EFFICIENT ESTIMATORS FOR THE POPULATION MEAN

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1 Hacettepe Journal of Mathematics and Statistics Volume 38) 009), 17 5 EFFICIENT ESTIMATORS FOR THE POPULATION MEAN Nursel Koyuncu and Cem Kadılar Received 31:11 :008 : Accepted 19 :03 :009 Abstract M. Khoshnevisan, R. Singh, P. Chauhan, N. Sawan and F. Smarandache A general family of estimators for estimating population mean using known value of some population parameters), Far East Journal of Theoretical Statistics, , 007) introduced a family of estimators using auxiliary information in simple random sampling. They showed that these estimators are more efficient than the classical ratio estimator and that the minimum value of the mean square error MSE) of this family is equal to the MSE of the regression estimator. In this paper we propose another family of estimators using the results of B. Prasad Some improved ratio type estimators of population mean and ratio in finite population sample surveys, Communications in Statistics: Theory and Methods 18, , 1989). Expressions for the bias and MSE of the proposed family are derived. Besides, considering the minimum cases of these MSE equations, a comparison of the efficiency conditions between the Khoshnevisan and proposed families are obtained. The proposed family of estimators is found to be more efficient than Khoshnevisan s family of estimators under certain conditions. Finally, these theoretical findings are illustrated by a numerical example with original data. Keywords: Ratio estimator, Product estimator, Regression estimator, Auxiliary information, Mean square error, Efficiency. 000 AMS Classification: 6 D05. Hacettepe University, Department of Statistics, Beytepe, Ankara, Turkey. N. Koyuncu) nkoyuncu@hacettepe.edu.tr C. Kadılar) kadilar@hacettepe.edu.tr Corresponding Author

2 18 N. Koyuncu, C. Kadılar 1. Introduction and notation When the study variable y is highly correlated with the auxiliary variable, the use of auxiliary information in the ratio and product estimators can increase the precision of the estimates. To obtain the most efficient estimator, many authors proposed ratio and product estimators using the standard deviation, coefficient of variation, skewness, kurtosis, correlation coefficient, etc. of the auxiliary variable. In this study, we suggest a new family of estimators to estimate the population mean of the study variable Y by using the estimators of Prasad 4 and Khoshnevisan et al., and the optimum cases of the suggested family of estimators are also obtained. Consider a finite population of size N from which a sample s of size n is drawn according to simple random sampling without replacement. Let Y i and X i denote the values of the study and auxiliary variables for the i-th unit, i = 1,,..., N), of the population. Further, let ȳ and x be the sample means of the study and auxiliary variables, respectively. To obtain the bias and MSE, let us define e 0 = ȳ Ȳ Ȳ notations, where and λ = N n Nn. E e 0) = E e 1) = 0, Ee 0) = λc y, Ee 1) = λc x, E e 0e 1) = λc yx = λρc yc x, and e 1 = x. Using these Cy = S y Ȳ, C x = S x Syx, Cyx = Ȳ, N Sy i=1 yi = Ȳ ) N, Sx i=1 xi = ) N i=1 yi, S yx = Ȳ ) x i Ȳ ) N 1 N 1 N 1. The suggested family of estimators Khoshnevisan et al. defined a family of estimators for the population mean in simple random sampling as a.1) t = ȳ g + b αa x + b) + 1 α) a + b ) where a 0), b are either real numbers or functions of the known parameters of the auxiliary variable x such as the standard deviation S x), coefficient of variation C x), skewness β 1 x)), kurtosis β x)) and the correlation coefficient ρ) of the population. Here, g and α are suitably chosen scalars such that the mean square error of t is minimum. The bias and MSE of this family of estimators are respectively given by g g + 1).) B t) = λȳ α υ Cx αυgc yx, and.3) MSE t) = λȳ C y + α υ g C x αυgc yx, where υ = a a +b. The minimum value of MSE for α = K υg.4) MSE min t) = Ȳ λcy 1 ρ ) which is equal to the MSE of the regression estimator. ) where K = ρ Cy C x is also given by

