ON IMPROVEMENT IN ETIMATING POPULATION PARAMETER() UING AUILIAR INFORMATION Raje ig Departmet of tatitic BHU Varaai (U. P.) Idia Editor Floreti maradace Departmet of Matematic Uiverit of New Meico Gallup UA Editor Etimator Bia ME Firt order ecod order Firt order ecod order t 0.0095 0.00 9.75 9.5 t 0-0.0006 9.75 (for g) 9.55 (for g) t -0.09-0.095 9.75 9.90 t 0.809-0.608 9.75 9.855 t5-0.07679-0.09 9.75 9.787
ON IMPROVEMENT IN ETIMATING POPULATION PARAMETER() UING AUILIAR INFORMATION Raje ig Departmet of tatitic BHU Varaai (U. P.) Idia Editor Floreti maradace Departmet of Matematic Uiverit of New Meico Gallup UA Editor Educatioal Publiig & Joural of Matter Regularit (Beijig) 0
Ti book ca be ordered o paper or electroic format from: Educatio Publiig eapeake Aveue olumbu Oio UA Tel. (6) 85-07 E-mail: ifo@edupublier.com oprigt 0 b EducatioPubliig & Joural of Matter Regularit (Beijig) Editor ad te Autor for teir paper Frot ad back cover b te Editor Peer Reviewer: Prof. Adria Olaru Uiverit POLITEHNIA of Bucaret (UPB) Bucaret Romaia. Dr. Lifa Mao Academ of Matematic ad tem iee Academ of ciece Beijig 0090 P. R. ia.. P. Liu Departmet of Applied Matematic Beijig Jiaotog Uiverit Beijig 000 P. R. ia. Prof. ige Li cool of Maagemet Nigbo Ititute of Tecolog Zejiag Uiverit Nigbo 500 P. R. ia. IBN: 978599705 Prited i te Uited tate of America
otet Preface:. Ue of auiliar iformatio for etimatig populatio mea i tematic amplig uder o- repoe: 5. ome improved etimator of populatio mea uig iformatio o two auiliar attribute: 7. tud of ome improved ratio tpe etimator uder ecod order approimatio: 5. Improvemet i etimatig te populatio mea uig dual to ratio-cum-product etimator i imple radom amplig: 5. ome improved etimator for populatio variace uig two auiliar variable i double amplig: 5
Preface Te purpoe of writig ti book i to ugget ome improved etimator uig auiliar iformatio i amplig ceme like imple radom amplig ad tematic amplig. Ti volume i a collectio of five paper writte b eigt coautor (lited i te order of te paper): Maoj K. audar aci Malik Raje ig Floreti maradace Hemat Verma Praa arma Olufadi uua ad Viplav Kumar ig from Idia Nigeria ad UA. Te followig problem ave bee dicued i te book: I capter oe a etimator i tematic amplig uig auiliar iformatio i tudied i te preece of o-repoe. I ecod capter ome improved etimator are uggeted uig auiliar iformatio. I tird capter ome improved ratio-tpe etimator are uggeted ad teir propertie are tudied uder ecod order of approimatio. I capter four ad five ome etimator are propoed for etimatig ukow populatio parameter() ad teir propertie are tudied. Ti book will be elpful for te reearcer ad tudet wo are workig i te field of fiite populatio etimatio.
Ue of Auiliar Iformatio for Etimatig Populatio Mea i tematic amplig uder No- Repoe Maoj K. audar aci Malik Raje ig ad Floreti maradace Departmet of tatitic Baara Hidu Uiverit Varaai-005 Idia Departmet of Matematic Uiverit of New Meico Gallup UA orrepodig autor Abtract I ti paper we ave adapted ig ad ukla (987) etimator i tematic amplig uig auiliar iformatio i te preece of o-repoe. Te propertie of te uggeted famil ave bee dicued. Epreio for te bia ad mea quare error (ME) of te uggeted famil ave bee derived. Te comparative tud of te optimum etimator of te famil wit ratio product dual to ratio ad ample mea etimator i tematic amplig uder o-repoe a alo bee doe. Oe umerical illutratio i carried out to verif te teoretical reult. Keword: Auiliar variable tematic amplig factor-tpe etimator mea quare error o-repoe. 5
. Itroductio Tere are ome atural populatio like foret etc. were it i ot poible to appl eail te imple radom amplig or oter amplig ceme for etimatig te populatio caracteritic. I uc ituatio oe ca eail implemet te metod of tematic amplig for electig a ample from te populatio. I ti amplig ceme ol te firt uit i elected at radom te ret beig automaticall elected accordig to a predetermied patter. tematic amplig a bee coidered i detail b Madow ad Madow (9) ocra (96) ad Lairi (95). Te applicatio of tematic amplig to foret urve a bee illutrated b Hael (9) Fie (98) ad Nair ad Bargava (95). Te ue of auiliar iformatio a bee permeated te importat role to improve te efficiec of te etimator i tematic amplig. Kuwaa ad ig (989) uggeted a cla of almot ubiaed ratio ad product tpe etimator for etimatig te populatio mea uig jack-kife tecique iitiated b Queouille (956). Later Baarai et al. (99) ig ad ig (998) ig et al. (0) ig et al. (0) ad ig ad olaki (0) ave made a attempt to improve te etimator of populatio mea uig auiliar iformatio i tematic amplig. Te problem of o-repoe i ver commo i urve ad coequetl te etimator ma produce bia reult. Hae ad Hurwitz (96) coidered te problem of etimatio of populatio mea uder o-repoe. Te propoed a amplig pla tat ivolve takig a ubample of o-repodet after te firt mail attempt ad te eumeratig te ubample b peroal iterview. El-Badr (956) eteded Hae ad Hurwitz (96) tecique. Hae ad Hurwitz (96) tecique i imple radom amplig i decribed a: From a populatio U (U U --- UN) a large firt pae ample of ize i elected b imple radom amplig witout replacemet (RWOR). A maller ecod pae ample of ize i elected from b RWOR. No-repoe occur o te ecod pae of ize i wic uit repod ad uit do ot. From te o-repodet b RWOR a ample of r / k; k > uit i elected. It i aumed tat all te r uit repod ti time roud. (ee ig ad Kumar (0009)). everal autor uc a ocra (977) odipo ad Obiea (007) Rao (987) Kare ad rivatava ( 997) ad Okafor ad Lee (000) ave tudied te problem of o-repoe uder R. 6
I te equece of improvig te etimator ig ad ukla (987) propoed a famil of factor-tpe etimator for etimatig te populatio mea i imple radom amplig uig a auiliar variable a ( A ) ( A fb) fb T α (.) were ad are te ample mea of te populatio mea ad repectivel. A B ad are te fuctio ofα wic i a calar ad coe o a te ME of te etimator Tα i miimum. Were A ( α )( α ) ( α )( α ) ( α )( α )( ) B α ; α > 0 ad f. N Remark : If we take α ad te reultig etimator will be ratio product dual to ratio ad ample mea etimator of populatio mea i imple radom amplig repectivel (for detail ee ig ad ukla (987) ). I ti paper we ave propoed a famil of factor-tpe etimator for etimatig te populatio mea i tematic amplig i te preece of o-repoe adaptig ig ad ukla (987) etimator. Te propertie of te propoed famil ave bee dicued wit te elp of empirical tud.. amplig trateg ad Etimatio Procedure Let u aume tat a populatio coit of N uit umbered from to N i ome order. If N k were k i a poitive iteger te tere will be k poible ample eac coitig of uit. We elect a ample at radom ad collect te iformatio from te uit of te elected ample. Let uit i te ample repoded ad uit did ot repod o tat. Te uit ma be regarded a a ample from te repoe cla ad uit a a ample from te o-repoe cla belogig to te populatio. Let u aume tat N ad N be te umber of uit i te repoe cla ad o-repoe 7
