Mixture Regression Estimators Using Multi-Auxiliary Variables and Attributes in Two-Phase Sampling

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1 Oen Journal of ascs Publshed Onlne Auus 04 n ces. h:// h://d.do.or/0.436/ojs Mure eresson Esaors Usn Mul-Aular Varables and Arbues n Two-Phase aln John Kun u Grace Chuba Leo Odono Dearen of Maheacs Kenaa Unvers Narob Kena Eal: johnunu08@ahoo.co chubarace@al.co eceved 3 Ma 04; revsed 6 June 04; acceed 0 Jul 04 Corh 04 b auhors and cenfc esearch Publshn Inc. Ths wor s lcensed under he Creave Coons Arbuon Inernaonal Lcense CC BY. h://creavecoons.or/lcenses/b/4.0/ Absrac In hs aer we have develoed esaors of fne oulaon ean usn Mure eresson esaors usn ul-aular varables and arbues n wo-hase saln and nvesaed s fne sale roeres n full aral and no nforaon cases. An ercal sud usn naural daa s ven o coare he erforance of he roosed esaors wh he esn esaors ha ulzes eher aular varables or arbues or boh for fne oulaon ean. The Mure eresson esaors n full nforaon case usn ulle aular varables and arbues are ore effcen han ean er un eresson esaor usn one aular varable or arbue eresson esaor usn ulle aular varable or arbues and Mure eresson esaors n boh aral and no nforaon case n wo-hase saln. A Mure eresson esaor n aral nforaon case s ore effcen han Mure eresson esaors n no nforaon case. Kewords eresson Esaor Mulle Aular Varables Mulle Aular Arbues Two-Phase aln B-eral Correlaon Coeffcen. Inroducon The hsor of usn aular nforaon n surve saln s as old as he hsor of surve saln. The wor of Nean [] a be referred o as he nal wor where aular nforaon has been used o esae oulaon araeers. Hansen and Hurwz [] also suesed he use of aular nforaon n selecn he sale wh varn robables. The conce of rao esaon was nroduced n sale surve b Cochran How o ce hs aer: Kun u J. Chuba G. and Odono L. 04 Mure eresson Esaors Usn Mul-Aular Varables and Arbues n Two-Phase aln. Oen Journal of ascs h://d.do.or/0.436/ojs

2 [3]; s referred when he sud varable s hhl osvel correlaed wh he aular varable. Wason [4] used he reresson esaor of leaf area on leaf weh o esae he averae area of he leaves on a lan. Oln [5] was he frs erson usn nforaon on ore han one suleenar characer whch s osvel correlaed wh he varable under sud usn a lnear cobnaon of rao esaor based on each aular varable. aj [6] suesed a ehod of usn ul-aular nforaon n sale surve. The conce of double saln was frs roosed b Nean [] n saln huan oulaons when he ean of aular varable was unnown. I was laer eended o ulhase b obson [7]. Abdul Zahoor and Hanf [8] also develoed a eneralzed ulvarae reresson esaor for ul-hase saln usn ulaular varables. Zahoor Abdul and Muhhaad [9] suesed a eneralzed reresson-cu-rao esaor for wo-hase saln usn ulle aular varables. I s advanaeous when he an n recson s subsanal as coared o he ncrease n he cos due o collecon of nforaon on he aular varae for lare sales. I was roved ha ou esaor n he roosed class of esaors was aroael euall effcen wh he usual based lnear reresson esaor. auddn and Hanf [0] nroduced rao and reresson esaon rocedures for esan oulaon ean n wo-hase saln for dfferen hree suaons deendn uon he avalabl of nforaon on wo aular varables for oulaon. The consdered hree suaons frs when nforaon on boh aular varables was no avalable second when nforaon on one aular varable was avalable and hrd when nforaon was avalable on boh aular varables. Jhajj hara and Grover [] roosed a fal of esaors usn nforaon on aular arbue. The used nown nforaon of oulaon rooron ossessn an arbue hhl correlaed wh sud varable Y. The ou esae of he roosed fal of ean was less based and ore effcen han ean er un esaor. The arbue s norall used when he aular varable s no avalable e.. an aoun of l roduced and a arcular breed of cow or an aoun of eld of whea and a arcular vare of whea. The esaor erfored beer han he usual sale ean and Na and Gua [] esaor. ajesh Panaj Nrala and Florenns [3] used he aular arbue n reresson-rao e eonenal esaor follown he wor of Bahl and Tueja [4]; he esaor was ore effcen coared o ean er un rao and roduc e eonenal esaor as well as Na and Gua [] esaor. Hanf Ha and hahbaz [5] roosed a eneral fal of esaors usn ulle aular arbue n snle and double hase saln. The esaor had a saller coared o ha of Jhajj hara and Grover []. The also eended her wor o rao and reresson esaor whch was eneralzaon of Na and Gua [] esaor n snle and double hase saln wh full nforaon aral nforaon and no nforaon. Moeen hahbaz and HanIf [6] roosed a class of ure rao and reresson esaors for snle hase saln for esan oulaon ean b usn nforaon on aular varables and arbues sulaneousl. Kun u and Odono [7] and [8] roosed rao-cu-roduc esaors usn ulle aular arbues n snle and wo-hase saln. In our aer we wll eend he ure reresson esaor roosed b Moeen hahbaz and HanIf [6] o wo-hase saln under full aral and no nforaon case sraees nroduced b auddn and Hanf [0] and also ncororae Arora and Bans [9] aroach n wrn down he ean suared error.. Prelnares.. Noaon and Assuon Consder a oulaon of N uns. Le Y be he varable for whch we wan o esae he oulaon ean and X X X are aular varables. For wo-hase saln desn le n and n n < n are sale szes for frs and second hase resecvel. and denoe he h aular varables for frs and second hase sales resecvel and denoe he varable of neres fro second hase. X and C denoe he oulaon eans and coeffcen of varaon of h aular varables resecvel and denoes he oulaon correlaon coeffcen of Y and X. Furher le θ θ θ < θ n N n N Y + e X + e X + e and.0 356

