A NEW CLASS OF EXPONENTIAL REGRESSION CUM RATIO ESTIMATOR IN TWO PHASE SAMPLING

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1 Hacettepe Journal of Mathematics and Statistics Volume 43 1) 014), A NEW CLASS OF EXPONENTIAL REGRESSION CUM RATIO ESTIMATOR IN TWO PHASE SAMPLING Nilgun Ozgul and Hulya Cingi Received 07 : 03 : 013 : Accepted 15 : 05 : 013 Two phase sampling, Auiliary variable, Eponential estimation, Effi- Keywords: ciency. Abstract In this paper, we propose a new class of eponential regression cum ratio estimator using the auiliary variable for the estimation of the finite population mean under two phase sampling scheme. The Bias and Mean Square Error MSE) equations of the proposed estimator are obtained and compared with the MSE equations of some eisting estimators in two phase sampling. We find theoretically the proposed estimator is always more efficient than classical ratio and regression estimators, Singh and Vishwakarma [17] ratio type eponential estimator in two phase sampling. In addition, theoric results are supported by a numerical eample using original data sets. 000 AMS Classification: 1. Introduction In the sampling theory, the use of auiliary information results in considerable improvement in the precision of estimators of population mean. The ratio and regression methods have been widely used when auiliary information is available. In literature, number of authors introduced many ratio and regression type estimators by using general linear transformation of the auiliary variable. For recent development, eponential estimators have been widely studied by several authors such as Bahl and Tuteja [], Singh et al. [19] and Grover and Kaur [6]. Under various sampling schemes, many eponential estimators, using the population information of the auiliary variable, have been proposed. However, the knowledge on the population mean of the auiliary variable is not always available. In this situation, two phase sampling method is the most popular sampling scheme in literature. Two Hacettepe University, Department of Statistics, Beytepe, 06800, Ankara, Turkey. N. Ozgul) nozgul@hacettepe.edu.tr H. Cingi) hcingi@hacettepe.edu.tr

2 13 N. Ozgul and H. Cingi phase sampling, first introduced by Neyman [13], is a cost effective technique in survey sampling. It is typically used when it is very epensive to collect data on the variables of interest, but it is relatively inepensive to collect data on variables that are correlated with the variables of interest. By these aspects two phase sampling is a powerful and cost economical procedure for finding the reliable estimate in first phase sample for the unknown parameters of the auiliary variable. Simply, a field survey is to be undertaken to determine the average value of some characters of a population. For eample, the amount of money families spend on food. As the collection of data requires long interviews by specially trained enumerators, the cost per family is quite high. The cost of survey is constrained within a specified amount but the sample does not appear to yield an estimate of desired precision because of the great variability of the character. Nevertheless, the character is correlated with a second character that can be determined at a lower cost per family so that a precise estimate of the distribution of this second character is readily obtained. Hence, a more precise estimate of the original character can be found by first estimating the distribution of the second character alone from a large random sample [10]. In literature, many authors improved ratio and regression estimators using at least one auiliary variable under two phase sampling scheme. Singh and Espejo [16] suggested a class of ratio-product estimators in two phase sampling with its properties and identified asymptotically optimum estimators from proposed class of estimators. Samiuddin and Hanif [14] proposed ratio and regression estimation procedures to estimate the population mean in two-phase sampling using idea of partial and no information cases. Ahmad [1] has proposed various estimators for two phase and multiphase sampling using information on several auiliary variables. Hanif et al. [7] proposed regression estimator using several auiliary variables. In recent years, eponential estimators have not been studied sufficiently in two phase sampling. Singh and Vishwakarma [17] adapted Bahl and Tuteja [] eponential ratio type estimator into two phase sampling. We, here, give the notations about two phase sampling and various estimators of the population mean in two phase sampling method in Section. We propose a class of eponential regression cum ratio estimator in Section 3. In Section 4, the proposed estimator is compared with other eisting estimators in two phase sampling and we obtain certain conditions that proposed estimator is found to be more efficient than other estimators. In Section 5, the theoretical results are supported by a numerical eample. In Section 6, we give conclusion.. Notations and Various Eisting Estimators Consider a finite population U = U 1, U,..., U N, of size N units. Let y denote the study variable taking the values y i on the unit U i, i = 1,,..., N) and Y is its unknown population mean. Let denotes the auiliary variable taking the values i on the unit U i, i = 1,,..., N) positively correlated with y and X is its unknown population mean. It is well known that when the population mean of the auiliary variable is not known, two phase sampling is used. Two phase sampling consists of two phase. In first phase, a sample of fied size is drawn by Simple Random Sampling Without Replacement SR- SWOR) from the finite population to estimate the mean of the auiliary variable. The sample is drawn in first phase is named as primary sample and epressed by s. The usual practice is to estimate the mean of the auiliary variable by sample mean. In second phase, a sample s s s ) of fied size n is drawn SRSWOR from the primary sample s ) to estimate the mean of the study variable. The sample is drawn in second phase is named as sub sample and epressed by s [14].

