A note on a difference type estimator for population mean under two phase sampling design

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1 Khan and Al Hossain SpringerPlus 0165:73 DOI /s RESEARCH Open Access A note on a dference type estimator for population mean under two phase sampling design Mursala Khan 1* and Abdullah Yahia Al Hossain *Correspondence: mursala.khan@yahoo.com 1 Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad 060, Pakistan Full list of author information is available at the end of the article Abstract In this manuscript, we have proposed a dference-type estimator for population mean under two-phase sampling scheme using two auxiliary variables. The properties and the mean square error of the proposed estimator are derived up to first order of approximation; we have also found some efficiency comparison conditions for the proposed estimator in comparison with the other existing estimators under which the proposed estimator performed better than the other relevant existing estimators. We show that the proposed estimator is more efficient than other available estimators under the two phase sampling scheme for this one example; however, further study is needed to establish the superiority of the proposed estimator for other populations. Keywords: Study variable, Auxiliary variable, Bias, Mean squared-error, Two phase sampling, Exponential chain-type estimator, Efficiency Background In survey sampling, the use of the auxiliary information at the estimation stage is widely used in order to obtain improved designs and the precision of an estimator of the unknown population parameter. When the knowledge of the auxiliary variable is used at the estimation stage, the ratio, product and regression methods of estimation are widely employed in these situations. The most important topic which is widely discussed in the various probability sampling schemes is the estimation of the population mean of the study variable. A large number of authors have paid their attention towards the formulation of new or modied estimators for the estimation of population mean, for the case, see Hansen and Hurwitz 1943, Sukhatme 196, Srivastava 1970, Chand 1975, Cochran 1977, Kiregyera 1980, 1984, Srivastava et al. 1990, Bahl and Tuteja 1991, Singh et al. 006, 007, 011, Singh and Choudhury 01, Khare et al. 013, Singh and Majhi 014 and Khan 015, 016 etc. Symbols and notations Let us consider a finite population of size N of dferent units U = {U 1, U, U 3,, U N }. Let y and x be the study and the auxiliary variable with corresponding values y i and x i respectively for the i-th unit i = {1,, 3,, N} defined in a finite population U with 016 The Authors. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate changes were made.

2 Khan and Al Hossain SpringerPlus 0165:73 Page of 7 means Ȳ = 1/N N i=1 y i and X = 1 / N N i=1 x i of the study as well as auxiliary variable respectively. Also let S y = 1 / N 1 N i=1 yi Ȳ and S x = 1 / N 1 N i=1 xi X be the population variances of the study and the auxiliary variable respectively and let C y and C x be the coefficient of variation of the study as well as auxiliary variable respectively, and ρ yx be the correlation coefficient between x and y. Let y and x be the study and the auxiliary variable in the sample with corresponding values y i and x i respectively for the i-th unit i = {1,, 3, n} in the sample with unbiased means ȳ = 1 / n n i=1 y i and x = 1 / n n i=1 x i respectively. Also let Ŝ y = 1 / n 1 n i=1 yi ȳ and Ŝ x = 1 / n 1 n i=1 x i x be the corresponding sample variances of the study as well as auxiliary variable respectively. Ni=1 y Let S yx = i Ȳx i X Ni=1 y N 1, S yz = i Ȳ z i Ni=1 x N 1 and S xz = i X z i N 1 be the co-variances between their respective subscripts respectively. Similarly b yxn = Ŝyx is the Ŝx corresponding sample regression coefficient of y on x based on a sample of size n. Also C y = S y Ȳ, C x = S x and C X z = S z are the coefficients of variations of the study and auxiliary variables respectively. Also θ = 1n N 1, θ 1 = 1n N 1 and θ = 1n n 1. Some existing estimators Let us consider a finite population U = {U 1, U, U 3,, U N } of size N units. To estimate the population mean Ȳ of the variable of interest say y taking values y i, in the existence of two auxiliary variables say x and z taking values x i and z i for the ith unit U i. We assume that there is a high correlation between y and x as compared to the correlation between y and z, i.e. ρ yx > ρ yz > 0. When the population X of the auxiliary variable x is unknown, but information on the other cheaply auxiliary variable say z closely related to x but compared to x remotely to y, is available for all the units in a population. In such a situation we use a two phase sampling. In the two phase sampling scheme a large initial sample of size n n < N is drawn from the population U by using simple random sample without replacement sampling SRSWOR scheme and measure x and z to estimate X. In the second phase, we draw a sample subsample of size n from first phase sample of size n, i.e. n < n by using SRSWOR or directly from the population U and observed the study variable y. The variance of the usual simple estimator t 0 =ȳ = 1 n ni=1 y i up to first order of approximation is, given by V t 0 = θs y 1 The classical ratio and regression estimators in two-phase probability sampling and their mean square errors up to first order of approximation are, given by t 1 = ȳ x x MSEt 1 = Ȳ ] θcy + θ Cx ρ yxc y C x t =ȳ + b yxn x x 3 4