3 Efficient Estimators for the Population Mean 19 The ratio estimators, which are given in Table 1, are in the same family.1), and we can express the Mean Square Error in.3) for these estimators as Ȳ λc.5) MSE t i) = y + Cx C yx), i = 1 Ȳ λ Cy υ i 1) C yx + υ i 1) Cx, i = 3, 5, 7,..., 17. Table 1. Some members of the family of estimators of t Ratio estimators g = 1) Product estimators g = 1) α a b X x t 1 = ȳ t = ȳ x X X + Cx t 3 = ȳ x + C x Sisodia and Dwivedi 8 βx) X + C x t 5 = ȳ β x) x + C x Upadhyaya and Singh 9 CxX + β x) t 7 = ȳ C x x + β x) Upadhyaya and Singh 9 X + Sx t 9 = ȳ x + S x β1x) X + S x t 11 = ȳ β 1x) x + S x βx) X + S x t 13 = ȳ β x) x + S x X + ρ t 15 = ȳ x + ρ Singh and Tailor 6 X + βx) t 17 = ȳ x + β x) Singh et al. 7 x + Cx t 4 = ȳ X + C x Pandey and Dubey 3 βx) x + C x t 6 = ȳ β x) X + C x Upadhyaya and Singh 9 Cx x + β x) t 8 = ȳ C xx + β x) Upadhyaya and Singh 9 x + Sx t 10 = ȳ X + S x Singh 5 β1x) x + S x t 1 = ȳ β 1x) X + S x Singh 5 βx) x + S x t 14 = ȳ β x) X + S x Singh 5 x + ρ t 16 = ȳ X + ρ Singh and Tailor 6 x + βx) t 18 = ȳ X + β x) Singh et al. 7 For the product estimators in Table 1, the MSE equation is 1 1 C x 1 β x) C x 1 C x β x) 1 1 S x 1 β 1x) S x 1 β x) S x 1 1 ρ 1 1 β x) Ȳ λ ) Cy + Cx + C yx, j =.6) MSE t j) = Ȳ λ Cy + υ j ) 1 Cyx + υ j ) 1 C x, j = 4,6, 8,..., 18,

4 0 N. Koyuncu, C. Kadılar where υ 1 = + C x, υ = β x) β x) + Cx, υ 3 = C x C x + βx), υ 4 = + S x, υ 5 = β 1x) β 1x) + Sx, υ 6 = β x) β x) + Sx, υ 7 = + ρ, υ8 = + β x). Motivated by Prasad 4 and Gandge et al. 1 we propose a new family of estimators as given below g a + b.7) η = κȳ αa x + b) + 1 α) a + b ) where κ is a suitable constant to be determined later. Expressing the estimator, η in terms of e i, i = 0, 1), we can write.7) as.8) η = κȳ 1 + e0) 1 + αυe1 g. Expanding the right hand side of.8) to a first order approximation and subtracting Ȳ from both sides we get.9) η Ȳ = κȳ g g + 1) 1 gαυe 1 + α υ e 1 + e 0 gαυe 0e 1 Ȳ. Taking the expectation on both sides of equation.9), we get the bias of the estimator η as g g + 1).10) B η) = κȳ λ α υ Cx gαυc yx + Ȳ κ 1) Squaring both sides of equation.9) gives.11) η Ȳ ) = κ Ȳ 1 gαυe 1 + g g + 1) α υ e 1 + e 0 g αυe 0e 1 + Ȳ κȳ 1 gαυe 1 + g g + 1) α υ e 1 + e 0 gαυe 0e 1 and then taking the expectation, we get the MSE of the estimator η, to a first order approximation, as.1) MSE η) = Ȳ {κ λc y + κ g + g ) κ g + g )) α υ λc x gαυ κ κ ) λc yx + κ 1) }. The minimum of MSEη) is obtained for the optimal value of κ, which is.13) κ = A B where A = g + g ) α υ λc x gαυλc yx +, B = λc y + g + g ) α υ λc x 4gαυλC yx + 1 Thus, the minimum MSE of the estimator η is obtained as { }.14) MSE min η) = Ȳ 1 A. 4B