3 A New Class of Eponential Regression When information is not available on the auiliary variable,, that is positively correlated with the study variable, y, the classical ratio estimator is a widely used estimator to estimate the population mean, Y, in two phase sampling as follows:.1) y R = y where is the primary sample mean of the auiliary variable, y and are the sub sample means of the study and auiliary variables, respectively. It is well known that the MSE equation of the classical ratio estimator is given by.) MSE y R ) = Y [ λc y + λ C 1 K y) ] C y where K y = ρ y ; λ = 1 C n 1 N ; λ = 1 n 1 n ; n is the primary sample size; n is the sub sample size; N is the number of units in the population; ρ y is the population correlation coefficient between the auiliary and the study variables, C and C y are the population coefficients of variation of the auiliary and study variables, respectively. When auiliary variable is correlated with the study variable, the classical unbiased regression estimator is used to estimate the population mean, in two phase sampling as follows: ).3) y lr = y + β y where β y is the regression coefficient between the auiliary and the study variables. It is well known that the variance of the classical regression estimator is given by.4) V ar y lr ) = Y Cy λ λ ρ y) Singh and Vishwakarma [17] suggested the following modified eponential ratio estimator in two phase sampling.5) y svr = y ep + The MSE equation of the estimator can be given by [ ].6) MSE y svr ) = Y λcy + λ C 4 ρycyc In sampling literature, the authors rarely consider the eponential estimators in two phase sampling scheme. For this reason, we improved a class of eponential regression cum ratio estimator in two phase sampling using the ratio and regression methods and their linear transformation in this study. 3. Suggested Eponential Estimator in Two Phase Sampling Replacing regression estimator instead of sample mean and using linear transformation in eponential term in Singh and Vishwakarma [17] eponential ratio estimator given in.5), we improve a class of eponential regression cum ratio estimator as follows: 3.1) y NH = ] [k 1y + k z z z + z )

4 134 N. Ozgul and H. Cingi where k 1 and k are some known constants, z is a transformation of the auiliary variable at first phase as z = a +b, and z is a transformation of the auiliary variable at second phase as z = a + b. Then, we have 3.) z = a + b z = a + b where a 0) and b are either any known constants or functions of any known population parameters of the auiliary variable, such as standard deviation σ ), coefficient of variation C ), coefficient of skewness β 1 ), coefficient of kurtosis β ), coefficient of correlation ρ y) [9]. The list of new eponential estimator generated from 3.1) is given in Table 1. To obtain the Bias and MSE equations for the proposed estimator, we define following notations: y Y X X 3.3) e 0 =, e 1 = Y X, e1 = X such that E e 0) = E e 1) = E e 1 = 0; E e0) = λc y ; E e1) = λc ; 3.4) E e 1 = λ C; E e 0e 1) = λρ yc yc ; E e 0e 1 = λ ρ yc yc ; E e 1e 1 = λ C where λ = 1 n 1 N, λ = 1 n N i X ) 1 N, C y = S y Y, C = S X, S y = N yi Y ) i X ) N yi Y ) i=1 N 1, S = i=1 N 1 i=1, ρ y = N i X ) N yi Y ) i=1 and we use Taylor series method [4] for two variables to solve the eponential term as f e 1, e 1 = f e 1, e 1) e1 =e + 1 f e 1, e 1 e1 1 =0 1! e =e 1 1 =0 + 1 f e 1, e 1 1! e e1 =e + 1 f e 1, e =0! e e1 =e 1 1 =0 3.5) + 1 f e 1, e 1! e e1 =e + 1 f e 1, e 1 1 =0! e 1 1e e1 =e 1 1 =0 + 1 f e 1, e 1 e1! e 1 e1 =e =0 Epressing 3.1) in terms of e s and using 3.5) for the eponential term, we have ) [ )] ax e 3.6) y NH = k 1Y 1 + e 0) + k X e 1 e 1 1 e 1 ep ax e 1 + e 1 + ) + b i=1

