America Joural of Mathematics ad Statistics 01, (4): 95-100 DOI: 10.593/j.ajms.01004.05 Modified Ratio s Usig Kow Media ad Co-Efficet of Kurtosis J.Subramai *, G.Kumarapadiya Departmet of Statistics, Podicherry Uiversity, R V Nagar, Kalapet, Puducherry,605014, Idia drjsubramai@yahoo.co.i, kumarstat88@gmail.com Abstract The preset paper deals with two modified ratio estimators for estimatio of populatio mea of the study variable usig the liear combiatio of the kow populatio values of the Media ad the Co-efficiet of Kurtosis of the auiliary variable. The biases ad the mea squared errors of the proposed estimators are derived ad are compared with that of eistig modified ratio estimators for certai atural populatios. Further we have also derived the coditios for which the proposed estimators perform better tha the eistig modified ratio estimators. From the umerical study it is also observed that the proposed modified ratio estimators perform better tha the eistig modified ratio estimators. Keywords Auiliary Variable, Bias, Modified Ratio s, Simple Radom Samplig 1. Itroductio 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 }. 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. 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 auiliary variable available. Sometimes i sample surveys, alog with the study variable Y, iformatio o auiliary variable X, correlated with Y is also collected. This iformatio o auiliary variable X, may be utilized to obtai a more efficiet estimator of the populatio mea. Ratio method of estimatio, usig auiliary iformatio is a attempt made i this directio. Before discussig further about the modified ratio estimators ad the proposed modified ratio estimators the otatios to be used are described. N Populatio size Sample size f = /N, Samplig fractio Y Study variable X Auiliary variable X, Y Populatio meas * Correspodig author: drjsubramai@yahoo.co.i (J. Subramai) Published olie at http://joural.sapub.org/ajms Copyright 01 Scietific & Academic Publishig. All Rights Reserved, 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 X) 3 (N 1)(N )S 3, Co-efficiet of skewess of the auiliary variable β = N(N+1) N i=1 (X i X) 4 3(N 1), Co-efficiet of (N 1)(N )(N 3)S 4 (N )(N 3) kurtosis of the auiliary variable M d Media of the auiliary variable B(. ) Bias of the estimator MSE(. ) Mea squared error of the estimator i ( pi ) Eistig (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 (1) where R = y = y is the estimate of R = Y X = Y X The Ratio estimator give i (1) is more precise tha the simple radom sample mea, whe there eists a positive correlatio betwee X ad Y. Se [8] has preseted the historical developmets of the ratio method of estimatio startig from the year 166. Hece the readers, who are iterested i kowig more details o the chroological developmets of the ratio methods of estimatio, are referred to Se [8] ad the refereces cited therei. Further improvemets are also achieved o the classical ratio estimator by itroducig a large umber of modified ratio estimators with the use of kow parameters like, Co-efficiet of Variatio, Co-efficiet of Kurtosis, Co-efficiet of Skewess, Populatio Correlatio Coefficiet ad Media. It is to be oted that the eistig modified ratio estimators
J.Subramai et al.: Modified Ratio s Usig Kow Media ad Co-Efficet of Kurtosis 96 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[10], Sigh ad Tailor[11,1], Sigh et.al[13], Sisodia ad Dwivedi[14], Srivastava [15,16], Subramai ad Kumarapadiya[17,18], Upadhyaya ad Sigh[19], Walsh [0], Ya ad Tia[1] ad the refereces cited there i. The lists 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 are give respectively i Table 1 ad Table respectively. 