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Ratio Estimator, Product Estimator, Two-Phase Sampling, Mean Square Error American Journal of Operational Research 2016, 6(3): 61-68 DOI: 10.5923/j.ajor.20160603.02 Modified Ratio and Product Estimators for Estimating Population Mean in Two-Phase Sampling Subhash Kumar Yadav1, Sat Gupta2, S. S. Mishra3,*, Alok Kumar Shukla4 1Department of Mathematics and Statistics (A Centre of Excellence on Advanced Computing), Dr. RML Avadh University, Faizabad, U.P., India 2Department of Mathematics and Statistics, University of North Carolina at Greensboro, Greensboro, NC, USA 3Department of Mathematics and Statistics (A Centre of Excellence on Advanced Computing), Dr. R. M. L. Avadh University, Faizabad, U.P., India 4Department of Statistics, D.A-V. College, Kanpur, U.P., India Abstract In the present article, we have proposed a modified class of ratio and product type estimators of population mean using auxiliary information in two-phase sampling. The expressions for the Bias and Mean Squared Error of the proposed estimators have been obtained up to the first order of approximation. An efficiency comparison has been made with some of the other ratio and product estimators of population mean under two- phase sampling. A numerical study is also carried out to evaluate the performance of proposed and existing ratio and product estimators. It has been shown that the proposed estimators have smaller mean squared errors. Keywords Ratio estimator, Product Estimator, Two-Phase Sampling, Mean Square Error propose the dual to ratio type estimator for improved 1. Introduction estimation of population mean of the study variable. Bahl and Tuteja [1] were the first to propose the exponential type It is well known that to estimate any parameter, a suitable ratio and product estimators of population mean using estimator is the corresponding statistic. Thus for estimating auxiliary information. In all of the estimators discussed population mean, sample mean is the most appropriate above, the mean X of the auxiliary variable is assumed estimator. Although it is unbiased, it has a large amount of known. When mean X of auxiliary variable is not known, variation. Therefore we seek an estimator which may be two-phase or double sampling, suggested by Neyman [9], is biased but has smaller man squared error as compared to used. Kumar and Bahl [6] were the first to propose dual to sample mean. This is achieved through the use of an ratio estimator of population mean in two-phase sampling. auxiliary variable that has strong positive or negative Singh and Choudhury [11] proposed the dual to product correlation with the study variable. When there is strong estimator of population mean in two-phase sampling. positive correlation between the study variable and the Exponential type ratio and product estimators of population auxiliary variable and the line of regression passes through mean in two-phase sampling have also been studied by Singh origin, then the ratio type estimators are used for improved and Vishwakarma [12]. Corresponding dual estimators in estimation of population mean. Product type estimators are two-phase sampling have been studies by Kalita and Singh used when there is strong negative correlation. The [4]. Our main motivation in this study is to improve further regression type estimators are used for the improved the estimators by Kalita and Singh [4]. estimation of population mean when the line of regression Let the finite population consist of N distinct and does not pass through the origin. identifiable units under study. A random sample of size n is Cochran [2] utilized the positively correlated auxiliary drawn using simple random sampling without replacement variable and for the first time proposed the usual ratio 1 N 1 N estimator of population mean of the study variable. Later (SRSWOR) technique. Let YY i and XX N N i Robson [10] and Murthy [7] proposed the traditional product i1 i1 estimator independently, using negatively correlated respectively be the population means of the study and the auxiliary variable. Srivenkataramana [13] was the first to 1 n 1 n auxiliary variables, and yy i and xx i * Corresponding author: n i1 n i1 [email protected] (S. S. Mishra) Published online at http://journal.sapub.org/ajor be the respective sample means. When X is not known, Copyright © 2016 Scientific & Academic Publishing. All Rights