S. (98) : A class of esimaors of e populaio mea i surve samplig usig auiliar iformaio, Biomerika, 68 (), Srivekaaramaa, T. (980) : A dual o raio esimaor i sample surves, Biomerika, 67 (), Sukame, B. V. (96) : Some raio-pe esimaors i wo-pase samplig, Joural of e America Saisical Associaio, 57,

43 A Famil of Media Based Esimaors i Simple Radom Samplig Hema K. Verma, Rajes Sig ad Florei Smaradace Deparme of Saisics, Baaras Hidu Uiversi Varaasi-005, Idia Deparme of Maemaics, Uiversi of ew Meico, Gallup, USA orrespodig auor, rsigsa@aoo.com Absrac I is paper we ave proposed a media based esimaor usig kow value of some populaio parameer(s) i simple radom samplig. Various eisig esimaors are sow paricular members of e proposed esimaor. Te bias ad mea squared error of e proposed esimaor is obaied up o e firs order of approimaio uder simple radom samplig wiou replaceme. A empirical sud is carried ou o judge e superiori of proposed esimaor over oers. Kewords: Bias, mea squared error, simple radom samplig, media, raio esimaor.. Iroducio osider a fiie populaio U {U, U,..., U } of disic ad ideifiable uis. Le be e sud variable wi value measured o U i, i,,3...,. Te problem is o esimae e i populaio mea i i. Te simples esimaor of a fiie populaio mea is e sample mea obaied from e simple radom samplig wiou replaceme, we ere is o auiliar iformaio available. Someimes ere eiss a auiliar variable wic is posiivel correlaed wi e sud variable. Te iformaio available o e auiliar variable ma be uilized o obai a efficie esimaor of e populaio mea. Te samplig eor describes a wide varie of eciques for usig auiliar iformaio o obai more efficie esimaors. Te raio esimaor ad e regressio esimaor are e wo impora esimaors available i e lieraure wic are usig e auiliar iformaio. To kow more abou e raio ad regressio esimaors ad oer relaed resuls oe ma refer o [-3]. 4

44 We e populaio parameers of e auiliar variable suc as populaio mea, coefficie of variaio, kurosis, skewess ad media are kow, a umber of modified raio esimaors are proposed i e lieraure, b eedig e usual raio ad Epoeial- raio pe esimaors. Before discussig furer abou e modified raio esimaors ad e proposed media based modified raio esimaors e oaios ad formulae o be used i is paper are described below: - Populaio size - Sample size - Sud variable - Auiliar variable μ3 r β Were μr (i ), oefficie of skewess of e auiliar variable 3 μ i ρ - orrelaio o-efficie bewee ad, - Populaio meas, - Sample meas M, - Average of sample medias of m - Sample media of β - Regressio coefficie of o B (.) - Bias of e esimaor V (.) - Variace of e esimaor MSE (.) - Mea squared error of e esimaor MSE(e) PRE (e,p) 00 - Perceage relaive efficiec of e proposed esimaor p MSE(e) wi respec o e eisig esimaor e. Te formulae for compuig various measures icludig e variace ad e covariace of e SRSWOR sample mea ad sample media are as follows: V (), f f (i ) S, V() (i ) S, V(m) (mi M) i i i ov(, ) f (i )(i ) (i )(i ), i i 43

45 ov(, m) V() ', i (m M)( ), V(m) ', M i i ov(, m) ', M ' mm m ov(, ) (i ), S (i ) i i Were f ; S, I e case of simple radom samplig wiou replaceme (SRSWOR), e sample mea is used o esimae e populaio mea. Ta is e esimaor of r wi e variace V( f (.) r ) S Te classical Raio esimaor for esimaig e populaio mea of e sud variable is defied as R. Te bias ad mea squared error of R are give as below: B( R MSE( ' ' { } ) (.) R ' ) V() R V() R ov(, ) were ' R ' (.3). Proposed esimaor Suppose M δ(m m ) 0,, ep were M am b, m am b αm ( α)m M m Suc a,, w, were w deoes e se of all possible raio pe esimaors for esimaig 0 e populaio mea. B defiiio e se w is a liear varie, if w for 0 i 0 w w i w w R i W, g (.) (.) were w i (i0,, ) deoes e saisical cosas ad R deoes e se of real umbers. Also, αm M ( α)m g, δ(m m ) ep M m 44

