Estimation of Population Mean Using Co-Efficient of Variation and Median of an Auxiliary Variable
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1 Iteratioal Joural of Probability ad Statistics 01, 1(4: DOI: /j.ijps Estimatio of Populatio Mea Usig Co-Efficiet of Variatio ad Media of a Auxiliary Variable J. Subramai *, G. Kumarapadiya Departmet of Statistics, Ramauja School of Mathematical Scieces, Podicherry Uiversity, R V Nagar, Kalapet, , Puducherry Abstract The preset paper deals with a class of modified ratio estimators for estimatio of populatio mea of the study variable usig the liear combiatio of the kow values of the Co-efficiet of Variatio ad the Media of the auxiliary variable. The biases ad the mea squared errors of the proposed estimators are derived ad are compared with that of existig modified ratio estimators. Further we have also derived the coditios for which the proposed estimators perform better tha the existig modified ratio estimators. The performaces of the proposed estimators are also assessed with that of the existig estimators for certai atural populatios. From the umerical study it is observed that the proposed modified ratio estimators perform better tha the existig modified ratio estimators. Keywords Mea Squared Error, Modified Ratio s, Natural Populatios, Simple Radom Samplig 1. Itroductio The simplest estimator of populatio mea is the sample mea obtaied by usig simple radom samplig without replacemet, whe there is o additioal iformatio o the auxiliary variable available. Sometimes i sample surveys, alog with the study variable Y, iformatio o auxiliary variable X, correlated with Y, is also collected. Th is iformatio o auxiliary variable X, may be utilized to obtai a more efficiet estimator of the populatio mea. Ratio method of estimatio is a attempt i this directio. This method of estimatio may be used whe (i X represets the same character as Y, but measured at some previous date whe a complete cout of the populatio was made ad (ii the character X is cheaply, quickly ad easily available. Cosider a fiite populatio U = {U 1, U,, U N } of N distict ad idetifiable uits. Let Y is a study variable with value Y i measured o U i, i = 1,,3,, N givig a vector Y = {Y 1, Y,, Y N } ad let X is a auxiliary variable which is readily available. The problem is to estimate the populatio mea Y = 1 N Y N i=1 i with some desirable properties o the basis of a radom sample selected from the populatio U usig auxiliary iformatio. Whe the populatio parameters of the auxiliary variable X such as Populatio Mea, Co-efficiet of Variatio, Co-efficiet of Kurtosis, Co-efficiet of Skewess, Media are kow, a umber of estimators such as ratio, product ad liear regressio estimators ad their modificatios are * Correspodig author: drjsubramai@yahoo.co.i (J. Subramai Published olie at Copyright 01 Scietific & Academic Publishig. All Rights Reserved proposed i the literature. Before discussig further about the modified ratio estimators ad the proposed modified ratio estimators the otatios to be used i this paper are described below: N Populatio size Sample size f = /N, Samplig fractio Y Study variable X Auxiliary variable X, Y Populatio meas x, y Sample meas S X, S y Populatio stadard deviatios