3 where e e and e are saln error and are ver sall. We assue ha E e E e E e 0. Consder a sale of sze n drawn b sle rando saln whou relaceen fro a oulaon of sze h N. Le j and denoes he observaons on varable and r resecvel for he j un where j n. In defnn he arbues we assue colee dchoo so ha; Le N A and a j j h h f j un of oulaon ossess aular arbue j 0 oherwse n j j. be he oal nuber of uns n he oulaon and sale resecvel os- A a sessn arbue. Le P and be he corresondn rooron of uns ossessn a secfc N n arbues and s he ean of he an varable a second hase. Le and denoe he h aular arbue for frs and second hase sales resecvel and denoe he varable of neres fro second hase. The ean of an varable of neres a second hase wll be denoed b. Also le us defne e Y e P e P.3 The coeffcen of varaon and correlaon coeffcen are ven b C C Y C Y P z z z Pb and z Then for sle rando saln whou relaceen for boh frs and second hases we wre b usn hase wse oeraon of eecaons as: θ θ θ θ θ θ θ Pb θ θ j θ θ e θ θ PC ; ; θ j j j θ j j j θ θ j ; j θ θ j j j j j θ j θ j j ; j j ; j j j ; θ θ j j j E e Y C E e e P C E e e X C E e e YX C C E e e YPC C E e e e YX C C E e e e X C E e e E e e X X C C j E e e PPC C j E e e e YX C C E e e e e PPC C j E e e e e XXC C E e e e j X X C C j E e e e PPC C j z z.4 T Aj d A A C.5 A j A Arora and La [9].6 357

4 The follown noaons wll be used n dervn he ean suare errors of roosed esaors Deernan of oulaon correlaon ar of varables and. r Deernan of h nor of corresondn o he h eleen of. Denoes he ulle coeffcen of deernaon of on r and r. r r Denoes he ulle coeffcen of deernaon of on and. Deernan of oulaon correlaon ar of varables r and r. Deernan of oulaon correlaon ar of varables and. Deernan of he correlaon ar of r and r. Deernan of he correlaon ar of and. j Deernan of he nor corresondn o of he correlaon ar of r j j r and r j. j Deernan of he nor corresondn o of he correlaon ar of j and.7.. Mean er Un n Two-Phase aln j j The sale ean usn sle rando saln whou relaceen n wo hase saln s ven b s ven b Whle s varance s ven Var n n j.0 θ Y C..3. eresson Esaors Usn One and Mulle Aular Varables and Arbues Le n j n j n j n j and be he unbased esaor of sale eans of Y and X resecvel n wo hase saln. The sle reresson esaor for nown X suesed b Wason [4] s Is ean suared error s ven b EX β + X. EX θy C In case of ulle aular varables reresson esaor s ven b Is ean suared error s ven b.3 MEX + α X.4 MEX θy C Na and Gua [] defned eresson esaor of oulaon when he ror nforaon of oulaon rooron of uns ossessn he sae arbue s varable as.5 358