cla repectivel i te populatio. Obvioul N ad N are ot kow but teir ubiaed etimate ca be obtaied from te ample a N ˆ N / ; N ˆ N /. Furter uig Hae ad Hurwitz (96) tecique we elect a ub-ample of ize from te o-repodet uit uc tat L ( L > ) ad gater te iformatio o all te uit elected i te ub-ample (for detail o Hae ad Hurwitz (96) tecique ee ig ad Kumar (009)). Let ad be te tud ad auiliar variable wit repective populatio mea ad. Let ij ( ij ) be te obervatio o te t j uit i te i t tematic ample uder tud (auiliar) variable ( i... k : j... ).Let u coider te ituatio i wic orepoe i oberved o tud variable ad auiliar variable i free from o-repoe. Te Hae-Hurwitz (96) etimator of populatio mea ad ample mea etimator of baed o a tematic ample of ize are repectivel give b ad ij j were ad are repectivel te mea baed o repodet uit ad orepodet uit. Obvioul repective variace of ad are ubiaed etimator of ad repectivel. Te ad are epreed a V ad N ( ) ( ) L { } W ρ (.) N ( ) N N V { ( ) ρ } (.) 8
were ρ ad ρ are te correlatio coefficiet betwee a pair of uit witi te tematic ample for te tud ad auiliar variable repectivel. ad repectivel te mea quare of te etire group for tud ad auiliar variable. are be te populatio mea quare of o-repoe group uder tud variable ad W i te orepoe rate i te populatio. Aumig populatio mea of auiliar variable i kow te uual ratio product ad dual to ratio etimator baed o a tematic ample uder o-repoe are repectivel give b R (.) ad P (.) ( N ) D ( N ). (.5) Obvioul all te above etimator R ad P are biaed. To derive te biae D ad mea quare error (ME) of te etimator approimatio let R P ad D uder large ample ( ) e 0 ( e ) uc tat E ( e 0 ) ( e ) V ( ) E ( ) e 0 V E ( ) ( ) e ad E 0 N N L ρ (.6) { ( ) } W N N { ( ) ρ } 9 (.7)
ov ( ) E ( ) e 0 e N N { ( ) ρ } { ( ) ρ } ρ (.8) were ad are te coefficiet of variatio of tud ad auiliar variable repectivel i te populatio (for proof ee ig ad ig(998) ad ig (00 pg. o. 8) ). Te biae ad ME of te etimator R approimatio uig (.6-.8) are repectivel give b ( R ) N N B ( ) ( R ) N N { ρ }( Kρ ) ME ( ) ( P ) N N { ρ } ρ ( Kρ ) B { ( ) ρ } Kρ ( P ) N N ME ( ) ( D ) N N B ( ) ( D ) N N ad P up to te firt order of (.9) [ ] { ρ } ρ ( Kρ ) { ρ }[ ] ρ K ME { ( ) ρ } D L W (.0) (.) [ ] ρ L W (.) (.) f f f f ρ K L W ( ) (.) were { ρ } ( ) ρ { ( ) ρ } ad K ρ. ( for detail of proof refer to ig et al.(0)). Te regreio etimator baed o a tematic ample uder o-repoe i give b 0
lr b( ) (.5) ME of te etimator lr i give b N ME ( lr ) ( ) ρ N { }[ K ]. Adapted Famil of Etimator ρ L W ( ) (.6) Adaptig te etimator propoed b ig ad ukla (987) a famil of factor-tpe etimator of populatio mea i tematic amplig uder o-repoe i writte a ( A ) ( A fb) fb Tα. (.) Te cotat A B ad f are ame a defied i (.). It ca eail be ee tat te propoed famil geerate te o-repoe verio of ome well kow etimator of populatio mea i tematic amplig o puttig differet coice ofα. For eample if we take α ad te reultig etimator will be ratio product dual to ratio ad ample mea etimator of populatio mea i tematic amplig uder o-repoe repectivel.. Propertie of T α Obvioul te propoed famil i biaed for te populatio mea. I order to fid te bia ad ME of i term of i e ( 0) T α we ue large ample approimatio. Epreig te equatio (.) i we ave ( e )( De ) [( A ) fb( e )] 0 Tα (.) A fb were D. A fb ice D < ad e i < eglectig te term of i ta two te equatio (.) ca be writte a e ( 0) i avig power greater
α ( A ){ e0 De D e De0e } T A fb [ { e ( D ) e D( D ) e ( D ) e e }] fb. (.) 0 Takig epectatio of bot ide of te equatio (.) we get 0 E [ T ] α ( fb) ( ) ( ) A fb A E e E e0e fb. Let φ ( α ) A fb fb ad φ ( α ) A fb te φ ( α ) φ ( α ) - ( α ) φ fb. A fb Tu we ave E [ T ] α ( α )[ φ ( α ) E( e ) E( e )] φ. (.) Puttig te value of ( ) equatio (.) we get te bia of ( ) T α N N 0e E ad ( ) e T α a B φ( α ) { ( ) ρ } φ ( α ) E e 0 e from equatio (.7) ad (.8) ito te [ ρ ] K. (.5) quarig bot te ide of te equatio (.) ad te takig epectatio we get E [ T ] α [ E( e ) φ ( α ) E( e ) φ ( α ) E( e e )] ubtitutig te value of E ( ) ( ) 0. (.6) e 0 E ad ( ) (.6) (.7) ad (.8) ito te equatio (.6) we get te ME of ( ) T α N N e 0 E e 0 e from te repective equatio T α a [ { ρ K} ] ME { ( ) ρ } ρ φ ( α ) φ ( α ) ( ) L W. (.7)
. Optimum oice of α I order to obtai te optimum coice of α we differetiate te equatio (.7) wit repect to α ad equatig te derivative to zero we get te ormal equatio a N N [ ρ ] K { ( ) ρ } φ ( α ) φ ( α ) φ ( α ) 0 (.8) were φ ( α ) i te firt derivative of ( α ) φ wit repect to α. Now from equatio (.8) we get ( α ) φ ρ K (.9) wic i te cubic equatio iα. Tu α a tree real root for wic te ME of propoed famil would attai it miimum. Puttig te value of φ ( α ) from equatio (.9) ito equatio (.7) we get ( T α ) mi N N ME ( ) { ρ }[ K ] ρ L W ( ) (.0) wic i te ME of te uual regreio etimator of populatio mea i tematic amplig uder o-repoe.. Empirical tud I te upport of teoretical reult we ave coidered te data give i Murt (967 p. -). Tee data are related to te legt ad timber volume for te block of te black moutai eperimetal foret. Te value of itracla correlatio coefficiet ρ ad ρ ave bee give approimatel equal b Murt (967 p. 9) ad Kuwaa ad ig (989) for te tematic ample of ize 6 b eumeratig all poible tematic ample after arragig te data i acedig order of trip legt. Te particular of te populatio are give below: N 76 6 8.66 6.99.6700 8.7600 ρ 0.870
8086.005. Table depict te ME ad variace of te etimator of propoed famil wit repect to o-repoe rate ( W ). Table : ME ad Variace of te Etimator for L. α W 0. 0. 0. 0. ( R ) 7.7 8. 597.5 70.8 ( P ) 908.8 0.85.89 7.9 ( D ) 06. 76.6 89.0 0. ( ) 0.69 5. 66.7 79.05 α opt ( T α ) ) 70.67 8.7 96.75 609.78 ( mi 5. ocluio I ti paper we ave adapted ig ad ukla (987) etimator i tematic amplig i te preece of o-repoe uig a auiliar variable ad obtaied te optimum etimator of te propoed famil. It i oberved tat te propoed famil ca geerate te o-repoe verio of a umber of etimator of populatio mea i tematic amplig o differet coice ofα. From Table we oberve tat te propoed famil uder optimum coditio a miimum ME wic i equal to te ME of te regreio etimator (mot of te cla of etimator i amplig literature uder optimum coditio attai ME equal to te ME of te regreio etimator). It i alo ee tat te ME or variace of te etimator icreae wit icreae i o repoe rate i te populatio.
