Joural of Mathematics Research; Vol. 6, No. 3; 014 ISSN 1916-9795 E-ISSN 1916-9809 Published by Caadia Ceter of Sciece ad Educatio Alterative Ratio Estimator of Populatio Mea i Simple Radom Samplig Ekaette Iyag Eag 1, Victoria Matthew Akpa 1 & Emmauel Joh Ekpeyog 1 Departmet of Mathematics/Statistics & Computer Sciece, Uiversity of Calabar, Calabar, Nigeria Departmet of Statistics, Michael Okpara Uiversity of Agriculture,Umudike, Abia State, Nigeria Correspodece: Ekaette Iyag Eag, Departmet of Mathematics/Statistics & Computer Sciece, Uiversity of Calabar, P.M.B. 1115, Calabar, Cross River State, Nigeria. Tel: 34-803-400-3354. E-mail: ekkaass@yahoo.com Received: March 4, 014 Accepted: Jue 1, 014 Olie Published: July 7, 014 doi:10.5539/jmr.v63p54 URL: http://dx.doi.org/10.5539/jmr.v63p54 Abstract A alterative ratio estimator is proposed for a fiite populatio mea of a study variable Y i simple radom samplig usig iformatio o the mea of a auxiliary variable X, which is highly correlated with Y. Expressios for the bias ad the mea square error of the proposed estimator are obtaied. Both aalytical ad umerical comparisos have show the proposed alterative estimator to be more efficiet tha some existig oes. The bias of the proposed estimator is also foud to be egligible for all populatios cosidered, idicatig that the estimator is as good as the regressio estimator ad better tha the other estimators uder cosideratio. Keywords: ratio estimator, efficiecy, bias, simple radom samplig, auxiliary variables, precisio 1. Itroductio 1.1 Backgroud of Study Estimatio theory is a importat part of statistical studies, whereby, populatio parameters are obtaied usig sample statistics. I ay survey work, the experimeters iterest is to make use of methods that will improve precisios of estimates of the populatio parameters both at the desig stage ad estimatio stage. These parameters ca be totals, meas or proportios of some desired characters. I sample surveys, auxiliary iformatio is used at selectio as well as estimatio stages to improve the desig as well as obtaiig more efficiet estimators. Icreased precisio ca be obtaied whe study variable Y is highly correlated with auxiliary variable X. Usually, i a class of efficiet estimators, the estimator with miimum variace or mea square error is regarded as the most efficiet estimator. A good estimator ca also be described by the value of its bias. A estimator with miimum absolute bias is regarded as a better estimator amog others i the class Rajesh et al., 011. Whe the populatios mea of a auxiliary variable is kow, so may estimators for populatio parameters of study variable have bee discussed i literature. The literature o survey samplig describes a great variety of techiques for usig auxiliary iformatio by meas of ratio, product ad regressio methods. If the regressio lie of the character of iterest Y o the auxiliary variable, X is through the origi ad whe correlatio betwee study ad auxiliary variables is positive high, the the ratio estimate of mea or total may be used Cochra 1940. O the other had, if the regressio lie used for the estimate does ot pass through the origi but makes a itercept alog the y-axis, the regressio estimatio is used Okafor, 00. Furthermore, whe correlatio betwee study variable ad auxiliary variable is egative, the product method of estimatio is preferred. Robso 1957, Murthy 1967, Perri 005, Muhammad et al. 009 ad Solaki et al. 01 had established that the regressio estimator is geerally more efficiet tha the ratio ad product estimators except whe the regressio lie of the study variable o the auxiliary variable passes through a suitable eighbourhood of the origi, i which case, the efficiecies of these estimators are almost equal. Whe the populatio parameters of the auxiliary variable X such as populatio mea, coefficiet of variatio, coefficiet of kurtosis, coefficiet of skewess, media are kow, a 54