46 ad M am b, m am b. To obai e bias ad MSE epressios of e esimaor, we wrie ( e0 ), m M( e) suc a E (e0) E(e ) 0, ( ) V( m) ov(,m) V E (e ), E(e ), E(e e ) M M 0 mm 0 m Epressig e esimaor i erms of e s, we ave ( e0 ) w 0 w am were υ. am b ( υαe ) g w υδe ep υe (.3) Epadig e rig ad side of equaio(.3) up o e firs order of approimaio, we ge υwe e0 υ w δ were w αgw w. Takig epecaios of bo sides of (.4) ad e subracig from bo sides, we ge e biases of esimaors,up o e firs order of approimaio as g(g ) B() υ w α αυ(g ) B() gαυ δυ δ υ B( ) 4 8 From (.4), we ave mm g(g ) α δ δ w 4 8 mm m δυ m δ δ w 4 8 mm υw e m υwe 0 e (.4) (.5) (.6) (.7) (.8) e (e0 υwe) (.9) Squarig bo sides of (.9) ad e akig epecaios, we ge e MSE of e esimaor, up o e firs order of approimaio as 45

47 MSE() V were R ( ) υ R w V( m) υrwov(, m). M (.0) MSE() will be miimum, we (, m) ov w k(sa) υr V m ( ) (.) Puig e value of w(k) i (.0), we ge e miimum MSE of e esimaor, as mi. MSE() V ( )( ρ ) (.) Te miimum MSE of e esimaor is same as a of radiioal liear regressio esimaor. From (.5) ad (.), we ave δ αgw w k (.3) From (.) ad (.3), we ave ol wo equaios i ree ukows. I is o possible o fid e uique values of w i s (i0,, ). I order o ge uique values for w i s, we sall impose e liear resricio ( ) w B( ) w B( ) 0 (.4) w B 0 Equaios (.), (.) ad (.4) ca be wrie i mari form as 0 0 αg B( ) w δ w B( ) w 0 k 0 (.5) Usig (.5) we ge e uique value of w i s (i0,, ) as 46

48 w w w 0 Δ 0 Δ r Δ Δ r Δ Δ r were Δ 0 δ Δ r αgb( ) B() δ B( )( αg k) B() k Δ kb( ) Δ kb( ) (.6) Table.: Some members of e proposed esimaor w 0 w w a b α g δ Esimaors q M q m 0 0 β ρ β M ρ - q3 βm ρ 0 0 ρ ρm β β - q 4 ρm β (M m) q 5 ep M m 0 0 β ρ - - q 6 ep β β (M m) ( M m) ρ 0 0 ρ β - - q 7 ep ρ ρ(m m) ( M m) β 0 β ρ q 8 β M ρ ep βm ρ β β (M m) ( M m) ρ 0 ρ β q 9 ρm β ep ρm β ρ ρ(m m) ( M m) β M (M m) 0 0 q 0 ep m M m 47

49 3. Empirical Sud For umerical illusraio we cosider: e populaio ad ake from [4] pageo.77, e populaio 3 is ake from [5] page o.04. Te parameer values ad cosas compued for e above populaios are give i e Table 3.. MSE for e proposed ad eisig esimaors compued for e ree populaios are give i e Table 3. wereas e PRE for e proposed ad eisig esimaors compued for e ree populaios are give i e Table 3.3. Table: 3. Parameer values ad cosas for 3 differe populaios Parameers For sample size 3 For sample size 5 Popl- Popl- Popl-3 Popl- Popl- Popl M β R V () V () V (m) ov (,m) ov (, ) ρ

50 Table: 3.. Variace / Mea squared error of e eisig ad proposed esimaors Esimaors For sample size 3 For sample size 5 Populaio- Populaio- Populaio-3 Populaio- Populaio- Populaio-3 q q q q q q q q q q (op) Table: 3.3. Perceage Relaive Efficiec of esimaors wi respec o Esimaors For sample size 3 For sample size 5 Populaio- Populaio- Populaio-3 Populaio- Populaio- Populaio-3 q q q q q q q q

