Mixture Regression-Cum-Ratio Estimator Using Multi-Auxiliary Variables and Attributes in Single-Phase Sampling

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1 Open Jounal of Saisics, 04, 4, Published Online Augus 04 in Scies. hp:// hp://dx.doi.og/0.436/os Mixue egession-um-aio Esimao Using Muli-Auxilia Vaiables and Aibues in Single-Phase Sampling Teesio Muembei, John Kung u, hisophe Ouma Depamen of Saisics and Acuaial Science, Kenaa Univesi, Naiobi, Kena eceived 5 Ma 04; evised 30 June 04; acceped 3 Jul 04 opigh 04 b auhos and Scienific eseach Publishing Inc. This wo is licensed unde he eaive ommons Aibuion Inenaional License ( BY. hp://ceaivecommons.og/licenses/b/4.0/ Absac In his pape, we have poposed a class of mixue egession-cum-aio esimao fo esimaing populaion mean b using infomaion on muliple auxilia vaiables and aibues simulaneousl in single-phase sampling and analzed he popeies of he esimao. An empiical was caied ou o compae he pefomance of he poposed esimao wih he exising esimaos of finie populaion mean using simulaed populaion. I was found ha he mixue egession-cumaio esimao was moe efficien han aio and egession esimaos using one auxilia vaiable and aibue, aio and egession esimaos using muliple auxilia vaiables and aibues and egession-cum-aio esimaos using muliple auxilia vaiables and aibues in singlephase sampling fo finie populaion. Kewods egession-um-aio Esimao, Muliple Auxilia Vaiables and Aibues, Single-Phase Sampling. Inoducion The wo of Neman [] ma be efeed o as he iniial wos whee auxilia infomaion has been used. Wason [] used he egession esimao of leaf aea on leaf weigh o esimae he aveage aea of he leaves on a plan. ochan [3] used auxilia infomaion in single-phase sampling o develop he aio esimao fo esimaion of populaion mean. In he aio esimao, he sud vaiable and he auxilia vaiable had a high posiive coelaion and he egession line was passing hough he oigin. Hansen and Huwiz [4] also suggesed he use of auxilia infomaion in selecing he sample wih vaing pobabiliies. How o cie his pape: Muembei, T., Kung u, J. and Ouma,. (04 Mixue egession-um-aio Esimao Using Muli-Auxilia Vaiables and Aibues in Single-Phase Sampling. Open Jounal of Saisics, 4, hp://dx.doi.og/0.436/os