www.ccseet.org/jmr Joural of Mathematics Research Vol. 6, No. 3; 014 umber of modified estimators such as modified ratio estimators, modified product estimators ad modified liear regressio estimators have bee proposed ad is widely acceptable i the literature Subramai & Kumarpadiya, 01. I samplig literature, may estimators have bee proposed whe a sigle auxiliary variable is ivolved. Uder some realistic coditios, they are foud to be more efficiet tha the sample mea, the ratio ad product estimators ad are as efficiet as the regressio estimator i the optimum case but the problem of the best estimator i terms of both efficiecy ad biasedess has ot bee fully addressed. This paper is aother attempt i solvig this problem. A alterative ratio estimators for populatio mea of the study variable, Y, which is more efficiet tha some of the existig estimators is proposed usig iformatio o oe auxiliary variable, X, that is highly correlated with study variable. 1. Summary of Some Existig Estimators To ehace effective compariso, we summarize below some existig estimators, their biases ad mea square errors. Cosider a fiite populatio of N distict ad idetifiable uits Π={U 1, U,...U N }. Let a sample of size be draw from the populatio by simple radom samplig without replacemet. Suppose that iterest is to obtai a ratio estimate of the mea of a radom variable Y from the sample usig a related variable X as supplemetary iformatio ad assumig that the total of X is kow from sources outside the survey. Table 1. Some existig estimators, their biases ad mea square error S/N Estimator Bias Mea Square Error 1. ȳ 0 Ȳ Cy. 3. 4. 5. ȳ cl = ȳ x Classical ratio ȳ ȳ ST = x + ρ + ρ Sigh ad Tailor 003 ȳ KC = ȳ + b x x Kadilar ad Cigi 004 ȳ reg = ȳ + b x Regressio estimator Ȳ[C x ρ C y ] Ȳ[wC x ρw C y ] 0 ȲC x Ȳ [C y + C x ρ C y ] Ȳ [C y + C xww θ] Ȳ [C x + C y1 ρ ] Ȳ C y1 ρ Where = S x ; C y = S y Ȳ ; the coefficiet of variatio of the auxiliary variable, X ad the respose variable, Y; ρ = S xy ; the correlatio coefficiet betwee X ad Y; S x S y w = + ρ, k = Ȳ ; B = S xy ; the regressio coefficiet; S x θ = ρc y ad f = N, where S N x = N 1 1 xi, S y = N 1 1 N yi Ȳ ; the populatio variaces of the auxiliary ad study variables respectively; S xy = N 1 1 N xi y i Ȳ ; the populatio covariace betwee X ad Y; = N 1 N x i, Ȳ = N 1 N y i ; populatio meas of the auxiliary ad study variables; x = 1 x i, ȳ = 1 y i ; sample meas of the auxiliary ad study variables are respectively defied wherever they appear.. Proposed Estimator The proposed ratio estimator is obtaied by formig liear combiatio of Sigh ad Tailor 003 ad Kadilar 55
