ISSN 1684-8403 Joural of Statistics Volume, 015. pp. 84-305 Abstract A Class of Modified Liear Regressio Type Ratio Estimators for Estimatio of Populatio Mea usig Coefficiet of Variatio ad Quartiles of a Auxiliary Variable Jambuligam Subramai 1, Gaasegara Kumarapadiya ad Samiatha Balamurali 3 The preset paper deals with a class of Modified Liear Regressio Type Ratio Estimators for estimatio of populatio mea of the study variable whe the populatio coefficiet of variatio ad quartiles of the auxiliary variable are kow. The Bias ad the Mea Squared Error of the proposed Estimators are derived ad are compared with that of Simple Radom Samplig With-Out Replacemet (SRSWOR) sample mea, the Ratio Estimator ad the existig Modified Ratio Estimators. As a result we have derived the coditios for which the proposed Estimators perform better tha the other existig Estimators. Further, the performace of the proposed Estimators with that of the existig Estimators are assessed for a atural populatio. From the umerical study, it is observed that the proposed Modified Ratio Estimators perform better tha the existig Estimators. Keywords First quartile, Iter-quartile rage, Simple radom samplig, Semi-quartile average, Semi-quartile rage 1 Departmet of Statistics, Ramauja School of Mathematical Scieces, Podicherry Uiversity, R V Nagar, Kalapet, Puducherry 605014, Idia. Email: drjsubramai@yahoo.co.i Idia Statistical Istitute (ISI), Cheai Cetre, MGR Kowledge City, CIT Campus, Cheai 600113, Idia. Email: kumarstat88@gmail.com 3 Departmet of Computer Applicatios, Kalasaligam Uiversity, Krisha Koil, Srivilliputhur via, Virudhuagar - 6616, Tamiladu, Idia. Email:sbmurali@rediffmail.com
A Class of Modified Liear Regressio Type Ratio Estimators for 85 Estimatio of Populatio Mea usig Coefficiet of Variatio ad Quartiles of a Auxiliary Variable 1. Itroductio Cosider a fiite populatio U = U 1, U,, U N of N distict ad idetifiable uits. Let, Y be a study variable with value Y i measured o U i, i = 1,, 3,, N givig a vectory = Y 1, Y,, Y N. The problem is to estimate the populatio mea Y = 1 N Y N i=1 i o the basis of a radom sample selected from the populatiou. The SRSWOR sample mea is the simplest Estimator for estimatig the populatio mea. If a auxiliary variablex, closely related to the study variable Y is available the oe ca improve the performace of the Estimator of the study variable by usig the kow values of the populatio parameters of the auxiliary variable. That is, whe the populatio parameters of the auxiliary variable X such as populatio mea, coefficiet of variatio, coefficiet of Kurtosis, coefficiet of Skewess are kow, a umber of Estimators such as Ratio, Product ad Liear Regressio Estimators ad their modificatios are available i literature ad are performig better tha the SRSWORsample mea uder certai coditios. Amog these Estimators the Ratio Estimator ad its modificatios are widely attracted may researchers for the estimatio of the mea of the study variable. Recetly, Al-Omari et al. (009), Subramai ad Kumarapadiya (01, 013) ad Ya ad Tia (010), have cosidered i modifyig the Ratio Estimator for estimatig populatio mea. I search of the best Estimators amog the classes of Estimators, Haif et al. (009), Subramai (013), Subramai et al. (014) ad Subramai ad Kumarapadiya (014) ad have suggested some Modified Ratio Estimators with kow parameters of the auxiliary variable. The mai objective of this study is to itroduce some Modified Liear Regressio Type Estimators which perform better tha the Modified Estimators already existig i the literature. Further, it has bee show that the proposed Estimators perform better tha existig Modified Ratio Estimators theoretically, aalytically ad umerically. The structure of rest of this paper is as follows: I the ext sectio the proposed Estimators are give i the two group format. The aalytical results i this sectio demostrate that the proposed Estimators are better tha the Estimators suggested by various authors (refer Table 1 ad Table ). Sectio 3 shows the efficiecy of the proposed Estimators while i Sectio 4 a empirical compariso of the proposed ad various exitig Estimators has bee preseted.