2 T. Muembei e al. Olin [5] was he fis peson o use infomaion on moe han one supplemena chaace, which is posiivel coelaed wih he vaiable unde sud, using a linea combinaion of aio esimao based on each auxilia vaiable. Shula [6] poposed ha egession esimao using muliple auxilia was moe efficien han egession esimao using single auxilia vaiable. a [7] suggesed a mehod of using muli-auxilia infomaion in sample suve. Singh [8] poposed a aio-cum-poduc esimao and is muli-vaiable expession which wee moe efficien han aio, poduc and mean pe uni esimaos. Jha, Shama and Gove [9] poposed a famil of esimaos using infomaion on auxilia aibue. The used nown infomaion of populaion popoion possessing an aibue ha is highl coelaed wih sud vaiable Y. The aibue is nomall used when he auxilia vaiable is no available e.g. an amoun of mil poduced and a paicula beed of cow o an amoun of ield of whea and a paicula vaie of whea. aesh, Pana, Nimala and Floenins [0] used he infomaion on auxilia aibue in aio esimao in esimaing populaion mean of he vaiable of inees using nown aibues such as coefficien of vaiaion, coefficien uosis and poin bi-seial coelaion coefficien. The esimao pefomed bee han he usual sample mean and Nai and Gupa [] esimao. aesh, Pana, Nimala and Floenins [0] also used he auxilia aibue in egession, poduc and aio pe exponenial esimao following he wo of Bahl and Tuea []. Hanif, Ha and Shahbaz [3] [4] poposed a geneal famil of esimaos using muliple auxilia aibue in single and double phase sampling. The esimao had a smalle compaed o ha of Jha, Shama and Gove [9]. The also exended hei wo o aio esimao which was genealizaion of Nai and Gupa [] esimao in single and double phase sampling wih full infomaion, paial infomaion and no infomaion. The concep of double sampling was fis poposed b Neman [] in sampling human populaions when he mean of auxilia vaiable was unnown. I was lae exended o muliphase b obson [5]. In mos suves he auxilia infomaion is alwas available and eve fom of auxilia infomaion should be used in developing sampling saegies. Samiuddin and Hanif [6] inoduced he following appoach using auxilia vaiable. Full infomaion case: infomaion fo all auxilia vaiables is available. No infomaion case: infomaion fo all auxilia vaiables is no available. 3 Paial infomaion case: infomaion fo some auxilia vaiable is available fo all populaion unis. Ahmad [7] genealized mulivaiae aio and egession esimaos fo muli-phase sampling. Zahoo, Muhhamad and Muni [8] suggesed a genealized egession-cum-aio esimao fo wo-phase sampling using muliple auxilia vaiables in full, paial and no infomaion case. Kung u and Odongo [9] and [0] poposed aio-cum-poduc esimaos using muliple auxilia aibues in single phase sampling and wo-phase sampling using muliple auxilia aibues in full, paial and no infomaion case. Moeen, Shahbaz and HanIf [] poposed a class of mixue aio and egession esimaos fo single-phase sampling fo esimaing populaion mean b using infomaion on auxilia vaiables and aibues simulaneousl. In his pape, we will incopoae boh muliple auxilia vaiables and aibues in egession-cum-aio esimao o fom mixue egession-cum-aio esimao in single-phase sampling and also incopoae Aoa and Bansi [] appoach in wiing down he mean suaed eo.. Peliminaies.. Noaion and Assumpion The following noaion will be used in his poec. onside a populaion of N unis. Le Y be he sud vaiable fo which we wan o esimae he populaion mean and X, X,, X ae auxilia vaiables and,,, ae auxilia aibues. Fo single-phase sampling design le n be sample sizes fo fis phase while x and denoe he h auxilia vaiables and auxilia aibue, and denoe he vaiable of inees fom fis phase. Le θ = n N and = Y + e, x = X + ex and p (,,, = P + e = p (.0 whee e, e x and e ae sampling eo and ae ve small. We assume ha E e = E e = E e = (. ( ( x ( x 0 In defining he aibues we assume complee dichoom so ha; 368

3 T. Muembei e al. Le A N = and i i= h h, if i uni of populaion possess auxilia aibue i = 0, ohewise a n i i= (. = be he oal numbe of unis in he populaion and sample especivel pos- A a sessing aibue. Le P = and p = be he coesponding popoion of unis possessing a specif- N n ic aibues and is he mean of he main vaiable a second phase. S The coefficien of vaiaions ae given b, S S x S =, x = = while = is he co- Y X P SS elaion coefficien beween sud vaiable and auxilia vaiables and S = is he bi-seial coelaion SS coefficien beween sud vaiable and auxilia vaiables. Then fo simple andom sampling wihou eplacemen fo boh fis and second phases we wie b using phase wise opeaion of expecaions as: E e = θy E e e = θyx ( ( x x ( x x = θ i x x xx i ( = θ ( = θ i = θ i i i E e e X X i E e P E e e YP E e e PP i p xp x (.3 T A d A A = ( = (.4 A i A ( p = Aoa and Lai [] (.5 The following noaions will be used in deiving he mean suae eos of poposed esimaos Deeminan of populaion coelaion maix of vaiables, x, x,, xp and xp. p i p p Deeminan of h i mino of p coesponding o he h i elemen of i. Denoes he muliple coefficien of deeminaion of on x, x,, x and x. x xp i i xp Denoes he muliple coefficien of deeminaion of on, x, x,, xp and xp. Deeminan of populaion coelaion maix of vaiables x, x,, x and x. Deeminan of populaion coelaion maix of vaiables x, x,, xp and x Deeminan of he coelaion maix of i, x, x,, x and x. Deeminan of he coelaion maix of i, x, x,, xp and xp. Deeminan of he mino coesponding o i x of he coelaion maix of i i,, x, x,, x and xi (. Deeminan of he mino coesponding o of he coelaion maix of i,, x, x,, xp i xp and xpi (... Mean pe Uni in Single-Phase Sampling The sample mean using simple andom sampling wihou eplacemen is given b, i p 369