C X, C y Co-efficiet of variatios ρ Co-efficiet of correlatio β 1 = N N i =1 ( X i X3 3, Co-efficiet of skewess of the (N 1(N S auxiliary variab le β = N (N+1 N i=1 (X i X4 (N 1(N (N 3S 3 (N 1 4 (N (N 3, Co-efficiet of kurtosis of the auxiliary variable M d Media of the auxiliary variable B(. Bias of the estimator MSE(. Mea squared error of the estimator i ( pi Existig (proposed modified ratio estimator of Y The Ratio estimator for estimatig the populatio mea Y of the study variable Y is defied as R = y X = R X x where R = y = y is the estimate of R = Y = Y (1 x x X X List of modified ratio estimators together with their biases, mea squared errors ad costats available i the literature are classified ito two classes amely Class 1, Class ad
2 11 J. Subramai et al.: Estimatio of Populatio Mea Usig Co-Efficiet of Variatio ad Media of a Auxiliary Variable are give respectively i Table 1 ad Table respectively. It is to be oted that the existig modified ratio estimators meas the list of modified ratio estimators to be cosidered i this paper uless otherwise stated. It does ot mea to the etire list of modified ratio estimators available i the literature. For a more detailed discussio o the ratio estimator ad its modificatios oe may refer to Cochra[1], Kadilar ad Cigi[,3], Koyucu ad Kadilar[4], Murthy[5], Prasad[6], Rao[7], Sigh ad Tailor[9, 11], Sigh et.al[10], Sisodia ad Dwivedi[1], Subramai ad Kumarapadiya [13,14,15,16,17], Upadhyaya ad Sigh[18], Ya ad Tia[19] ad the refereces cited there i. The modified ratio estimators give i Table 1 ad Table are biased but have miimum mea squared errors compared to the classical ratio estimator. The list of estimators give i Table 1 ad Table uses the kow values of the parameters like X, C x, β 1, β, ρ, M d ad their liear combiatios. However, it seems, o attempt is made to use the liear combiatio of kow values of the Co-efficiet of variatio ad Media of the auxiliary variable to improve the ratio estimator. The poits discussed above have motivated us to itroduce modified ratio estimators usig the liear combiatio of the kow values of Co-efficiet of variatio ad Media of the auxiliary variable. It is observed that the proposed estimators perform better tha the existig modified ratio estimators listed i Table 1 ad Table. Table 1. Existig modified ratio estimators (Class 1 with their biases, mea squared errors ad their costats Bias - BB(. Mea squared error MMMMMM(. Costat θθ ii 1 = y X + C x x + C x Sisodia ad Dwivedi[1] Y (θ 1 C x θ 1 C x C y ρ Y (C y + θ 1 C x θ 1 C x C y ρ θ 1 = X X + C x = y X + β x + β Sigh et.al[10] Y (θ C x θ C x C y ρ Y (C y + θ C x θ C x C y ρ θ = X X + β 3 = y X + β 1 x + β 1 Ya ad Tia[19] Y (θ 3 C x θ 3 C x C y ρ Y (C y + θ 3 C x θ 3 C x C y ρ θ 3 = X X + β 1 4 = y X + ρ x + ρ Sigh ad Tailor[9] 5 = y X C x + β x C x + β Upadhyaya ad Sigh[18] 6 = y X β + C x x β + C x Upadhyaya ad Sigh[18] Y (θ 4 C x θ 4 C x C y ρ Y (C y + θ 4 C x θ 4 C x C y ρ θ 4 = X X + ρ Y (θ 5 C x θ 5 C x C y ρ Y (C y + θ 5 C x θ 5 C x C y ρ θ 5 = X C x X C x + β Y (θ 6 C x θ 6 C x C y ρ Y (C y + θ 6 C x θ 6 C x C y ρ θ 6 = X β X β + C x 7 = y X β 1 + β x β 1 + β Ya ad Tia[19] Y (θ 7 C x θ 7 C x C y ρ Y (C y + θ 7 C x θ 7 C x C y ρ θ 7 = X β 1 X β 1 + β 8 = y X C x + β 1 x C x + β 1 Ya ad Tia[19] Y (θ 8 C x θ 8 C x C y ρ Y (C y + θ 8 C x θ 8 C x C y ρ θ 8 = X C x X C x + β 1 9 = y X + M d x + M d Subramai ad Kumarapadiya[15] Y (θ 9 C x θ 9 C x C y ρ Y (C y + θ 9 C x θ 9 C x C y ρ θ 9 = X X + M d
