Estimation of Population Ratio in Post-Stratified Sampling Using Variable Transformation
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1 Ope Joural o Statistics, 05, 5, -9 Published Olie Februar 05 i SciRes. Estimatio o Populatio Ratio i Post-Stratiied Samplig Usig Variable Trasormatio Alo Chijioke Oeka, Chieaka Hostesia Izuobi, Iheai Slvester Iwueze Departmet o Statistics, Federal Uiversit o Techolog, Owerri, Nigeria alooeka@uto.edu.g, chieaka007@ahoo.com, isiwueze@ahoo.com Received 8 December 04; accepted 6 Jauar 05; published 0 Jauar 05 Copright 05 b authors Scietiic Research Publishig Ic. This work is licesed uder the Creative Commos Attributio Iteratioal icese CC BY). Abstract Extedig the work carried out b [], this paper proposes six combied-tpe estimators o populatio ratio o two variables i post-stratiied samplig scheme, usig variable trasormatio. Properties o the proposed estimators were obtaied up to irst order approximatios, o ), both or achieved sample coiguratios coditioal argumet) over repeated samples o ixed size ucoditioal argumet). Eiciec coditios were obtaied. Uder these coditios the proposed combied-tpe estimators would perorm better tha the associated customar combied-tpe estimator. Furthermore, optimum estimators amog the proposed combiedtpe estimators were obtaied both uder the coditioal ucoditioal argumets. A empirical work coirmed the theoretical results. Kewords Variable Trasormatio, Combied-Tpe Estimator, Ratio, Product Regressio-Tpe Estimators, Mea Suared Error. Itroductio The use o iormatio o auxiliar character to improve estimates o populatio parameters o the stud variable is a commo practice i sample surve, sometimes, iormatio o several variables is used to estimate or predict a characteristic o iterest. The ivestigators ote collect observatios rom more tha oe variable, icludig the variable o iterest some auxiliar variables x. The use o these variables kow as auxiliar iormatio i sample surve desig) ote results i eiciet estimate o populatio parameters e.g. How to cite this paper: Oeka, A.C., Izuobi, C.H. Iwueze, I.S. 05) Estimatio o Populatio Ratio i Post-Stratiied Samplig Usig Variable Trasormatio. Ope Joural o Statistics, 5, -9.
2 mea, ratio, proportio, etc.) uder some realistic coditios, especiall whe there is a strog correlatio betwee the stud variables the auxiliar variables. Ma authors have made cotributios i this regard, icludig [] [3]. I this cotext, ratio, product regressio methods o estimatio are good examples. Ratio product-tpe estimators take advatage o the correlatio betwee the auxiliar variable the stud variable, to improve the estimate o the characteristic o iterest. For example, whe iormatio is available o the auxiliar variable that is highl positivel correlated with the stud variable, the ratio method o estimatio proposed b [4] is a suitable estimator to estimate the populatio mea, whe the correlatio is egative, the product method o estimatio, as evisaged b [5] [6], is appropriate. However, i some studies, the ratio o the populatio meas or totals) o the stud auxiliar variables might be o great sigiicace, hece the eed to estimate such ratios. The customar estimator o the populatio ratio R = Y ) o the populatio meas o two variables, x, uder the simple rom samplig scheme, is give as ˆR = x, which is the ratio o the sample meas o the two variables [] [7]). The estimator, ˆR = x, uses iormatio o ol two variables, amel the stud variable ) oe auxiliar variable x ). However, several authors, like [7] [8], have cotributed to the problem o estimatig the populatio ratio o two meas, ote utilizig additioal iormatio o oe or more auxiliar variables, sa,, ) zi i =. While it is possible to record icreased eiciec b itroducig such additioal auxiliar variables, it is obvious that extra cost is ivolved i order to obtai iormatio o such additioal auxiliar variables. Reereces [] [9] have argued that such extra cost could be avoided b usig variable trasormatio o