4 T. Muembei e al. While is vaiance is given, n i n = = (.6 θ V = Y (.7.3. aio and egession Esimao Using Auxilia Vaiable Le n i n = = and n xi n i = x = be he unbiased esimao of populaion means Y and X especivel. Then he classical aio esimao b ochan [3] and egession esimao b Wason [] ae defined especivel b, X = 0 x x e whee X, he populaion mean of he auxilia vaiable X is nown whee (.8 = + X x (.9 ae opimum values of aio and egession esimao especivel. The minimum of and e up o he fis ode of appoximaion ae, ( θy ( x x ( e θy ( = and x Y = X = + (.9 = (.0.4. aio and egession Esimao Using Muliple Auxilia Vaiables In case of muliple auxilia vaiables, he aio and egession esimaos Ahmad [7] ae given b, e X = (. = x β (. = = + X x +. whee = ( ae he opimum values β x suaed eo of and e up o he fis ode of appoximaion ae, ( θy ( Y +. =. The minimum mean X x = (.3 ( e θy ( = (.4.5. egession-um-aio Esimao Using Muliple Auxilia Vaiables The egession-cum-aio esimao b Zahoo [8] using muliple auxilia vaiables is given b, X e = + β X x = = + x + s= (.5 x 370

5 T. Muembei e al. The of = e + x x and β Y = X up o he fis ode of appoximaion ae, x + x ( e = θy.(, x s.6. aio and egession Esimao Using Auxilia Aibue ae he opimum values. The minimum (.6 In ode o have an esimae of he populaion mean Y he sud vaiable, assuming he nowledge of he populaion popoion P, Nai and Gupa [] defined aio and egession esimaos of populaion when he pio infomaion of populaion popoion of unis, possessing he same aibue is vaiable. Using (.8 and (.9 Nai and Gupa [] poposed following esimaos: The minimum of and e whee P = p (.7 e = + β ( P p (.8 up o he fis ode of appoximaion ae = fy + (.9 ( ( ( e Y β P β PY = + (.0 Y b = and β = ae opimum values of aio and egession esimao especivel. P.7. aio and egession Esimao Using Muliple Auxilia Aibues. The aio and egession esimaos b Hanif, Ha and Shahbaz [4] fo single-phase sampling using infomaion on muliple auxilia aibues ae given b, P P P p p p = (. = + β P p + β P p + + β P p (. e The of he ( and e( up o he fis ode of appoximaion ae, = = ψ = ( e = θ Y + P PY pb + P Pθ = = ϕ = ( = θy + pb + Q ψ (.3 ϕ ϕ ϕ ϕ (.4.8. egession-um-aio Esimao Using Muliple Auxilia Aibues The egession-cum-aio esimao using muliple auxilia aibues is given b, P e = + β P p = = + p + s= (.5 The + = x and β Y = X x + ae he opimum values o he fis ode 37