86 Jambuligam Subramai, Gaasegara Kumarapadiya ad Samiatha Balamurali Notatios to be used i this paper has bee give i Appedix A. Further, SRSWOR sample mea, the Ratio ad the Liear Regressio Estimator for estimatig populatio mea have bee discussed i Appedix B. The Ratio Estimator ad the Liear Regressio Estimator are used for improvig the precisio of estimate of the populatio mea compared to SRSWOR whe there is a positive correlatio betwee X ady. It has bee established; i geeral that the Liear Regressio Estimator is more efficiet tha the Ratio Estimator wheever the Regressio lie of the study variable o the auxiliary variable does ot passes through the eighborhood of the origi. Thus, the Liear Regressio Estimator is more precise tha the Ratio Estimator uless β = R (See page 196 i Cochra (1977)). The Ratio Estimator ad Modified Ratio Estimators ca be applicable i the followig practical situatio. A atioal park is partitioed ito N uits. Y = the umber of aimals i the i th uit X = the size of the i th uit A certai city has N bookstores. Y = the sales of a give book title at the i th bookstore X = the size of the i th bookstore A forest that has N trees. Y = the volume of the tree X = the diameter of the tree Kadilar ad Cigi (004) have suggested a class of Modified Ratio Estimators i the form of Liear Regressio Estimator as give below: Y 1 = y + b X x X x (1.1) Kadilar adcigi (004) have show that the Modified Liear Regressio Type Ratio Estimator give i equatio (1.1) perform well compare to the Ratio Estimator uder certai coditios. Further Estimators are proposed by modifyig the Estimator give i equatio (1.1) by usig the kow coefficiet of variatio, coefficiet of Kurtosis, coefficiet of Skewess etc. together with some other Modified Ratio Estimators i Kadilar ad Cigi (004, 006) ad Ya ad Tia (010). The list of existig Modified Liear Regressio Type Ratio Estimators together with the costat, the Bias ad the Mea Squared Error are give i Table 1.
A Class of Modified Liear Regressio Type Ratio Estimators for Estimatio of Populatio Mea usig Coefficiet of Variatio ad Quartiles of a Auxiliary Variable Al-Omari et al. (009) have suggested Modified Ratio Estimators usig the first quartile Q 1 ad the third quartile Q 3 of the auxiliary variable. Subramai ad Kumarapadiya (01) have suggested a class of Modified Ratio Estimators usig the iter-quartile rage Q r, semi-quartile rage Q d ad semi-quartile average Q a of the auxiliary variable. The class of Modified Ratio Estimators together with the costat, the Bias ad the Mea Squared Error are give i Table. 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. The list of Modified Liear Regressio Type Ratio Estimators give i Table 1 uses the kow values of the parameters like X, C x, β 1, β, ρad their Liear combiatios whereas the list give i Table uses quartiles ad their Liear combiatios of a auxiliary variable like iter-quartile rage, semi-quartile rage ad semi-quartile average to improve the Ratio Estimator i estimatio of populatio mea. I this paper, a attempt is made to use the Liear combiatio of the kow values of the coefficiet of variatio ad quartiles of the auxiliary variable to itroduce a class of Modified Liear Regressio Type Ratio Estimators for estimatig populatio mea i lie with Kadilar ad Cigi (004).. Proposed Modified Liear Regressio Type Ratio Estimators I this sectio, a class of Modified Liear Regressio Type Ratio Estimators is proposed for estimatig the fiite populatio mea ad also derived the Bias ad the Mea Squared Error of the proposed Estimators. The proposed Estimators are classified ito two groups:group 1 ad Group..1 Proposed Estimators (Group 1): The proposed Estimators belog to Group 1 is represeted asy pj ; j = 1,,3,4 ad 5 ad it is defied as give below: 87 X + Q. Y pj = y + b X x x + Q. where Q. = Q 1, Q 3, Q r, Q d ad Q a ; j = 1,,3,4 ad 5
