AN IMPROVEMENT IN ESTIMATING THE POPULATION MEAN BY USING THE CORRELATION COEFFICIENT

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1 Hacettepe Journal of Mathematics and Statistics Volume , AN IMPROVEMENT IN ESTIMATING THE POPULATION MEAN BY USING THE ORRELATION OEFFIIENT em Kadılar and Hülya Çıngı Received 10 : 03 : 005 : Accepted 0 : 06 : 006 Keywords: Efficiency. Abstract We propose a class of ratio estimators for the estimation of population mean by adapting the estimators in Upadhyaya and Singh 7] to the estimator in Singh and Tailor 6]. We obtain mean square error MSE equations for all proposed estimators and find theoretical conditions that make each proposed estimator more efficient than the traditional estimators and ratio estimator in Singh and Tailor 6]. In addition, these conditions are satisfied with an application with original data. Ratio estimator, Auxiliary variable, Simple random sampling, 000 AMS lassification: 6 D Introduction The classical ratio estimator for the population mean Y of the study variable y is defined by 1 ȳ r = y x X, where y and x are the sample means of the study and auxiliary variables, respectively, and it is assumed that the population mean X of the auxiliary variable x is known ]. The MSE of this estimator is as follows: MSE ȳ r = 1 f n Y y + x 1 θ ]. Here f = n y ; n is the sample size; N is the number of units in the population; θ = ρ N x ; x and y are the population coefficients of variation of auxiliary and study variables, respectively 1]. Hacettepe University, Faculty of Science, Department of Statistics, Beytepe, Ankara, Turkey. . Kadılar kadilar@hacettepe.edu.tr H. Çıngı hcingi@hacettepe.edu.tr

2 104. Kadılar, H. Çıngı Singh and Tailor 6] suggested the following ratio estimator: 3 y ST = y X + ρ, x + ρ where ρ is the correlation coefficient between auxiliary and study variables. The MSE of this ratio estimator is as follows: 4 MSE ȳ ST = 1 f n Y y + xω ω θ ], where ω = X X+ρ.. The Suggested Estimators Motivated by the estimators in Upadhyaya and Singh 7], we propose ratio estimators using the correlation coefficient as follows: y y pr1 = Xx + ρ x x + ρ y y pr = Xρ + x xρ + x y y pr3 = X β x + ρ xβ x + ρ y y pr4 = Xρ + β x, xρ + β x where β x is the population coefficient of the kurtosis of the auxiliary variable. We assume that x, ρ, and β x are known. We obtain the MSE and bias equations for these proposed estimators as 9 MSE ȳ pri = 1 f n Y y + xλ i λ i θ ], i = 1,, 3, 4, B ȳ pri = 1 f n Y xλ i λ i θ, i = 1,, 3, 4, respectively, where λ 1 = Xx ; X λ = x+ρ details, please see the Appendix Xρ Xρ+ x ; λ 3 = Xβ x Xβ x+ρ and λ4 = Xρ Xρ+β x. for 3. Efficiency omparisons In this section, we try to obtain the efficiency conditions for the proposed estimators by comparing the MSE of the proposed estimators with the MSE of the sample mean, traditional ratio estimator and the ratio estimator suggested by Singh and Tailor 6]. It is well known that under simple random sampling without replacement SRSWOR the variance of the sample mean is 10 V y = 1 f n S y = 1 f n Y y.

3 An Improvement in Estimating the Population Mean 105 omparing the MSE of the proposed estimators, given in 9, with the variance of this sample mean, we have the following conditions: 11 1 MSEy pri < V y, i = 1,, 3, 4., xλ iλ i θ < 0 λ i < θ if λ i > 0 λ i > θ if λ i < 0. When either of the restrictions 11 or 1 is satisfied, the proposed estimators are more efficient than the sample mean. omparing the MSE of the proposed estimators with the MSE of the classical ratio estimator, given in, we have the following conditions: MSE y pri < MSE yr i = 1,, 3, 4., λ i λ i θ < 1 θ, λ i λ iθ + θ < 1 λ i + θ 1 λ i < 1 λ i + 1 > θ if λ i < 1, λ i + 1 < θ if λ i > 1. When either of the conditions 13 or 14 is satisfied, the proposed estimators are more efficient than the traditional ratio estimator. omparing the MSE of the proposed estimators with the MSE of the estimator in Singh and Tailor 6], given in 4, we have the following conditions: MSE y pri < MSE yst, i = 1,, 3, 4, λ i λ i θ < ω ω θ, λ i λ iθ < ω ωθ, λ i ω < λ iθ ωθ, λ i ω λ i + ω < θ λ i ω, λ i + ω < θ if λ i > ω, λ i + ω > θ if λ i < ω. When either of the conditions 15 or 16 is satisfied, the proposed estimators are more efficient than the ratio estimator suggested by Singh and Tailor 6]. omparing the MSE of one proposed estimator with another, we have the following conditions: MSE y pri < MSE yprj, i = 1,, 3, 4; j = 1,, 3, 4 and i j, λ i λ i θ < λ j λ j θ, λ i λ j < θ λ i λ j, λ i λ j λ i + λ j < θ λ i λ j, λ i + λ j < θ if λ i > λ j, λ i + λ j > θ if λ i < λ j. When either of the conditions 17 or 18 is satisfied, the ith proposed estimator is more efficient than the jth proposed estimator.