5 Is ean suared error s ven b EP α + P.6 EP θy C.7 YC b α are ou for eresson esaor. C P r In case of ulle aular varables reresson esaor s ven b Is ean suared error s ven b r Pb s he b-seral correlaon coeffcen. MEP + α r P.8 MEX θy C The ure rao esaor based on ulle aular varables and arbues b Moeen hahbaz and HanIf [6] s ven b: EXP.9 α β X + P + I s norall nown ha he above esaors are based bu he bas ben of he order n can be assued nelble n lare sales. I s assued ha he sale of sze n s lare enouh so ha he bases of hese esaors are nelble. Our rojec wll eend he ure reresson esaor roosed b Moeen hahbaz and Hanf [6] o wohase saln under full aral and no nforaon case sraees nroduced b auddn and Hanf [0]. 3. Mehodolo 3.. Proosed Mure eresson Esaor n Two-Phase aln Full Inforaon Case If we esae a sud varable when nforaon on all aular varables and arbues s avalable fro oulaon s ulzed n he for of her eans. B an he advanae of Mure eresson esaor echnue for wo-hase saln a eneralzed esaor for esan oulaon ean of sud varable Y wh he use of ul aular varables and arbues s suesed as: P P P α X + α X + + α X M 3.0 ubsun Euaon.0 and.3 n 3.0 we e The ean suared error of M 3.0 s ven b + β + β + + β e + Y α e β e M 3.0 r + M 3.0 M h+ α β E Y E e e e We dfferenae he Euaon 3.3 arall wh resec o α and β + + hen euae o zero usn and.7 we e + YC α 3.4 XC 359

6 + YC β PC h Usn noral euaon ha s used o fnd he ou values ven 3.3 we can wre 3.3 as h 3.5 h+ M 3.0 E e e α e β e + + h M 3.0 α β E e E e e E e e Tan eecaon n 3.7 and subsun.4 we e h YC + YC h M 3.0 θ Y C + X YCC + PYC C Pb XC PC I h M 3.0 θ Y C h h Pb + h M 3.0 θy C θy C Usn.6 n 3.3 we e θy C M M 3.0 θy C θ M 3.0 Y C. 3.. Mure eresson Esaor n Two-Phase aln Paral Inforaon Case In hs case suose we have no nforaon on all aular varables and h aular arbues fro oulaon. Consdern Mure eresson esaor echnue he oulaon ean of sud varable Y can be es- 360

7 aed for wo-hase saln usn ul-aular varables and arbues as: δ X α α α P P P P β + P β P + α + α + + α + δ X + δ X M β + β + + β γ γ + + γ + β ubsun.0 and.3 n 3.5 we e α δ α β + + e + Y + e e e + e e + e e γe + β e e Mean suared error of M 3. esaor s ven b e e e e e E M 3. Ee + α e e δe + α e e + + β γ r + β We dfferenae he Euaon 3.4 wh resec o α r β r α + + γ + + λ + + r r h h euae o zero and use.6 and.7. The ou value s as follows 3.7 γ h+ h+ and YC + + YC s α δ XC XC s r r YC YC + + s z β α XC PC s z h C γ Y YC + β + PC PC h h + + h h+ h+ Usn noral euaon ha are used o fnd he ou values ven 3.7 we can wre e e e e e M 3. EEe e + α e e δe + α e e + + β γ r + β

8 E e M 3. + α EE e e e δe e e + δee e e e Usn.4 n 3.8 we e + α E E e e e γ E e e + δ E E e e e θ Y C M 3. + θ θ α XYCC θ δ XCC + θ θ XCC α + + θ θ β PYCC θ γ PCC θ θ β PCC Pb Pb Pb Y C + + θ θ θ + θ j j θθ + θθ + + θ + j j + θ θ Pb γ Pb + h+ + Y C θ θ θ θ θ θ + θ θ + θ θ θ θ + j + + Pb Pb j j j + + θ Pb + θθ γ Pb + + M 3. j Y C θ θ + + Pb r+ + + j r + + θ + Pb + Pb