6 T. Muembei e al. of appoximaion. The minimum of e up o he fis ode of appoximaion ae, ( e = θy.(, x s (.6.9. Mixue aio and egession Using Muliple Auxilia Vaiables and Aibues The mixue aio esimao based on muliple auxilia vaiables and aibues b Moeen, Shahbaz and HanIf [] is given b: M(3 β X P (.7 = x p = = + β (.8 = + X x + P p M e(3 = = The minimum of M and e M up o he fis ode of appoximaion ae ( ( M(3 = θy.(, x s ( e(4 = θy.(, M x s (.9 (.30 In geneal hese esimaos have a bias of ode n. Since he sandad eo of he esimaes is of ode n, he uani bias/s.e is of ode and becomes negligible as n becomes lage. In pacice, his uani is n usuall unimpoan in samples of modeae and lage sizes. In his pape, we have combined mixue aio and mixue egession esimao o fom mixue egessioncum-aio esimao unde single-phase sampling and sudied he popeies of he poposed esimao. 3. Mehodolog 3.. Mixue egession-um-aio Esimao Using Muli-Auxilia Vaiables and Aibues in Single-Phase Sampling If we esimae a sud vaiable when infomaion on all auxilia vaiables is available fom populaion, i is uilized in he fom of hei means. B aing he advanage of mixue egession-cum-aio esimao echniue fo single-phase sampling, a genealized esimao fo esimaing populaion mean of sud vaiable Y wih he use of muli auxilia vaiables and aibues is suggesed as: ( ( + ( ( + + = + X x + X x + + X x + β P p + β P p M λ+ λ+ λ γ+ γ+ γ X X + + X P P P β( P p x + x + x p+ p+ p Using (.0 in (.0, we ge, λ e x x β = = + = + X = + e M = e e + + P Ignoing he second and highe ems fo each expansion of poduc and afe simplificaion, we wie, e x e M = e + Y ex β e λ Y γ Y X P The mean suaed eo of M is given b, ( M ( M γ (.0 (. (. = = + = = + e x e x β λ γ (.3 = = + = X = + P = E Y = E e e e Y Y 37

7 T. Muembei e al. We diffeeniae he Euaion (.3 paiall wih espec o ( =,,,, (,,, λ ( = + + +,, and (,,, β = + +, γ = + + and euae o zeo. The opimum value age given b, Y + = ( =,,, (.4 X x Y + β = ( = +, +,, (.5 P + λ = ( = +, +,, (.6 x + γ = ( = +, +,, (.7 x Using nomal euaion ha is used o find he opimum values given (3.8 we can wie, O e e M = E e e e e Y Y x x β λ γ (.8 = = + = X = + P ( M ( ( ( x E( ee E ee = E e E e e β E e e λ Y γ Y Taing expecaion of (3.49, we ge, (.9 x = = + = X = + M = θ Y YX β YP x = = + λy x γ Y = + = + Subsiuing he opimum (.4 o (.7 in (.0 and afe simplificaion we ge, ( M = θy + + = = + ( + + = + = + P (.0 (. O ( M = Y + + = = + ( = + = (. 373

8 T. Muembei e al. O θ Y + + M x, (, x = = + (, x = + = (.3 θ Y M = x, (, (, (, x x x + +, x, x (, x + (, + (, x + x + + x, x, x, ( + ( + + ( Using (.8 in (.4, we ge, Using (.5 in (.5, we ge, Y M = θ ( M = θy (. (.4 (.5 ( Bias and onsisenc of Mixue egession-um-aio Esimao These mixue egession-cum-aio esimao using muliple auxilia vaiables and aibues in single-phase sampling ae biased. Howeve, hese biases ae negligible fo modeae and lage samples. I s easil shown ha he mixue egession-cum-aio esimao using muliple auxilia vaiables and aibues is a consisen esimao since i is a linea combinaion of consisen esimaos i follows ha i is also consisen. 4. Simulaion, esul and Discussion In his secion, we caied ou daa analsis o compae he pefomance of mixue egession-cum-aio esimao using muliple auxilia vaiables and aibues wih alead exising esimao namel mean pe uni, aio and egession esimaos using one auxilia vaiable and aibue, aio and egession esimaos using wo auxilia vaiables and aibues and egession-cum-aio esimaos using fou auxilia vaiables and aibues in single-phase sampling fo finie populaion. In he simulaed populaion, he sud vaiable is nomall disibued while auxilia vaiables and aibues ae also nomall disibued and songl posiivel coelaed wih he sud vaiable. Sud vaiable N = 350, n = 50 mean = 50 sandad deviaion = 0 Fo aio esimao he auxilia vaiable and aibues ae posiivel coelaed wih he sud vaiable and he line passes hough he oigin. N = 350, n = 50 = = 0.73 = = Fo egession esimao he auxilia vaiable and aibues ae posiivel coelaed wih he sud vaiable and he line passes does no pass in he neighbohood of he oigin. N = 350, n = 50 = 76 = = =