4 106. Kadılar, H. Çıngı 4. An Application In this section we use the data set in Kadilar and ingi 4]. We apply the traditional ratio estimator, given in 1, the Singh-Tailor ratio estimator, given in 3, and the proposed estimators, given in 5-8, to data concerning the level of apple production as the study variable and the number of apple trees as the auxiliary variable 1 unit=1000 trees in 104 villages in the East Anatolia region of Turkey in 1999 Source: Institute of Statistics, Republic of Turkey. We take the sample size as n = 0 and use simple random sampling 1]. The MSE of these estimators are computed as given in, 4 and 9 and these estimators are compared to each other with respect to their MSE values. Table 1. Data Statistics N = 104 y = λ 1 = n = 0 x = λ = ρ = β x = 17.5 λ 3 = X = θ = λ 4 = Y = ω = 0.94 We observe in Table 1 statistics about the population. Note that the correlation between the variables is 87%. When we examine the conditions determined in Section 3 for this data set, they are satisfied for the first and third proposed estimators as follows: λ 1 = : θ = 1.95 λ 3 = : λ 1, λ 3 > 0 λ = 1.96 : θ = 1.95 λ =.00 : λ 1, λ 3 < 1 λ 1 + ω = 1.91 : θ = 1.95 λ 3 + ω = 1.94 : λ 1, λ 3 > ω = ondition 11 is satisfied. = ondition 13 is satisfied. = ondition 15 is satisfied. For the MSE comparison between the first and third proposed estimators, the condition 18 is satisfied as follows: λ 1 + λ 3 = 1.96 : θ = 1.95 : λ 1 < λ 3. Thus, the first proposed estimator is more efficient than the third. As a result, we suggest that we should apply the first and third estimators to this data set. In Table, the values of the MSE are given. As expected, it is seen that the first and third proposed estimators have a smaller MSE than the sample mean, the traditional ratio and the Singh-Tailor ratio estimators. It is obvious that the proposed estimators are more efficient than the other estimators. It is worth pointing out that the classical ratio estimator is more efficient than the ratio estimator, suggested by Singh and Tailor 6], for this data set. Table. MSE Values of the Ratio Estimators

5 An Improvement in Estimating the Population Mean 107 sample mean y traditional y r Singh-Tailor y ST proposed 1 y pr proposed 3 y pr onclusion We develop some ratio estimators using the correlation coefficient and give a theoretical argument to show that the proposed estimators have a smaller MSE than the traditional and the Singh-Tailor ratio estimators under certain conditions. These theoretical conditions are also satisfied by the results of an application with original data. In future work, we hope to adapt the ratio estimators, presented here, to ratio estimators suggested by Kadilar and ingi 3] and to ratio estimators in stratified random sampling as in Kadilar and ingi 4,5]. Appendix To the first degree of approximation, the MSE of the third proposed estimator can be found using the Taylor series method defined by A1 MSE y = pr d Σ d where ha,b d = a Y, X Σ = 1 f ] S y S yx n S xy ha,b b S x ] Y, X see Wolter 8]. Here h a, b = h y, x = y pr3 in 7; Sy and Sx denote the population variances of the study and the auxiliary variables, respectively, and S yx = S xy denotes the population covariance between the study and the auxiliary variables. According to this definition, we obtain d for the third proposed estimator as ] d = 1 Y β x Xβ x+ρ We obtain the MSE equation of the third proposed estimator using A1 as follows: MSE y pr3 n n Y n Y Sy Y β x Xβ x + ρ Syx + Y β x Xβ x + ρ ] S x y β x Xβ x + ρ ] Y Syx + β x Xβ x + ρ ] S x { } y β x X β x X Sx + Xβ x + ρ Xβ x + ρ ] X Syx XY n Y y + λ 3 λ3 x ρ y x ] n Y y + xλ 3 λ 3 θ ].

6 108. Kadılar, H. Çıngı We would like to remark that the MSE equations of the other proposed estimators can easily be obtained in the same way. In general, Taylor series method for k variables can be given as where and h x 1, x,..., x k = h k X 1, X,..., X k + d j xj X j + Rk X k, a d j = h a1, a,..., a k a j R k X k, a = k k j=1 i=1 j=1 1 h X 1,X,..., X k xj X j xi X i + Ok.! X i X j Here O k represents the terms in the expansion of the Taylor series of more than the second degree. When we omit the term R k Xk, a, we obtain Taylor series method for two variables as follows: A hx, y hx, Y hc, d = hc, d c x X + X, Y d y Y. X, Y Instead of using A, we can also obtain the MSE of the proposed estimator using A1, an alternative method to A 8]. If we do not omit the term R k X k, a, we obtain the bias of the proposed estimators by the following equation: A3 B 1 y pri = d11v y + d1cov y, x + d1cov x, y + dv x], where d 11 = h y pri = 0; y y d 1 = d 1 = h y pri = h y pri x y y x Using these equalities, we can write A3 as B y = 1 pri λi 1 f X n Syx + R X λ i n Y λi λi x yx = λi X ; d = hy pri = R x x X λ i. 1 f n S x n Y λi x λ i θ ; i = 1,, 3, 4. where yx = ρ y x. References 1] ingi, H. Örnekleme Kuramı Hacettepe University Press, ] ochran, W. G. Sampling Techniques John Wiley and Sons, New-York, ] Kadilar,. and ingi, H. Ratio estimators in simple random sampling, Applied Mathematics and omputation 151, , ] Kadilar,. and ingi, H. Ratio estimators in stratified random sampling, Biometrical Journal 45, 18 5, ] Kadilar,. and ingi, H. A new ratio estimator in stratified random sampling, ommunications in Statistics: Theory and Methods 34 3, ] Singh, H. P. and Tailor, R. Use of known correlation coefficient in estimating the finite population mean, Statistics in Transition 6 4, , 003.

7 An Improvement in Estimating the Population Mean 109 7] Upadhyaya, L. N. and Singh, H. P. Use of transformed auxiliary variable in estimating the finite population mean, Biometrical Journal 41 5, , ] Wolter, K. M. Introduction to Variance Estimation Springer-Verlag, 003.