51 q q (op) oclusio From empirical sud we coclude a e proposed esimaor uder opimum codiios perform beer a oer esimaors cosidered i is paper. Te relaive efficiecies ad MSE of various esimaors are lised i Table 3. ad 3.3. Refereces. Mur M.. (967). Samplig eor ad meods. Saisical Publisig Socie, alcua, Idia.. ocra, W. G. (977): Samplig Teciques. Wile Easer Limied. 3. Kare B.B. ad Srivasava S.R. (98): A geeral regressio raio esimaor for e populaio mea usig wo auiliar variables. Alig. J. Sais.,: Sisodia, B.V.S. ad Dwivedi, V.K. (98): A modified raio esimaor usig co-efficie of variaio of auiliar variable. Joural of e Idia Socie of Agriculural Saisics 33(), Sig G.. (003): O e improveme of produc meod of esimaio i sample surves. Joural of e Idia Socie of Agriculural Saisics 56 (3), Sig H.P. ad Tailor R. (003): Use of kow correlaio co-efficie i esimaig e fiie populaio meas. Saisics i Trasiio 6 (4), Sig H.P., Tailor R., Tailor R. ad Kakra M.S. (004): A improved esimaor of populaio mea usig Power rasformaio. Joural of e Idia Socie of Agriculural Saisics 58(), Sig, H.P. ad Tailor, R. (005): Esimaio of fiie populaio mea wi kow coefficie of variaio of a auiliar. STATISTIA, ao LV,.3, pp Kadilar. ad igi H. (004): Raio esimaors i simple radom samplig. Applied Maemaics ad ompuaio 5, Koucu. ad Kadilar. (009): Efficie Esimaors for e Populaio mea. Haceepe Joural of Maemaics ad Saisics, Volume 38(), Sig R., Kumar M. ad Smaradace F. (008): Almos ubiased esimaor for esimaig populaio mea usig kow value of some populaio parameer(s). Pak.j.sa.oper.res., Vol.IV, o., pp

52 . Sig, R. ad Kumar, M. (0): A oe o rasformaios o auiliar variable i surve samplig. Mod. Assis. Sa. Appl., 6:, 7-9. doi 0.333/MAS Sig R., Malik S., audar M.K., Verma H.K., ad Adewara A.A. (0): A geeral famil of raio-pe esimaors i ssemaic samplig. Jour. Reliab. Sa. Ssci., 5(): Sig, D. ad audar, F. S. (986): Teor ad aalsis of surve desigs. Wile Easer Limied. 5. Mukopada, P. (998): Teor ad meods of surve samplig. Preice Hall. 5

53 DIFFREE-TPE ESTIMATORS FOR ESTIMATIO OF MEA I THE PRESEE OF MEASUREMET ERROR Viplav Kr. Sig, Rajes Sig ad Florei Smaradace Deparme of Saisics, Baaras Hidu Uiversi Varaasi-005, Idia Deparme of Maemaics, Uiversi of ew Meico, Gallup, USA orrespodig auor, rsigsa@aoo.com Absrac I is paper we ave suggesed differece-pe esimaor for esimaio of populaio mea of e sud variable i e presece of measureme error usig auiliar iformaio. Te opimum esimaor i e suggesed esimaor as bee ideified alog wi is mea square error formula. I as bee sow a e suggesed esimaor performs more efficie e oer eisig esimaors. A empirical sud is also carried ou o illusrae e meris of proposed meod over oer radiioal meods. Ke Words: Sud variable, Auiliar variable, Measureme error, Simple radom Samplig, Bias, Mea Square error.. PERFORMAE OF SUGGESTED METHOD USIG SIMPLE RADOM SAMPLIG 5