3 Iteratioal Joural of Probability ad Statistics 01, 1(4: Ta ble. Existig modified ratio estimators (Class with their biases, mea squared errors ad their costats Bias-BB(. Mea squared error MMMMMM(. Costat RR ii 1 = y + b(x x X x Kadilar ad Cigi[] Y R S y (1 ρ R 1 = Y X = y + b(x x (X + C (x + C x x Kadilar ad Cigi[] 3 = y + b(x x (X + β (x + β Kadilar ad Cigi[] 4 = y + b(x x (x β + C x (X β + C x Kadilar ad Cigi[] 5 = y + b(x x (x C x + β (X C x + β Kadilar ad Cigi[] 6 = y + b(x x (X + β (x + β 1 1 Ya ad Tia[19] 7 = y + b(x x (X + ρ (x + ρ Kadilar ad Cigi[3] 8 = y + b(x x (xc x + ρ (X C x + ρ Kadilar ad Cigi[3] 9 = y + b(x x (xρ+ C x (X ρ + C x Kadilar ad Cigi[3] Y R + S y (1 ρ R = Y X + C x Y R S y (1 ρ R 3 = Y X + β Y R S y (1 ρ R 4 = Y β Xβ + C x Y R S y (1 ρ R 5 = Y C x XC x + β Y R S y (1 ρ R 6 = Y X + β 1 Y R S y (1 ρ R 7 = Y X + ρ Y R S y (1 ρ R 8 = Y C x XC x + ρ Y R S y (1 ρ R 9 = Y ρ Xρ + C x 10 = y + b(x x (xβ + ρ (X β + ρ Kadilar ad Cigi[3] Y R S y (1 ρ R 10 = Y β Xβ + ρ 11 = y + b(x x (xρ+ β (X ρ + β Kadilar ad Cigi[3] Y R S y (1 ρ R 11 = Y ρ Xρ + β. Proposed Modified Ratio s I this sectio, we have suggested a class of modified ratio estimators usig the liear combiatio of Co-efficiet of variatio ad Media of the auxiliary variable. The proposed modified ratio estimators for estimatig the populatio mea Y together with the first degree of approximatio, the biases ad mea squared errors ad the costats are give below:
4 114 J. Subramai et al.: Estimatio of Populatio Mea Usig Co-Efficiet of Variatio ad Media of a Auxiliary Variable p1 = y X C x + M d x C x + M d Ta ble 3. Proposed modified ratio estimators with their biases, mea squared errors ad their costats Bias BB(. Mea squared errors MMMMMM(. Y (θ p1 C x θ p1 C x C y ρ Costats θθ ii oooo RR ii Y (C y + θ p1 C x θ p1 C x C y ρ θ p1 = X C x X C x + M d p p3 = y + b(x x (X + M (x + M d d = y + b(x x (x C x + M d (X C x + M d Y R p Y R p3 p S x + S y (1 ρ R p = Y X + M d p3 S x + S y (1 ρ R p3 = Y C x X C x + M d 3. Efficiecy Compariso For wat of space; for the sake of coveiece to the readers ad for the ease of comparisos, the modified ratio estimators give i Table 1, Table are represeted ito two classes as give below: Class 1: The biases, the mea squared errors ad the costats of the modified ratio type estimators 1 to 9 listed i the Table 1 are represeted i a sigle class (say, Class 1, which will be very much useful for comparig with that of proposed modified ratio estimators ad are give below: B i = (1 f Y (θ i C x θ i C x C y ρ MSE i = ( 1 f Y C y + θ i C x θ i C x C y ρ; where θ 1 = θ 5 = X C x X X C x i = 1,, 3,..., 9 ( X X+C x, θ = X β, θ C x +β 6 =, θ β +C 7 = x X X, θ X+β 3 =, θ X+β 4 = 1 X X β 1, X β 1 +β X X+ρ, θ 8 = ad θ X C x +β 9 = X 1 X+M d Class : The biases, the mea squared errors ad the costats of the 11 modified ratio estimators 1 to 11 listed i the Table are represeted i a sigle class (say, Class, which will be very much useful for comparig with that of proposed modified ratio