88 Jambuligam Subramai, Gaasegara Kumarapadiya ad Samiatha Balamurali. Proposed Estimators (Group ): The proposed Estimators belog to Group is represeted as Y pj ad is defied as give below: C x X + Q (.) Y pj = y + b X x ; j = 6,7,8,9 ad 10 C x x + Q (.) where Q. = Q 1, Q 3, Q r, Q d ad Q a The Bias ad Mea Squared Error of the proposed Estimators have bee derived (see Appedix C). The proposed Estimators (Group 1 ad Group ) are combied ito a sigle Class (say Class 3) ad it is defied as Y pj ; j = 1,,,10 for estimatig the populatio mea Y together with the costat, the Bias ad the Mea Squared Error are preseted i Table 3. 3. Efficiecy of the proposed Estimators The Variace of SRSWOR sample mea y srs is give below: V y srs = 1 f S y (3.1) The Mea Squared Error of theratio Estimator Y R to the first degree of approximatio is give below: MSE Y R = 1 f Y C y + C x ρc x C y (3.) The Modified Ratio Estimators give i Table 1, Table ad the proposed Modified Ratio Estimators give i Table 3 are represeted i three classes as give below: 3.1 Class 1: The Mea Squared Error ad the costat of the Modified Ratio Estimators Y 1 to Y 1 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: MSE Y i = 1 f R i + S y 1 ρ ; i = 1,, 3,..., 1 (3.1.1) 3. Class : The Mea Squared Error ad the costat of the Modified Ratio Estimators Y 13 to Y 17 listed i Table are represeted i a sigle class (say, Class
A Class of Modified Liear Regressio Type Ratio Estimators for Estimatio of Populatio Mea usig Coefficiet of Variatio ad Quartiles of a Auxiliary Variable ), which will be very much useful for comparig with that of proposed Modified Ratio Estimators ad are give below: MSE Y i = 1 f Y C y + θ i C x ρθ i C x C y ; i = 13,14,15,16 ad 17 (3..1) 3.3 Class 3: The Mea Squared Error of the 10 proposed Modified Ratio Estimators Y p1 to Y p10 listed i the Table 3 are represeted i a sigle class (say, Class 3), which will be very much useful for comparig with that of existig Modified Ratio Estimators (give i Class1 ad Class ) are give below: MSE Y pj = 1 f R pj S x + S y 1 ρ ; j = 1,, 3,..., 10 (3.3.1) From the expressios give iequatio (3.1), (3.), (3.1.1), (3..1) ad (3.3.1) we have derived the coditios (see AppedixD) for which the proposed Estimator Y pj are more efficiet tha the Simple Radom Samplig With-Out Replacemet (SRSWOR) sample mea y srs ad are give below: MSE Y pj < V y srs if R pj ρ S y ; j = 1,, 3,..., 10 (3.3.) From the expressios give i equatio (3.), (3.1.1), (3..1) ad (3.3.1) we have derived the coditios (see Appedix E) for which the proposed Estimators Y pj are more efficiet tha the Ratio Estimator ad are give below: MSE Y pj MSE Y R if Y C x ρc y R pj Y ρc y C x or Y ρc y C x R pj Y C x ρc y (3.3.3) From the expressios give i equatio(3.1.1), (3..1) ad (3.3.1) we have derived the coditios (see Appedix F) for which the proposed Estimators Y pj ; j = 1,,3, 10 are more efficiet tha the existig Modified Ratio Estimators give i Class 1, Y i ; i = 1,, 3,..., 1 ad are give below: MSE Y pj < MSE Y i if R pj < R i ; i = 1,, 3,..., 1: j = 1,, 3,..., 10 (3.3.3) From the expressios give i equatio (3..1) ad (3.3.1) we have derived the coditios (see Appedix G) for which the proposed Estimators Y pj ; j = 89
90 Jambuligam Subramai, Gaasegara Kumarapadiya ad Samiatha Balamurali 1,,3, 10 are more efficiet tha the existig Modified Ratio Estimators give i Class, Y i ; i = 13, 14, 15, 16 ad 17 ad are give below: MSE Y pj MSE Y i if Y θ i C x ρc y R S pj Y ρc y θ i C x or x Y ρc y θ i C x R S pj Y θ i C x ρc y (3.3.4) x 4. Empirical study The performace of the proposed Modified Liear Regressio Type Ratio Estimators are assessed with that of the SRSWOR sample mea, the Ratio Estimator ad the existig modified ratio Estimators for a atural populatio cosidered by Kadilar ad Cigi (004). The populatio cosists of data o apple productio amout (as study variable) ad umber of apple trees (as auxiliary variable) i 106 villages of Aegea Regio i 1999. The parameters computed from the above populatio