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, 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 Media ad Co-efficiet of Kurtosis of the auiliary variable to improve the ratio estimator. The poits discussed above have motivated us to itroduce two modified ratio estimators usig the liear combiatio of the kow values of Media ad Co-efficiet of Kurtosis of the auiliary variable. It is observed that the proposed estimators perform better tha the eistig modified ratio estimators listed i Table 1 ad Table.. Proposed Modified Ratio s I this sectio, we have suggested two modified ratio estimators usig the liear combiatio of Media ad Co-efficiet of Kurtosis of the auiliary variable. The proposed modified ratio estimators for estimatig the populatio mea Y together with the first degree of approimatio, the biases ad mea squared errors ad the costats are give below: p1 = y X β + M d β + M d B p1 = MSE p1 = Y (θ p1 C θ p1 C C y ρ) Y C y + θ p1 C θ p1 C C y ρ where θ p1 = X β () Xβ +M d p = y + b(x ) ( β + M d ) (X β + M d ) S B p = Y R p MSE p = where R p = 3. Efficiecy Compariso R p S + S y (1 ρ ) Yβ Xβ +M d (3) 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. Further it is to be oted that the proposed estimator p1 is compared with that of the modified ratio estimators listed i Class 1 whereas the proposed estimator p is compared with that of the modified ratio estimators listed i Class. Table 1. Eistig 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 + β C + β Upadhyaya ad Sigh[19] = y X β + C β + C Upadhyaya ad Sigh[19] 3 = y X β + β 1 β + β 1 Ya ad Tia[1] 4 = y X β 1 + β β 1 + β Ya ad Tia[1] Y (θ 1 C θ 1 C C y ρ) Y (θ C θ C C y ρ) Y (θ 3 C θ 3 C C y ρ) Y (θ 4 C θ 4 C C y ρ) Y (C y + θ 1 C θ 1 C C y ρ) θ 1 = X C X C + β Y (C y + θ C θ C C y ρ) θ = X β X β + C Y (C y + θ 3 C θ 3 C C y ρ) θ 3 = X β X β + β 1 Y (C y + θ 4 C θ 4 C C y ρ) θ 4 = X β 1 X β 1 + β
97 America Joural of Mathematics ad Statistics 01, (4): 95-100 Table. Eistig modified ratio estimators (Class ) with their biases, mea squared errors ad their costats Bias-BB(. ) Mea squared error MMMMMM(. ) Costat RR ii 5 = y+b(x ) (X β ( β +C ) + C ) Kadilar ad Cigi[] 6 = y+b(x ) (X C ( C +β ) + β ) Kadilar ad Cigi[] 7 = y+b(x ) (X β ( β 1 +β ) 1 + β ) Ya ad Tia[1] 8 = y+b(x ) (β +ρ) (X β + ρ) Kadilar ad Cigi[3] 9 = y+b(x ) (ρ+β ) (X ρ + β ) Kadilar ad Cigi[3] Y R 5 Y R 6 Y R 7 Y R 8 Y R 9 R 5 S + S y (1 ρ ) R 5 = Y β Xβ +C R 6 S + S y (1 ρ ) R 6 = Y C XC +β R 7 S + S y (1 ρ ) R 7 = Y β 1 Xβ 1 +β R 8 S + S y (1 ρ ) R 8 = Y β Xβ +ρ R 9 S + S y (1 ρ ) R 9 = Y ρ Xρ+β Class 1:The biases, the mea squared errors ad the costats of the modified ratio estimators 1 to 4 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 = Y (θ i C θ i C C y ρ) MSE i = Y C y + θ i C θ i C C y ρ; i = 1,,3,4 X C where θ 1 =, θ C +β =, θ β +C 3 = ad θ β +β 4 = 1 X X X β 1 X β X X β (4) X β 1 +β Class :The biases, the mea squared errors ad the costats of the remaiig 5 modified ratio estimators 5 to 9 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: B i = Y R i MSE i = R i S + S y (1 ρ ) ; i = 5,6,7,8,9 where R 5 = Y β, R Xβ +C 6 = Y C, R XC +β 7 = Y β 1, R Xβ 1 +β 8 = Yβ ad R Xβ +ρ 9 = Y ρ (5) Xρ+β As derived earlier i sectio, the biases, the mea squared errors ad the costats of two proposed modified ratio estimators are give below: B p1 = MSE p1 = MSE p = S Y (θ p1 C θ p1 C C y ρ) Y C y + θ p1 C θ p1 C C y ρ where θ p1 = X β Xβ +M d (6) B p = Y R p S + S y (1 ρ ) where R p = R p S Yβ Xβ +M d (7) From the epressios give i (4) ad (6) we have derived the coditios for which the proposed estimator p1 is more efficiet tha the eistig modified ratio estimators give i Class 1, i ; i = 1,, 3, 4 ad are give below: MSE p1 < MSE i if ρ < θ p 1+θ i C ; i = 1,, 3, 4 (8) C y From the epressios give i (5) ad (7) we have derived the coditios for which the proposed estimator p is more efficiet tha the eistig modified ratio estimators give i Class, i ; i = 5, 6, 7, 8 ad 9 ad are give below: MSE p < MSE i if R p < R i ; i = 5, 6, 7, 8 ad 9 (9) 4. Numerical Study The performaces of the proposed modified ratio estimators are assessed with that of eistig modified ratio estimators listed i Table 1 ad Table for certai atural populatios. I this coectio, we have cosidered four atural populatios for the assessmet of the performaces of the proposed modified ratio estimators with that of eistig modified ratio estimators. The populatio 1 ad are