Calibration Approach Separate Ratio Estimator for Population Mean in Stratified Sampling

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1 Article International Journal of Modern Mathematical Sciences, 015, 13(4): International Journal of Modern Mathematical Sciences Journal homepage: ISSN: X Florida, USA Calibration Approach Separate Ratio Estimator for Population Mean in Stratified Sampling Etebong P. Clement Department of Mathematics and Statistics, University of Uyo, Uyo, Nigeria Article history: Received 8 January 015, Received in revised form 5 March 015, Accepted 10 May 015, Published 17 October 015. Abstract: Calibration approach in survey sampling provides an important class of technique for the efficient combination of data sources to improve the precision of parameter estimates. This paper introduces calibration approach separate ratio estimator for population mean Y of the study variable y using auxiliary variable x in stratified sampling. The variance and variance estimator of the proposed estimator have been derived using analytical approaches. An empirical study to evaluate the relative performances of the proposed estimator against members of its class was carried out. Results of analysis showed that the proposed estimator is substantially more efficient than members of its class under consideration with appreciable efficiency gain eywords: auxiliary variable, calibration approach, separate ratio estimator, stratified sampling, study variable Mathematics Subject Classification (010):6D05, 6G05, 6H1 1. Introduction Sampling survey established that the linear regression estimator is generally more efficient than both the ratio and product estimators except when the regression line of the study variable (Y) on the Copyright 015 by Modern Scientific Press Company, Florida, USA

2 Int. J. Modern Math. Sci. 015, 13(4): auxiliary variable (X) passes through the origin and the variance of the study (Y) variable about this line is proportional to the auxiliary variable; in which case the efficiency of the estimators are the same (Scheaffer et al, 1990). However, in most practical situations, the regression line does not pass through the origin, in which case the ratio and the product estimators do not perform equally well as the conventional linear regression estimator. Thus, the ratio and product estimators most practically have the limitation of having efficiency not exceeding that of the regression estimator. To address this problem, most survey statisticians have carried out researches towards the modification of the existing ratio and product estimators to provide better alternatives and improve their precision. Among the authors who have proposed various improved ratio and product estimators include adilar and Cingi (003), Singh et al (009), Sharma and Tailor (010), Onyeka (01), Choudhury and Singh (01)and Singh and Audu (013) It has been observed that in most of these previous works that the performances of the proposed estimators depend on some optimality conditions that need to be satisfied to guarantee better estimates. An important technique to address these problems is by calibration. Calibration estimation is a method that uses auxiliary variables to adjust the original design weights to improve the precision of survey estimates of population parameters. Deville and Sarndal (199) first presented calibration estimators in survey sampling and calibration estimation has been studied by many survey statisticians. A few key references are Wu and Sitter (001), Montanari and Ranalli (005), Farrel and Singh (005), Arnab and Singh (005), Estavao and Sarndal (006), ott (006), Singh (006a, 006b), Sarndal (007), im and Park (010) and Clement et al (014a) defined some calibration estimators using different constraints. In stratified random sampling, calibration approach is used to obtain optimum strata weights. im, Sungur and Heo (007), oyuncu and adilar (013) defined some calibration estimators in stratified random sampling for population characteristics and Clement et al (014b) defined calibration estimators for domain totals in stratified random sampling. In this paper, calibration approach is used to modify ratio estimator in the context of stratified random sampling design.. Calibration Approach to Ratio Estimation in Stratified Sampling Consider a finite population U of N elements U = (U 1, U,, U N ) (1) Suppose the finite population of equation (1) consists of strata with N j units in the jth stratum from which a simple random sample of size n j is taken without replacement. The total population size be N = N j and the sample size n = n j, respectively. Associated with the ith element of the jth stratum Copyright 015 by Modern Scientific Press Company, Florida, USA

3 Int. J. Modern Math. Sci. 015, 13(4): are y ji and x ji with x ji > 0 being the covariate; where y ji is the y value of the ith element in stratum j, and x ji is the x value of the ith element in stratum j, j = 1,,, and i = 1,,, N j where y and x are the study variable and auxiliary variable respectively. For the jth stratum, let w j = N j N be the stratum weights, d j = N j n j be the design weights and f j = n j N j, the sample fraction. Let the j th stratum means of the study variable y and auxiliary variable x (y j = y ji n j ; x j = x ji n j ) be the unbiased estimator of the population mean (Y j = y ji N j ; X j = x ji N j ) of y and x respectively, based on n j observations. In this paper, we proposed calibration approach separate ratio estimator for population mean in stratified sampling using auxiliary information. Calibration ratio estimator under the stratified sampling is given by y R = w j y j Let R j = y j x j ; x j 0 be the estimate of the ratio R j = Y j X j ; X j 0 of the jth stratum in the population. This estimator is only efficient if the variables are strongly positively correlated. Then y j = R jx j (3) Following from () and (3) we defined a calibration separate ratio estimator in stratified sampling as: y RS () = w j R jx j (4) with the new weights w j called the calibration weights. The calibration weights w j are chosen such that a chi-square-type loss functions of the form: L = N j d j ( w j 1) q j d j is minimum subject to a calibration constraints of the form: w j x j = X (6) Minimizing the loss function (5) subject to the calibration constraints (6) leads to the calibration weights for stratified sampling given by (5) w j = d j + (X d j x j) q j d j x j q j d j x j (7) Let Copyright 015 by Modern Scientific Press Company, Florida, USA