54 ITRODUTIO Te prese sud deals wi e impac of measureme errors o esimaig populaio mea of sud variable () i simple radom samplig usig auiliar iformaio. I eor of surve samplig, e properies of esimaors based o daa are usuall presupposed a e observaios are e correc measureme o e caracerisic beig sudied. We e measureme errors are egligible small, e saisical iferece based o observed daa coiue o remai valid. A impora source of measureme error i surve daa is e aure of variables (sud ad auiliar). Here aure of variable sigifies a e eac measureme o variables is o available. Tis ma be due o e followig ree reasos:. Te variable is clearl defied bu i is ard o ake correc observaio a leas wi e currel available eciques or because of oer pes of pracical difficulies. Eg: Te level of blood sugar i a uma beig.. Te variable is cocepuall well defied bu observaio ca obai ol o some closel relaed subsiues kow as Surrogaes. Eg: Te measureme of ecoomic saus of a perso. 3. Te variable is full compreesible ad well udersood bu i is o irisicall defied. Eg: Ielligece, aggressiveess ec. Some auors icludig Sig ad Karpe (008, 009), Salab(997), Alle e al. (003), Maisa ad Sig (00, 00), Srivasava ad Salab (00), Kumar e al. (0 a,b), Malik ad Sig (03), Malik e al. (03) ave paid eir aeio owards e esimaio of populaio mea μ of sud variable usig auiliar iformaio i e presece of measureme errors. Fuller (995) eamied e imporace of measureme errors i esimaig parameers i sample surves. His major cocers are esimaio of populaio mea or oal ad is sadard error, quarile esimaio ad esimaio roug regressio model. SMBOLS AD SETUP 53

55 Le, for a SRS sceme ( i, i ) be e observed values isead of rue values, ) o wo caracerisics (, ), respecivel for all i(,, ) ad e observaioal or ( i i measureme errors are defied as u v i i ( ) () i i ( ) () i i were ui ad vi are socasic i aure wi mea 0 ad variace For e sake of coveiece, we assume a u ad v respecivel. u i ' s ad v i ' s are ucorrelaed aloug i ' s ad ' s are correlaed.suc a specificaio ca be, owever, relaed a e i cos of some algebraic complei. Also assume a fiie populaio correcio ca be igored. Furer, le e populaio meas ad variaces of (, ) be ( μ, μ ) ad (, ). ad ρ be e populaio covariace ad e populaio correlaio coefficie bewee ad respecivel. Also le ad μ μ are e populaio coefficie of variaio ad is e populaio coefficies of covariace i ad. LARGE SAMPLE APPROIMATIO Defie: e 0 μ ad μ e μ μ were, e 0 ad e are ver small umbers ad e i < (i 0,). Also, E(e i ) 0(i 0,) u ad, E(e0 ) θ δ 0, 54

56 55 v ) (e E δ θ, 0 ) e (e E θρ, were θ.. EISTIG ESTIMATORS AD THEIR PROPERTIES Usual mea esimaor is give b i i (3) Up o e firs order of approimaio e variace of is give b u Var() θμ (4) Te usual raio esimaor is give b μ R (5) were μ is kow populaio mea of. Te bias ad MSE ( R ), o e firs order of approimaio, are respecivel, give ρ θμ v R ) ( B (6) ρ θμ v u R ) ( MSE (7) Te radiioal differece esimaor is give b ) k( d μ (8) were, k is e cosa wose value is o be deermied. Miimum mea square error of d a opimum value of

57 56 v k μ ρ μ, is give b ρ θ μ v u u d ) ( MSE (9) Srivasava (967) suggesed a esimaor S μ (0) were, is a arbirar cosa. Up o e firs of approimaio, e bias ad miimum mea square error of S a opimum value of v ρ are respecivel, give b θρ θ μ v S ) ( ) ( B () ρ θ μ v u u S ) ( MSE () Wals (970) suggesed a esimaor w μ μ w ) ( (3) were, is a arbirar cosa.

58 57 Up o e firs order of approimaio, e bias ad miimum mea square error of w a opimum value of v ρ, are respecivel, give b ρ θ μ v w ) ( B (4) ρ θ μ v u u w ) ( MSE (5) Ra ad Saai (979) suggesed e followig esimaor μ 3 3 RS ) ( (6) were, 3 is a arbirar cosa. Up o e firs order of approimaio, e bias ad mea square of RS a opimum value of ρ v 3 are respecivel, give b 3 RS ) ( B ρ μ θ (7) ρ θ μ v u u RS ) ( MSE (8) 3. SUGGESTED ESTIMATOR Followig Sig ad Solaki (03), we sugges e followig differece-pe class of esimaors for esimaig populaio mea of sud variable as