the alread observed auxiliar variable, istead o itroducig additioal ew) auxiliar variables. However, the works carried out b [] [9] were restricted to estimatio o populatio ratio i simple rom samplig scheme. The preset stud is ecessitated b the eed to exted to poststratiied samplig scheme, the works o ratio estimatio carried out b [] [9] uder the simple rom samplig scheme. This is i order to exted to other samplig schemes, the obvious advatage o reduced cost i the use o variable trasormatio istead o itroducig additioal ew) auxiliar variables whe estimatig populatio ratio o two populatio parameters.. The Proposed Combied-Tpe Estimators et uits be draw rom a populatio o N uits usig simple rom samplig method let the sampled uits be allocated to their respective strata, where h is the umber o uits that all ito stratum h such that h =. et hi x hi be the i th observatio o the stud auxiliar variables, respectivel. h= Cosider the ollowig variable trasormatio o the auxiliar variable, x, uder post-stratiied samplig scheme. N xhi xhi =, h =,,, i =,,, N.) N A euivalet o the trasormatio.), i simple rom samplig scheme, has bee used b authors like [] [8]-[3]. The associated sample mea estimator o the trasormed variable.), i post-stratiied samplig scheme, ca be writte as x = ω x h h h= x = π) πx, where π = N = ω are sample mea estimators based o x hi h h h=.) hi respectivel. Usig the sample meas, x x, assumig that the populatio mea, o the auxiliar variable x hi, is kow, we proposed six combied-tpe estimators o the populatio ratio R = Y i post stratiied samplig scheme as Rˆ c = x b x ).3)
3 x ˆ c = = x x x R ˆ 3c = = x x x x R Rˆ Rˆ 4c 5c x.4).5) =.6) = x b x ) ˆ x R6c = =. x x x.. Coditioal Properties o the Proposed Estimators Reerece [4] deied that uder the coditioal argumet, that is, or the achieved sample coiguratio, =,, 3,, ) the post stratiied estimator, is ubiased or the populatio mea, Y, with variace S ω S V S N.7).8) h h h ) = ωh h ) = ωh h.9) h= h h= h h= where V reers to coditioal variace S h is the populatio variace o i stratum h. Similarl, Oeka 0) obtaied the coditioal variace o x the coditioal covariace o x respectivel as: S ω S V x S N xh h xh ) = ωh h ) = ωh xh.0) h= h h= h h= S ω S C x S N xh h xh, ) = ωh h ) = ωh xh.) h= h h= h h= where S xh is the populatio variace o x i stratum h, S xh is the covariace o x i stratum h, C reers to coditioal covariace. et The, uder the coditioal argumet, Y x e0 = e =..) Y ) ) E e = E e =.3) 0 0 ) V S h E e ) = = h).4) 0 ω h Y Y h= ) V x Sxh E e ) = = h).5) ω h h= h h 3
4 ) C, x S E ee 0 ) = =..6) ˆ C xh ωh h) Y Y h= h Usig.), the irst proposed estimator, R, give i.3), ca be re-writte up to irst order approxima- o, i expected value, as tio, ) ) ) ) ) Rˆ c R = R e0 + bπ e + bπ e0e+ + bπ e c = 0+ + π) + π) 0 Rˆ R R e b e b ee..7).8) We take coditioal expectatio o.7).8), use.3) to.6) to make the ecessar substitutios. This gives the coditioal bias mea suare error o C respectivel as where ˆ C ) = + π) + π) B R b RA b A ˆ C ) = + + π) + π) MSE R A b R A b RA.9).0) ) ω ) h h Sh ωh h S ω xh h h ) Sxh,,..) A = A = A = h= h h= h h= h Followig similar procedure, we obtai the coditioal biases mea suare errors o the six proposed estimators, together with those o the customar combied-tpe estimator, C = x, o populatio ratio o, as: ) R, i post-stratiied samplig, up to irst order approximatio, ) B RˆC ) [ RA A ] =.) ˆ C ) π) π) B R = + b + b RA A.3) ˆ C ) π) π) B R = + + RA A.4) ˆ C ) = π + π ) π) B R3 RA A ˆ ) C B R = 4 RA A π + π ˆ ) ) C ).5).6) B R = 5 b b RA A π + π + +.7) ˆ C ) π)[ π ] B R6 = + RA + A.8) ˆ C ) MSE R = A R A RA + ˆ C ) = + + π) + π) MSE R A b R A b RA.9).30) 4