Referece. Baarai Kuwaa.N.. ad Kuwaa K.. (99): A cla of ratio product ad differece (RPD) etimator i tematic amplig Microelectro. Reliab. 55 57.. ocra W. G. (96): Relative accurac of tematic ad tratified radom ample for a certai cla of populatio AM 7 6-77.. Fie D.J. (98): Radom ad tematic amplig i timber urve Foretr 6-99.. Hae M. H. ad Hurwitz W. N. (96) : Te problem of o-repoe i ample urve Jour. of Te Amer. tat. Aoc. 57-59. 5. Hael A. A. (9): Etimatio of volume i timber tad b trip amplig AM 79-06. 6. Kuwaa K.. ad ig H.P. (989): la of almot ubiaed ratio ad product etimator i tematic amplig Jour. Id. oc. Ag. tatitic 9 05. 7. Lairi D. B. (95): O te quetio of bia of tematic amplig Proceedig of World Populatio oferece 6 9-6. 8. Madow W. G. ad Madow L.H. (9): O te teor of tematic amplig I. A. Mat. tatit. 5 -. 9. Murt M.N. (967): amplig Teor ad Metod. tatitical Publiig ociet alcutta. 0. Nair K. R. ad Bargava R. P. (95): tatitical amplig i timber urve i Idia Foret Reearc Ititute Deradu Idia foret leaflet 5.. Queouille M. H. (956): Note o bia i etimatio Biometrika 5-60.. ig R ad ig H. P. (998): Almot ubiaed ratio ad product tpe- etimator i tematic amplig Quetiio 0-6.. ig R. Malik. audar M.K. Verma H. ad Adewara A. A. (0) : A geeral famil of ratio tpe- etimator i tematic amplig. Jour. Reliab. ad tat. tud.5() 7-8).. ig R. Malik. ig V. K. (0) : A improved etimator i tematic amplig. Jour. Of cie. Re. 56 77-8. 5
5. ig H.P. ad Kumar. (009) : A geeral cla of d etimator of populatio ratio product ad mea i te preece of o-repoe baed o te ub-amplig of te o-repodet. Pak J. tatit. 6() 0-8. 6. ig H.P. ad olaki R.. (0) : A efficiet cla of etimator for te populatio mea uig auiliar iformatio i tematic amplig. Jour. of tat. Ter. ad Pract. 6() 7-85. 7. ig. (00) : Advaced amplig teor wit applicatio. Kluwer Academic Publier. 8. ig V. K. ad ukla D. (987): Oe parameter famil of factor-tpe ratio etimator Metro 5 (-) 7-8. 6
ome Improved Etimator of Populatio Mea Uig Iformatio o Two Auiliar Attribute Hemat Verma Raje ig ad Floreti maradace Departmet of tatitic Baara Hidu Uiverit Varaai-005 Idia Departmet of Matematic Uiverit of New Meico Gallup UA orrepodig autor Abtract I ti paper we ave tudied te problem of etimatig te fiite populatio mea we iformatio o two auiliar attribute are available. ome improved etimator i imple radom amplig witout replacemet ave bee uggeted ad teir propertie are tudied. Te epreio of mea quared error (ME ) up to te firt order of approimatio are derived. A empirical tud i carried out to judge te bet etimator out of te uggeted etimator. Ke word: imple radom amplig auiliar attribute poit bi-erial correlatio pi correlatio efficiec. Itroductio Te role of auiliar iformatio i urve amplig i to icreae te preciio of etimator we tud variable i igl correlated wit auiliar variable. But we we talk about qualitative peomea of a object te we ue auiliar attribute itead of auiliar variable. For eample if we talk about eigt of a pero te e will be a good auiliar attribute ad imilarl if we talk about particular breed of cow te i ti cae milk produced b tem will be good auiliar variable. Mot of te time we ee tat itead of oe auiliar variable we ave iformatio o two auiliar variable e.g.; to etimate te ourl wage we ca ue te iformatio o marital tatu ad regio of reidece (ee Gujrati ad ageeta (007) page-). 7
I ti paper we aume tat bot auiliar attribute ave igificat poit bi-erial correlatio wit te tud variable ad tere i igificat pi-correlatio (ee ule (9)) betwee te auiliar attribute. oider a ample of ize draw b imple radom amplig witout replacemet (RWOR) from a populatio of ize N. let j φ ij (i) deote te obervatio o variable ad φ i (i) repectivel for te j t uit (i N). We ote tat φ ij if j t uit poee attribute φ ij 0 oterwie. Let A i φij a i φ N j j ij ; i deote te total umber of uit i te populatio ad ample repectivel poeig attribute φ. imilarl let A i Pi ad N p i a i ;(i ) deote te proportio of uit i te populatio ad ample repectivel poeig attribute φ i (i). I order to ave a etimate of te tud variable aumig te kowledge of te populatio proportio P Naik ad Gupta (996) ad ig et al. (007) repectivel propoed followig etimator: t t P p p P (.) (.) P p t ep P p (.) t p ep p P P (.) Te bia ad ME epreio of te etimator t i (i ) up to te firt order of approimatio are repectivel give b B ( t ) f [ ] p K pb ( t ) f K pb p B (.5) (.6) 8
K pb p ( ) f B t K pb p ( ) f B t ME ( t ) f [ ( )] p K pb ME ( t ) f [ ( )] p K pb ME ( t ) f p K pb ME ( t ) f p K pb (.7) (.8) (.9) (.0) (.) (.) N N N N were f - φ ( ji Pj ) ( i )( ji Pj ) j φ φ j φ i N i ρ pb j φ j φ j p j φ P j j ; (j ) K pb ρ pb p K pb ρ pb p. φφ φ ( )( ) φ φ p p ad i φi ρφ i φ φ be te ample pi-covariace ad picorrelatio betwee φ ad φ repectivel correpodig to te populatio pi-covariace N φ φ i i P ad N ρ φφ φ. i φ φ ad pi-correlatio ( φ P )( φ ) I ti paper we ave propoed ome improved etimator of populatio mea uig iformatio o two auiliar attribute i imple radom amplig witout replacemet. A comparative tud i alo carried out to compare te optimum etimator wit repect to uual mea etimator wit te elp of umerical data. 9
. Propoed Etimator Followig Olki (958) we propoe a etimator ta t 5 w were w P P w p p ad w are cotatuc tat w oider aoter etimator t6 a [ K K ( P p )] P p t 6 6 6 ep P p were K ad K are cotat. 6 6 w. Followig aoo et al. (99) we propoe aoter etimator t7 a ( P p ) K ( P p ) t 7 K 7 7 were K ad K are cotat. 7 7 (.) (.) (.) Bia ad ME of etimator t 5 t 6 ad t 7 : To obtai te bia ad ME epreio of te etimator t i (i 567) to te firt degree of approimatio we defie e 0 e p P P e p P P uc tat E(e i ) 0 ; i 0. Alo E(e ) f E(e ) f p E(e ) f p 0 E(e0e) fk pb E(e0e ) fk pb E(e e ) f K p p φ p K pb ρpb K pb ρpb K φ ρφ p p p p Epreig (.) i term of e we ave t 5 ( e ) 0 w P P w P ( e ) ( ) P e 0
t [ ] ( e ) w ( e ) w ( ) 5 0 e (.) Epadig te rigt ad ide of (.) ad retaiig term up to ecod degree of e we ave t [ e w e w e w e w e w e e w e e ] 5 0 0 0 (.) Takig epectatio of bot ide of (.) ad te ubtractig from bot ide we get te bia of etimator t Bia(t 5 upto te firt order of approimatio a [ w ( K ) w ( )] 5 ) f p pb p K pb (.) From (.) we ave ( t ) [ e w e w ] 5 0 e (.) quarig bot ide of (.) ad te takig epectatio we get te ME of t 5 up to te firt order of approimatio a ME(t [ w w w K w K w w K ] 5 ) f p p pb p pb pb φ p (.5) Miimizatio of (.5) wit repect to w ad w we get te optimum value of w ad w a w w (opt) (opt) K pb p K φ - K K pb p K φ K w p p p p [ K ] K (opt) pb φp φ p φ p w p p w ( a) ( a) imilarl we get te bia ad ME epreio of etimator t 6 ad t 7 repectivel a Bia(t 8 (.6) 6 ) K 6 f p K pb K Pf K φ p Bia(t 7 ) 0 (.7) Ad