9 T. Muembei e al. All he esuls wee obained afe caing ou seveal andom sample and aing he aveage. In ode o evaluae he efficienc gain we could achieve b using he poposed esimaos, we have calculaed he vaiance of mean pe uni and he mean suaed eo of all esimaos we have consideed. We have hen calculaed pecen elaive efficienc of each esimao in elaion o vaiance of mean pe uni. We have hen compaed he pecen elaive efficienc of each esimao, he esimao wih he highes pecen elaive efficienc is consideed o be he mos efficien han he ohe esimao. The pecen elaive efficienc is calculaed using he following fomulae. ( Y ˆ ( ˆ Va eff = 00 (3.0 The Table shows pecen elaive efficienc of mean pe uni, aio and egession esimaos using one auxilia vaiable and aibue, aio and egession esimaos using wo auxilia vaiables and aibues and egession-cum-aio esimaos using fou auxilia vaiables and aibues and mixue egession-cum-aio esimao using muliple auxilia vaiables and aibues wih espec o mean pe uni esimao fo singlephase sampling. I is obseved ha ou poposed mixue egession-cum-aio esimao using muliple auxilia vaiables and aibues using muliple auxilia vaiables and aibues is he mos efficien of he welve esimaos since i has he highes pecen elaive efficienc. 5. onclusion Accoding o Table, he poposed mixue egession-cum-aio esimao using muliple auxilia vaiables and aibues using muliple auxilia vaiables and aibues has he highes pecen elaive efficienc compaed o mean pe uni, aio and egession esimaos using one auxilia vaiable and aibue, aio and egession esimaos using wo auxilia vaiables and aibues and egession-cum-aio esimaos using fou auxilia vaiables and aibues in single-phase sampling fo finie populaion. This means ha he mixue ( Y ˆ Table. elaive efficienc of exising and poposed esimaos wih espec o mean pe uni esimao fo single-phase sampling. Esimaos Pecen elaive efficienc wih espec o mean pe uni e 9 06 e e 5 45 e 34 e 49 e 40 M e( M e( (poposed M

