Research Article An Improved Class of Chain Ratio-Product Type Estimators in Two-Phase Sampling Using Two Auxiliary Variables

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1 robability and Statistics, Article ID , 6 pages esearch Article An Improved Class of Chain atio-roduct Type Estimators in Two-hase Sampling Using Two Auxiliary Variables Gajendra K. Vishwakarma and Manish Kumar Department of Applied Mathematics, Indian School of Mines, Dhanbad, Jharkhand , India Correspondence should be addressed to Manish Kumar; manishstats88@gmail.com eceived 13 September 2013; Accepted 23 January 2014; ublished 6 March 2014 Academic Editor: Zhidong Bai Copyright 2014 G. K. Vishwakarma and M. Kumar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper presents a technique for estimating finite population mean of the study variable in the presence of two auxiliary variables using two-phase sampling scheme when the regression line does not pass through the neighborhood of the origin. The properties of the proposed class of estimators are studied under large sample approximation. In addition, bias and efficiency comparisons are carried out to study the performances of the proposed class of estimators over the existing estimators. It has also been shown that the proposed technique has greater applicability in survey research. An empirical study is carried out to demonstrate the performance of the proposed estimators. 1. Introduction The use of auxiliary information for estimating population mean of the study variable has greater applicability in survey research. It is utilized at the estimation stage and design stagetoobtainanimprovedestimatorcomparedtothose not utilizing auxiliary information. The use of ratio and product strategies in survey sampling solely depends upon the knowledge of population mean X of the auxiliary variable X. The ratio estimator was developed by Cochran [1] to estimate the population mean Y of the study variable Y by using information on auxiliary variable X, positively correlated with Y.Theratioestimatorismosteffectivewhen the relationship between Y and X is linear through the origin and the variance of Y is proportional to X.obson[2] defined a product estimator that was revisited by Murthy [3]. The product estimator is used when the auxiliary variable X is negatively correlated with the study variable Y. When the population mean X of the auxiliary variable X is not known before the start of a survey, then a firstphase sample of size n is selected from the population of size onwhichonlytheauxiliaryvariablex is measured in order to furnish a good estimate of X. And then a second-phase sample of size n is selected from the first-phase sample of size n on which both the study variable Y and the auxiliary variable X are measured. This procedure of selecting the samples from the given population is known as two-phase sampling (or double sampling). The concept of double sampling was first introduced by eyman [4]. Some contribution to two-phase sampling has been made by Sukhatme [5], Hidiroglou and Sarndal [6], Fuller [7], Hidiroglou [8], Singh and Vishwakarma [9], and Sahoo et al. [10]. We can use either one or two (or more than two) auxiliary variables while estimating population mean of the study variable; keeping this fact, Chand [11] introducedchain ratio estimators. This led various authors including Kiregyera [12], Singh and Upadhyaya [13], rasad et al. [14], Singh et al. [15], Singh and Choudhury [16], and Vishwakarma and Gangele [17] to modify the chain type estimators and discuss their properties. When the population mean Z of another auxiliary variable Z which has a positive correlation with X (i.e., ρ XZ >0) is known and if ρ YX >ρ YZ >0, then it is advisable to estimate X by X=x (Z/z ), which would provide a better estimate of X as compared to x.