Reserved double sampling or two-phase sampling is used to estimate 62 Subhash Kumar Yadav et al.: Modified Ratio and Product Estimators for Estimating Population Mean in Two-Phase Sampling the population mean of the study variable y. Under double 1 N . sampling scheme the following procedure is used for the yx(y i Y )( x i X ) selection of the required sample: N i1 Case I: A large sample S of size n ()nN is drawn Singh and Choudhury [11] proposed the dual to product from the population by SRSWOR and the observations are estimator of population mean in two- phase sampling as taken only on the auxiliary variable x to estimate the d x population mean X of the auxiliary variable. tyP (1.7) x1 Case II: A sample S of size n ()nN is drawn either from S or directly from the population of size N d The mean squared error of tP , up to the first order of to observe both the study variable and the auxiliary variable. approximation for both the case-I and Case-II respectively The most suitable estimator for the population mean is the are, corresponding sample mean given by d 2 2 ** 2 MSE( tP ) I Y [ C y C x (1 2 C )] (1.8) ty0 (1.1) d 2 2 *** 2 2 The variance of t0 , up to the first order of approximation, MSE( tP ) II Y [ C y C x 2 CC x ] (1.9) is given by Singh and Vishwakarma [12] proposed the exponential 22 type ratio and product estimators of population mean of V() t0 Y Cy (1.2) study variable in two-phase sampling respectively as, where, d xx1 11 Sy 1 N ty exp (1.10) , C and 22. Re y Syi() y Y xx1 nN Y N 1 i1 Cochran [2] proposed the classical ratio type estimator of d xx 1 tyPe exp (1.11) population mean utilizing the auxiliary information under xx simple random sampling as 1 d X The mean squared errors of both the estimators tRe and tyR (1.3) x d tPe , up to the first order of approximation for both the Kumar and Bahl [6] proposed the usual ratio estimator of Case-I and Case-II respectively are, population mean in two- phase sampling as d 2 2 ** 2 1 MSE( tRe )I Y [ C y C x ( C )] (1.12) d x1 4 tyR (1.4) x d 2 21 *** 2 2 MSE()[] tRe II Y C y C x C x C (1.13) 1 n 4 where xx1 i is an unbiased estimator of population d 2 2 ** 2 1 n i1 MSE( t ) Y [ C C ( C )] (1.14) Pe I y x 4 mean X of the auxiliary variable based on the sample S of size n . d 2 21 *** 2 2 MSE()[] tPe II Y C y C x C x C (1.15) d 4 The mean squared error of tR , up to the first order of approximation, for Case-I and Case-II respectively are, Kumar and Bahl [6] proposed the following dual to ratio estimator of population mean under two-phase sampling as d 2 2 ** 2 MSE( tR ) I Y [ C y C x (1 2 C )] (1.5) *d *d x ty (1.16) d 2 2 *** 2 2 R x MSE( tR ) II Y [ C y C x 2 CC x ] (1.6) 1 where, *d The mean squared error of tR , up to the first order of * 11 ** 11 *** * approximation for Case-I and Case-II respectively are, , , , nN' nn' *d 2 2 ** 2 MSE( tR ) I Y [ C y g C x ( g 2 C )] (1.17) C N y Sx 221 C C , S() x X and *d 2 2 2 *** yx x xi MSE( t ) Y [ C gC ( g 2 C )] (1.18) Cx X N 1 i1 R II y x American Journal of Operational Research 2016, 6(3): 61-68 63 n exponential dual to ratio and product-type estimators in where, g . nn' double sampling. The large sample properties have been studied up to the first order of approximation. Singh and Choudhury [11] proposed the following dual to product estimator of population mean under two-phase sampling as, 2. Proposed Estimators x *d 1 Using the estimators of Kalita and Singh [4], we propose tyP (1.19) x*d two generalized estimators of population mean as exponential dual to ratio and exponential dual to *d The mean squared error of tP , up to the first order of product-type estimators respectively, as given below: approximation for Case-I and Case-II respectively are, **dd Re yt (1 ) Re (2a) *d 2 2 ** 2 MSE( t ) Y [ C g C ( g 2 C )] (1.20) P I y x **dd Pe yt (1 ) Pe (2b) *d 2 2 2 *** MSE( t ) Y [ C gC ( g 2 C )] (1.21) P II y x where, and are the characterizing scalars which are Kalita and Singh [4] proposed the following exponential obtained by minimizing the mean squared errors of the dual to ratio and exponential dual to product estimator in proposed estimators. two-phase sampling respectively as The Bias and MSE of the proposed estimators are obtained for the following two cases. *d Case I: When the second phase sample of size n is a *d xx 1 tyRe exp (1.22) xx*d subsample of the first phase sample of size n ' .
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