9 j θ θ θ Y C Usn.6 n 3.7 we e.. Y C M 3. θ θ + θ M 3. θ. + θ.. Y C Mure eresson Esaor n Two-Phase aln No Inforaon Case If we esae a sud varable when nforaon on all aular varables s unavalable fro oulaon s ulzed n he for of her eans. B an he advanae of Mure eresson esaor echnue for wohase saln a eneralzed esaor for esan oulaon ean of sud varable Y wh he use of ul aular varables and arbues s suesed as: α α α M 3. + β + β + + β + + ubsun euaon.0 and. n 3.9 we e β e + Y + α e e + e e M 3. + The ean suared error of P 3. s ven b M 3. M α h+ β 3.3 E Y E e + e e + e e + We dfferenae he Euaon 3.3 arall wh resec o α and β + + hen euae o zero usn and.7 we e + YC P α 3.3 XC + YC β PC h Usn noral euaon ha s used o fnd he ou values ven 3.3 we can wre h+ M 3. α + h β E e e + e e + e e h MP 3. α β + E e + E e e e E e e e + Tan eecaon and subsun 3.3 and 3.33 and we e

10 YC M 3. θ Y C + θ θ X YCC XC + h YC h + θ θ PYC C PC + h + h h M 3. Y C θ + θ θ + θ θ Pb + h Usn.6 n 3.38 we e lfn 3.38 we e. Y C M 3. θ θ + θ Y C M 3. θ θ. + θ M 3. θ. + θ. Y C Bas and Conssenc of Mure eresson Esaors These ure reresson esaors usn ulle aular varables n wo hase saln are based. However hese bases are nelble for oderae and lare sales. I s easl shown ha he ure reresson esaors are conssen esaors usn ulle aular varables snce he are lnear cobnaons of conssen esaors follows ha he are also conssen. 4. esul and Dscusson In hs secon we carred ou soe daa analss usn sascal acae o coare he erforance of ure reresson esaors wh alread esn esaor n wo-hase saln for fne oulaon ha uses one or ulle aular varables or arbues. In he naural oulaon he sud varable was bod fa and aular varables are Thh crcuference and ches crcuference whle arbues were abdoen and h crcuference. N 5 n 80 n Pb Pb Poulaon: The sulaed oulaon was a norall dsrbued wh he follown araeers N 600 n 90 n 36 ean 75 sandard devaon Pb Pb All he resuls were obaned afer carrn ou several rando sale and an he averae. In order o evaluae he effcenc an we could acheve b usn he roosed esaors we have calculaed he varance of ean er un and he ean suared error of all esaors we have consdered. We have hen calculaed ercen relave effcenc of each esaor n relaon o varance of ean er un. We have hen coared he ercen relave effcenc of each esaor he esaor wh he hhes ercen relave effcenc s consdered o be he os effcen han he oher esaor. The ercen relave effcenc s calculaed usn he follown forulae. 364

11 Y ˆ ˆ Y ˆ J. Kun u e al. Var eff The Table shows ercen relave effcenc of roosed and esn esaor wh resec o ean er un esaor for wo hase saln. I s observed ha eresson esaors usn one aular varables and arbues are ore effcen han ean er un n he wo oulaons. Aan eresson esaors usn ulle aular varables and arbues are ore effcen han ean er un and eresson esaors. Fnall Mure eresson esaors usn ulle aular varables and arbues s he os effcen of he fve esaors n he wo oulaons snce has he hhes ercen relave effcenc. Fnall Table coares he effcenc of full nforaon case and aral case o no nforaon case and full o aral nforaon case. I s observed ha he full nforaon case and aral nforaon case are ore effcen han no nforaon case because he have hher ercen relave effcenc han no nforaon case. In addon he full nforaon case s ore effcen han he aral nforaon case because has a hher ercen relave effcenc han aral nforaon case. 5. Conclusons The ercen relave effcenc s used n sale surve o coare he effcenc of dfferen esaors. The esaor wh he hhes ercen relave effcenc wh resec o ean er un s norall consdered o be ore effcen coared o he oher esaors. Accordn o Table he roosed Mure eresson esaors usn ulle aular varables and arbues n wo-hase saln has he hhes ercen relave effcenc coared o ean er un eresson esaors usn one aular varable and arbues eresson esaors usn ulle aular varables and arbues. Ths eans ha he rao-cu-roduc esaor n wo-hase saln s he os effcen esaor coared o he esaors ha ulze aular varables and arbues. The Mure eresson esaors were hen eended o wo-hase saln n aral and no nforaon case. In Table we coared he effcenc of full and aral nforaon case o no nforaon case and Table. elave effcenc of suesed esaor wh resec o ean er un esaor for wo hase saln. Esaors elave effcenc of suesed esaor wh resec o ean er un esaor for wo hase saln Poulaon I Poulaon II EX EP 7 30 MEX MEP roosed M Table. Coarsons of full aral and no nforaon cases for roosed ure reresson esaor. Poulaon Percen relave effcenc of full and aral o no nforaon Percen relave effcenc of full o aral n foraon case Esaors M 3. M 3. M 3.0 M 3. M