59 [ α α ( α α ) μ ] α μ p (9) were α, ) are suiabl cose scalars suc a MSE of e proposed esimaor is ( α miimum, ( η λ), μ ημ λ) wi (, λ) are eier cosas or fucio of some ( kow populaio parameers. Here i is ieresig o oe a some eisig esimaors ave bee sow as e members of proposed class of esimaors p for differe values of ( λ α, α, α, η, ), wic is summarized i Table. Table : Members of suggesed class of esimaors Values of osas Esimaors α α α η λ [Usual ubiased] R [Usual raio] 0 0 d [Usual differece] α 0 - μ S [Srivasava (967)] 0 α 0 DS [Dube ad Sig] α α 0 0 Te properies of suggesed esimaor are derived i e followig eorems. Teorem.: Esimaor p i erms of e i ;i 0, epressed as: p [ μ αaeμ Bμ e α 0 0 { αae Be e μ αaμ e e } α { e α }] ημ Ae 58

60 r s igorig e erms E(ei e j ) for (rs)>,were r,s0,,... ad i 0, ; j (firs order of approimaio). were, ημ α( α ) A, B A ad μ μ. ημ λ Proof p [ α α ( α α ) μ ] μ α Or [ α ( e ) α η μ e ( α ) μ ][ Ae ] α (0) p 0 We assume e <, so a e erm A α ( Ae) is epadable. Epadig e rig ad side (0) ad eglecig e erms of e s avig power greaer a wo, we ave p μ αae μ Bμ e α { αae Be e μ αaμ e e } 0 0 α { e α } ημ Ae Teorem:. Bias of e esimaor p is give b B( p Proof: [ Bμ δ α { Bδ αaμ δ } α ημ Aαδ ] ) () B( p ) E( p μ ) 0 [ 0 0 E μ μ αae μ Bμ e α { αae Be e μ αaμ e e } [ Bμ δ α { Bδ αaμ δ } α ημ Aαδ ] 0 α { e α }] ημ Ae were, δ 0, δ adδ0 are alread defied i secio 3. 59

61 Teorem.3: MSE of e esimaor p, up o e firs order of approimaio is MSE( p ) α { μ δ0 δ( α A B ) 4αAμ δ0} α η μ δ { δ( α A μ Bμ )} α{ δ( B Bμ α A μ ) δ0αaμ ( μ ) } α ημ αaδ ( μ ) α α ημ ( μ δ Aα δ ) () Proof: 0 MSE( p ) E( p μ ) [ 0 0 e E α { Aαe e μ Be αaμ e e } α ημ { e Aα } αae ] μ Bμ e Squarig ad e akig epecaios of bo sides, we ge e MSE of e suggesed esimaor up o e firs order of approimaio as MSE( p ) α { μ δ0 δ( α A B ) 4αAμ δ0} α η μ δ { δ( α A μ Bμ )} α{ δ( B Bμ α A μ ) δ0αaμ ( μ ) } α ημ αaδ ( μ ) α α ημ ( μ δ Aα δ ) 0 Equaio () ca be wrie as: MSE ( ) α ϕ α ϕ α ϕ α ϕ α α ϕ ϕ (3) p Differeiaig (3) wi respec o α, ) ad equaig em o zero, we ge e opimum values of α, ) as ( α ( α ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ α(op) ad α (op) ϕϕ ϕ5 ϕϕ ϕ5 were, ϕ μ δ0 δ( α A B ) 4αAμ δ0 ϕ η μ δ 60

62 ( B Bμ α A μ ) δ αaμ ( μ ) ϕ 3 δ 0 ( μ ) ϕ 4 ημ αaδ ϕ ( μ δ Aα ). 5 ημ 0 δ ϕ δ ( α A μ Bμ ) I e Table some esimaors are lised wic are paricular members of e suggesed class of esimaors p for differe values of ( α, η, λ). Table : Paricular members of e suggesed class of esimaors p Esimaors Values of cosas α η λ μ [ α α ( α α ) μ ] - 0 μ [ α α ( ) ( α α )( μ ) ] μ [ α α ( ) ( α α )( μ ) ] - μ ρ [ α α ( ρ) ( α α )( μ ρ) ] - ρ [ α α ( ) ( α α )( μ )] ρ - μ 6