ME(t ) K were 6 A A A 6 f f p fk A pb K p 6 p P K K A φ p K pb 6 K 6 P A ( K ) (.8) 6 Ad te optimum value of K 6 ad K 6 are repectivel give a K K 6(opt) 6(opt) A A A A P A K ( A A A ) 6 ( a) K 6 ( a) ME(t ) f 7 K 7 K K 7 P f P P f K 7 φ p K p 7 P f p K 7 P f K pb p K 7 P f K pb p (.9) Ad te optimum value of K 7 ad K 7 are repectivel give a K K 7(opt) 7(opt) K pb p P p K pb p P p K K K K K K pb φ p φp pb φ p φp K K 7 7 ( a) ( a). Empirical tud Data: (ource: Govermet of Pakita (00)) Te populatio coit rice cultivatio area i 7 ditrict of Pakita. Te variable are defied a: rice productio (i 000 toe wit oe toe 0.98 to) durig 00 P productio of farm were rice productio i more ta 0 toe durig te ear 00 ad P proportio of farm wit rice cultivatio area more ta 0 ectare durig te ear 00. For ti data we ave N7 6. P 0.7 P 0.5 ρ pb 0.6 ρ pb 0.67 ρ φ 0.889. 7. φ 0.590 φ 0.8
Te percet relative efficiec (PRE ) of te etimator t i (i 7) wit repect to uuual ubiaedetimator ave bee computed ad give i Table.. Table. : PRE of te etimator wit repect to Etimator PRE 00.00 t 6.765 t 8.787 t.5899 t 60.8 t 5 65.8780 t 6 97.7008 t 7 8.7 ocluio I ti paper we ave propoed ome improved etimator of populatio mea uig iformatio o two auiliar attribute i imple radom amplig witout replacemet. From te Table. we oberve tat te etimator t 6 i te bet followed b te etimator t 7. Referece Govermet of Pakita 00 rop Area Productio b Ditrict (Miitr of Food Agriculture ad Livetock Diviio Ecoomic Wig Pakita). Gujarati D. N. ad ageeta ( 007): Baic ecoometric. Tata McGraw Hill.
Malik. ad ig R. (0): A famil of etimator of populatio mea uig iformatio o poit bi-erial ad pi correlatio coefficiet. IJE Naik V. D ad Gupta P..(996): A ote o etimatio of mea wit kow populatio proportio of a auiliar caracter. Jour. Id. oc. Agri. tat. 8() 5-58. Olki I. (958): Multivariate ratio etimatio for fiite populatio Biometrika 5 5 65. ig R. aua P. awa N. ad maradace F.( 007): Auiliar iformatio ad a priori value i cotructio of improved etimator. Reaiace Hig pre. ig R. aua P. awa N. ad maradace F. (008): Ratio Etimator i imple Radom amplig Uig Iformatio o Auiliar Attribute. Pak. Jour. tat. Oper. Re. Vol. IV No. pp. 7-5. ig R. Kumar M. ad maradace F. (00): Ratio etimator i imple radom amplig we tud variable i a attribute. WAJ (5): 586-589. ule G. U. (9): O te metod of meaurig aociatio betwee two attribute. Jour. of te Roal oc. 75 579-6.
tud of ome Improved Ratio Tpe Etimator Uder ecod Order Approimatio Praa arma Raje ig ad Floreti maradace Departmet of tatitic Baara Hidu Uiverit Varaai-005 Idia Departmet of Matematic Uiverit of New Meico Gallup UA orrepodig autor Abtract akrabart (979) Koevia et al. (007) aai ad Ra (980) Imail et al. (0) ad olaki et al. (0) propoed etimator for etimatig populatio mea. Up to te firt order of approimatio ad uder optimum coditio te miimum mea quared error (ME) of all te above etimator i equal to te ME of te regreio etimator. I ti paper we ave tried to foud out te ecod order biae ad mea quare error of tee etimator uig iformatio o auiliar variable baed o imple radom amplig. Fiall we ave compared te performace of tee etimator wit ome umerical illutratio. Keword: imple Radom amplig populatio mea tud variable auiliar variable epoetial ratio tpe etimator epoetial product etimator Bia ad ME.. Itroductio Let U (U U U..Ui.UN ) deote a fiite populatio of ditict ad idetifiable uit. For etimatig te populatio mea of a tud variable let u coider be te auiliar variable tat are correlated wit tud variable takig te correpodig value of te uit. Let a ample of ize be draw from ti populatio uig imple radom amplig witout replacemet (RWOR) ad i i (i.. ) are te value of te tud variable ad auiliar variable repectivel for te i-t uit of te ample. 5
I amplig teor te ue of uitable auiliar iformatio reult i coiderable reductio i ME of te ratio etimator. Ma autor uggeted etimator uig ome kow populatio parameter of a auiliar variable. Upadaa ad ig (999) ig ad Tailor (00) Kadilar ad igi (006) Koevia et al. (007) ig et al. (007) ig et al. (008) ad ig ad Kumar (0) uggeted etimator i imple radom amplig. Mot of te autor dicued te propertie of etimator alog wit teir firt order bia ad ME. Hoai et al. (006) tudied ome etimator i ecod order approimatio. I ti tud we ave tudied propertie of ome etimator uder ecod order of approimatio.. ome Etimator i imple Radom amplig For etimatig te populatio mea of akrabart (979) propoed ratio tpe etimator - t ( α) α (.) were i ad. i i i Koevia et al. (007) ratio tpe etimator i give b t g ( ) β β (.) were β ad g are cotat. aai ad Ra (980) propoed a etimator t a W t (.) Imail et al. (0) propoed ad etimator t for etimatig te populatio mea of a 6
t ( ) ( ) p a b (.) were p a ad b are cotat. Alo for etimatig te populatio mea of olaki et al. (0) propoed a etimator t5 a ( ) λ δ t 5 ep (.5) ( ) were λ ad δ are cotat uitabl coe b miimizig mea quare error of te etimator t 5.. Notatio ued Let u defie e 0 ad e te E (e0 ) E(e) 0. For obtaiig te bia ad ME te followig lemma will be ued: Lemma. (i) N V (e 0 ) E{(e 0 ) } 0 L 0 N (ii) N V (e) E{(e) } 0 L0 N N (iii) OV (e 0 e ) E{(e 0 e )} L N Lemma. (iv) (N ) (N ) E {(e e0 )} L (N ) (N ) (v) (N ) (N ) E {(e )} 0 L 0 (N ) (N ) Lemma. (vi) E (e e 0 ) L L 0 (vii) (N ))(N N 6N 6 ) E {(e )} 0 L0 L0 (N )(N )(N ) (viii) E (e e 0 ) L 0 L 0 Were (N ))(N N 6N 6 ) N(N ))(N )( ) L L (N )(N )(N ) (N )(N )(N ) 7
p q (i - ) (i - ) ad pq. p q Proof of tee lemma are traigt forward b uig RWOR (ee ukatme ad ukatme (970)).. Firt Order Biae ad Mea quared Error Te epreio for te biae of te etimator t t t t ad t5 are repectivel give b Bia (t) αl 0 αl (.) ( g ) g Bia (t ) L 0 gβl (.) ( w ) w Bia (t ) L 0 wl (.) ( a)( p ) D b Bia (t ) bdl 0 DL L 0 (.) ( ) K K Bia (t 5 ) L 0 KL (.5) ( δ λ) were D p (b-a) ad k. Epreio for te ME of te etimator t t t t ad t5 are repectivel give b ME(t ME(t ME(t ME(t ME(t [ L0 α L0 αl ] [ L0 g β L0 gβl ] [ L w L wl ] ) (.6) ) (.7) ) 0 0 (.8) [ L DL DL ] ) 0 0 (.9) [ L k L kl ] 5 ) 0 0 (.0) 5. ecod Order Biae ad Mea quared Error Epreig etimator ti (i) i term of e (i0) we get t { } ( e )( α) α( ) 0 e Or 8