5 ˆ C ) = + + π) + π) MSE R A R A RA ˆ c ) = + π) π) MSE R3 A R A RA ˆ ) C π MSE R4 = A + R A + πra ˆ C ) = + π + ) + π + ) MSE R5 A b R A b RA ˆ C ) = + + π) + + π) MSE R6 A R A RA. Geerall, we have or the proposed six combied-tpe estimators, where =,, 6 ˆ ) θ MSE Rc = A + R A θ RA b ) ) ) b) ) ).3).33).34).35).36) θ = + π, θ = + π, θ = π, θ = π, θ = π +, θ = + π..37).. Ucoditioal Properties o the Proposed Estimators Followig [4] we obtai the ollowig ucoditioal) variaces covariace, or repeated samples o ixed size. = h h h= ) ω V S.38) = h xh h= ) ω S V x.39) Cov, x ) = ωhsxh.40) h= where = N is the populatio samplig ractio. B takig ucoditioal expectatios o.7).8), usig.38)-.40) to make the ecessar substitutios, we obtai the ucoditioal bias mea suare errors o the irst proposed estimator, ˆ c o, as: where R, up to irst order approximatio, ) ˆ C ) π) π) B R = + b + b RA A ˆ C ) = + + π) + π) MSE R A R A RA = ω h h = ω h xh = ωh xh h= h= h= A S, A S, A S..4).4).43) Followig similar procedure, we obtai the ucoditioal biases mea suare errors o the six proposed estimators, together with those o the customar combied-tpe estimator, C = x, o populatio ratio o, as: ) R, i post-stratiied samplig, up to irst order approximatio, ) B RˆC ) = [ RA A ].44) 5
6 , ˆ C ) π) π) B R = + b + b RA A ˆ C ) π) π) B R = + + RA A ˆ C ) = π + π ) π) B R3 RA A ˆ C ) π B R4 = R A + πa ˆ C ) π) π) B R5 = b + b + RA + A ˆ 6C ) = + π)[ π + ] B R RA A ˆ C ) MSE R = A + R A RA ˆ C ) π ) MSE R = A + R + b A RA ˆ C ) = + + π) + π) MSE R A R A RA ˆ C ) = + π) π) MSE R3 A R A RA ˆ C ) π MSE R4 = A + RA + RπA ˆ C ) = + + π) + + π) MSE R5 A b R A b RA ˆ C ) = + + π) + + π) MSE R6 A R A RA. Geerall, the ucoditioal mea suare errors o the proposed combied-tpe estimators is obtaied as where θ, =,, 6 is as give i.37). 3. Eiciec Compariso ˆ ) θ MSE R C = A R A θ RA +.45).46).47).48).49).50).5).5).53).54).55).56).57).58) The eiciecies o the six proposed combied-tpe estimators are irst compared with that o the customar combied ratio estimator R ˆC i estimatig the populatio ratio R o two populatio meas uder the coditioal ucoditioal argumets i post-stratiied rom samplig scheme. Secodl, the perormaces o the proposed estimators amog themselves are ivestigated. Furthermore, the optimum estimators amog the proposed estimators are also obtaied. The eiciec compariso is carried out usig the mea suare errors o the estimators the results are show i Table. 4. Numerical Illustratio Here, we use the ial ear GPA ) the level o abseteeism x ) o 0/03 graduatig studets o 6
7 Statistics Departmet, Federal Uiversit o Techolog Owerri to illustrate the properties o the estimators proposed i the preset stud. Abseteeism is measured as the average umber o das abset rom lectures i a moth. The class cosists o 50 studets, with 3 8 studets respectivel allig ito low-abseteeism 0-3 das per moth) high-abseteeism 4-6 das per moth) grou or strata. Our iterest is to estimate the ratio o ial ear GPA to abseteeism rom lectures, based o a post-stratiied sample o 0 out o the 50 graduatig studets i the class. The data statistics, cosistig mail o populatio parameters are show i Table. Table 3 shows the percetage relative eiciecies PRE-) o the proposed combied-tpe estimators, R ˆc, Table. Eiciec coditios uder coditioal ucoditioal argumets. Estimator Coditioal argumet Ucoditioal argumet R is better tha c R is better tha kc R is optimum i: c R i: c R i: jc ) θ < β < R or ) θ > β > R θ < θ θ < β R j ) j k or θ > θ θ > β R j ) j k θ 0 = β Where β = A A, β = A A θ, =,, 6 is as give i.37). R ) θ < β < R or ) θ > β > R θ < θ θ < β R j ) j k or θ > θ θ > β R j ) j k θ = β R Table. Data statistics or ial ear GPA ) abseteeism rom lectures x ). 0 Populatio/sample parameters Stratum low-abseteeism) Stratum high-abseteeism) N = 50 N = 3 N = 8 = 0 = = 8 ) = 0.60 ) = 0.65 ) = Y =.98 Y = 3.6 Y =.65 = 3.6 =.03 = 5.7 R = 0.94 R =.56 R = 0.5 π = 0.67 ω = 0.64 S = 0.4 S = x S = S = x S = 0.4 S = 0.06 x x ω = 0.64 ω = 0.36 Table 3. Percetage relative eiciecies uder coditioal ucoditioal argumets. Estimator c θ Coditioal argumet Ucoditioal argumet MSE PRE- %) PRE- %) MSE PRE- %) PRE- %) c c c c c c ˆ 0 c R