63 μ [ α α ( ) ( α α )( μ )] 6-7 [ α α ( ) ( α α )( μ )] μ EMPIRIAL STUD Daa saisics: Te daa used for empirical sud as bee ake from Gujarai (007) Were, True cosumpio epediure, i i True icome, i Measured cosumpio epediure, i Measured icome. μ μ ρ u v Te perceage relaive efficiecies (PRE) of various esimaors wi respec o e mea per ui esimaor of, a is,ca be obaied as Var() PRE (.) 00 MSE(.) Table 3: MSE ad PRE of esimaors wi respec o Esimaors Mea Square Error Perce Relaive Efficiec R d

64 S DS PERFORMAE OF SUGGESTED ESTIMATOR I STRATIFIED RADOM SAMPLIG SMBOLS AD SETUP osider a fiie populaio U (u, u,...,u ) of size ad le ad respecivel be e auiliar ad sud variables associaed wi eac ui u j (j,,..., ) of populaio. Le e populaio of be sraified i o L sraa wi e sraum coaiig L uis, were,,,3,.,l suc a i. A simple radom size is drow wiou replaceme from e sraum suc a L i. Le, ) of wo ( i i caracerisics (,) o i ui of e sraum, were i,,,,. I addiio le ( i, i ), i i ( s W, s W i i ), 63

65 ( μ i, μ i i L Ad ( μ W μ, μ i i ), W μ ) be e samples meas ad populaio meas of(,) respecivel, were W is e sraum weig. Le e observaioal or measureme errors be u i (4) i i v i (5) i i Were ui ad vi are socasic i aure ad are ucorrelaed wi mea zero ad variaces V ad U respecivel. Furer le ρ be e populaio correlaio coefficie bewee ad i e sraum. I is also assumed a e fiie populaio correcio erms f ) ad ( f ) ca be igored were f ad ( Le, LARGE SAMPLE APPROIMATIO μ ( e ), ad μ ( e ) s 0 suc a, E(e ) E(e ) 0, 0 s f. E(e 0 U ) 0, θ E(e V ), θ E(e. 0e ) ρ 0 were,,, μ U V θ ad θ. μ U 64 V

66 EISTIG ESTIMATORS AD THEIR PROPERTIES s is usual ubiased esimaor i sraified radom samplig sceme. Te usual combied raio esimaor i sraified radom samplig i e presece of measureme error is defied as- T R μ s (6) s Te usual combied produc esimaor i e presece of measureme error is defied as- T PR s s (7) μ ombied differece esimaor i sraified radom samplig is defied i e presece of measureme errors for a populaio mea, as T D d( μ ) (8) s s Te variace ad mea square erm of above esimaors, up o e firs order of approimaio, are respecivel give b U Var ( s ) (9) MSE(T MSE(T W L R ) R θ θ W L P ) R θ θ ( R β θ ) ( R β θ ) (30) (3) L L L W W W MSE (T D ) d d β (3) θ θ 65

67 were, d op L L W β W θ 6. SUUGESTED ESTIMATOR AD ITS PROPERTIES Le B(.) ad M(.) deoe e bias ad mea square error (M.S.E) of a esimaor uder give samplig desig. Esimaor p defied i equaio (9) ca be wrie i sraified radom samplig as [ β β ( β β ) μ ] β μ T P s s (33) s were α, ) are suiabl cose scalars suc a MSE of proposed esimaor is ( α miimum, ( η λ), μ ημ λ) wi (, λ) are eier cosas or fucios of s s ( some kow populaio parameers. Here i is ieresig o oe a some eisig esimaors ave bee foud paricular members of proposed class of esimaors Tp for differe values of α, α, α, η, ), wic are summarized i Table 4. ( λ Table 4: Members of proposed class of esimaors T p Values of osas Esimaors α α α η λ s [Usual ubiased] T R [Usual raio] 0 0 T PR [Usual produc] 0-0 T D [Usual differece] α 0 - μ 66