are give below: N = 106 = 40 Y = 1.5943 X = 741.6981 S y = 11496.910 C y = 5.1961 = 57188.930 C x =.0855 ρ = 0.8560 β x = 34.573 β 1 x = 5.138 b = 0.171 Q 1 = 387.5 Q 3 = 6700 Q r = 431.5 Q d = 1156.5 Q a = 14543.75 The costat, the Bias ad the Mea Squared Error of the existig ad proposed Modified Ratio Estimators are give i Table 4. From the values of Table 4, it is observed that Mea Squared Error of the proposed Modified Ratio Estimators are less tha the variace of SRSWOR sample mea, Mea Squared Error of the Ratio Estimator ad all the 17 existig Modified Ratio Estimators. The Percet Relative Efficiecies (PRE s) of the proposed Estimators with respect to the existig Estimators computed by the formula as give below: PRE Y pj = MSE. MSE Y p j 100; j = 1,,3,,10 ad are preseted i Table 5. From the values of Table 5, it is observed that the PRE s of the proposed Estimators with respect to the existig Estimators are ragig from 10.93 to 34.0. This shows that proposed Estimators are more efficiet ad perform better tha the SRSWOR sample mea, the Ratio Estimator ad the existig Modified Ratio Estimators. It has bee show that all the te proposed Estimators
A Class of Modified Liear Regressio Type Ratio Estimators for Estimatio of Populatio Mea usig Coefficiet of Variatio ad Quartiles of a Auxiliary Variable perform better tha the existig Estimators. We wat to kow that amog the 10 proposed Estimators which proposed Estimator is more efficiet. To fid that, PRE s of each proposed Estimator with respect to remaiig 9 proposed Estimators respectively are computed ad preseted i Table 6. From the values of Table 6, it has bee observed that PRE s of proposed EstimatorY p with respect to remaiig ie proposed Estimators is more tha 100 ad is ragig from 101.7 to 130.77. This shows that the proposed EstimatorY p is more efficiet amog the all 10 proposed Estimators. 5. Coclusio I this paper, we have proposed a class of Modified Ratio Estimators usig the Liear combiatio of the coefficiet of variatio, first quartile, third quartile, iter-quartile rage, semi-quartile rage ad semi-quartile average of the auxiliary variable. The Bias ad Mea Squared Error of the proposed Estimators are obtaied ad compared with that of the existig Estimators. Further, we have derived the coditios for which the proposed Estimators are more efficiet tha the SRSWOR sample mea, the Ratio Estimator ad the existig Modified Ratio Estimators. We have also assessed the performace of the proposed Estimators with that of the existig Estimators for a atural populatio. From the umerical study it is observed that the Mea Squared Error of the proposed Estimators are less tha variace of the SRSWOR sample mea ad themea Squared Error of the Ratio Estimator ad the existig Modified Ratio Estimators. Hece, we strogly recommed that the proposed Modified Ratio Estimators may be preferred over the existig Estimators for the use of practical applicatios. Ackowledgemet The authors are thakful to the editor ad the referees for their costructive commets, which have improved the presetatio of the paper. The first author wishes to record his gratitude ad thaks to Uiversity Grats Commissio for the fiacial assistace received through UGC-Major Research Project, New Delhi. 91
9 Jambuligam Subramai, Gaasegara Kumarapadiya ad Samiatha Balamurali Table 1: Existig Modified Ratio Estimators (Class 1) together with the costat, Bias ad Mea Squared Error Estimators Costat R i Bias-B. MSE. Y 1 = y + b X x X x Kadilar ad Cigi (004) X + C x Y = y + b X x x + C x Kadilar ad Cigi (004) X + β Y 3 = y + b X x x + β Kadilar ad Cigi (004) R 1 = Y X () Y R 1 R = Y () X + C x Y R R 3 = Y () X + β Y R 3 R 1 + S y 1 ρ R + S y 1 ρ R 3 + S y 1 ρ Y 4 β X + C x = y + b X x β x + C x Kadilar ad Cigi (004) R 4 = β () Y β X + C x Y R 4 R 4 + S y 1 ρ Y 5 C x X + β = y + b X x C x x + β Kadilar ad Cigi (004) R 5 = C () xy C x X + β Y R 5 R 5 + S y 1 ρ Y 6 = y + b X x Ya ad Tia (010) X + β 1 x + β 1 R 6 = Y () X + β 1 Y R 6 R 6 + S y 1 ρ Y 7 β 1 X + β = y + b X x β 1 x + β Ya ad Tia (010) X + ρ Y 8 = y + b X x x + ρ Kadilar ad Cigi (006) C x X + ρ Y 9 = y + b X x C x x + ρ Kadilar ad Cigi (006) R 7 = β () 1Y β 1 X + β R 8 = Y X + ρ R 9 = C xy C x X + ρ Y R 7 () Y R 8 () Y R 9 R 7 + S y 1 ρ R 8 + S y 1 ρ R 9 + S y 1 ρ Y 10 ρx + C x = y + b X x ρx + C x Kadilar ad Cigi (006) β X + ρ Y 11 = y + b X x β x + ρ Kadilar ad Cigi (006) R 10 = ρy () ρx + C x Y R 10 R 11 = β Y β X + ρ () Y R 11 R 10 S x + S y 1 ρ R 11 S x + S y 1 ρ