take from Sigh ad Chaudhary[9] give i page 177, populatio 3 is take from Sigh ad Chaudhary[9] give i page 141 ad populatio 4 is take from Cochra[1] give i page 15. The populatio parameters ad the costats computed from the above populatios are give below: The costats of the eistig ad proposed modified ratio estimators for the above populatios are give i the Table 4 ad Table 5: The biases of the eistig ad proposed modified ratio estimators for the above populatios are give i the Table 6 ad Table 7: The mea squared errors of the eistig ad proposed modified ratio estimators for the above populatios are give i the Table 8 ad Table 9: From the values of Table 6 ad Table 7, it is observed that
J.Subramai et al.: Modified Ratio s Usig Kow Media ad Co-Efficet of Kurtosis 98 the bias of the proposed modified ratio estimator p1 is less tha the biases of the eistig modified ratio estimators i ; i = 1,, 3, 4 give i Class 1 ad the bias of the proposed modified ratio estimator p is less tha the biases of the eistig modified ratio estimators i ; i = 5, 6, 7, 8 ad 9 give i Class. Similarly from the values of Table 8 ad Table 9, it is observed that the mea squared error of the Table 3. Parameters ad Costats of the Populatios proposed modified ratio estimator p1 is less tha the mea squared errors of the eistig modified ratio estimators i ; i = 1,, 3, 4 give i Class 1 ad the mea squared error of the proposed modified ratio estimator p is less tha the mea squared errors of the eistig modified ratio estimators i ; i = 5, 6, 7, 8 ad 9 give i Class. Parameters Populatio 1 Populatio Populatio 3 Populatio 4 N 34 34 49 0 0 5 0 Y 856.4117 856.4117.609 116.1633 X 08.883 199.441 1467.5455 98.6735 ρ 0.4491 0.4453 0.90 0.6904 S y 733.1407 733.1407 33.0469 98.886 C y 0.8561 0.8561 1.4609 0.8508 S 150.5059 150.150 56.1449 10.9709 C 0.705 0.753 1.7459 1.0436 β 0.0978 1.0445 13.3694 5.9878 β 1 0.978 1.183 3.3914.44 M d 150.0000 14.5000 534.5000 64.0000 Table 4. The costats of the (Class 1) eistig ad proposed modified ratio estimators Costats θθ ii Populatio 1 Populatio Populatio 3 Populatio 4 1 Upadhyaya ad Sigh[18] 0.9994 0.9931 0.9948 0.9450 Upadhyaya ad Sigh[18] 0.9658 0.9964 0.9999 0.998 3 Ya ad Tia[1] 0.954 0.9944 0.9998 0.9959 4 Ya ad Tia[1] 0.9995 0.9956 0.9973 0.9756 p1 (Proposed estimator)* 0.1195* 0.5938* 0.9735* 0.903* Table 5. The costats of the (Class ) eistig ad proposed modified ratio estimators Costats RR ii Populatio 1 Populatio Populatio 3 Populatio 4 5 Kadilar ad Cigi[] 3.9598 4.786 0.0154 1.175 6 Kadilar ad Cigi[] 4.0973 4.644 0.0153 1.116 7 Ya ad Tia[1] 4.0980 4.751 0.0154 1.1485 8 Kadilar ad Cigi[3] 4.0115 4.849 0.0154 1.1759 9 Kadilar ad Cigi[3] 4.0957 4.441 0.0153 1.081 p (Proposed estimator)* 0.4899*.5499* 0.0150* 1.06* Table 6. The biases of the (Class 1) eistig ad proposed modified ratio estimators Bias BB(. ) Populatio 1 Populatio Populatio 3 Populatio 4 1 Upadhyaya ad Sigh[19] 4.607 4.8369.543 1.3519 Upadhyaya ad Sigh[19] 3.81 4.8860.6106 1.668 3 Ya ad Tia[1] 3.673 4.8556.6095 1.6144 4 Ya ad Tia[1] 4.69 4.8739.5763 1.5070 p1 (Proposed estimator)* 0.459* 0.507*.674* 1.146* Table 7. The biases of the (Class ) eistig ad proposed modified ratio estimators Bias BB(. ) Populatio 1 Populatio Populatio 3 Populatio 4 5 Kadilar ad Cigi[] 8.5387 9.9303 10.6540 3.730 6 Kadilar ad Cigi[] 9.141 9.8646 10.5456 3.3433 7 Ya ad Tia[1] 9.145 9.9143 10.5989 3.567 8 Kadilar ad Cigi[3] 8.769 9.9597 10.6549 3.7347 9 Kadilar ad Cigi[3] 9.1349 9.7711 10.4439 3.1630 p (Proposed estimator)* 0.1307* 3.569* 10.098* 3.0475*