2 2 robability and Statistics The usual chain type ratio and product estimators of Y under double sampling scheme using two auxiliary variables X and Z are given, respectively, by = Z yx x z, = y x z x Z. Singh and Choudhury [16] suggested the following exponential chain type ratio and product estimators of Y under double sampling scheme using two auxiliary variables X and Z: e = y exp {(x /z ) Z x (x /z ) Z+x }, e = y exp {x (x /z ) Z x+(x /z ) Z }, where x and z are the sample means of X and Z,respectively, basedonthefirst-phasesampleofsizen drawn from the population of size with the help of Simple andom Sampling Without eplacement (SSWO) scheme. Also, y and x are the sample means of Y and X,respectively,basedon the second-phase sample of size n drawn from the first-phase sample of size n with the help of SSWO scheme. 2. roposed Estimator It has been theoretically established that, in general, the linear regression estimator is more efficient than the ratio (product) estimator except when the regression line of Y on X passes through the neighborhood of the origin, in which the efficiencies of these estimators are almost equal. However, owing to stronger intuitive appeal, survey statisticians favour theuseofratioandproductestimators.further,wenotethat, in many practical situations, the regression line does not pass through the neighborhood of the origin. In these situations, the ratio estimator does not perform well as the linear regression estimator. Considering this fact, Singh and uiz Espejo [18] made an attempt to improve the performance of these estimators and suggested the following ratio-product type estimator forpopulationmeany under double sampling scheme using single auxiliary variable X: y d = y[αx x + (1 α) x ], (3) x where α is a real constant. We propose the following exponential chain ratioproducttypeestimatorforpopulationmeany under double sampling scheme using two auxiliary variables X and Z: = y[αexp {(x /z ) Z x (x /z ) Z+x } + (1 α) exp { x (x /z ) Z x+(x /z ) Z }], (1) (2) (4) where α is a real constant to be determined such that the Mean Square Error (MSE) of the proposed estimator is minimum. For α = 1, e,whereas,forα = 0, ydc e. emark. It is noted that the proposed estimator in (4) isa special case of the class of estimators y class = yh(x, z ) proposed by Srivastava [19], where H( ) is a parametric function such that H(x s 1, Z) = 1 and satisfies certain regularity conditions defined in Srivastava [19]. 3. Bias and MSE of the roposed Estimator To obtain the Bias and Mean Square Error (MSE) of the proposed estimator, we consider y=y(1+e 0 ), x=x(1+e 1 ), x = X(1+e 1 ), z = Z(1+e 2 ), (5) such that E(e 0 )=E(e 1 )=E(e 1 )=E(e 2 )=0, (6) where e 0 <1, e 1 <1, e 1 <1, e 2 <1. Let C Y, C X,andC Z be the coefficients of variation of Y, X, andz, respectively.also,letρ YX, ρ YZ,andρ XZ be the correlation coefficients between Y and X, Y and Z,andXand Z,respectively.Then,wehave E(e 2 0 )=f 1C 2 Y, E(e2 1 )=f 1C 2 X, E(e 2 1 )=f 2C 2 X, E(e 2 2 )=f 2C 2 Z, E(e 0 e 1 )=f 1 ρ YX C Y C X, E(e 0 e 1 )=f 2ρ YX C Y C X, where E(e 0 e 2 )=f 2ρ YZ C Y C Z, E(e 1 e 1 )=f 2C 2 X, E(e 1e 2 )=f 2ρ XZ C X C Z, E(e 1 e 2 )=f 2ρ XZ C X C Z, f 1 =( 1 n 1 ), f 2 =( 1 n 1 ), f 3 =f 1 f 2 =( 1 n 1 ), n C 2 Y = S2 Y Y 2, ρ YX = S YX S Y S X, C2 X = S 2 X X 2, ρ YZ = S YZ S Y S Z, C2 Z = S 2 Z Z 2, ρ XZ = S XZ S X S Z, S 2 Y = 1 (Y ( 1) i Y) 2, S 2 X = 1 (X ( 1) i X) 2, S 2 Z = 1 (Z ( 1) i Z) 2, (7)

3 robability and Statistics 3 1 S YX = (Y ( 1) i Y) (X i X), 1 S YZ = (Y ( 1) i Y) (Z i Z), 1 S XZ = (X ( 1) i X) (Z i Z). ow, expressing the estimator in terms of e 0, e 1, e 1, and e 2 and neglecting the terms of e 0, e 1, e 1,ande 2 involving degree greater than two, we get (8) B( e )=Y[3 8 {f 3C 2 X +f 2C 2 Z } 1 2 {f 3ρ YX C Y C X +f 2 ρ YZ C Y C Z }], B( e )=Y[ 1 8 {f 3C 2 X +f 2C 2 Z } {f 3ρ YX C Y C X +f 2 ρ YZ C Y C Z }], B(y d )=Y[αf 3C 2 X (2α 1) f 3ρ YX C Y C X ], MSE ( )=Y2 [f 1 C 2 Y +f 3C 2 X +f 2C 2 Z (12) = Y[1+α(e 1 e 2 e 1 +e 0 e 1 e 0e 2 e 0e 1 ) α 2 (e 2 1 e 2 2 e2 1 )+e (e 1 e 2 e 1 +e 0 e 1 e 0e 2 e 0e 1 ) (e 2 1 e 2 2 e2 1 e 1 e 2 +e 1e 2 e 1e 1 ) (e 2 1 +e 2 2 +e2 1 )] Tothefirstdegreeofapproximation,theBiasandMean Square Error (MSE) of the proposed estimator are given by (9) 2f 3 ρ YX C Y C X 2f 2 ρ YZ C Y C Z ], MSE ( )=Y2 [f 1 C 2 Y +f 3C 2 X +f 2C 2 Z +2f 3 ρ YX C Y C X +2f 2 ρ YZ C Y C Z ], MSE ( e )=Y2 [f 1 C 2 Y {f 3C 2 X +f 2C 2 Z } {f 3 ρ YX C Y C X +f 2 ρ YZ C Y C Z }], MSE ( e )=Y2 [f 1 C 2 Y {f 3C 2 X +f 2C 2 Z } +{f 3 ρ YX C Y C X +f 2 ρ YZ C Y C Z }], B( ) MSE (y d )=Y2 [f 1 C 2 Y +4α2 f 3 C 2 X (4α 1) = Y[ 8 {f 3 C 2 X +f 2C 2 1) Z } (2α 2 (10) 4αf 3 {C 2 X +ρ YXC Y C X } +f 3 {C 2 X +2ρ YXC Y C X }]. (13) MSE ( ) {f 3 ρ YX C Y C X +f 2 ρ YZ C Y C Z }], = Y 2 [f 1 C 2 Y + (2α 1)2 4 {f 3 C 2 X +f 2C 2 Z } (2α 1) {f 3 ρ YX C Y C X +f 2 ρ YZ C Y C Z }]. (11) To the first degree of approximation, the expressions for Bias and Mean Square Error (MSE) of the estimators, ydc, e, ydc e,andyd are, respectively, given by B( )=Y[f 3C 2 X +f 2C 2 Z f 3ρ YX C Y C X f 2 ρ YZ C Y C Z ], B( )=Y[f 3ρ YX C Y C X +f 2 ρ YZ C Y C Z ], 3.1. Optimum Value of α. As we know, α is determined so as to minimize the Mean Square Error (MSE) of the estimators y d and.so,theoptimumvaluesofα, forwhichmse(yd ) and MSE( ) are minimum, are obtained by using the following conditions: α MSE (yd )=0, (14) MSE (ydc α )=0. The optimum value of α, which minimizes the Mean Square Error (MSE) of the estimator y d,isgivenby α opt = 1 2 [1 + ρ YX C Y C X ]. (15)