e α α α α t e 0 e αe0e αe0e e e0e e (5.) 6 6 Takig epectatio we get te bia of te etimator t up to te ecod order of approimatio a Bia α α α ) L 0 αl L 0 αl 0 6 6 ( L L ) (t α (L 0 L 0 ) (5.) imilarl we get te biae of te etimator t t t ad t5 up to ecod order of approimatio repectivel a Bia g(g ) g(g ) g(g )(g ) 6 (t ) β L 0 gβl β L β L 0 g(g )(g ) β 6 ( L L ) g(g )(g )(g ) β (L 0 L 0 ) 0 (5.) Bia w(w ) w(w ) w(w )(w ) 6 (t ) L 0 wl L L 0 w(w )(w ) 6 ( L L ) 0 w(w )(w )(w ) (L 0 L 0 ) (5.) Bia (D bd) (bd D ) (b D bd D ) (t ) L 0 DL L L 0 ( b D bd) ( L L ) 0 ( b D b D bd D) (L 0 L 0 ) (5.5) 9
were D (b D a)(p ) (b a)(p ) D. D Bia k(k ) k(k ) ) L 0 kl L ML 0 M L 0 0 )] N(L L (5.6) 0 ( L ) (t 5 ( δ 6δ ) ( α( δ δ) Were M δ 8 k ( δ λ) δ δ 8 λ( λ ) λ( λ )( λ ) δ α( δ 6δ λ( λ ) δ 6 λ( λ )( λ )( λ ) δ ( ) ( ) N ( ). Te ME of te etimator t t t t ad t5 up to te ecod order of approimatio are repectivel give b ME (t) [ L0 α L0 αl α L 0 (α α)l ( L L ) α 0 5 ( L L ( )) α (L L α( α ) 0 0 0 0 ) (5.7) ME (t ) [ L0 g β L 0 βgl β g (g )L 0 g(g ) β L 7g 9g g βgl g(g ) β β 0 ( L L ( )) 0 0 ( L L ) g 9g 0g 6 β (L 0 L 0 (5.8) 0
ME (t ) [ L0 w L 0 wl w (w )L 0 w(w )L wl 5w w L w w ( L L ) 0 7w 8w w ( L ( )) (L L ) 0 0 0 0 (5.9) ME (t ) [ L 0 D L 0 DL DDL 0 (bd D D )L DL { D b D DD bd bd }( L L ) { D D bd}( L L ( 00 ) ) { b D D DD bdd }(L L )] (5.0) 0 0 0 ME (t 5 ) [ L 0 k L 0 kl kl kl k (k )L 0 k (k ) ( L L ) k( L L ( )) 0 0 0 ( k k) (L0 L 0 ) (5.) 6. Numerical Illutratio For a atural populatio data we ave calculated te biae ad te mea quare error of te etimator ad compare tee biae ad ME of te etimator uder firt ad ecod order of approimatio. Data et Te data for te empirical aali are take from 98 Utter Prade Ditrict eu Hadbook Aligar. Te populatio coit of 0 village uder koil police tatio wit Number of agricultural labour i 98 ad Area of te village (i acre) i 98. Te followig value are obtaied
7.76765 9.0 N 0 70 0 70 00.76 0.667 0.69 0.077 0.589 00.7877 0. 0.885 07.75 0.8 0.05 Table 6.: Biae ad ME of te etimator Etimator Bia ME Firt order ecod order Firt order ecod order t 0.0095 0.00 9.75 9.5 t 0-0.0006 9.75 (for g) 9.55 (for g) t -0.09-0.095 9.75 9.90 t 0.809-0.608 9.75 9.855 t5-0.07679-0.09 9.75 9.787 I te Table 6. te biae ad ME of te etimator t t t t ad t5 are writte uder firt order ad ecod order of approimatio. For all te etimator t t t t ad t5 it wa oberved tat te value of te biae decreaed ad te value of te ME icreaed for ecod order approimatio. ME of te etimator up to te firt order of approimatio uder optimum coditio are ame. From Table 6. we oberve tat uder
ecod order of approimatio te etimator t5 i bet followed b tad t amog te etimator coidered ere for te give data et. 7. Etimator uder tratified radom amplig I urve amplig it i well etablied tat te ue of auiliar iformatio reult i ubtatial gai i efficiec over te etimator wic do ot ue uc iformatio. However i plaig urve te tratified amplig a ofte proved eedful i improvig te preciio of etimate over imple radom amplig. Aume tat te populatio U coit of L trata a UU U UL. Here te ize of te tratum U i N ad te ize of imple radom ample i tratum U i were ---L. Te akrabart(979) ratio- tpe etimator uder tratified radom amplig i give b were t ( α) t αt (7.) t i i i i L L t w t w ad w. L Koevia et al. (007) ratio- tpe etimator uder tratified radom amplig i give b g t t (7.) β t ( β) were g i a cotat for g t i ame a covetioal ratio etimator werea for g - it become covetioal product tpe etimator. aai ad Ra (980) etimator t uder tratified radom amplig i give b W t t t (7.) Imail et al. (0) etimator uder tratified radom amplig t i give b
( t ) ( ) p a t t (7.) b t olaki et al. (0) etimator uder tratified radom amplig i give a λ ( ) t δ t t 5 t ep (7.5) ( t ) were λ ad δ are te cotat uitabl coe b miimizig ME of te etimator t 5. 8. Notatio ued uder tratified radom amplig Let u defie e 0 t ad e t te E (e0 ) E(e) 0. To obtai te bia ad ME of te propoed etimator we ue te followig otatio i te ret of te article: uc tat ad V r L W r E r [( ) ( ) ] Alo L w γ 0 ) V 0 E (e L w γ ) V 0 E(e
5 L 0 V w ) e (e E γ Were N ) ( N i N ) ( N i N ) )( ( N i f γ N f. N w ome additioal otatio for ecod order approimatio ( ) ( ) [ ] r r L r r E W V Were ( ) ( ) [ ] N i r () r N L () () k W V L () () k W V L 0() () 0 k W V L 0() () 0 k W V L 0() 0() () () () k k W V L 0() () 0() () 0 k k W V ( ) L () 0() 0() () () () k k W V Were
k () (N )(N ) (N )(N ) k () (N (N )(N )N )(N 6 )(N (N ) ) k () (N )N (N (N )(N )(N )( ) ) 9. Firt Order Biae ad Mea quared Error Te biae of te etimator t t t t ad t 5 are repectivel give b Bia (t ) αlv0 αv (9.) ( g ) g Bia (t ) β V0 gβv (9.) ( w ) w Bia (t ) V0 wv (9.) ( a)( p ) D b Bia (t ) bdv0 DV V0 (9.) ( ) K K Bia (t 5 ) V0 KV (9.5) Were D p(b-a) ad ( δ λ) k. Te ME of te etimator t t t t ad t 5 are repectivel give b ME(t ME(t ME(t ME(t ME(t [ V0 α V0 α ] [ V0 g β V0 gβ ] [ V w V ] (9.6) ) V (9.7) ) V (9.8) ) 0 0 wv [ V DV ] (9.9) ) 0 0 DV [ V k V ] (9.0) 5 ) 0 0 kv 6
0. ecod Order Biae ad Mea quared Error Epreig etimator ti (i) i term of e (i0) we get t { } ( e )( α) α( ) 0 e Or e α α α α t e 0 e αe0e αe0e e e0e e (0.) 6 6 Takig epectatio we get te bia of te etimator t up to te ecod order of approimatio a α α α α Bia (t) V0 αv V0 αv V V0 (0.) 6 6 imilarl we get te Biae of te etimator t t t ad t 5 up to ecod order of approimatio repectivel a Bia g(g ) g(g ) g(g )(g ) 6 (t ) β V0 gβv β V β V0 g(g )(g ) g(g )(g )(g ) β V β V0 6 (0.) Bia w(w ) w(w ) w(w )(w ) 6 (t ) V0 wv V V0 w(w )(w ) V 6 w(w )(w )(w ) V0 (0.) (D bd) Bia (t ) V0 DV (bd D)V (b D bd D ) V0 D )V (0.5) ( b D bd)v (b D b D bd 0 ] Were D p(b-a) D (b D a)(p ) D D (b a)(p ) k(k ) k(k ) Bia (t 5 ) V0 kv V MV0 MV NV 0 (0.6) 7
( δ 6δ ) ( α( δ δ) Were M λ( λ ) λ( λ )( λ ) δ k ( δ λ) ( δ δ δ ) ( α( δ 6δ) N 8 8 6 λ( λ ) ( δ λ( λ )( λ )( λ ) δ) Followig are te ME of te etimator t t t t ad t 5 up to ecod order of approimatio ME ME (t ) [ V0 α V0 αv α V0 (α α) V 5 α V α( α )V α V (0.7) 0 (t ) [ V0 g β V0 βgv β g (g )V 0 g(g ) β V 7g 9g g βgv β β V g(g ) V g 9g 0g β V 6 0 (0.8) ME (t ) [ V0 w V0 wv w (w )V 0 w(w )V wv 5w w w 7w 8w w V wv V0 (0.9) ME (t ) [ V0 D V0 DV DDV0 (bd D D )V DV { D b D DD bd bd } V { D D bd} V { b D D DD bdd } V ] (0.0) 0 ME (t 5 ) [ V0 k V0 kv kv kv k (k ) V0 (k k) k (k )V kv V (0.) 0 8