99 America Joural of Mathematics ad Statistics 01, (4): 95-100 Table 8. The mea squared errors of the (Class 1) eistig ad proposed modified ratio estimators Mea Squared Error MMMMMM(. ) Populatio 1 Populatio Populatio 3 Populatio 4 1 Upadhyaya ad Sigh[18] 10534.5417 1090.7384 45.894 14.7486 Upadhyaya ad Sigh[18] 1098.443 10930.3879 45.8857 33.6573 3 Ya ad Tia[1] 100.4736 10913.804 45.8758 3.7813 4 Ya ad Tia[1] 10535.7860 1093.6103 45.5814 5.956 p1 (Proposed estimator)* 10178.990* 8937.406* 4.931* 01.363* Table 9. The mea squared errors of the (Class ) eistig ad proposed modified ratio estimators Mea Squared Error MMMMMM(. ) Populatio 4 Populatio Populatio 3 Populatio 4 5 Kadilar ad Cigi[] 16146.614 17376.0389 7.4185 584.5606 6 Kadilar ad Cigi[] 16663.3064 17319.7468 69.9654 539.610 7 Ya ad Tia[1] 16665.9758 1736.58 71.1716 565.0981 8 Kadilar ad Cigi[3] 16338.6465 17401.1397 7.4393 585.0781 9 Kadilar ad Cigi[3] 16657.1867 1739.6579 67.6660 518.6688 p (Proposed estimator)* 8945.887* 1189.074* 59.8459* 505.469* 5. Coclusios I this paper we have proposed two modified ratio estimators usig liear combiatio of Media ad Co-efficiet of Kurtosis of the auiliary variable. The biases ad mea squared errors of the proposed estimators are obtaied ad compared with that of eistig modified ratio estimators. Further we have derived the coditios for which the proposed estimators are more efficiet tha the eistig modified ratio estimators. We have also assessed the performaces of the proposed estimators for some kow populatios. 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 eistig modified ratio estimators for certai kow populatios. Hece we strogly recommed that the proposed modified estimators may be preferred over the eistig modified ratio estimators for the use of practical applicatios. ACKNOWLEDGEMENTS The authors are thakful to the referees whose costructive commets led to improvemet i the paper. The secod author wishes to record his gratitude ad 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, 893-90 [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), 103-109 [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, 379 39 [7] Rao, T.J. (1991). O certai methods of improvig ratio ad regressio estimators, Commuicatios i Statistics: Theory ad Methods 0 (10), 335 3340 [8] Se, A.R. (1993): Some early developmets i ratio estimatio, Biometric Joural 35(1), 3-13 [9] Sigh, D. ad Chaudhary, F.S. (1986). Theory ad Aalysis of Sample Survey Desigs, New Age Iteratioal Publisher [10] Sigh, G.N. (003). O the improvemet of product method of estimatio i sample surveys, Joural of the Idia Society of Agricultural Statistics 56 (3), 67 65 [11] Sigh, H.P. ad Tailor, R. (003). Use of kow correlatio co-efficiet i estimatig the fiite populatio meas, Statistics i Trasitio 6 (4), 555-560 [1] Sigh, H.P. ad Tailor, R. (005). Estimatio of fiite populatio mea with kow co-efficiet of variatio of a auiliary, STATISTICA, ao LXV,.3, pp 301-313 [13] Sigh, H.P., Tailor, R., Tailor, R. ad Kakra, M.S. (004). A Improved of populatio mea usig Power trasformatio, Joural of the Idia Society of Agricultural Statistics 58(), 3-30 [14] Sisodia, B.V.S. ad Dwivedi, V.K. (1981). A modified ratio estimator usig co-efficiet of variatio of auiliary variable, Joural of the Idia Society of Agricultural Statistics 33(1), 13-18 [15] Srivastava, S.K. (1967). A estimator usig auiliary ifor-
J.Subramai et al.: Modified Ratio s Usig Kow Media ad Co-Efficet of Kurtosis 100 matio i sample surveys, Bulleti/Calcutta Statistical Associatio, vol. 16, pp.11-13 [16] Srivastava, S.K. (1980). A Class of estimators usig auiliary iformatio i sample surveys, The Caadia Joural of Statistics, vol. 8, o., pp.53-54 [17] Subramai, J. ad Kumarapadiya, G. (01). A Class of almost ubiased modified ratio estimators for populatio mea with kow populatio parameters, Eliir Statistics 44, 7411-7415 [18] Subramai, J. ad Kumarapadiya, G. (01). Modified Ratio for Populatio Mea Usig Media of the Auiliary Variable, Proceedigs of Natioal Coferece o Recet developmets i the Applicatios of Reliability Theory ad Survival Aalysis held o d ad 3rd February 01 at the Departmet of Statistics, Podicherry Uiversity [19] Upadhyaya, L.N. ad Sigh, H.P. (1999). Use of trasformed auiliary variable i estimatig the fiite populatio mea, Biometrical Joural 41 (5), 67-636 [0] Walsh, J.E. (1970). Geeralizatio of ratio estimator for populatio total, Sakhya, A, 3, 99-106 [1] Ya, Z. ad Tia, B. (010). Ratio Method to the Mea Estimatio Usig Co-efficiet of Skewess of Auiliary Variable, ICICA 010, Part II, CCIS 106, pp. 103 110