www.ccseet.org/jmr Joural of Mathematics Research Vol. 6, No. 3; 014 ad Cigi 004 estimators as show below: ȳ pr = αȳ + ρ x + ρ + β ȳ + b x x 1 such that α + β = 1.1 Bias ad Mea Square Error of the Proposed Estimator To obtai the approximate expressio for the bias ad the mea squared error for the proposed ratio estimator, let x = 1 + e x ; ȳ = Ȳ 1 + e y where So that, e x = x, e y = ȳ Ȳ Ȳ E e x = E e y = 0, E ex = E e x e y = ρ C y = θ C x, E e y = C y Therefore, expressig 1 i terms of, we obtai ȳ pr = αȳ 1 + e y + ρ + ρ Ȳ [ 1 + e y + b 1 + e x ] + 1 α + ρ 1 + e x 3 = αȳ1 + e y + ρ + ρ + e x + 1 α [ Ȳ 1 + e y 1 + ex 1 b e x 1 + e x 1] = αȳ 1 + e y 1 + wex 1 + Ȳ 1 e x + e x + e y e y e x Ȳ α αe x + αe x + αe y αe y e x b e x + b e x + αb e x αb e x By Taylor Series approximatio up to order, the expressio becomes ȳ pr = αȳ 1 we x + w e x + e y we y e x + Ȳ 1 e x + e x + e y e y e x + Ȳ αe x α αe x αe x + αe y e x B Ȳ e x + B Ȳ e x + α Ȳ e x αb Ȳ e x = Ȳ[α αwe x + αw e x + αe y αwe y e x + 1 e x + e x + e y e y e x α + αe x αe x αe y + αe y e x B Ȳ e x + B Ȳ e x + αb Ȳ e x αb Ȳ e x] The expressio for the Bias of this estimator to first order approximatio is obtaied as follows: Bȳ pr = Eȳ pr Ȳ = E[Ȳ 1 + e y + αbk BK α αw e x + α αw 1 e y e x + αw + 1 α + BK αbk e x Ȳ ] = Ȳ[α αw 1 ρ C y + αw + 1 α + BK αbk C x The MSE of this estimator is give by: 4 MS E ȳ pr = Eȳpr Ȳ = E[Ȳ 1 + e y + αbk BK α αw e x + α αw 1 e y e x + αw + 1 α + BK αbk e x Ȳ ] = Ȳ [C y + αbk BK α αw ρc y + αbk BK α αw C x 56 5
www.ccseet.org/jmr Joural of Mathematics Research Vol. 6, No. 3; 014. Optimal Coditios for the Proposed Estimator To obtai the value of α that miimizes the MSE, we take partial derivative of Equatio 5 with respect to α ad equate it to zero as follows: MS E ȳ pr = Ȳ [ BK + 1 w ρc y + BK + 1 wαbk BK α αw C α x] = 0 ρ C y + α BK + 1 w C x BK + 1 C x = 0 6 α = BK + 1 C x ρc y BK + 1 w C x Substitutig for 6 i 5 gives the optimal MSE for ȳ pr as: MS E ȳ pr = Ȳ [ C ] y 1 ρ 7 3. Efficiecy Compariso I order to compare the efficiecy of the various existig estimators with that of proposed estimator, we require the expressios of mea square error of these estimators, up to first order of approximatio. A aalytical compariso of the proposed estimator with three of the existig estimators amely: the classical, Sigh ad Tailor 003 ad Kadilar ad Cigi 004 estimators are carried out. 3.1 Efficiecy Compariso of ȳ pr ad ȳ cl I this sectio, the aalytical coditio uder which the proposed estimator will be more efficiet tha the classical ratio estimator is established. MS E y pr MS Eȳcl = Ȳ [ Cy = = [ = 1 ρ ] Ȳ C y C yρ C y C x + ρ C y Ȳ ρ C y C x Cyρ Ȳ ] C y ρ Ȳ [C y + C x ρ C y ] Sice the expressio i the square bracket is always positive, we coclude that the proposed estimator will always be more efficiet tha the classical ratio estimator. 3. Efficiecy Compariso of ȳ pr ad ȳ ST 8 MS E y pr MS EȳST = = = Ȳ [ C ] y 1 ρ Ȳ [Cy + C xw w θ] Ȳ [Cy Cy C xw w ρc y ] Ȳ [ Cyρ C xw w ρc y ] { = Ȳ [ C xρ + C xw w θ ]} 9 Therefore, for the proposed estimator to be more efficiet tha Sigh ad Tailor 003, the terms i the secod bracket must be positive. This implies that: C yρ + C xw w θ > 0 10 57
www.ccseet.org/jmr Joural of Mathematics Research Vol. 6, No. 3; 014 3.3 Efficiecy Compariso of ȳ pr ad ȳ KC Sice the expressio i the square bracket of Equatio 11 is always positive, it therefore meas that the proposed estimator will always be more efficiet tha Kadilar ad Cigi 004 estimator of populatio mea. MS E ȳ pr MS EȳKC = Ȳ [ C ] y 1 ρ Ȳ [C x + C y 1 ρ ] = [ ] 11 = Ȳ C x Sice the expressio i the square bracket of Equatio 11 is always positive, it therefore meas that the proposed estimator will always be more efficiet tha Kadilar ad Cigi 004 estimator of populatio mea. 4. Numerical Compariso I this sectio, to study the performace of the estimator preseted i this work, we cosider four empirical populatios used