A Class of Modified Liear Regressio Type Ratio Estimators for Estimatio of Populatio Mea usig Coefficiet of Variatio ad Quartiles of a Auxiliary Variable Estimators Costat R i Bias-B. MSE. 93 Y 1 ρx + β = y + b X x ρx + β Kadilar ad Cigi (006) R 1 = ρy () ρx + β Y R 1 R 1 S x + S y 1 ρ Table : Existig Modified Ratio Estimators (Class ) together with the costat, the Bias ad the Mea Squared Errors Estimators Costat θ i Bias - B. MSE. Y 13 = y X + Q 1 x + Q 1 Y 14 = y X + Q 3 x + Q 3 Y 15 = y X + Q r x + Q r Y 16 = y X + Q d x + Q d Y 17 = y X + Q a x + Q a θ 13 = X X + Q 1 θ 14 = X X + Q 3 θ 15 = X X + Q r θ 16 = X X + Q d θ 17 = X X + Q a () Y θ 13 C x θ 13 C x C y ρ () Y (θ 14 C x θ 14 C x C y ρ) () Y (θ 15 C x θ 15 C x C y ρ) () Y (θ 16 C x θ 16 C x C y ρ) () Y (θ 17 C x θ 17 C x C y ρ) Y (C y + θ 13 C x Y (C y + θ 14 C x Y (C y + θ 15 C x Y (C y + θ 16 C x Y (C y + θ 17 C x θ 13 C x C y ρ) θ 14 C x C y ρ) θ 15 C x C y ρ) θ 16 C x C y ρ) θ 17 C x C y ρ) Table 3: Proposed Modified Ratio Estimators with the costat, Bias ad Mea Squared Error Costat Estimators R Bias-B. MSE. pj Y p1 = y + b X x X + Q 1 x + Q 1 R p1 = Y X + Q 1 () Y R p 1 R p1 S x + S y 1 ρ Y p = y + b X x X + Q 3 x + Q 3 R p = Y X + Q 3 () Y R p R p S x + S y 1 ρ Y p3 = y + b X x X + Q r x + Q r R p3 = Y X + Q r () Y R p 3 R p3 S x + S y 1 ρ
94 Jambuligam Subramai, Gaasegara Kumarapadiya ad Samiatha Balamurali Costat Estimators R Bias-B. MSE. pj Y p4 = y + b X x X + Q d x + Q d R p4 = Y X + Q d () Y R p 4 R p4 S x + S y 1 ρ Y p5 = y + b X x X + Q a x + Q a R p5 = Y X + Q a () Y R p 5 R p5 S x + S y 1 ρ Y p6 = y + b X x C x X + Q 1 C x x + Q 1 R p6 = C xy C x X + Q 1 () Y R p 6 R p6 S x + S y 1 ρ Y p7 = y + b X x C x X + Q 3 C x x + Q 3 R p7 = C xy C x X + Q 3 () Y R p 7 R p7 S x + S y 1 ρ Y p8 = y + b X x C x X + Q r C x x + Q r R p8 = C xy C x X + Q r () Y R p 8 R p8 S x + S y 1 ρ Y p9 = y + b X x C x X + Q d C x x + Q d R p9 = C xy C x X + Q d () Y R p 9 R p9 S x + S y 1 ρ Y p10 = y + b X x C x X + Q a C x x + Q a R p10 = C xy C x X + Q a () Y R p 10 R p10 S x + S y 1 ρ Table 4: Costat, Bias ad Mea Squared Error of the existig ad proposed Estimators Estimators Costat B. MSE. SRSWOR Sample mea y srs - - 057484.9094 Ratio Estimator Y R 0.0807 169.683 975171.1057 Liear Regressio Estimator Y lr - - 549896.3301 Existig Modified Ratio Estimators (Class 1) Y 1 0.0807 149.801 881345.509 Y 0.0807 149.7784 88195.0933 Y 3 0.0806 149.441 880511.3 Y 4 0.0807 149.8005 881344.0447 Y 5 0.0806 149.60 880945.1187 Y 6 0.0807 149.745 8811.6738 Y 7 0.0807 149.775 88118.4501 Y 8 0.0807 149.7918 88134.8108
A Class of Modified Liear Regressio Type Ratio Estimators for Estimatio of Populatio Mea usig Coefficiet of Variatio 95 ad Quartiles of a Auxiliary Variable Y 9 0.0807 149.7967 881335.5808 Y 10 0.0807 149.7745 88186.6144 Y 11 0.0807 149.8009 881344.9044 Y 1 0.0806 149.3609 880371.30 Existig Modified Ratio Estimators (Class ) Proposed Modified Ratio Estimators (Class 3) Table 5: PREs of the proposed Estimators Y 13 0.9199 167.186 1037436.9350 Y 14 0.5067 13.4148 14667.1098 Y 15 0.5300 17.538 140144.5947 Y 16 0.699 149.446 137068.067 Y 17 0.6534 144.7988 17506.4165 Y p1 0.074 16.7661 830378.3611 Y p 0.0409 38.4557 634983.497 Y p3 0.048 4.0870 643017.8780 Y p4 0.0559 71.9113 709006.9180 Y p5 0.057 63.9617 691417.6597 Y p6 0.0775 138.0351 85531.0137 Y p7 0.0550 69.6186 