. Numerical Illutratio For te atural populatio data we all calculate te bia ad te mea quare error of te etimator ad compare Bia ad ME for te firt ad ecod order of approimatio. Data et- To illutrate te performace of above etimator we ave coidered te atural Data give i ig ad audar (986 p.6). Te data were collected i a pilot urve for etimatig te etet of cultivatio ad productio of fre fruit i tree ditrict of Uttar- Prade i te ear 976-977. Table.: Biae ad ME of te etimator Etimator Bia ME Firt order ecod order Firt order ecod order t -0.870790 -.65765 99.09 7.9068 t -0.870790 6.5758 99.09 67.586 t -7.05776-7.0658 99.09 7.785 t t 5.6955975 -.8569 99.09 605.7605 -.85709 -.9500 99.09 0.697 9
From Table. we oberve tat te ME of te etimator t t t t ad t 5 are ame up to te firt order of approimatio but te biae are differet. Te ME of te etimator t i miimum uder ecod order of approimatio followed b te etimator t ad oter etimator. ocluio I ti tud we ave coidered ome etimator woe ME are ame up to te firt order of approimatio. We ave eteded te tud to ecod order of approimatio to earc for bet etimator i te ee of miimum variace. Te propertie of te etimator are tudied uder RWOR ad tratified radom amplig. We ave oberved from Table 6. ad Table. tat te beavior of te etimator cage dramaticall we we coider te term up to ecod order of approimatio. REFERENE Bal. ad Tuteja R.K. (99) : Ratio ad product tpe epoetial etimator. Iformatio ad Optimizatio ciece III 59-6. akrabart R.P. (979) : ome ratio etimator Joural of te Idia ociet of Agricultural tatitic () 9 57. Hoai M.I. Rama M.I. ad Tareq M. (006) : ecod order biae ad mea quared error of ome etimator uig auiliar variable. RN. Imail M. abaz M.Q. ad Haif M. (0) : A geeral cla of etimator of populatio mea i preece of o repoe. Pak. J. tatit. 7() 67-76. Koevia M. ig R. aua P. awa N. ad maradace F. (007). A geeral famil of etimator for etimatig populatio mea uig kow value of ome populatio parameter() Far Eat Joural of Teoretical tatitic 8 9. Ra.K. ad aai A (980) : Efficiet familie of ratio ad product tpe etimator Biometrika 67() 5. ig D. ad udar F.. (986): Teor ad aali of ample urve deig. Wile Eater Limited New Deli. 0
ig H.P. ad Tailor R. (00). Ue of kow correlatio coefficiet i etimatig te fiite populatio mea. tatitic i Traitio 6 555-560. ig R. aua P. awa N. ad maradace F. (007): Auiliar Iformatio ad A Priori Value i otructio of Improved Etimator. Reaiace Hig Pre. ig R. aua P. ad awa N. (008): O liear combiatio of Ratio-product tpe epoetial etimator for etimatig fiite populatio mea. tatitic i Traitio9()05-5. ig R. Kumar M. ad maradace F. (008): Almot Ubiaed Etimator for Etimatig Populatio Mea Uig Kow Value of ome Populatio Parameter(). Pak. J. tat. Oper. Re. () pp6-76. ig R. ad Kumar M. (0): A ote o traformatio o auiliar variable i urve amplig. MAA 6: 7-9. olaki R.. ig H. P. ad Ratour A. (0) : A alterative etimator for etimatig te fiite populatio mea uig auiliar iformatio i ample urve. IRN Probabilit ad tatitic doi:0.50/0/65768 rivatava.k. (967) : A etimator uig auiliar iformatio i ample urve. al. tat. A. Bull. 5:7-. ukatme P.V. ad ukatme B.V. (970): amplig teor of urve wit applicatio. Iowa tate Uiverit Pre Ame U..A. Upadaa L. N. ad ig H. P. (99): Ue of traformed auiliar variable i etimatig te fiite populatio mea. Biom. Jour. 67-66.
IMPROVEMENT IN ETIMATING THE POPULATION MEAN UING DUAL TO RATIO-UM-PRODUT ETIMATOR IN IMPLE RANDOM AMPLING Olufadi uua Raje ig ad Floreti maradace Departmet of tatitic ad Matematical ciece Kwara tate Uiverit P.M.B 50 Malete Nigeria Departmet of tatitic Baara Hidu Uiverit Varaai (U.P.) Idia Departmet of Matematic Uiverit of New Meico Gallup UA orrepodig autor ABTRAT I ti paper we propoe a ew etimator for etimatig te fiite populatio mea uig two auiliar variable. Te epreio for te bia ad mea quare error of te uggeted etimator ave bee obtaied to te firt degree of approimatio ad ome etimator are ow to be a particular member of ti etimator. Furtermore compario of te uggeted etimator wit te uual ubiaed etimator ad oter etimator coidered i ti paper i carried out. I additio a empirical tud wit two atural data from literature i ued to epoud te performace of te propoed etimator wit repect to oter. Keword: Dual-to-ratio etimator; fiite populatio mea; mea quare error; multiauiliar variable; percet relative efficiec; ratio-cum-product etimator. INTRODUTION It i well kow tat te ue of auiliar iformatio i ample urve deig reult i efficiet etimate of populatio parameter (e.g. mea) uder ome realitic coditio. Ti iformatio ma be ued at te deig tage (leadig for itace to tratificatio
tematic or probabilit proportioal to ize amplig deig) at te etimatio tage or at bot tage. Te literature o urve amplig decribe a great variet of tecique for uig auiliar iformatio b mea of ratio product ad regreio metod. Ratio ad product tpe etimator take advatage of te correlatio betwee te auiliar variable ad te tud variable. For eample we iformatio i available o te auiliar variable tat i poitivel (ig) correlated wit te tud variable te ratio metod of etimatio i a uitable etimator to etimate te populatio mea ad we te correlatio i egative te product metod of etimatio a eviaged b Robo (957) ad Murt (96) i appropriate. Quite ofte iformatio o ma auiliar variable i available i te urve wic ca be utilized to icreae te preciio of te etimate. I ti ituatio Olki (958) wa te firt autor to deal wit te problem of etimatig te mea of a urve variable we auiliar variable are made available. He uggeted te ue of iformatio o more ta oe upplemetar caracteritic poitivel correlated wit te tud variable coiderig a liear combiatio of ratio etimator baed o eac auiliar variable eparatel. Te coefficiet of te liear combiatio were determied o a to miimize te variace of te etimator. Aalogoul to Olki ig (967) gave a multivariate epreio of Murt (96) product etimator wile Raj (965) uggeted a metod for uig multi-auiliar variable troug a liear combiatio of igle differece etimator. More recetl Abu- Dae et al. (00) Kadilar ad igi (00 005) Perri (00 005) Diaa ad Perri (007) Malik ad ig (0) amog oter ave uggeted etimator for uig iformatio o everal auiliar variable. Motivated b rivekataramaa (980) Badopada (980) ad ig et al. (005) ad wit te aim of providig a more efficiet etimator; we propoe i ti paper a ew etimator for we two auiliar variable are available.