by others. The source of populatios ad the values of requisite populatio parameters are give. We compare the efficiecy of the proposed estimator with the existig estimators usig the four kow populatio data. 4.1 Data Statistics for Populatio 1 From the above, other statistics derived are Source: Kadilar ad Cigi 004. N = 00 Ȳ = 500 = 50 = 5 ρ = 0.90 θ = ρ C y = 6.75 = BK C y = 15 w = + ρ = 0.97 = S y = 7500 S x = 5 = 50 K = Ȳ = 0.05 B = ρs y = 135 S x R = Ȳ = 0 Table. Estimators, biases, MSE ad relative bias usig oe auxiliary variable i Populatio 1 Estimator MSE Bias of Estimator %Relative Bias = Bȳ. /MS Eȳ. ȳ cl 656,50-17.5 9.9 ȳ ST 66,6.3-167.51 0.58 ȳ KC 175,31.50 30 7.16 ȳ pr 160,31.5 0.8888 0. 4. Data Statistics for Populatio N = 106 Ȳ = 1536.77 = 0 = 4375.59 ρ = 0.8 θ = ρ C y = 1.69683 = BK C y = 4.18 w = + ρ = 0.99997 =.0 58
www.ccseet.org/jmr Joural of Mathematics Research Vol. 6, No. 3; 014 From the above, other statistics derived are Source: Kadilar ad Cigi 003. S y = 645.09 S x = 49189.08 K = Ȳ = 15.86157 R = Ȳ = 0.063045 B = ρs y S x = 0.107109 Table 3. Estimators, biases, MSE ad relative bias usig oe auxiliary variable i Populatio Estimator MSE Bias of Estimator %Relative Bias = Bȳ. /MS Eȳ. ȳ cl 738,19.6-177.565 0.63 ȳ ST 738,10.9-177.591 0.63 ȳ KC 939,89.6 54,3749 6.4 ȳ pr 548,373.9-0.13436 0.0 4.3 Data Statistics for Populatio 3 From the above, other statistics derived are Source: Murthy 1967. N = 80 Ȳ = 51.864 = 0 = 11.64 ρ = 0.9413 θ = ρ C y = 0.44413 = BK C y = 0.354 w = + ρ = 0.98 = 0.7507 S y = 18.3569 S x = 5 = 8.4563 K = Ȳ = 0.17353 B = ρs y =.04337 S x R = Ȳ = 4.6008 Table 4. Estimators, biases, MSE ad relative bias usig oe auxiliary variable i Populatio 3 Estimator MSE Bias of Estimator %Relative Bias = Bȳ. /MS Eȳ. ȳ cl 18.9731 0.6985 9.9 ȳ ST 14.4507 0.483917 1.73 ȳ KC 58.0311 1.09555 14.36 ȳ pr 1.4399958-0.077576 6.46 4.4 Data Statistics for Populatio 4 N = 78 Ȳ = 39.068 = 48 = 5.111 ρ = 0.713 θ = ρ C y = 0.6435 = BK C y = 1.4451 w = + ρ = 0.97078 = 1.6198 59
www.ccseet.org/jmr Joural of Mathematics Research Vol. 6, No. 3; 014 From the above, other statistics derived are Source: Das 1988. S y = 56.457 S x = 40.6748 K = Ȳ = 0.64751 R = Ȳ = 1.55581 B = ρs y S x = 1.001175 Table 5. Estimators, biases, MSE ad relative bias usig oe auxiliary variable i Populatio 4 Estimator MSE Bias of Estimator %Relative Bias = Bȳ. /MS Eȳ. ȳ cl 35.1791 0.6985 9.9 ȳ ST 33.80755 0.564309 1.73 ȳ KC 95.3807 1.766793 14.36 ȳ pr 6.355636-0.04141 0.47 5. Discussio The optimal Mea square error MSE of the proposed estimator give i Equatio 7 has the same expressio as the MSE of the regressio estimator which is kow to be more efficiet tha the ratio ad product estimators. The aalytical compariso of the proposed estimator with the three existig oes i Equatios 8, 9 ad11 show that the proposed estimator will always be more efficiet tha the classical ad Kadilar ad Cigi 004 estimators ad be preferred to Sigh ad Tailor 003 estimator whe the coditio stated i Equatio 10 is satisfied. The empirical results preseted i Tables, 3, 4 ad 5 show that the MSE of the proposed estimator is cosistetly less tha those of classical ratio estimator, Sigh ad Tailor 003 ad Kadilar ad Cigi 004 estimators i populatio for all four populatios uder cosideratio, showig that the estimator, ȳ pr is more efficiet tha all the other estimators uder cosideratio. This is due to the fact that the proposed estimator is equally as efficiet as the regressio estimator