5 Efficient Estimators for the Population Mean 1 For the ratio estimators as given in Table, we can express the Mean Square Error given in equation.1) by the following equation Ȳ {κ λcy + 3κ κ ) λcx κ κ )λc yx + κ 1) },.15) MSE η i) = Ȳ {κ + λcy + 3κ + κ + )υ i 1) λcx υ i 1) κ + κ + )λc yx + κ + 1) }, i = 1 i = 3, 5,..., 17 and for product estimators, the Mean Square Error is given by the following equation Ȳ {κ λc y + κ λc x + κ κ ) λc yx + κ 1) },.16) MSE η j) = Ȳ {κ τ λcy + κ τ υ j ) 1 λc x + υ j κ τ κ τ) λcyx + κ τ 1) }. ) 1 j = j = 4, 6,..., 18 Table. Some members of the family of estimators of η Ratio estimators g = 1) Product estimators g = 1) α a b X x η 1 = κȳ η = κȳ x X Prasad 4 Gandge et al. 1 X + Cx x + Cx η 3 = κȳ η 4 = κȳ 1 1 C x + C x x X + C x βx) X + C x η 5 = κȳ β x) x + C x CxX + β x) η 7 = κȳ C x x + β x) X + Sx η 9 = κȳ x + S x β1x) X + S x η 11 = κȳ β 1x) x + S x βx) X + S x η 13 = κȳ β x) x + S x X + ρ η 15 = κȳ x + ρ X + βx) η 17 = κȳ x + β x) βx) x + C x η 6 = κȳ β x) X + C x Cx x + β x) η 8 = κȳ C xx + β x) x + Sx η 10 = κȳ X + S x β1x) x + S x η 1 = κȳ β 1x) X + S x βx) x + S x η 14 = κȳ β x) X + S x x + ρ η 16 = κȳ X + ρ x + βx) η 18 = κȳ X + β x) 1 β x) C x 1 C x β x) 1 1 S x 1 β 1x) S x 1 β x) S x 1 1 ρ 1 1 β x)

6 N. Koyuncu, C. Kadılar The expressions MSE η i) and MSE η j) are minimized for the optimal values of κ given by κ = κ + = 1 + λc x λc yx 1 + 3λC x 4λC yx + λc y υ i 1) λc y + 3υ i 1) = A B, λcx υ i 1) λc yx + 1 λc x 4υ i 1) λc yx + 1 = A+ B +, κ 1 + λc yx = = A 1 + λcy + λcx + 4λC yx B, υ κ τ j ) 1 λcyx + 1 = λcy + υ ) 1 j λc x + 4υ j ) 1 λcyx + 1 = A τ B. τ Substituting these optimal values in.15) and.16) we get the minimum MSE s as { } Ȳ 1 A, i = 1 B.17) MSE min η i) = { } Ȳ 1 A+, i = 3, 5,..., 17.18) B + { Ȳ 1 A B MSE min η j) = { Ȳ 1 Aτ 3. Efficiency comparisons B τ }, j =, }, j = 4,6,..., 18. The t-family of estimators is more efficient than the classical ratio estimator if, 3.1) MSE t i) < MSE t 1), i = 3, 5,7,..., 17, υ i 1) υ i 1) < Cyx C x > Cyx C x 1 for υ i 1) 1 for υ i 1) > 1 < 1 When condition 3.1) is satisfied, we can infer that the t-family is more efficient than the classical ratio estimator. The suggested family of estimators as given in Table is more efficient than the classical ratio estimator if 3.) MSE min η i) < MSE t 1), i = 3, 5,7,..., 17, } {1 A+ < λ ) Cy + Cx C yx B + When condition 3.) is satisfied, we can infer that the suggested family is more efficient than the classical ratio estimator. The suggested family of estimators is more efficient than the ratio estimator proposed by Prasad 4 if MSE min η i) < MSE min η 1), i = 3,5, 7,..., 17,