703933.9646 Y p8 0.0566 73.757 713091.0075 Y p9 0.0665 101.8835 77533.185 Y p10 0.0643 95.143 760566.8439 Estimators Y p1 Y p Y p3 Y p4 Y p5 Y p6 Y p7 Y p8 Y p9 Y p10 y srs 53.83 47.78 34.0 319.97 90.19 97.57 40.55 9.8 88.53 65.37 Y R 10.31 117.44 153.57 151.66 137.54 141.04 114.01 138.53 136.75 15.78 Y 1 106.14 138.80 137.06 14.31 17.47 103.04 15.0 13.60 113.67 115.88 Y 106.13 138.79 137.06 14.30 17.46 103.04 15.0 13.59 113.67 115.87 Y 3 106.04 138.67 136.93 14.19 17.35 10.95 15.08 13.48 113.57 115.77 Y 4 106.14 138.80 137.06 14.31 17.47 103.04 15.0 13.59 113.67 115.88 Y 5 106.09 138.74 137.00 14.5 17.41 103.00 15.15 13.54 113.6 115.83 Y 6 106.1 138.78 137.04 14.9 17.45 103.03 15.19 13.58 113.66 115.86 Y 7 106.1 138.77 137.04 14.8 17.45 103.0 15.18 13.57 113.65 115.86 Y 8 106.14 138.79 137.06 14.30 17.47 103.04 15.0 13.59 113.67 115.88 Y 9 106.14 138.80 137.06 14.31 17.47 103.04 15.0 13.59 113.67 115.88 Y 10 106.13 138.79 137.05 14.30 17.46 103.04 15.19 13.59 113.67 115.87 Y 11 106.14 138.80 137.06 14.31 17.47 103.04 15.0 13.60 113.67 115.88 Y 1 106.0 138.64 136.91 14.17 17.33 10.93 15.06 13.46 113.55 115.75
96 Jambuligam Subramai, Gaasegara Kumarapadiya ad Samiatha Balamurali Y 13 14.94 163.38 161.34 146.3 150.04 11.9 147.38 145.48 133.81 136.40 Y 14 171.76 4.61 1.81 01.16 06.8 166.75 0.61 00.01 183.96 187.53 Y 15 168.75 0.67 17.9 197.63 0.66 163.83 199.06 196.50 180.73 184.4 Y 16 148.98 194.8 19.38 174.48 178.9 144.63 175.74 173.48 159.56 16.65 Y 17 153.57 00.83 198.3 179.86 184.43 149.09 181.15 178.83 164.47 167.67 Table 6: PREs amog the proposed Estimators Estimators Y p1 Y p Y p3 Y p4 Y p5 Y p6 Y p7 Y p8 Y p9 Y p10 Y p1 100.0 130.77* 19.14 117.1 10.10 97.08 117.96 116.45 107.10 109.18 Y p 76.47 100.00* 98.75 89.56 91.84 74.4 90.0 89.05 81.90 83.49 Y p3 77.44 101.7* 100.00 90.69 93.00 75.18 91.35 90.17 8.94 84.54 Y p4 85.38 111.66* 110.6 100.00 10.54 8.89 100.7 99.43 91.45 93. Y p5 83.7 108.89* 107.53 97.5 100.00 80.84 98. 96.96 89.18 90.91 Y p6 103.0 134.70* 133.0 10.64 13.70 100.00 11.50 119.94 110.3 11.46 Y p7 84.77 110.86* 109.47 99.8 101.81 8.30 100.00 98.7 90.79 9.55 Y p8 85.88 11.30* 110.90 100.58 103.13 83.37 101.30 100.00 91.97 93.76 Y p9 93.37 1.10* 10.58 109.35 11.14 90.65 110.14 108.73 100.00 101.94 Y p10 91.59 119.78* 118.8 107.7 110.00 88.9 108.05 106.66 98.10 100.00 *idicates the miimum Mea squared error Refereces 1. Al-Omari, A. I., Jemai, A. A. ad Ibrahim, K. (009). New Ratio Estimators of the mea usig Simple Radom Samplig ad Raked Set Samplig methods. Revista Ivestigació Operacioal, 30(), 97-108.. Cochra, W.G. (1977). Samplig Techiques, Third Editio. Wiley Easter Limited, New Delhi. 3. Haif, M., Hammad, N. ad Shahbaz, M. Q. (009). A Modified Regressio Type Estimator i survey samplig. World Applied Sciece Joural, 7(1), 1559-1561. 4. Kadilar, C. ad Cigi, H. (004). Ratio Estimators i Simple Radom Samplig. Applied Mathematics ad Computatio, 151, 893-90. 5. Kadilar, C. ad Cigi, H. (006). A improvemet i estimatig the populatio mea by usig the correlatio coefficiet. Hacettepe Joural of Mathematics ad Statistics, 35(1), 103-109.
A Class of Modified Liear Regressio Type Ratio Estimators for Estimatio of Populatio Mea usig Coefficiet of Variatio ad Quartiles of a Auxiliary Variable 6. Subramai, J. (013). Geeralized Modified Ratio Estimator for estimatio of fiite populatio mea. Joural of Moder Applied Statistical Methods, 1, 11-155 7. 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, (), 19-3. 8. Subrmai, J. ad Kumarapadiya, G. (013). Estimatio of populatio mea usig coefficiet of variatio ad quartiles of a auxiliary variable. Iteratioal Joural of Iformatio ad Maagemet Scieces, 4(3), 13-4. 9. Subramai, J. ad Kumarapadiya, G. (014). Estimatio of populatiomea usig kow correlatio coefficiet ad media. Joural of Statistical Theory ad Applicatios, 13(4), 333-343. 