68 Teorem.: Esimaor r s T i erms of e i ; i 0, b igorig e erms E( e i e ) for P j (rs)>,were r,s0,,... ad i 0, ; j, ca be wrie as T P [ μ βa' eμ B' μ e β 0 0 { ' βa' e B' ' e e μ βa' μ e e } β { e βa' }] ημ e were, ημ β( β ) A ', B' A' ad ' μ μ. ημ λ Proof T P [ β β ( β β ) μ ] μ s s s β [ β e ) β η μ e ( β ) μ ][ A' ] β ( (34) 0 e We assume A ' e <, so a e erm A' e ) ( β is epadable. Tus b epadig e rig ad side (0) ad eglecig e erms of e s avig power greaer a wo, we ave [ 0 0 Tp μ βa' e μ B' μ e β { ' βa' ' e B' ' e e μ βa' μ e e } Teorem:. Bias of T p is give b P [ B' μ β { B' ' βa' μ } β ημ A' β ] 0 β { e βa' }] ημ e B(T ) (35) Proof: B(T ) E(T P P μ ) [ 0 0 E μ μ βa' e μ B' μ e β { ' βa' ' e B' ' e e μ βa' μ e e } β { e βa' }] ημ e 67

69 [ B' μ β { B' ' βa' μ } β βημ A' ] 0 were, 0, ad 0 are alread defied i secio 3. Teorem:.3 Mea square error of Tp, up o e firs order of approimaio is give b MSE(T ) β P { ' μ 0 ( β A' ' B'' ) 4βA'' μ 0} β η μ { ' ( β A' μ B' ' μ )} β{ ' ( B' ' B' ' μ β A' ' μ ) 0βAμ ( μ ' )} β ημ βa ( μ ' ) β β ημ ( μ A' β ) (36) Proof: 0 ' MSE(T MSE(T ) β P ) E(TP μ ) P { ' μ 0 ( β A' ' B'' ) 4βA'' μ 0} β η μ { ' ( β A' μ B'' μ )} β{ ' ( B'' B'' μ β A' ' μ ) 0βAμ ( μ ' )} β ημ βa ( μ ' ) β β ημ ( μ A' β ) 0 ' MSE(Tp) ca also be wrie as MSE (T ) β χ β χ β χ β χ β β χ χ (37) P Differeiaig equaio (37) wi respec o β, ) ad equaig i o zero, we ge e opimum values of β, ) respecivel, as ( β ( β χ χ χ χ χ χ χ β(op) adβ (op) χχ χ 5 χχ χ 5 χ were, χ ' μ 0 ( β A' ' B' ' ) 4βA' ' μ 0 χ η μ ( B B' ' μ β A' ' μ ) βa' μ ( μ ' ) χ 3 ' 0 ( μ ' ) χ 4 ημ βa' 68

70 χ ( μ A' β ' ). 5 ημ 0 χ ' ( β A' μ B' ' μ ) Wi e elp of ese values, we ge e miimum MSE of e suggesed esimaor Tp. 7. DISUSSIO AD OLUSIO I e prese sud, we ave proposed differece-pe class of esimaors of e populaio mea of a sud variable we iformaio o a auiliar variable is kow i advace. Te asmpoic bias ad mea square error formulae of suggesed class of esimaors ave bee obaied. Te asmpoic opimum esimaor i e suggesed class as bee ideified wi is properies. We ave also sudied some radiioal meods of esimaio of populaio mea i e presece of measureme error suc as usual ubiased, raio, usual differece esimaors suggesed b Srivasava(967), dube ad sig( 00), wic are foud o be paricular members of suggesed class of esimaors. I addiio, some ew members of suggesed class of esimaors ave also bee geeraed i simple radom samplig case. A empirical sud is carried o row lig o e performace of suggesed esimaors over oer eisig esimaors usig simple radom samplig sceme. From e Table 3, we observe a suggesed esimaor 3 performs beer a e oer esimaors cosidered i e prese sud ad wic reflecs e usefuless of suggesed meod i pracice. REFEREES Alle, J., Sig, H. P. ad Smaradace, F. (003): A famil of esimaors of populaio mea usig muliauiliar iformaio i presece of measureme errors. Ieraioal Joural of Social Ecoomics 30 (7), A.K. Srivasava ad Salab (00). Effec of measureme errors o e regressio meod of esimaio i surve samplig. Joural of Saisical Researc, Vol. 35, o., pp Bal, S. ad Tueja, R. K. (99): Raio ad produc pe epoeial esimaor. Iformaio ad opimizaio scieces (), adok, P.K., & Ha,.P.(990):O e efficiec of e raio esimaor uder Midzuo sceme wi measureme errors. Joural of e Idia saisical Associaio,8,