12 found ha he wo are ore effcen han he no nforaon case. We also coared he effcenc of full nforaon case o aral nforaon case and found ha he full nforaon case s ore effcen han he aral nforaon case. The roosed Mure eresson esaor usn ulle aular varables and arbues n wo-hase saln s recoended o esae he fne oulaons ean for full nforaon case as ouerfors all he oher esn esaors for full nforaon usn one aular or ulle aular varables and arbues. I also ouerfors Mure eresson esaors usn ulle aular varables and arbues n aral and no nforaon cases. When soe aular varables are unnown he wo-hase saln s recoended. If soe aular varables are nown he Mure eresson esaors usn ulle aular varables and arbues n aral nforaon case should be used bu f all he aular varables and arbues are unnown. Mure eresson esaors usn ulle aular varables n no nforaon case should be used o esae he fne oulaon ean. eferences [] Nean J. 938 Conrbuon o he Theor of aln Huan Poulaons. Journal of he Aercan ascal Assocaon h://d.do.or/0.080/ [] Hansen M.H. and Hurwz W.N. 943 On he Theor of aln fro Fne Poulaons. Annals of Maheacal ascs h://d.do.or/0.4/aos/ [3] Cochran W.G. 940 The Esaon of he Yelds of he Cereal Eerens b aln for he ao of Gran o Toal Produce. Journal of Arculural cence h://d.do.or/0.07/ [4] Wason D.J. 937 The Esaon of Leaf Areas. Journal of Arculural cence h://d.do.or/0.07/ x [5] Oln I. 958 Mulvarae ao Esaon for Fne Poulaon. Boera h://d.do.or/0.093/boe/ [6] aj D. 965 On a Mehod of Usn Mul-Aular Inforaon n ale urves. Journals of he Aercan ascal Assocaon h://d.do.or/0.080/ [7] obson D.. 95 Mulle aln of Arbues. Journal of he Aercan ascal Assocaon h://d.do.or/0.080/ [8] Zahoor A. Muhhaad H. and Munr A. 009 Generalzed Mulvarae ao Esaor Usn Mulle Aular Varables for Mul-Phase aln. Pasan Journal of asc [9] Zahoor A. Muhhaad H. and Munr A. 009 Generalzed eresson-cu-ao Esaors for Two Phase aln Usn Mulle Aular Varables. Pasan Journal of ascs [0] uddn M. and Hanf M. 007 Esaon of Poulaon Mean n nle and Two Phase aln wh or whou Addonal Inforaon. Pasan Journal of ascs [] Jhajj H.. hara M.K. and Grover L.K. 006 A Fal of Esaor of Poulaon Mean Usn Inforaon on Aular Arbues. Pasan Journal of ascs [] Na V.D. and Gua P.C. 996 A Noe on Esaon of Mean wh Known Poulaon of Aular Characer. Journal of he Indan oce of Arculural ascs [3] ajesh. Panaj C. Nrala. and Florenns. 007 ao-produc Te Eonenal Esaor for Esan Fne Poulaon Mean Usn Inforaon on Aular Arbues. enassance Hh Press UA. [4] Bahl. and Tueja.K. 99 ao and Produc Te Esaor. Inforaon and Ozaon cence h://d.do.or/0.080/ [5] Hanf M. Ha I.U. and hahbaz M.Q. 009 On a New Fal of Esaor Usn Mulle Aular Arbues. World Aled cence Journal [6] Moeen M. hahbaz Q. and HanIf M. 0 Mure ao and eresson Esaors Usn Mul-Aular Varable and Arbues n nle Phase aln. World Aled cences Journal [7] Kun u J. and Odono L. 04 ao-cu-produc Esaor Usn Mulle Aular Arbues n nle Phase aln. Oen Journal of ascs h://d.do.or/0.436/ojs [8] Kun u J. and Odono L. 04 ao-cu-produc Esaor Usn Mulle Aular Arbues n Two-Phase aln. Oen Journal of ascs h://d.do.or/0.436/ojs [9] Arora. and Bans Lal. 989 New Maheacal ascs. aa Praashan New Delh. 366

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