8 over the customar combied-tpe estimator, R ˆc, uder the coditioal uder the ucoditioal argumets. The table also shows the percetage relative eiciec PRE-) o the proposed combied-tpe estimators, c, over the other combied-tpe estimators, uder the coditioal uder the ucoditioal argumets. Table 3 shows that apart rom the estimators, c 6 c, the remaiig our proposed combied-tpe estimators, uder the coditioal uder the ucoditioal argumets, are more eiciet tha the customar combied-tpe estimator, c, or the data uder cosideratio, their gais i eiciec PRE-) are relativel large. Also, usig PRE-, we observe that the proposed combied-tpe estimator, c, is more eiciet tha the estimators, c, 6 c, c, uder the coditioal ucoditioal argumets. The optimum estimator, as expected, has the highest gai i eiciec, both uder the coditioal ucoditioal argumets. However, the customar combied-tpe estimator, o the other h, is oud to be more eiciet tha some o the proposed combied-tpe estimators or the give set o data. This coirms the theoretical results, which showed that the proposed estimators are ot alwas more eiciet tha the customar combied-tpe estimator. Notice that β = 0.6 R = 0.94 showig that β < R rom the theoretical results i Table, the proposed estimators would be more eiciet tha the customar combied-tpe estimator, uder the ucoditioal argumet, i θ <. The empirical results i Table 3 show that θ > θ 6 >, the proposed estimators ˆR PRE- = 44%) ˆR 6 PRE- = 65%) uder the ucoditioal argumet, are less eiciet tha the customar combied-tpe estimator, R ˆc. Hece the empirical results coirm the theoretical results. 5. Cocludig Remarks The stud exteds use o variable trasormatio i estimatig populatio ratio i simple rom samplig scheme to post-stratiied samplig scheme. Eiciec coditios or preerrig the proposed estimators to the customar combied-tpe estimator are obtaied. The stud shows that i a give surve, these eiciec coditios should be emploed i order to determie the appropriate proposed combied-tpe estimators to use or the purpose o estimatig the populatio ratio o two variables i post-stratiied samplig scheme, usig variable trasormatio. Reereces [] Oeka, A.C., Nlebedim, V.U. Izuobi, C.H. 03) Estimatio o Populatio Ratio i Simple Rom Samplig Usig Variable Trasormatio. Global Joural o Sciece Frotier Research, 3, [] Sukhatme, P.V. Sukhatme, B.V. 970) Samplig Theor o Surves with Applicatios. Iowa State Uiversit Press, Ames. [3] Cochra, W.G. 977) Samplig Techiues. 3rd Editio, Joh Wile & Sos, New York. [4] Cochra, W.G. 940) The Estimatio o the Yields o the Cereal Experimets b Samplig or the Ratio o Grai to Total Produce. The Joural o Agricultural Sciece, 30, [5] Robso, D.S. 957) Applicatio o Multivariate Polkas to the Theor o Ubiased Ratio-Tpe Estimatio. Joural o the America Statistical Associatio, 5, [6] Murth, M.N. 964) Product Method o Estimatio. Sakha Series A, 6, [7] Sigh, M.P. 965) O the Estimatio o Ratio Product o the Populatio Parameters. Sakha Series B, 7, [8] Upadhaa,.N., Sigh, G.N. Sigh, H.P. 000) Use o Trasormed Auxiliar Variable i the Estimatio o Populatio Ratio i Sample Surve. Statistics i Trasitio, 4, [9] Oeka, A.C., Nlebedim, V.U. Izuobi, C.H. 04) A Class o Estimators or Populatio Ratio i Simple Rom Samplig Usig Variable Trasormatio. Ope Joural o Statistics, 4, [0] Srivekataramaa, T. 980) A Dual o Ratio Estimator i Sample Surves. Biometrika, 67, [] Sigh, H.P. Tailor, R. 005) Estimatio o Fiite Populatio Mea Usig Kow Correlatio Coeiciet betwee Auxiliar Characters. Statistica, 4, [] Tailor, R. Sharma, B.K. 009) A Modiied Ratio-Cum-Product Estimator o Fiite Populatio Mea Usig Kow Coeiciet o Variatio Coeiciet o Kurtosis. Statistics i Trasitio New Series, 0,
9 [3] Sharma, B. Tailor, R. 00) A New Ratio-Cum-Dual to Ratio Estimator o Fiite Populatio Mea i Simple Rom Samplig. Global Joural o Sciece Frotier Research, 0, 7-3. [4] Oeka, A.C. 0) Estimatio o Populatio Mea i Post-Stratiied Samplig Usig Kow Value o Some Populatio Parameters). Statistics i Trasitio New Series, 3,
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