7 Efficient Estimators for the Population Mean 3 A 3.3) B A+ B < 0 + When condition 3.3) is satisfied we can infer that the suggested family of estimators is more efficient than the ratio estimator proposed in 4. The suggested family of estimators as given in Table is more efficient than the t-family of estimators given in Table 1 if 3.4) also 3.5) MSE min η 1) < MSE t 1), } {1 A < λ Cy + Cx ) C yx, B MSE min η i) < MSE t i), i = 3,5, 7,..., 17, {1 A+ B + } λ C y υ i 1) C yx + υ i 1) Cx < 0. It is clear that for the product estimators similar comparisons can be made and the related conditions can also be obtained. We would also like to note that the comparison between the minimum MSE of the proposed and t-families of estimators is obtained as MSE min η) < MSE min t), } {1 A < λc y 1 ρ ). 4B 4. A numerical example In this section, we use data concerning primary and secondary schools for 93 districts of Turkey in 007 Source: Ministry of Education, Republic of Turkey), taking the number of teachers as study variable and the number of students as auxiliary variable in both primary and secondary schools. Note that we take a sample of size n = 180, and we observe that the correlations between auxiliary and study variables are positive. Therefore, we use ratio estimators for the estimation of the population mean in this section. The summary statistics about the population is given in Table 3. Table 3. Data Statistics N = 93 n = 180 S y = S x = Y = X = ρ = β x) = β 1x) = The MSE values of the t and the η estimators have been obtained using.5) and.17), respectively. These values are given in Table 4.

8 4 N. Koyuncu, C. Kadılar Table 4. Mean square error of the t and η families t-family η-family Estimator MSE Estimator MSE t η t η t η t η t η t η t η t η t η represents the most efficient estimator among the t i estimators. represents the most efficient estimator among the η i estimators. Table 5. Efficiency Conditions t, η 3.1) 3.) 3.3) 3.4) 3.5) t < t 3 t 5 t t t t 13 t 15 t 17 η η E E-04 η E E-04 η E E-04 η E-05 η E-07 η E-05 η E E-04 η E E < υ i 1) < 1 < < 0 < 0 shows that the stated condition is satisfied.

9 Efficient Estimators for the Population Mean 5 When we examine Table 4, we observe that the 13 th t estimator t 13) and the proposed estimator η 13) have the smallest MSE values within their own family of estimators. From this result, we can infer that the 13 th t and η estimators are more efficient than both the classical ratio k 1) and the Prasad 4 estimator η 1) for this data set. When we further examine Table 4, we see that MSEη 13) < MSE t i), where i = 1, 3,5,..., 17. From this result, we can conclude that the proposed estimators are more efficient than the adapted estimators for this data set. However, these results are expected as the conditions 3.1) 3.5) are satisfied, as shown in Table 5. Khoshnevisan et al. have found that the minimum value of the MSE of the t- family is equal to the value of the MSE of the regression estimator when α takes the value α = K Cy where K = ρ υg C x ). For example, we can also obtain the minimum MSE when g = 1 ratio estimator) and αυ = K. For these values, the MSE of t is equal to the MSE of the regression estimator 4.619), and when we take the same values for the η estimator we get the MSE of estimator 4.355), which is slightly less than the MSE of the regression estimator. Moreover, there are various combinations of g and αυ that we can have smaller MSE values than the regression estimator. Consequently, under various conditions, the MSE of our proposed estimators can be smaller than the MSE of the regression estimator. References 1 Gandge, S. N., Varghese, T. and Prabhu-Ajgaonkar, S. G. A note on modified product estimator, Pakistan Journal of Statistics 93B, 31 36, Khoshnevisan, M., Singh R., Chauhan P., Sawan N. and Smarandache, F. A general family of estimators for estimating population mean using known value of some population parameters), Far East Journal of Theoretical Statistics, , Pandey, B. N. and Dubey, V. Modified product estimator using coefficient of variation of auxiliary variate, Assam Statistical Rev. ), 64 66, Prasad, B. Some improved ratio type estimators of population mean and ratio in finite population sample surveys, Communications in Statistics: Theory and Methods 18, , Singh G. N. On the improvement of product method of estimation in sample surveys, Jour. Ind. Soc. Agri. Statistics 563), 67 65, Singh H. P. and Tailor, R. Use of known correlation coefficient in estimating the finite population mean, Statistics in Transition 6, , Singh H.P., Tailor, R. and Kakran, M.S. An estimator of population mean using power transformation, J. I. S. A. S. 58), 3 30, Sisodia, B.V.S. and Dwivedi, V.K. A modified ratio estimator using coefficient of variation of auxiliary variable, Journal of Indian Society Agricultural Statistics 33, 13 18, Upadhyaya, L. N. and Singh H. P. Use of transformed auxiliary variable in estimating the finite population mean, Biometrical Journal 41, , 1999.