3 Khan and Al Hossain SpringerPlus 0165:73 Page 3 of 7 MSEt = S y ] θ 1 ρyx + θ 1 ρyx 5 Chand 1975, suggested the following chain ratio-type estimator the suggested estimator is, given by t 3 = ȳ x x z The mean square error of the suggested estimator is, given as MSEt 3 = Ȳ θc y + θ C x ρ yxc y C x + θ 1 C z ρ yzc y C z ] 6 7 Khare et al. 013, proposed a generalized chain ratio in regression estimator for population mean, the recommended estimator is given by t 4 =ȳ + b yx { α x z x} 8 where α is the unknown constant, and the minimum mean square error at the optimum value of α = ρ yzc x ρ yx c z is, given by ] MSEt 4 = Ȳ Cy θ θ ρyx θ 1ρyz 9 Recently Singh and Mahji 014, suggested a chain-type exponential estimators for Ȳ given by t 5 = ȳ z x x exp + z 10 } t 6 =ȳ + b yxn { x z exp + z x 11 t 7 =ȳ exp x x x + x z 1 The mean square errors of the suggested estimators, up to first order of approximation are, given as follows MSEt 5 = Ȳ θc y + θ Cx ρ yxc y C x + θ 1 C ] z ρ yzc y C z ] MSEt 6 = Ȳ Cy θ 1 ρyx + θ ρ yx Cz C z 4 Cx ρ yx ρ yz C x MSEt 7 = Ȳ θcy + θ Cx 4 4ρ yxc y C x + θ 1 C ] z ρ yzc y C z 14 15

4 Khan and Al Hossain SpringerPlus 0165:73 Page 4 of 7 The proposed estimator On the lines of Khare et al. 013, we propose a dference-type estimator for population mean under two-phase sampling scheme using two auxiliary variables; the suggested estimator is, given by t m =ȳ + k 1 x z x + k x x z 16 where k 1 and k are the unknown constants, To obtain the properties of the proposed estimator we define the following relative error terms and their expectations. Let e 0 = ȳ Ȳ Ȳ, e 1 = x X X, e 1 = x X, e X = z and e = z, such that Ee 0 = Ee i = E e i = 0, for i = 1,. E e0 = θcy, E e1 = θcx, E e 1 = θ 1 Cx, E e 1 e 1 = θ1 Cx, E E e 0 e = θ1 C yz, Ee 0 e 1 = θc yx, E e 0 e 1 = θ1 C yx, Ee 0 e = θc yz, E e 1 e = E e 1 e = θ1 C xz, Ee 1 e = θc xz, E = E e e = θ1 Cz. e e = θcz, Rewriting 16, in terms of e s, we have t m = 1 Ȳ 1 + e0 + k 1 X + e e e1 1 ] + k + e e e Expanding the right hand side of the above equation, and neglecting terms of e s having power greater than two, we have ] t m Ȳ = Ȳe0 k 1 X e 1 e 1 + e + e + e 1 e k e 1 e 1 + e e1 + e 1e 1 17 On squaring and taking expectation on both sides of Eq. 17, and keeping terms up to second order, we have MSEt m = E Ȳ e0 + k 1 X e1 + e 1 + e e 1e 1 + e 1e e 1 e + k e1 + e 1 + e e 1e 1 + e 1e e 1 e + k 1 k X e1 + e 1 e 1e 1 + e 1e e 1 e + e e 1 e 1 e 1 + e 1 e k 1 Ȳ X e 0 e 1 e 0 e 1 + e 0e k Ȳ e 0 e 1 e 0 e 1 + e ] 0e Further simplying, we get MSEt m = Ȳ θcy + k 1 X θ 1 Cz + θ Cx + k θcz + θ Cx + θ C xz + k 1 k X θ Cx + θ 1Cz + θ C xz k 1 Ȳ X θ C yx + θ 1 C yz k Ȳ ] θ C yx + θc yz 18

5 Khan and Al Hossain SpringerPlus 0165:73 Page 5 of 7 Now to find the minimum mean squared error of t m, we dferentiate Eq. 18 with respect to k 1 and k respectively and putting it equal to zero, that is MSEt m k 1 = 0 and MSEt m k = 0 k 1opt = Ȳ BC DE X AB E and k Ȳ AD CE opt = AB E. where A = θ 1 C z + θ C x, B = θc z + θ C x + θ C xz, C = θ C yx + θ 1 C yz, D = θ C yx + θc yz and E = θ 1 C z + θ C x + θ C xz. On substituting the optimum values of k 1 and k in Eq. 18 we get the minimum mean square error MSE of the proposed estimator t m up to order one is, given as AD MSEt m min = Ȳ θcy + BC CDE ] AB E Efficiency comparison In this section, we have compare the propose estimator with the other existing estimators By Eqs. 1 and 19, MSEt m min < MSEt 0 AD + BC CDE AB E ] > 0.. By Eqs. 3 and 19 MSEt m min < MSEt 1 AD + BC CDE ] AB E + θ Cx ρ yxc y C x > By Eqs. 5 and 19, MSEt m min < MSEt AD + BC CDE ] AB E θ Cy ρ yx > By Eqs. 7 and 19, MSEt m min < MSEt 3 AD θ + BC CDE ] C x Cx ρ yx C y + θ1 C z Cz ρ yz C y + > 0. AB E 5. By Eqs. 9 and 19, MSEt m min < MSEt 4 AD + BC CDE AB E θ ρ yx + θ 1ρ yz ] Cy > By Eqs. 13 and 19, MSEt m min < MSEt 5 θ Cx C xy + θ 1 4 Cz 4C yz AD + BC CDE ] + > 0. AB E