4 4 robability and Statistics The optimum value of α, which minimizes the Mean Square Error (MSE) of the estimator,isgivenby α opt = f 3 (2ρ YX C Y C X +C 2 X )+f 2 (2ρ YZ C Y C Z +C 2 Z ) 2(f 3 CX 2 +f 2CZ 2 ). (16) Substituting the value of α from (15) in(13), we get the minimum MSE of y d as MSE (y d ) = min Y2 [f 1 C 2 Y f 3ρ 2 YX C2 Y ]. (17) Substituting the value of α from (16) in(11), we get the minimum MSE of as MSE ( ) min = Y 2 [f 1 C 2 Y (f 3ρ YX C Y C X +f 2 ρ YZ C Y C Z ) 2 f 3 CX 2 +f 2CZ 2 ]. 4. Efficiency Comparisons (18) It is well known that the Bias and variance of the usual unbiased estimator y for population mean in SSWO are B(y) = 0, (19) V(y) = f 1 S 2 Y =f 1Y 2 C 2 Y. (20) From (11), (13), and (20), we have (i) MSE( )<V(y),if α< 4(f 3ρ YX C Y C X +f 2 ρ YZ C Y C Z )+f 3 C 2 X +f 2C 2 Z 2(f 3 C 2 X +f 2C 2 Z ), (21) (ii) MSE( )<MSE(ydc ),if α< 4(f 3ρ YX C Y C X +f 2 ρ YZ C Y C Z ) (f 3 C 2 X +f 2C 2 Z ) 2(f 3 C 2 X +f 2C 2 Z ), (iii) MSE( )<MSE(ydc ),if α< 4(f 3ρ YX C Y C X +f 2 ρ YZ C Y C Z )+3(f 3 C 2 X +f 2C 2 Z ) 2(f 3 C 2 X +f 2C 2 Z ), (iv) MSE( )<MSE(ydc e ),if (22) (23) α< 2(f 3ρ YX C Y C X +f 2 ρ YZ C Y C Z ) f 3 CX 2 +f 2CZ 2, (24) (v) MSE( )<MSE(ydc e ),if α< 2(f 3ρ YX C Y C X +f 2 ρ YZ C Y C Z )+f 3 C 2 X +f 2C 2 Z f 3 CX 2 +f 2CZ 2, (25) (vi) MSE( )<MSE(yd ),if α< 4(f 3ρ YX C Y C X f 2 ρ YZ C Y C Z ) + (3f 3 C 2 X f 2C 2 Z ) 2(3f 3 C 2 X f 2C 2 Z ). (26) The range of α provides enough scope for choosing many estimators that are more efficient than the above considered estimators. 5. Empirical Study To examine the merits of the proposed estimator of Y,wehave considered the following natural population datasets. opulation I (source: Cochran [20]) is shown as follows: Y: number of placebo children, X: number of paralytic polio cases in the placebo group, Z:numberofparalyticpoliocasesinthe notinoculated group =34, n =15, n=10, Y = 4.92, X=2.59,and Z = 2.91, ρ YX = , ρ YZ = , ρ XZ = , C 2 Y = , C 2 X = ,and C2 Z = opulation II (source: Murthy [21]) is shown as follows: Y: area under wheat in 1964, X: area under wheat in 1963, Z:cultivatedareain1961, =34, n =10, n=7, Y = , X = ,and Z = , ρ YX = , ρ YZ = , ρ XZ = , C 2 Y = , C 2 X = ,andC2 Z = Here,wehavecomputed (i) the Absolute elative Bias (AB) of different suggested estimators of Y using the formula Bias ( ) AB ( ) = Y, (27) (ii) the ercentage elative Efficiencies (Es) of different suggested estimators of Y with respect to y using the formula E (, y) = V(y) 100. (28) MSE ( )