estimators ad are give below: where R 1 = Y B j = (1 f R Y j MSE j = ( 1 f j S x + S y (1 ρ ; j = 1,, 3,..., 11 (3 Y, R X =, R X+C 3 = x Y C x Y, X+β Y Y, X+ρ R 4 = Y β, R Xβ +C 5 =, R x XC x +β 6 =, R X+β 7 = 1 R 8 = Y C x, R XC x +ρ 9 = Y ρ, R Xρ+C 10 = Y β ad R x Xβ +ρ 11 = Y ρ Xρ +β As derived earlier i sectio, the biases, the mea squared errors ad the costats of the proposed modified ratio estimators are give below: B p 1 = (1 f Y (θ p 1 C x θ p 1 C x C y ρ MSE p 1 = ( 1 f Y C y + θ p 1 C x θ p 1 C x C y ρ X C x where θ p 1 = (4 XC x +M d B p = (1 f MSE p = ( 1 f B p 3 = (1 f MSE p 3 = ( 1 f S x R Y p p S x R Y p 3 p 3 + S y (1 ρ Y where R p = (5 X+M d + S y (1 ρ where R p 3 = Y C x (6 XC x +M d From the expressios give i ( ad (4 we have derived the coditios for which the proposed estimator p 1 is more efficiet tha the existig modified ratio estimators give i Class 1, i ; i = 1,, 3,..., 9 ad are give below: MSE p 1 < MSE i if ρ < θ p1+θ i C x ; (7 C y i = 1,, 3,..., 9 From the expressios give i (5, (6 ad (4 we have derived the coditios for which the proposed estimator pi ; i =,3 is more efficiet tha the existig modified ratio estimators give i Class, j ; j = 1,, 3,..., 11 ad are give below: i =,3 ; j = 1,, 3,..., 11 MSE pi < MSE j if R pi < R j ; (8 4. Numerical Study The performaces of the proposed modified ratio estimators listed i Table 3 are assessed with that of existig modified ratio estimators listed i Table 1 ad Table for certai atural populatios. I this coectio, we have cosidered three atural populatios for the assessmet of the performaces of the proposed modified ratio estimators with that of existig modified ratio estimators. The populatio 1 ad populatio are take from Sigh ad Chaudhary[8] give i page 177 ad populatio 3 is take fro m Murthy[5] give i page 8. The populatio parameters ad the costats computed from the above populatios are give below: The costats, biases ad mea squared errors of the existig ad proposed modified ratio estimators for the above populatios are give from Table 5 to Table 10:
5 Iteratioal Joural of Probability ad Statistics 01, 1(4: Ta ble 4. Parameters ad Costats of the Populatios Paramete rs Populatio 1 Populatio Populatio 3 N Y X ρ S y C y C x β β M d Ta ble 5. The costats of the (Class 1 existig ad proposed modified ratio estimators Costats θθ ii Populatio 1 Populatio Populatio 3 1 Sisodia ad Dwivedi[1] Sigh et.al[10] Ya ad Tia[19] Sigh ad Tailor[9] Upadhyaya ad Sigh[18] Upadhyaya ad Sigh[18] Ya ad Tia[19] Ya ad Tia[19] Subramai ad Kumarapadiya[15] p1 (Proposed estimator* * 0.513* * Ta ble 6. The costats of the (Class existig ad proposed modified ratio estimators Costats RR jj Populatio 1 Populatio Populatio 3 1 Kadilar ad Cigi[] Kadilar ad Cigi[] Kadilar ad Cigi[] Kadilar ad Cigi[] Kadilar ad Cigi[] Ya ad Tia[19] Kadilar ad Cigi[3] Kadilar ad Cigi[3] Kadilar ad Cigi[3] Kadilar ad Cigi[3] Kadilar ad Cigi[3] p (Proposed estimator*.3863*.5046* * p3 (Proposed estimator*.0534*.036* *