71 Dube, V. ad Sig, S.K. (00). A improved regressio esimaor for esimaig populaio mea, J. Id. Soc. Agri. Sais., 54, p Gujarai, D.. ad Sageea (007): Basic ecoomerics. Taa McGraw Hill. Koucu,. ad Kadilar,. (00): O e famil of esimaors of populaio mea i sraified samplig. Pakisa Joural of Saisics. Pak. J. Sa. 00 vol 6 (), Kumar, M., Sig, R., Sig, A.K. ad Smaradace, F. (0 a): Some raio pe esimaors uder measureme errors. WASJ 4() :7-76. Kumar, M., Sig, R., Sawa,. ad aua, P. (0b): Epoeial raio meod of esimaors i e presece of measureme errors. I. J. Agricul. Sa. Sci. 7(): Malik, S. ad Sig, R. (03) : A improved class of epoeial raio- pe esimaor i e presece of measureme errors. OTOGO Maemaical Magazie,,, Malik, S., Sig, J. ad Sig, R. (03) : A famil of esimaors for esimaig e populaio mea i simple radom samplig uder measureme errors. JRSA, (), Maisa ad Sig, R. K. (00): A esimaio of populaio mea i e presece of measureme errors. Joural of Idia Socie of Agriculural Saisics 54(), 3 8. Maisa ad Sig, R. K. (00): Role of regressio esimaor ivolvig measureme errors. Brazilia joural of probabili Saisics 6, Salab (997): Raio meod of esimaio i e presece of measureme errors. Joural of Idia Socie of Agriculural Saisics 50(): Sig, H. P. ad Karpe,. (008): Raio-produc esimaor for populaio mea i presece of measureme errors. Joural of Applied Saisical Scieces 6, Sig, H. P. ad Karpe,. (009): O e esimaio of raio ad produc of wo populaios meas usig supplemear iformaio i presece of measureme errors. Deparme of Saisics, Uiversi of Bologa, 69(), Sig, H. P. ad Viswakarma, G. K. (005): ombied Raio-Produc Esimaor of Fiie Populaio Mea i Sraified Samplig. Meodologia de Ecuesas 8:

72 Sig, H. P., Raour, A., Solaki, R. S.(03): A improveme over differece meod of esimaio of populaio mea. JRSS, 6(): Sig H. P., Raour A., Solaki R.S. (03): a improveme over differece meod of esimaio of populaio mea. JRSS, 6(): Srivasava, S. K. (967): A esimaor usig auiliar iformaio i e sample surves. alcua saisical Associaio Bullei 6,-3. Wals, J.E.(970). Geeralisaio of raio esimae for populaio oal. Saka A.3,

73 Te purpose of wriig is book is o sugges some improved esimaors usig auiliar iformaio i samplig scemes like simple radom samplig, ssemaic samplig ad sraified radom samplig. Tis volume is a collecio of five papers, wrie b ie co-auors (lised i e order of e papers): Rajes Sig, Mukes Kumar, Maoj Kr. audar, em Kadilar, Praas Sarma, Florei Smaradace, Ail Prajapai, Hema Verma, ad Viplav Kr. Sig. I firs paper dual o raio-cum-produc esimaor is suggesed ad is properies are sudied. I secod paper a epoeial raio-produc pe esimaor i sraified radom samplig is proposed ad is properies are sudied uder secod order approimaio. I ird paper some esimaors are proposed i wo-pase samplig ad eir properies are sudied i e presece of o-respose. I four caper a famil of media based esimaor is proposed i simple radom samplig. I fif paper some differece pe esimaors are suggesed i simple radom samplig ad sraified radom samplig ad eir properies are sudied i presece of measureme error.