10. Subramai, J., Kumarapadiya, G.ad Balamurali. S. (014). Some Modified Liear Regressio Type Ratio Estimators for estimatio of populatio mea usig kow parameters of a auxiliary variable. Joural of Buildigs Research, 1(1), 8-4. 11. Ya, Z. ad Tia, B. (010), Ratio method to the mea estimatio usig coefficiet of Skewess of auxiliary variable. Iteratioal Coferece o Ifmatio Computig ad Applicatios, Part II, CCIS, 106, 103 110. Appedix A Notatios used i this paper are give below: Symbol Explaatio N Populatio size Sample size f = /N Samplig fractio Y Study variable X Auxiliary variable x, y Sample totals X, Y Populatio meas x, y Sample meas, S y Populatio stadard deviatios y Populatio covariace betwee X ad Y C x, C y Coefficiet of variatios ρ Coefficiet of correlatio betwee X ad Y 97
98 Jambuligam Subramai, Gaasegara Kumarapadiya ad Samiatha Balamurali β 1 = μ 3 Coefficiet of skewess of the auxiliary variable μ 3 β = μ 4 μ b = β = y Q 1 Q 3 Q r = (Q 3 Q 1 ) Q d = Q 3 Q 1 Q a = Q 3 + Q 1 B. MSE. Y i (Y pj ) Coefficiet of kurtosis of the auxiliary variable Regressio coefficiet of Y o X First (lower) quartile of Auxiliary Variable Third (upper) quartile of Auxiliary Variable Iter-quartile rage of Auxiliary Variable Semi-quartile rage of Auxiliary Variable Semi-quartile average of Auxiliary Variable Bias of the Estimator Mea squared error of the Estimator i th Existig (j th proposed) modified Ratio Estimator of Y Appedix B I case of Simple Radom Samplig With-Out Replacemet (SRSWOR), the sample mea y srs is used to estimate populatio meaywhich is a ubiased Estimator ad its variace is give below: V y srs = 1 f S y The Ratio Estimator for estimatig the populatio mea Y of the study variable Y is defied as: Y R = y x X The Bias ad Mea Squared Error of Y R to the first order of approximatio are give below: B Y R = (1 f) Y C x ρc x C y MSE Y R = 1 f Y C y + C x ρc x C y The Liear Regressio Estimator together with its variace is give below: Y lr = y + β X x V Y lr = 1 f S y 1 ρ
A Class of Modified Liear Regressio Type Ratio Estimators for 99 Estimatio of Populatio Mea usig Coefficiet of Variatio ad Quartiles of a Auxiliary Variable Appedix C We have derived the Bias ad Mea Squared Error of the proposed EstimatorY pj ; j = 1,,3,,10 to first order of approximatio ad are give below: Let e 0 = y Y Y ad e 1 = x X X. Further, we ca write y = Y 1 + e 0 ad x = X 1 + e 1 ad from the defiitio of e 0 ad e 1 we obtai: E e 0 = E e 1 = 0 E e 0 = 1 f C y ; E e 1 = 1 f C x ; E e 0 e 1 = 1 f ρc y C x The proposed Estimators Y pj i the form of e 0 ad e 1 is give below: Y pj where = Y 1 + e 0 + b X x X + λ j X 1 + e 1 + λ j ; j = 1,,3,.,10 λ 1 = Q 1, λ = Q 3, λ 3 = Q r, λ 4 = Q d, λ 5 = Q a, λ 6 = Q 1 C x, λ 7 = Q 3 C x, λ 8 = Q r C x λ 9 = Q d C x ad λ 10 = Q a C x Y pj Y pj where = Y 1 + e 0 + b X x = Y 1 + e 0 + b X x 1 + θ pj e 1 θ pj = X X + λ j ; j = 1,,3,.,10 Y pj X + λ j X + λ j 1 + e 1X X+λ j = Y 1 + e 0 + b X x 1 + θ pj e 1 1 Y pj = Y 1 + e 0 + b X x 1 θ pj e 1 + θ pj e 1 θ 3 pj e 3 1 + Neglectig the terms higher tha third order, we will get:
300 Jambuligam Subramai, Gaasegara Kumarapadiya ad Samiatha Balamurali e 1 Y pj = Y + Ye 0 bxe 1 Yθ pj e 1 Yθ pj e 0 e 1 + bxθ pj e 1 + Yθ pj Y pj Y = Ye 0 bxe 1 Yθ pj e 1 Yθ pj e 0 e 1 + bxθ pj e 1 + Yθ pj e 1 (A) Takig expectatio o both sides of (A), we get: E Y pj Y = YE e 0 + bxe e 1 Yθ pj E e 1 Yθ pj E e 0 e 1 + bxθ pj E e 1 Bias Y pj = + Yθ pj E e 1 Bias Y pj = 1 f Yθ S pj C xy y x θ pj C + θ pj C x x Yθ pj C x ; j = 1,, 3,,10 (B) Substitute the value of θ pj i (B), we get the Bias of the proposed EstimatorY pj ; j = 1,, 3,,10 as give below: Bias Y pj = 1 f Y X X+λ j C x (C) Multiply ad divide by Y i (3.19), we get: Bias Y pj = 1 f Y R p j where R pj = Y ; j = 1,, 3,,10 X+λ j Squarig both sides of (A), eglectig the terms more tha d order ad takig expectatio o both sides, we get: E Y pj Y = E Y e 0 + b X E e 1 + Y θ pj E e 1 byxe e 0 e 1 Y θ pj E e 0 e 1 + bxyθ pj E e 1 (D) MSE Y pj = Y C y + b X C x + Y θ pj C x byxρc y C x Y θ pj ρc y C x + bxyθ j C x MSE Y pj = y Y C y + y S + Y θ pj C x x y S y S y Yθ pj y S y S y C x + y Yθ pj C x