6 116 J. Subramai et al.: Estimatio of Populatio Mea Usig Co-Efficiet of Variatio ad Media of a Auxiliary Variable Ta ble 7. The biases of the (Class 1 existig ad proposed modified ratio estimators Bias BB(. Populatio 1 Populatio Populatio 3 1 Sisodia ad Dwivedi[1] Sigh et.al[10] Ya ad Tia[19] Sigh ad Tailor[9] Upadhyaya ad Sigh[18] Upadhyaya ad Sigh[18] Ya ad Tia[19] Ya ad Tia[19] Subramai ad Kumarapadiya[15] p1 (Proposed estimator* 0.150* * * Ta ble 8. The biases of the (Class existig ad proposed modified ratio estimators Bias BB(. Populatio 1 Populatio Populatio 3 1 Kadilar ad Cigi[] Kadilar ad Cigi[] Kadilar ad Cigi[] Kadilar ad Cigi[] Kadilar ad Cigi[] Ya ad Tia[19] Kadilar ad Cigi[3] Kadilar ad Cigi[3] Kadilar ad Cigi[3] Kadilar ad Cigi[3] Kadilar ad Cigi[3] p (Proposed estimator* 5.34* * * p3 (Proposed estimator* * * * Ta ble 9. The mea squared errors of the (Class 1 existig ad proposed modified ratio estimators Mea Squared Error MMMMMM(. Populatio 1 Populatio Populatio 3 1 Sisodia ad Dwivedi[1] Sigh et.al[10] Ya ad Tia[19] Sigh ad Tailor[9] Upadhyaya ad Sigh[18] Upadhyaya ad Sigh[18] Ya ad Tia[19] Ya ad Tia[19] Subramai ad Kumarapadiya[15] p1 (Proposed estimator* * * *
7 Iteratioal Joural of Probability ad Statistics 01, 1(4: Ta ble 10. The mea squared errors of the (Class existig ad proposed modified ratio estimators Mea Squared Error MMMMMM(. Populatio 1 Populatio Populatio 3 1 Kadilar ad Cigi[] Kadilar ad Cigi[] Kadilar ad Cigi[] Kadilar ad Cigi[] Kadilar ad Cigi[] Ya ad Tia[19] Kadilar ad Cigi[3] Kadilar ad Cigi[3] Kadilar ad Cigi[3] Kadilar ad Cigi[3] Kadilar ad Cigi[3] p (Proposed estimator* * * * p3 (Proposed estimator* * * * From the values of Table 7 ad Table 8, it is observed that the bias of the proposed modified ratio estimator p 1 is less tha the biases of the (Class 1 existig modified ratio estimators i ; i = 1,, 3,..., 9 ad the biases of the proposed modified ratio estimators pi ; i =,3 are less tha the biases of the (Class existig modified ratio estimators j ; j = 1,, 3,..., 11. Similarly from the values of Table 9 ad Table 10, it is observed that the mea squared error of the proposed modified ratio estimator p 1 is less tha the mea squared errors of the (Class 1 existig modified ratio estimators i ; i = 1,, 3,..., 9 ad the mea squared errors of the proposed modified ratio estimators pi ; i =,3 are less tha the mea squared errors of the (Class existig modified ratio estimators j ; j = 1,, 3,..., Coclusios The existig modified ratio estimators with the kow values of the parameters like X, C x, β 1, β, ρ ad M d are biased but have miimum mea squared errors compared to the classical ratio estimator. I this paper, we use the liear combiatio of kow values of the Co-efficiet of variatio ad Media of the auxiliary variable to improve the ratio estimator. The biases ad mea squared errors of the proposed estimators are obtaied. Further we have derived the coditios for which the proposed estimators are more efficiet tha the existig modified ratio estimators. We have also assessed the performaces of the proposed estimators for some kow populatios give i the text books [5] ad [8]. It is observed that the biases ad mea squared errors of the proposed estimators are less tha the biases ad mea squared errors of the existig modified ratio estimators. Hece we strogly recommed that the proposed modified estimators may be preferred over the existig modified ratio estimators for the use i practical applicatios. ACKNOWLEDGEMENTS The secod author wishes