5 A New Class of Eponential Regression ) ax e where fe 1, e 1 e 1 1) = ep ax e 1 + e 1 + ) and we solve the eponential term from + b 3.5) as [ )] y NH = k 1Y 1 + e 0) + k X e 1 e 1 1 θ e 1 e 1 + 3θ e 1 θ e 1 θ e 1e ax where θ = ax + b ). Assuming e 1 < 1, epanding the right hand side of 3.6), and retaining terms up to the second degree of e s, we have 3.7) y NH Y = Y [ k 1 1) k 1θ e 1 e 1 3θ )] [ k 1θ e 0e 1 e 0e 1 + k X e 1 e 1 + θ e 1 e 1 e 1 e 1 ) + k 1e 0 )] Squaring both sides of 3.7), retaining terms of e s up to the second degree and taking epectation, we get the Bias and MSE Equations of y NH to the second degree of approimation as 3.8) Bias y NH ) = E y NH Y ) [ = Y k1 1) + k 1λ θc ] 3θ Ky 3.9) MSE y NH ) = E y NH Y ) = Ȳ [ k 1 1) + k 1 λc y + 4λ θc θ K y) + k 1λ θc K y 3θ) + k λ X C + k X Y λ C k 1 θ K y) To obtain the minimum MSE y NH ), we get 3.10) k i MSE y NH ) = 0; i = 1,. Solving two equations simultaneously, the optimum values of k 1 and k are respectively, 3.11) k 1 = 1 λ θ C 1 + λ λ ρ y 3.1) k = Y X θ 1) + λ θ C 1 + θ K y) λ λ ρ y k 1 and k quantities can be guessed quite accurately through a pilot sample survey or sample data or eperience gathered in due course of time, see Das and Tripathi [5], Singh and Ruiz-Espejo [16], Singh, H.P. et al. [18] and Koyuncu and Kadilar [11].

6 136 N. Ozgul and H. Cingi When k 1 and k are replaced in 3.9), the minimum MSE of the proposed estimator can be written as 3.13) MSE min y NH ) = Y C y λ λ ρ y 1 λ θ C) λ θ 4 C C y λ λ ρy) = V ar y lr) 1 λ θ C λ Y θ 4 C 4 Y 4 Y + V ar y lr ) Table 1. Some Members of the Suggested Estimator y NH A subset of y NH a b ] y NH1 = [k 1y + k ] y NH = [k 1y + k ] ) y NH3 = [k 1y + k 1 β + ) +β ) ] y NH4 = [k 1y + k β ) ) β ) 1 y NH5 = y NH6 = y NH7 = y NH8 = y NH9 = ] [k 1y + k β )+ )+ C ) C + )+β ) ] [k 1y + k β ) ) β )+ )+C ] [k 1y + k ρ y ) ρ y+ )+β ) ] [k 1y + k β ) ) β )+ )+ρ y ] [k 1y + k C ) C + )+ρ y C β ) ρ y β ) C β ) C β ) ρ y ρ y 4. Efficiency Comparisons in Two Phase Sampling In this section, we obtain the efficiency conditions for the proposed estimator by comparing the MSE of the proposed estimators with the MSE of classical ratio and regression estimators and the eponential ratio estimator suggested by Singh and Vishwakarma [17]. We compare the MSE of the proposed estimator, y NH, given in 3.13), with the MSE of the eisting estimators, y R, y lr, y svr. From.) and 3.13), we have the condition MSE y NH ) < MSE y R ) Y C y λ λ ρ y 1 λ θ C) λ θ 4 C 4 4 < Y [ λc 1 + Cy λ λ ρ y + λ C 1 K ] y) y