A Class of Modified Liear Regressio Type Ratio Estimators for Estimatio of Populatio Mea usig Coefficiet of Variatio ad Quartiles of a Auxiliary Variable MSE Y pj = MSE Y pj = MSE Y pj = MSE Y pj = MSE Y pj = Y C y + Y θ pj C x + y S y x S Yθ j x + Yθ pj y C x Y θ pj C x + Y C y y S x Y θ pj C x + S y ρ S x S y sice ρ = y S y Y θ pj C x + S y ρ S y Y θ pj C x + S y 1 ρ Substitute the value of θ pj i the above expressio, we will get: y C x 301 MSE Y pj = Y X X + λ j C x + S y 1 ρ MSE Y pj = Y X + λ j X C x + S y 1 ρ The Mea Squared Error of the proposed Estimators Y pj is give below: MSE Y pj = 1 f R pj S x + S y 1 ρ where R pj = Y ; j = 1,, 3,,10 X+λ j (E) Appedix D The coditios for which the proposed Modified Ratio Estimators (Class 3) perform better tha the SRSWOR sample mea are derived ad are give below: For MSE Y pj V y srs
30 Jambuligam Subramai, Gaasegara Kumarapadiya ad Samiatha Balamurali R pj S x + S y 1 ρ R pj S x + S y 1 ρ S y R pj S x + S y S y ρ S y 0 R pj S x S y ρ 0 R pj S x S y ρ R pj S y ρ ρ R pj S y R pj ρ S y That is MSE Y pj V y srs if R pj ρ S y Appedix E S y The coditios for which the proposed Modified Ratio Estimators (Class 3) perform better tha the Ratio Estimator are derived ad are give below: For MSE Y pj MSE Y R R pj S x + S y 1 ρ Y R pj S x + C y 1 ρ Y C y + C x ρc x C y Y C y + C x ρc x C y where R pj = R p j Y R pj S x + C y 1 ρ C y C x + ρc x C y 0 R pj S x + C y ρ C y C y C x + ρc x C y 0 R pj S x ρ C y C x + ρc x C y 0 C x ρc y Rpj S x 0 C x ρc y + R pj C x ρc y R pj 0
A Class of Modified Liear Regressio Type Ratio Estimators for 303 Estimatio of Populatio Mea usig Coefficiet of Variatio ad Quartiles of a Auxiliary Variable Coditio 1: C x ρc y + R pj 0 ad C x ρc y R pj 0 C x + R pj ρc y ad C x R pj ρc y R pj C x ρc y where R pj ρc y C x R pj = R p j Y ad R pj ρc y C x C x ρc y Y C x ρc y R pj Y ρc y C x Coditio : C x ρc y + R pj 0 ad C x ρc y R pj 0 C x + R pj ρc y ad C x R pj ρc y R pj ρc y C x ρc y C x R pj ad R pj C x ρc y C x ρc y Y ρc y C x R pj Y C x ρc y That is MSE Y pj MSE Y R if Y C x ρc y R pj Y ρc y C x or Y ρc y C x R pj Y C x ρc y Appedix F The coditios for which the proposed Modified Ratio Estimators (Class 3) perform better tha the existig Modified Ratio Estimators (Class 1) are derived ad are give below: For MSE Y pj MSE Y i ; i = 1,,3, 1 ad j = 1,,3, 10 R pj S x + S y 1 ρ R i S x + S y 1 ρ R pj S x + S y 1 ρ R i S x + S y 1 ρ R pj S x R i
304 Jambuligam Subramai, Gaasegara Kumarapadiya ad Samiatha Balamurali R pj R i R pj R i That is MSE Y pj < MSE Y i if R pj < R i ; i = 1,, 3,..., 1: j = 1,, 3,..., 10 Appedix G The coditios for which the proposed Modified Ratio Estimators (Class 3) perform better tha the existig Modified Ratio Estimators (Class ) are derived ad are give below: For MSE Y pj MSE Y i ; i = 13,14,15,16 ad 17 R pj S x + S y 1 ρ Y R pj S x + C y 1 ρ Y C y + θ i C x θ i ρc x C y Y C y + θ i C x θ i ρc x C y where R pj = R p j Y R pj S x + C y 1 ρ C y θ i C x + θ i ρc x C y 0 R pj S x + C y ρ C y C y θ i C x + θ i ρc x C y 0 R pj S x ρ C y θ i C x + θ i ρc x C y 0 θ i C x ρc y Rpj S x 0 θ i C x ρc y + R pj θ i C x ρc y R pj 0 Coditio 1: θ i C x ρc y + R pj 0 ad θ i C x ρc y R pj 0 θ i C x + R pj ρc y ad θ i C x R pj ρc y R pj θ ic x ρc y where R pj ρc y θ i C x = R p j Y R pj ad R pj ρc y θ i C x θ ic x ρc y Y θ ic x ρc y R pj Y ρc y θ i C x
A Class of Modified Liear Regressio Type Ratio Estimators for 305 Estimatio of Populatio Mea usig Coefficiet of Variatio ad Quartiles of a Auxiliary Variable Coditio : θ i C x ρc y + R pj 0 ad θ i C x ρc y R pj 0 θ i C x + R pj ρc y ad θ i C x R pj ρc y R pj ρc y θ i C x ρc y θ i C x R pj ad R pj θ ic x ρc y θ ic x ρc y Y ρc y θ i C x R pj Y θ ic x ρc y That is MSE Y pj MSE Y i if Y θ i C x ρc y R pj Y ρc y θ i C x or Y ρc y θ i C x R pj Y θ ic x ρc y