to record his gratitude ad sicere thaks to the Vice Chacellor, Podicherry Uiversity ad other Uiversity authorities for havig give the fiacial assistace to carry out this research work through the Uiversity Fellowship. REFERENCES [1] Cochra, W. G. (1977. Samplig techiques, Third Editio, Wiley Easter Limited] [] Kadilar, C. ad Cigi, H. (004: Ratio estimators i simple radom samplig, Applied Mathematics ad Computatio 151, [3] Kadilar, C. ad Cigi, H. (006: A improvemet i estimatig the populatio mea by usig the correlatio co-efficiet, Hacettepe Joural of Mathematics ad Statistics Volume 35 (1, [4] Koyucu, N. ad Kadilar, C. (009: Efficiet s for the Populatio mea, Hacettepe Joural of Mathematics ad Statistics, Volume 38(, 17-5 [5] Murthy, M.N. (1967: Samplig theory ad methods, Statistical Publishig Society, Calcutta, Idia [6] Prasad, B. (1989: Some improved ratio type estimators of populatio mea ad ratio i fiite populatio sample surveys, Commuicatios i Statistics: Theory ad Methods 18, ,
8 118 J. Subramai et al.: Estimatio of Populatio Mea Usig Co-Efficiet of Variatio ad Media of a Auxiliary Variable [7] Rao, T.J. (1991: O certai methods of improvig ratio ad regressio estimators, Commuicatios i Statistics: Theory ad Methods 0 (10, [8] Sigh, D. ad Chaudhary, F.S. (1986: Theory ad Aalysis of Sample Survey Desigs, New Age Iteratioal Publisher [9] Sigh, H.P. ad Tailor, R. (003: Use of kow correlatio co-efficiet i estimatig the fiite populatio meas, Statistics i Trasitio 6 (4, [10] Sigh, H.P., Tailor, R., Tailor, R. ad Kakra, M.S. (004: A improved estimator of populatio mea usig power trasformatio, Joural of the Idia Society of Agricultural Statistics 58(, 3-30 [11] Sigh, H.P. ad Tailor, R. (005: Estimatio of fiite populatio mea with kow co- efficiet of variatio of a auxiliary, STATISTICA, ao LXV,.3, pp [1] Sisodia, B.V.S. ad Dwivedi, V.K. (1981: A modified ratio estimator usig co-efficiet of variatio of auxiliary variable, Joural of the Idia Society of Agricultural Statistics 33(1, [13] Subramai, J. ad Kumarapadiya, G. (01: A class of almost ubiased modified ratio estimators for populatio mea with kow populatio parameters, Elixir Statistics 44, [14] Subramai, J. ad Kumarapadiya, G. (01: Almost ubiased modified liear regressio estimators for estimatio of populatio mea, Bofrig Iteratioal Joural of Idustrial Egieerig ad Maagemet Sciece, Vol. (, 4-7 [15] Subramai, J. ad Kumarapadiya, G. (01: Modified ratio estimator for populatio mea usig media of the auxiliary variable, Submitted for Publicatio. [16] Subramai, J. ad Kumarapadiya, G. (01: Modified ratio estimators for populatio mea usig fuctio of quartiles of auxiliary variable, Bofrig Iteratioal Joural of Idustrial Egieerig ad Maagemet Sciece, Vol. (, 19-3 [17] Subramai, J. ad Kumarapadiya, G. (01: Modified ratio estimators usig kow media ad co-efficiet of kurtosis, America Joural of Mathematics ad Statistics, Vol. (4, [18] Upadhyaya, L.N. ad Sigh, H.P. (1999: Use of trasformed auxiliary variable i estimatig the fiite populatio mea, Biometrical Joural 41 (5, [19] Ya, Z. ad Tia, B. (010: Ratio method to the mea estimatio usig co-efficiet of skewess of auxiliary variable, ICICA, Part II, CCIS 106, pp
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