. BAKGROUND TO THE UGGETED ETIMATOR oider a fiite populatio P ( P P )... P N of N uit. Let a ample of ize be draw from ti populatio b imple radom amplig witout replacemet (RWOR). Let ad z ) repreet te value of a repoe variable ad two auiliar variable i ( i i ( z) are available. Te uit of ti fiite populatio are idetifiable i te ee tat te are uiquel labeled from to N ad te label o eac uit i kow. Furter uppoe i a urve problem we are itereted i etimatig te populatio mea of aumig tat te populatio mea ( Z ) for are give a of ( z) are kow. Te traditioal ratio ad product etimator R ad P z Z repectivel were i i i i ad z z i i are te ample mea of ad z repectivel. ig (969) improved te ratio ad product metod of etimatio give above ad uggeted te ratio-cum-product etimator for a I literature it a bee ow b variou autor; ee for eample Redd (97) ad z Z rivekataramaa (978) tat te bia ad te mea quare error of te ratio etimator R ca be reduced wit te applicatio of traformatio o te auiliar variable. Tu autor like rivekataramaa (980) Badopada (980) Trac et al. (996) ig et al. (998) ig et al. (005) ig et al. (007) Bartku ad Plikua (009) ad ig et al. (0) ave improved o te ratio product ad ratio-cum-product metod of etimatio uig te traformatio o te auiliar iformatio. We give below te traformatio emploed b tee autor:
( g) ad zi ( g) Z gzi for i... N () i g i were g. N Te clearl ( g) g ad z ( g) Z gz are alo ubiaed etimate of ad Z repectivel ad orr ( ) ρ ad orr ( z ) ρ z. It i to be oted tat b uig te traformatio above te cotructio of te etimator for require te kowledge of ukow parameter wic retrict te applicabilit of tee etimator. To overcome ti retrictio i practice iformatio o tee parameter ca be obtaied approimatel from eiter pat eperiece or pilot ample urve iepeivel. Te followig etimator R P ad E are referred to a dual to ratio product ad ratio-cum-product etimator ad are due to rivekataramaa (980) Badopada (980) ad ig et al. (005) repectivel. Te are a preeted: Z P z R ad E Z z It i well kow tat te variace of te imple mea etimator uder RWOR deig i V ( ) λ ad to te firt order of approimatio te Mea quare Error (ME) of R P R P ad E are repectivel give b ( ) λ ( R R ) ME R ( ) λ ( R R ) ME ME P ( ) λ [ D ] ( ) λ ( g R gr ) ME R ( ) λ ( g R gr ) ME P z z z z 5
ME ( ) λ ( g gd) E were f λ N N N f ( i ) ( i )( i ) N i N i ρ R R Z R RR z R z D R R z ad j for ( j z) repreet te variace of ad z repectivel; wile z ad z deote te covariace betwee ad ad z ad z ad repectivel. Note tat ρ z ρ z z z ad z are defied aalogoul ad repective to te ubcript ued. More recetl arma ad Tailor (00) propoed a ew ratio-cum-dual to ratio etimator of fiite populatio mea i imple radom amplig teir etimator wit it ME are repectivel give a T α ( α ) ( ) λ ( ρ ) ME T.. PROPOED DUAL TO RATIO-UM-PRODUT ETIMATOR Uig te traformatio give i () we ugget a ew etimator for a follow: PR Z θ z z ( θ ) Z We ote tat we iformatio o te auiliar variable z i ot ued (or variable z take te value `uit) ad θ te uggeted etimator PR reduce to te `dual to ratio etimator R propoed b rivekataramaa (980). Alo PR reduce to te `dual to product etimator P propoed b Badopada (980) etimator if te iformatio o te auiliar variate i ot ued ad θ 0. Furtermore te uggeted etimator reduce 6
to te dual to ratio-cum-product etimator uggeted b ig et al. (005) we θ ad iformatio o te two auiliar variable ad z are bee utilized. I order to tud te propertie of te uggeted etimator PR (e.g. ME) we write ( ); ( ); z Z ( ) k 0 wit E ( k ) E( k ) E( k ) 0 ad E E 0 λ k k ; λ λ z ( k 0 ) ; E( k ) ; E( k ) ( k k ) λ Z z Now epreig PR i term of k we ave PR ; E( k0 k ) ; E( k k ) Z λ [ ] ( k ) ( gk )( gk ) ( θ )( gk ) ( gk ) 0 λ Z z 0 ; θ () We aume tat gk < ad gk < o tat te rigt ad ide of () i epadable. Now epadig te rigt ad ide of () to te firt degree of approimatio we ave PR [ k ( ) g( k k k k k k ) g ( k k k α( k k )] 0 α () 0 Takig epectatio o bot ide of () we get te bia of PR to te firt degree of approimatio a were B [ gda g ( R R R ( R R )] ( PR ) λ z θ z A θ two we ave quarig bot ide of () ad eglectig term of ( ) [ k Agk Agk ] PR 0 [ k Agk k Agk k A g k k A g k A g k ] 0 0 0 0 k ivolvig power greater ta () 7
Takig epectatio o bot ide of () we get te ME of PR to te firt order of approimatio a ME ( ) [ AgD A g ] PR λ (5) Te ME of te propoed etimator give i (5) ca be re-writte i term of coefficiet of variatio a ME ( ) [ ] λ Ag D A g PR were z ρ z z ad D ρ ρ z z z z Z Te ME equatio give i (5) i miimized for D g θ θ 0 (a) g We ca obtai te miimum ME of te uggeted etimator PR b uig te [ ] optimal equatio of θ i (5) a follow: mi. ME( ) λ F( D F ) PR were F g E ad E D g. EFFIIEN OMPARION I ti ectio te efficiec of te uggeted etimator PR over te followig etimator R P R P E ad T are ivetigated. We will ave te coditio a follow: D g ME PR if θ > g (a) ( ) V ( ) < 0 (b) ( ) ME( ) < 0 ME if PR ( D Ag) < R ( ) R Ag R provided (c) ( ) ME( ) < 0 ME if PR P < R 8
( D Ag) < R ( R ) Ag z z provided < z R z D ME PR if < provided Ag (d) ( ) ME( ) < 0 (e) ( ) ME( ) < 0 ME if PR R ( ) g < A ( θg g D) < gr R R R gr θ z z z (f) ( ) ME( ) < 0 ME if PR P ( θg g D) < gr R R R gr θ z z (g) ( ) ME( ) < 0 ME if PR E D g < provided A < ( A ) () ME ( ) ME( ) < 0 if ( ) PR. NUMERIAL ILLUTRATION T Ag D Ag < ρ To aalze te performace of te uggeted etimator i compario to oter etimator coidered i ti paper two atural populatio data et from te literature are beig coidered. Te decriptio of tee populatio are give below. () Populatio I [ig (969 p. 77]; a detailed decriptio ca be ee i ig (965) : Number of female emploed : Number of female i ervice z : Number of educated female N 6 0 7. 6 5. Z 79 8. 088 6. 76 z 08.95 ρ 0. 777 ρ 0. 070 ρ z 0. 00 z () Populatio II [ource: Joto 97 p. 7]; A detailed decriptio of tee variable i ow i Table. : Percetage of ive affected b dieae : Mea Jauar temperature 9