5 robability and Statistics 5 Table 1: Absolute elative Bias (AB) of different estimators of Y. Estimators opulation I opulation II y e e y d Table 2: ercentage elative Efficiencies (Es) of different estimators of Y with respect to y. Estimators opulation I opulation II y e e y d Data is not applicable. 6. Conclusion It is observed from Table 1 that, (i) for population I, AB (y) < AB ( e e ) (29) < AB (y d ), (ii) for population II, AB (y) < AB (y d e ) (30) < AB ( e ). From Table 2, we see that the ercentage elative Efficiency (E) of the proposed estimator, for populations I and II, is more as compared to all other existing estimators, that is, usual unbiased estimator y, chain type ratio estimator, chain type product estimator ydc, exponential chain type ratio estimator e, exponential chain type product estimator e, and ratio-product type estimator yd. Finally, from Tables 1 and 2, we conclude that the proposed estimator (based on two auxiliary variables X and Z) is a more appropriate estimator in comparison to other existing estimators as it has appreciable efficiency as well as lower relative bias. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments The authors are grateful to the editor rofessor Zhidong Bai and the learned referee for their comments leading to the improvement of the paper. eferences [1] W. G. Cochran, The estimation of the yields of the cereal experiments by sampling for the ratio of grain to total produce, The Agricultural Science, vol. 30, pp , [2] D.S.obson, Applicationsofmultivariatepolykaystothetheory of unbiased ratio-type estimation, the American Statistical Association,vol.52,pp ,1957. [3] M.. Murthy, roduct method of estimation, The Indian Statistics A, vol. 26, pp , [4] J. eyman, Contribution to the theory of sampling human populations, American Statistical Association,vol.33, pp , [5] B. V. Sukhatme, Some ratio-type estimators in two-phase sampling, the American Statistical Association, vol. 57, pp , [6] M. A. Hidiroglou and C. E. Sarndal, Use of auxiliary information for two phase sampling, Survey Methodology, vol. 24, pp , [7] W. A. Fuller, Two-phase sampling, in roceedings of the Annual Meeting of the Survey Methods Section of the Statistical Society of Canada, pp , [8] M. A. Hidiroglou, Double sampling, Survey Methodology, vol. 27, pp , [9] H.. Singh and G. K. Vishwakarma, Modified exponential ratio and product estimators for finite population mean in double sampling, Austrian Statistics, vol.36,no.3,pp , [10] L.. Sahoo, G. Mishra, and S.. ayak, On two different classes of estimators in two-phase sampling using multiauxiliary variables, Model Assisted Statistics and Applications, vol.5,no.1,pp.61 68,2010. [11] L. Chand, Some ratio type estimators based on two or more auxiliary variables [h.d. dissertation], IowaStateUniversity, Ames, Iowa, USA, [12] B. Kiregyera, A chain ratio-type estimator in finite population double sampling using two auxiliary variables, Metrika,vol.27, no. 4, pp , [13] G.. Singh and L.. Upadhyaya, A class of modified chaintype estimators using two auxiliary variables in two phase sampling, Metron,vol.53,no.3-4,pp ,1995. [14] B.rasad,.S.Singh,andH..Singh, Somechainratio-type estimators for ratio of two population means using two auxiliary characters in two phase sampling, Metron, vol.54,no.1-2,pp , [15] S.Singh,H..Singh,andL..Upadhyaya, Chainratioand regression type estimators for median estimation in survey sampling, Statistical apers,vol.48,no.1,pp.23 46,2007. [16] B. K. Singh and S. Choudhury, Exponential chain ratio and product type estimators for finite population mean under double sampling scheme, Global Science Frontier esearch,vol.12,no.6,2012. [17] G. K. Vishwakarma and. K. Gangele, A class of chain ratiotype exponential estimators in double sampling using two

6 6 robability and Statistics auxiliary variates, Applied Mathematics and Computation, vol. 227, pp , [18] H.. Singh and M. uiz Espejo, Double sampling ratio-product estimator of a finite population mean in sample surveys, Applied Statistics,vol.34,no.1-2,pp.71 85,2007. [19] S. K. Srivastava, A generalized estimator for the mean of a finite population using multi-auxiliary information, American Statistical Association,vol.66,pp ,1971. [20] W. G. Cochran, Sampling Techniques,JohnWiley&Sons,ew York, Y, USA, [21] M.. Murthy, Sampling Theory and Methods, Statisticalublishing Society, Calcutta, India, 1967.

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