10 T. Muembei e al. egession-cum-aio esimao using muliple auxilia vaiables and aibues using muliple auxilia vaiables and aibues is he mos efficien esimao compaed o he esimaos ha uilize auxilia vaiables and aibues. The poposed mixue egession-cum-aio esimao using muliple auxilia vaiables and aibues using muliple auxilia vaiables and aibues in single-phase sampling is ecommended o esimae he finie populaion mean as i oupefoms all he ohe namel mean pe uni, aio and egession esimaos using one auxilia vaiable and aibue, aio and egession esimaos using wo auxilia vaiables and aibues and egession-cum-aio esimaos using fou auxilia vaiables and aibues in single-phase sampling. efeences [] Neman, J. (938 onibuion o he Theo of Sampling Human Populaions. Jounal of he Ameican Saisical Associaion, 33, 0-6. hp://dx.doi.og/0.080/ [] Wason, D.J. (937 The Esimaion of Leaf Aeas. Jounal of he Agiculual Science, 7, hp://dx.doi.og/0.07/s x [3] ochan, W.G. (940 The Esimaion of he Yields of he eeal Expeimens b Sampling fo he aio of Gain o Toal Poduce. Jounal of he Agiculual Science, 30, hp://dx.doi.og/0.07/s [4] Hansen, M.H. and Huwiz, W.N. (943 On he Theo of Sampling fom Finie Populaions. Annals of Mahemaical Saisics, 4, hp://dx.doi.og/0.4/aoms/ [5] Oliin, I. (958 Mulivaiae aio Esimaion fo Finie Populaion. Biomeia, 45, hp://dx.doi.og/0.093/biome/ [6] Shula, G.K. (965 Mulivaiae egession Esimae. Jounal of he Indian Saisical Associaion, 3, 0-. [7] a, D. (965 On a Mehod of Using Muli-Auxilia Infomaion in Sample Suves. Jounals of he Ameican Saisical Associaion, 60, hp://dx.doi.og/0.080/ [8] Singh, M.P. (967 aio-um-poduc Mehod of Esimaion. Meia,, hp://dx.doi.og/0.007/bf06348 [9] Jha, H.S., Shama, M.K. and Gove, L.K (006 A Famil of Esimao of Populaion Mean Using Infomaion on Auxilia Aibues. Paisan Jounal of Saisics,, [0] aesh, S., Pana,., Nimala, S. and Floenins, S. (007 aio-poduc Tpe Exponenial Esimao fo Esimaing Finie Populaion Mean Using Infomaion on Auxilia Aibues. enaissance High Pess, USA. [] Nai, V.D. and Gupa, P.. (996 A Noe on Esimaion of Mean wih Known Populaion of Auxilia haace. Jounal of he Indian Socie of Agiculual Saisics, 48, [] Bahl, S. and Tuea,.K. (99 aio and Poduc Tpe Esimao. Infomaion and Opimizaion Science,, hp://dx.doi.og/0.080/ [3] Hanif, M., Ha, I.U. and Shahbaz, M.Q. (009 On a New Famil of Esimao Using Muliple Auxilia Aibues. Wold Applied Science Jounal,, [4] Hanif, M., Ha, I. and Shahbaz, M.Q. (00 aio Esimaos Using Muliple Auxilia Aibues. Wold Applied Sciences Jounal, 8, [5] obson, D.S. (95 Muliple Sampling of Aibues. Jounal of he Ameican Saisical Associaion, 47, hp://dx.doi.og/0.080/ [6] Samiuddin, M. and Hanif, M. (007 Esimaion of Populaion Mean in Single and Two-Phase Sampling wih o wihou Addiional Infomaion. Paisan Jounal of Saisics, 3, [7] Ahmad, Z. (008 Genealized Mulivaiae aio and egession Esimaos fo Muli-Phase Sampling. Ph.D. Thesis, Naional ollege of Business Adminisaion and Economics, Lahoe. [8] Zahoo, A., Muhhamad, H. and Muni, A. (009 Genealized egession-um-aio Esimaos fo Two-Phase Sampling Using Muliple Auxilia Vaiables. Paisan Jounal of Saisics, 5, 93. [9] Kung u, J. and Odongo, L. (04 aio-um-poduc Esimao Using Muliple Auxilia Aibues in Single Phase Sampling. Open Jounal of Saisics, 4, hp://dx.doi.og/0.436/os [0] Kung u, J. and Odongo, L. (04 aio-um-poduc Esimao Using Muliple Auxilia Aibues in Two-Phase Sampling. Open Jounal of Saisics, 4, hp://dx.doi.og/0.436/os [] Moeen, M., Shahbaz, Q. and HanIf, M. (0 Mixue aio and egession Esimaos Using Muli-Auxilia Vaiable and Aibues in Single Phase Sampling. Wold Applied Sciences Jounal, 8, [] Aoa, S. and Bansi, L. (989 New Mahemaical Saisics. Saa Paashan, New Delhi. 376

On The Estimation of Two Missing Values in Randomized Complete Block Designs

On The Estimation of Two Missing Values in Randomized Complete Block Designs Mahemaical Theoy and Modeling ISSN 45804 (Pape ISSN 505 (Online Vol.6, No.7, 06 www.iise.og On The Esimaion of Two Missing Values in Randomized Complee Bloc Designs EFFANGA, EFFANGA OKON AND BASSE, E.