7 A New Class of Eponential Regression Table 1 Continued: Some Members of the Suggested Estimator y NH A subset of y NH a b ] y NH10 = [k 1y + k ρy ) y NH11 = y NH1 = y NH13 = y NH14 = y NH15 = y NH16 = ρ y+ )+C ] [k 1y + k σ ) σ + )+ρ y ] [k 1y + k ρy ) ρ y+ )+σ ] [k 1y + k β ) ) β )+ )+σ ] [k 1y + k ] [k 1y + k ] [k 1y + k σ ) σ + )+β ) β 1 ) ) β 1 )+ )+β ) β ) ) β )+ )+β 1 ) ρ y σ ρ y β ) σ β 1 ) β ) C ρ y σ σ β ) β ) β 1 ) Note: In addition to estimators listed in Table 1, a large number of estimators can also be generated from 3.1) by putting 1, C, β ), ρ y, σ, β 1 ) values for a and b. 4.1) C y λ λ ρ y 1 λ θ C) λ θ 4 C 4 4 < λc 1 + Cy λ λ ρy) y + λ C ρ yc y) λ ρ yc y λ θ C + V ar y lr ) Y + λ C ρ yc y) 1 + V ar y lr ) Y > 0 The condition 4.1) is always satisfied, the proposed estimator, y NH, is always more efficient than the classical ratio estimator, y R. From.4) and 3.13), we have the condition MSE y NH ) < V ar y lr ) 4.) V ar y Y lr ) 1 λ θ C λ Y θ 4 C 4 4 Y + V ar y lr ) V ar ylr ) + λ θ C > 0 Y < V ar y lr ) The condition 4.) is always satisfied, the proposed estimator, y NH, is always more efficient than the classical regression estimator, y lr. From.6) and 3.13), we have the condition

8 138 N. Ozgul and H. Cingi MSE y NH ) < MSE y svr ) Y C y λ λ ρ y 1 λ θ C) λ θ 4 C 4 [ ] 4 < Y λc 1 + Cy λ λ ρ y + λ C y 4 ρycyc C y λ λ ρ y 1 λ θ C) λ θ 4 C 4 4 < λc 1 + Cy λ λ ρ y +λ C y ρycy ) λ ρ yc y V ar ylr ) 4.3) Y + λ θ C + λ C ρycy 1 + V ar y lr) Y > 0 The condition 4.3) is always satisfied, the proposed estimator, y NH, is always more efficient than Singh and Vishwakarma [17] eponential ratio estimator, y svr. Thus, finally, we conclude from the efficiency comparisons that the class of eponential regression cum ratio estimator, y NH, is always more efficient than the estimators, y R, y lr and y svr. 5. Numerical Eample To show the performance of the proposed estimator in comparison to other estimators in two phase sampling, four original data sets used by other authors in literature has been considered. The descriptions of the populations are given below. Population I : Cingi et. al. [3], y : the number of teachers : the number of student in both primary and secondary school for 93 districts N = 93, n = 400, n = 00, Y = 436, 3, X = 11440, 50, C y = 1, 7, C = 1, 86, ρ y = 0, 955. Population II : Sukhatme and Sukhatme [0], y: No. of villages in the circle. : A circle consisting more than five villages. N = 89, n = 30, n = 0, Y = 3, 360, X = 0, 14, C y = 0, 604, C =, 190, ρ y = 0, 766. Population III : Kadilar and Cingi [9], y: Level of apple production. : No. of apple trees. N = 104, n = 40, n = 0, Y = 65, 37, X = 13, 930, C y = 1, 866, C = 1, 653, ρ y = 0, 865. Population IV : Murthy [1], y: Output : fied capital N = 80, n = 40, n = 0, Y = 51, 86, X = 11, 65, C y = 0, 354, C = 0, 751, ρ y = 0, 9413.