ad cofirms Cochra 194, Robso 1957, Murthy 1967 ad Perri 005 assertio that the regressio estimator is geerally more efficiet tha ratio ad product estimators Aalyses of biases have also show the proposed estimator to have the smallest bias compared to all the existig estimators for all populatios cosidered. The relative bias of the proposed estimator show i the four tables is 10 % for all the populatios uder cosideratio showig that the proposed estimator is almost ubiased. This agrees with Okafor 00 assertio that ay estimator with a relative bias of less tha 10 % is cosidered to have a egligible bias. 6. Coclusio I coclusio, sice the proposed estimator gives the same precisio as the regressio estimator ad is cosistetly better i terms of bias ad efficiecy tha the three estimators uder cosideratio, ȳ pr ca always be used as a alterative to the regressio estimator ad gives a better replacemet to some existig ratio estimators. Refereces Cochra, W. G. 1940. The Estimatio of the Yields of the Cereal Experimets by Samplig for the Ratio of Grai to Total Produce. The Joural of Agric Sciece, 30, 6-75. http://dx.doi.org/10.1017/s00185960004801 Das, A. K. 1988. Cotributio to the Theory of Samplig Strategies based o Auxiliary Iformatio. Ph.D Thesis Submitted to Bidha Chadra, Idia. Kadilar, C., & Cigi, H. 003. A study o the chai Ratio-Type Estimator. Hacettepe Joural of Mathematics ad Statistics, 3, 105-108. Kadilar, C., & Cigi, H. 004. Ratio Estimators i Simple Radom Samplig. Mathematics ad Computatio, 151, 893-90. http://dx.doi.org/10.1016/s0096-30030300803-8 Muhammad, H., Naqvi, H., & Muhammad, Q. 009. A modified Regressio-type estimator i survey samplig. Applied Scieces Joural, 71, 1559-1561. Murthy, M. N. 1967. Samplig Theory ad Methods. Calcutta, Statistics Publishig Society. 60
www.ccseet.org/jmr Joural of Mathematics Research Vol. 6, No. 3; 014 Okafor, F. C. 00. Sample Survey Theory with Applicatios 1st ed.. Nsukka, Nigeria, Afro-Orbis. Perri, P. F. 005. Combiig two Auxiliary Variables i Ratio-cum-product type Estimators. Proceedigs of Italia Statistical Society. Itermediate meetig o Statistics ad Eviromet, Messia, 1-3 September, 005, 193-196. Rajesh, T., Rajesh, P., & Jog-Mid, K. 011. Ratio-cum-product Estimators of Populatio mea usig kow Parameters of Auxiliary Variables. Commuicatios of the Korea Statistical Society, 18, 155-164. Robso, D. S. 1957. Applicatio of Multivariate Polykays to the Theory of ubiased ratio-type estimatio. Joural of the America Statistical Associatio, 5, 511-5. http://dx.doi.org/10.307/81700 Sigh, H. P., & Tailor, R. 003. Use of kow Correlatio Coefficiet i Estimatig the Fiite Populatio Mea. Statistics i Trasitio, 64, 555-560. Steel, R. G. D., & Torrie, J. H. 1960. Priciples ad Procedures of Statistics. New York, NY: MC Graw Hill. Subramai, J., & Kumarapadiya, G. 01a. Estimatio of Populatio Mea usig kow Media ad Coefficiet of Skewess. America Joural of Mathematics ad Statistics, 5, 101-107. Subramai, J., & Kumarapadiya, G. 01b. A class of almost ubiased modified ratio Estimators of Populatio parameters. Elixir statistics, 44, 7411-7415. Udofia, G. A. 006. Sample Survey Theory ad Methods I 1st ed.. Calabar: Uiversity of Calabar Press. Copyrights Copyright for this article is retaied by the authors, with first publicatio rights grated to the joural. This is a ope-access article distributed uder the terms ad coditios of the Creative Commos Attributio licese http://creativecommos.org/liceses/by/3.0/. 61