4 Int. J. Modern Math. Sci. 015, 13(4): w j = (d j + (X d j x j) q j d j x j q j d j x j ) (8) and setting the tuning parameter q j = x j 1 ; then w j = ( X x st ) (9) where x st = d j x j substituting (7) into () we obtained the calibration ratio estimator under the stratified sampling as: y R = d j y j + q jd j x jy j q j d j x j (X d j x j) (10) Similarly, substituting (7) into (4) we obtained the calibration separate ratio estimator under the stratified sampling as: y RS = R jd j x j + R jq j d j q j d j x j x j (X d j x j ) (11) 3. Variance Estimator for the Calibration Approach Separate Ratio Estimators In this section the estimator of variance of the calibration approach separate ratio estimator in stratified sampling is derived. The general estimator of variance of the calibration ratio estimator of (10) is given by Var(y R ) = w j γ j S jy Var(y R ) = ( X x st ) γ j S jy where γ j = (1 f j) ; S n jy = 1 (y j N j 1 ji y j) is the j-th stratum variance. Let Y R = y x X be the ratio estimates of the population mean Y under the simple random sampling of size n (n large), then; (1 f) 1 Var(Y R) = n N 1 (y i Rx i ) N i=1 Let Y Rj = y j X be the ratio estimates of the population mean Y under the Stratified random sampling of x j size n j (n j large), then; (1) Copyright 015 by Modern Scientific Press Company, Florida, USA

5 Int. J. Modern Math. Sci. 015, 13(4): Var (Y Rj ) = w 1 j γ j N j 1 (y ji R j x ji ) Var (Y Rj ) = w 1 j γ j N j 1 [(y ji y j) R j (x ji x j)] (13) Var (Y Rj ) = w 1 j γ j N j 1 [ (y ji y j) + R j (x ji x j) R j (y ji y j)(x ji x j) ] Var (Y Rj ) = w j γ j (S jy + R j S jx R j S jxy ) (14) where γ j = ( 1 1 ) ; S n j N jy = 1 (y j N j 1 ji y j) ; S jx = 1 (x N j 1 ji x j) ; S jxy = 1 N j 1 (x ji x j)(y ji y j) Var(y RS ) = w j γ j (S jy + R j S jx R j S jxy ) Note that since y RS = y Rj and sampling is independent in each stratum, then Var(y RS ) = Var(y Rj ) and the result in (15) follows. Substituting (9) into (15) the variance estimator for the calibration approach separate ratio estimator is obtained as follows: Var(y RS ) = ( X ) x st γ j (S jy + R j S jx R j S jxy ) (15) (16) 4. Empirical Study To judge the relative performances of the proposed calibration approach separate ratio estimator over members of its class, data set from Vishwakarma and Singh (011 pp. 66) given in table 1 was considered. Two measuring criteria; variance and percent relative efficiency (PRE) were used to compare the performance of each estimator. Var(y st ) = w j γ j S jy = Var(y RC ) = w j γ j (S jy + R S jx RS jxy ) = Copyright 015 by Modern Scientific Press Company, Florida, USA

6 Int. J. Modern Math. Sci. 015, 13(4): Var(y RS ) = w j γ j (S jy + R j S jx R j S jxy ) = Var(y RS ) = ( X ) x st γ j (S jy + R j S jx R j S jxy ) = Table 1. Population adapted from Vishwakarma and Singh (011) Parameter Stratum 1 Stratum Stratum 3 Total N j N = 5 n j n = 10 X j X = Y j Y = S jx S x = S jy S y = S jxy S xy = ρ jxy 0, ρ = γ j R = w j ρ = The percent relative efficiency (PRE) of an estimator θ with respect to the usual unbiased estimator in stratified sampling (y st ) is defined by PRE(θ, y st ) = Var(y st) Var(θ) 100 The percent relative efficiency of the usual unbiased estimator in stratified sampling (y st ), the usual combined ratio estimator in stratified sampling (y RC ), the usual separate ratio estimator in stratified sampling (y RS ) and the calibration approach separate ratio estimator in stratified sampling (y RS ) with respect to y st were computed and presented in table. Table. Performance of estimators from analytical study Estimator Variance PRE(t, y st ) y st y RC y RS y RS Copyright 015 by Modern Scientific Press Company, Florida, USA