9 A New Class of Eponential Regression We compute the MSE values of classical ratio and regression estimators, Singh and Vishwakarma [17] estimator and proposed estimator using the equations,.),.4), X.6), and 3.13), respectively. We have taken a = b = 1, that is, θ =, just for X + 1 the sake of simplicity. These MSE values are shown in Table. We observe that the most efficient estimator is the proposed eponential regression cum ratio estimator as compared to those eisting ones. Table. MSE Values of Estimators in Two Phase Sampling Population Estimators I II III IV Classical Ratioȳ R) 807,59 0, ,75 1,64 Classical Regressionȳ lr ) 780,89 1, ,17 16,87 Singh and Vishwakarma ȳ svr) 1045,59 0, ,14 5,9 Proposed Est.ȳ NH) 774,71 0,1 6960,89 5,1 6. Conclusion We propose a class of regression cum estimator using the eponential function for the population mean in two phase sampling improving the eponential ratio estimator suggested in Singh and Vishwakarma [17]. Theoretically, we demonstrate that the proposed estimator is always the most efficient estimator in two phase sampling and numerically, for various specific data sets, we show that the proposed estimator has small MSE value according to other estimators. In future work, we will improve the proposed estimator, presented here, with using several auiliary variables and adding more parameters for other sampling schemes. References [1] Ahmad, Z. Generalized Multivariate Ratio and Regression Estimators for Multi-phase Sampling PhD thesis, National College of Business Administration and Economics, Lahore, Pakistan, 008). [] Bahl, S. and Tuteja, R.K. Ratio and product eponential estimators, Journal of Information and Optimization Sciences, 1 1), , [3] Cingi, H., Kadilar, C. and Kocberber, G. Determination of the Opportunities of the Primary and the Secondary Schools in Turkey and Suggestions to Solve the Determined Problems TUBITAK, SOBAG106K077.in Turkish), 007). [4] Cingi, H. and Kadilar, C. Advances in Sampling Theory-Ratio Method of Estimation Bentham Science Publishers, 009). [5] Das, A.K. and Tripathi, T.P. Use of auiliary information in estimating the finite population variance, Sankhya C 40 ): , [6] Grover, L.K. and Kaur, P. An improved estimator of the finite population mean in simple random sampling, Model Assisted Statistics and Applications 6 1), 47-55, 011. [7] Hanif, M., Shahbaz, M.Q. and Ahmad, Z. Some Improved Estimators in Multiphase Sampling, Pak. J. Statist. 6 1), 195-0, 010. [8] Kadilar, C. and Cingi, H. Ratio Estimators in Stratified Random Sampling, Biometrical Journal 45 ), 18-5, 003.

10 140 N. Ozgul and H. Cingi [9] Kadilar, C. and Cingi, H. An Improvement in Estimating The Population Mean By Using The Correlation Coefficient, Hacettepe Journal of Mathematics and Statistics 35 1), , 006. [10] Keen, K. J. Two-Phase Sampling, Encyclopedia of Biostatistics, 005. [11] Koyuncu, N. and Kadilar, C. On the family of estimators of population mean in stratified random sampling, Pak. Jour. Statist. 6 ), , 010. [1] Murthy, M.N. Sampling Theory and methods Statistical Publishing Society, Calcutta, India, 1967). [13] Neyman, J. Contribution to the theory of sampling human populations, Journal of the American Statistical Association 33, , [14] Ozgul, N. Mean Estimators in Two Phase Sampling MSc.Thesis in Turkish), Department of Statistics, Hacettepe University, Ankara, Turkey, 007). [15] Samiuddin, M. and Hanif, M. Estimation of population mean in single and two phase sampling with or without additional information, Pak. J. Stat. 3 ), , 007. [16] Singh, H.P. and Ruiz-Espejo, M. On linear regression and ratio-product estimation of a finite population mean, The Statistician 5 1), 59-67, 003. [17] Singh, H.P. and Vishwakarma, G.K. Modified eponential ratio and product estimators for finite population mean in two phase sampling, Austrian Journal of Statistics 36 3), 17-5, 007. [18] Singh, H.P., Tailor, R. and Tailor, R. On ratio and product methods with certain known population parameters of auiliary variable in sample surveys, SORT 34 ), , 010. [19] Singh, R., Chauhan, P., Sawan, N. and Smarandache, F. Improvement in estimating the population mean using eponential estimator in simple random sampling, Bulletin of Statistics and Economics 3 9), 13-18, 009. [0] Sukhatme, P.V. and Sukhatme, B.V. Sampling theory of surveys with applications Second Edition) Asia Publishing House, Bombay, India, 1970).