z : Date of flowerig of a particular ummer pecie (umber of da from Jauar ) N 0 5 Z 00 65. 9776 9. 9880 8 ρ 0.8 ρ 0. 9 ρ 0. 07 z z For tee compario te Percet Relative Efficiecie (PRE) of te differet etimator are computed wit repect to te uual ubiaed etimator uig te formula ( ) ( ) (). V PRE. 00 ME ad te are a preeted i Table. Table : Decriptio of Populatio II. z 9 5 00 0 5 8 6 0 5 0 0 59 9 5 9 6 6 88 55 50 96 6 50 90 z Table ow clearl tat te propoed dual to ratio-cum-product etimator PR a te iget PRE ta oter etimator; terefore we ca coclude baed o te tud populatio tat te uggeted etimator i more efficiet ta te uual ubiaed etimator te traditioal ratio ad product etimator ratio-cum-product etimator b ig (969) 50
rivekataramaa (980) etimator Badopada (980) etimator ig et al. (005) etimator ad arma ad Tailor (00). Table : PRE of te differet etimator wit repect to Etimator Populatio I Populatio II 00 00 05 77 R 0 87 P 95 5 9 R 05 50 P 6 0 E 50 78 T 79 57 PR 5. ONLUION We ave developed a ew etimator for etimatig te fiite populatio mea wic i foud to be more efficiet ta te uual ubiaed etimator te traditioal ratio ad product etimator ad te etimator propoed b ig (969) rivekataramaa (980) Badopada (980) ig et al. (005) ad arma ad Tailor (00). Ti teoretical iferece i alo atified b te reult of a applicatio wit origial data. I future we ope to eted te etimator uggeted ere for te developmet of a ew etimator i tratified radom amplig. 5
REFERENE. ABU-DAEH W. A. AHMED M.. AHMED R. A. ad MUTTLAK H. A. (00): ome etimator of fiite populatio mea uig auiliar iformatio Applied Matematic ad omputatio 9 87 98.. BANDOPADHA. (980): Improved ratio ad product etimator aka erie 5-9.. DIANA G. ad PERRI P.F. (007): Etimatio of fiite populatio mea uig multiauiliar iformatio. Metro Vol. LV Number 99-. JOHNTON J. (97): Ecoometric metod (d ed) McGraw-Hill Toko. 5. KADILAR. ad INGI H. (00): Etimator of a populatio mea uig two auiliar variable i imple radom amplig Iteratioal Matematical Joural 5 57 67. 6. KADILAR. ad INGI H. (005): A ew etimator uig two auiliar variable Applied Matematic ad omputatio 6 90 908. 7. Malik. ad ig R. (0) : ome improved multivariate ratio tpe etimator uig geometric ad armoic mea i tratified radom amplig. IRN Prob. ad tat. Article ID 50986 doi:0.50/0/50986. 8. MURTH M. N. (96): Product metod of etimatio. aka A 6 9-07. 9. OLKIN I. (958): Multivariate ratio etimatio for fiite populatio Biometrika 5 5 65. 0. PERRI P. F. (00): Alcue coiderazioi ull efficieza degli timatori rapporto-cumprodotto tatitica & pplicazioi o. 59 75.. PERRI P. F. (005): ombiig two auiliar variable i ratio-cum-product tpe etimator. Proceedig of Italia tatitical ociet Itermediate Meetig o tatitic ad Eviromet Meia - eptember 005 9 96.. RAJ D. (965): O a metod of uig multi-auiliar iformatio i ample urve Joural of te America tatitical Aociatio 60 5 65.. REDD V. N. (97): O a traformed ratio metod of etimatio aka 6 59 70.. HARMA B. ad TAILOR R. (00): A New Ratio-um-Dual to Ratio Etimator of Fiite Populatio Mea i imple Radom amplig. Global Joural of ciece Frotier Reearc Vol. 0 Iue 7-5. INGH M.P. (965): O te etimatio of ratio ad product of te populatio parameter aka erie B 7-8. 6. INGH M. P. (967): Multivariate product metod of etimatio for fiite populatio Joural of te Idia ociet of Agricultural tatitic 75 78. 7. INGH M.P. (969): ompario of ome ratio-cum-product etimator aka erie B 75-78. 5
8. INGH H.P. INGH R. EPEJO M.R. PINEDA M.D ad NADARAJAH. (005): O te efficiec of a dual to ratio-cum-product etimator i ample urve. Matematical Proceedig of te Roal Iri Academ 05A () 5-56. 9. INGH R. INGH H.P AND EPEJO M.R. (998): Te efficiec of a alterative to ratio etimator uder a uper populatio model. J..P.I. 7 87-0. 0. INGH R. HAUHAN P. AND AWAN N. (007): O te bia reductio i liear variet of alterative to ratio-cum-product etimator. tatitic i Traitio8()9-00.. INGH R. KUMAR M. HAUHAN P. AWAN N. AND MARANDAHE F. (0): A geeral famil of dual to ratio-cum-product etimator i ample urve. IT_ewerie () 587-59.. RIVENKATARAMANA T. (978): age of origi ad cale i ratio ad differece metod of etimatio i amplig aad. J. tat. 6 79 86.. RIVENKATARAMANA T. (980): A dual to ratio etimator i ample urve Biometrika 67() 99 0.. TRA D.. INGH H.P. AND INGH R. (996): A alterative to ratio-cum-product etimator i ample urve. J..P.I.575-87. 5
ome Improved Etimator for Populatio Variace Uig Two Auiliar Variable i Double amplig Viplav Kumar ig Raje ig ad Floreti maradace Departmet of tatitic Baara Hidu Uiverit Varaai-005 Idia Departmet of Matematic Uiverit of New Meico Gallup UA orrepodig autor Abtract I ti article we ave propoed a efficiet geeralied cla of etimator uig two auiliar variable for etimatig ukow populatio variace of tud variable.we ave alo eteded our problem to te cae of two pae amplig. I upport of teoretical reult we ave icluded a empirical tud.. Itroductio Ue of auiliar iformatio improve te preciio of te etimate of parameter.out of ma ratio ad product metod of etimatio are good eample i ti cotet. We ca ue ratio metod of etimatio we correlatio coefficiet betwee auiliar ad tud variate i poitive (ig) o te oter ad we ue product metod of etimatio we correlatio coefficiet betwee auiliar ad tud variate i igl egative. Variatio are preet everwere i our da-to-da life. A agriculturit eed a adequate udertadig of te variatio i climatic factor epeciall from place to place (or time to time) to be able to pla o we ow ad were to plat i crop. Te problem of etimatio of fiite populatio variace of te tud variable wa dicued b Iaki (98) ig ad ig (00 00 00) ig et al. (008) Grover (00) ad ig et al. (0). 5
Let ad z are auiliar variate avig value z ) ad i te tud variate avig ( i i value ( i ) repectivel. Let V i (i...n) i te populatio avig N uit uc tat i poitivel correlated ad egativel correlated wit z. To etimate we aume tat ad are kow were z N N (i ) i N ad z (z i Z). N N N ( i ) i Aume tat N i large o tat te fiite populatio correctio term are igored. A ample of ize i draw from te populatio V uig imple radom ample witout replacemet. i Uual ubiaed etimator of populatio variace i were (i ). ( ) i Up to te firt order of approimatio variace of i give b var( ) 00 (.) were 00 00 μ pqr pqr p / q / r / μ μ μ 00 00 00 ad μ N p q r pqr (i ) ( i ) (z i Z) N i. Eitig Etimator Let ( e ) ( e ) ad ( e ) 0 z z (z i z) ( ) i ( ) i were ( i ) z ad i z z i. Alo let i E(e ) E(e ) 0 i ; p q r beig te o-egative iteger. 55