7 Int. J. Modern Math. Sci. 015, 13(4): Conclusion This paper introduced calibration estimation to separate ratio estimator in stratified sampling. We proposed calibration approach separate ratio estimator in stratified sampling and derived the estimator of variance for the proposed estimator. Analysis showed that the estimator of variance of the calibration approach separate ratio estimator in stratified sampling is more efficient than the estimators of variance of the standard ratio estimator in stratified sampling (y st ), combined ratio estimator in stratified sampling (y RC ) and the separate ratio estimator in stratified sampling (y RS ). It is observed that the new calibration approach separate ratio estimator is very attractive as it does not depend on any optimality conditions. Consequently, it should be preferred in practice as it provides consistent and more precise parameter estimates. References [1] Arnab, R. and Singh, S., A note on variance estimation for the generalized regression predictor. Australia and New Zealand Journal of Statistics. 47()(005): [] Choudhury, S. and Singh, B.. A class of chain ratio-cum-dual to ratio type estimator with two auxiliary characters under double sampling in sample surveys. Statistics in Transition-New Series, 13(3)(01): [3] Clement, E. P., Udofia, G.A. and Enang, E. I., Estimation for domains in stratified random sampling in the presence of non-response. American Journal of Mathematics and Statistics, 4()(014a): [4] Clement, E. P., Udofia, G.A. and Enang, E. I., Sample design for domain calibration estimators. International Journal of Probability and Statistics, 3(1)(014b): [5] Deville, J.C. and Sarndal, C. E., Calibration estimators in survey sampling. Journal of the American Statistical Association, 87(199): [6] Estavao, V. M. and Sarndal, C.E., Survey estimates by calibration on complex auxiliary information. International Statistical Review, 74(006): [7] Farrell, P.J. and Singh, S., Model-assisted higher order calibration of estimators of variance. Australia and New Zealand Journal of Statistics, 47(3)(005): [8] adilar, C and Cingi, H A study on the chain ratio-type estimator. Hacettepe Journal of Mathematics and Statistics. 3 (003): [9] im, J.M. Sungur, E.A. and Heo T.Y., Calibration approach estimators in stratified sampling, Statistics and Probability Letters, 77(1)(007): Copyright 015 by Modern Scientific Press Company, Florida, USA

8 Int. J. Modern Math. Sci. 015, 13(4): [10] im, J.. and Park, M., Calibration estimation in survey sampling, International Statistical Review, 78(1)(010): 1-9. [11] ott, P.S., Using calibration weighting to adjust for non-response and coverage errors. Survey Methodology, 3(006): [1] oyuncu, N. and adilar, C., Calibration estimators using different measures in stratified random sampling. International Journal of Modern Engineering Research, 3(1)(013): [13] Montanari, G. E. and Ranalli, M.G., Nonparametric model calibration estimation in survey sampling. Journal of the American Statistical Association, 100(005): [14] Onyeka, A, C Estimation of population mean in post-stratified sampling using known value of some population parameter (s). Statistics in Transition-New Series, 13(1)(01): [15] Sarndal, C-E., The calibration approach in survey theory and practice. Survey Methodology, 33(007): [16] Sarndal, C-E and Lundstrom, S., Estimation in surveys with non-response, John Wiley, New York, 005. [17] Scheaffer, R.L., William,M. and Lyman, O., Elementary survey sampling. PWS-ent Publishing Company, Boston, [18] Sharma,B and Tailor,R A new ratio-cum-dual to ratio estimator of finite population mean in simple random sampling. Global Journal of Science. 10(1) (010): [19] Singh, R, and Audu, A. Efficiency of ratio estimator in stratified random sampling using information on auxiliary attribute. International Journal of Engineering and Innovative Technology. (1) (013): [0] Singh R, Chauhan, P, Sawan, N and Smarandache, F. Improvement in estimating the population mean using exponential estimator in simple random sampling. Bulletin of Statistics and Economics 3(9)(009): [1] Singh, S. Survey statistician celebrate golden jubilee year- 003 of the linear regression estimator. Metrika. 006:1-18. [] Singh, S. Calibrated empirical likelihood estimation using a displacement function: Sir R. A. Fisher s Honest Balance. Presented at INTERFACE 006. Pasadena, CA, USA, 006. [3] Vishwakarma, G.. and Singh H.P. Separate ratio-product estimator for estimating population mean using auxiliary information. Journal of Statistical Theory and Application, 10(4)(011): [4] Wu, C. and Sitter, R.R., A Model-calibration approach to using complete auxiliary information from survey data, Journal of the American Statistical Association, 96(001): Copyright 015 by Modern Scientific Press Company, Florida, USA