6 Khan and Al Hossain SpringerPlus 0165:73 Page 6 of 7 7. By Eqs. 14 and 19, MSEt m min < MSEt 6 ρ θ 1 Cy yx Cz C z AD ρ yx ρ yz θ ρ yx 4 C Cy + + BC CDE ] > 0. x AB E C x 8. By Eqs. 15 and 19, MSEt m min < MSEt 7 AD + BC CDE AB E + θ Cx 4 4C xy + θ 1 Cz yz ] C > 0. Numerical comparison To examine the performance of the proposed estimator with various existing estimators, we have considered a real data set from the literature the description of the population are, given by Population Source, Cochran y: Number of placebo children; x: Number of paralytic polio cases in the placebo group; z: Number of paralytic polio cases in the not inoculated group. N = 34, n = 15, n = 10, Ȳ = 4.9, X =.59, =.91, C y = 1.048, C x = , C z = 1.149, C yx = , C yz = , ρ yx = 0.736, ρ yz = , ρ xz = Table 1. We have use the following expression for Percentage Relative Efficiency PRE ] Vart 0 PRE = MSE 100, for j = 0, 1,, 3, 4, 5, 6, 7 and m. t j or Vartj Table 1 The mean square errors MSE s and the Percent relative efficiencies PRE s of the estimators with respect to t 0 Population Estimator MSE s PRE t 0,t j t t t t t t t t t m

7 Khan and Al Hossain SpringerPlus 0165:73 Page 7 of 7 Conclusion From the above table, we have observed that the proposed estimator has smaller mean square error and has higher percent relative efficiency than the other existing estimators. However, although the proposed estimator has the highest percent relative efficiency than other existing estimators for this one example, it could have lower relative efficiency for other populations. Further work is needed before it can be recommended for general use in practical surveys. Authors contributions The authors contributed equally and signicantly in writing this article. Both authors read and approved the final manuscript. Author details 1 Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad 060, Pakistan. Department of Mathematics, Faculty of Science, Jazan University, Jazan 097, Saudi Arabia. Acknowledgements The authors are very thankful to the editor and the anonymous learned referees for their valuable suggestions regarding the improvement of the paper. Competing interests The authors declare that they have no competing interests. Received: 9 September 015 Accepted: 18 May 016 References Bahl S, Tuteja RK 1991 Ratio and product type exponential estimator. Inf Optim Sci 1: Chand L 1975 Some ratio type estimator based on two or more auxiliary variables. Unpublished Ph.D. dissertation, Lowa State University, Ames, Lowa Cochran WG 1977 Sampling techniques. Wiley, New-York Hansen MH, Hurwitz WN 1943 On the theory of sampling from finite populations. Ann Math Stat 14: Khan M 015 Improvement in estimating the finite population mean under maximum and minimum values in double sampling scheme. J Stat Appl Probab Lett :1 7 Khan M 016 A ratio chain-type exponential estimator for finite population mean using double sampling. SpringerPlus 5:1 9 Khare BB, Srivastava U, Kumar K 013 A generalized chain ratio in regression estimator for population mean using two auxiliary characters in sample survey. J Sci Res Banaras Hindu Univ Varanasi 57: Kiregyera B 1980 A chain ratio-type estimator in finite population mean in double sampling using two auxiliary variables. Metrika 7:17 3 Kiregyera B 1984 Regression-type estimator using two auxiliary variables and model of double sampling from finite populations. Metrika 31:15 3 Singh BK, Choudhury S 01 Exponential chain ratio and product-type estimators for finite population mean under double sampling scheme. Glob J Sci Front Res Math Decis Sci 16: Singh GN, Majhi D 014 Some chain-type exponential estimators of population mean in two-phase sampling. Stat Trans 15:1 30 Singh HP, Singh S, Kim JM 006 General families of chain ratio type estimators of the population mean with known coefficient of variation of the second auxiliary variable in two phase sampling. J Korean Stat Soc 354: Singh R, Chuhan P, Swan N 007 Families of estimators for estimating population mean using known correlation coefficient in two phase sampling. Stat Trans 81:89 96 Singh R, Chuhan P, Swan N, Smarandache F 011 Improved exponential estimator for population variance using two auxiliary variables. Ital J Pure Appl Math 8: Srivastava SK 1970 A two phase estimator in sampling surveys. Austr J Stat 1:3 7 Srivastava SR, Khare BB, Srivastava SR 1990 A generalized chain ratio estimator for mean of finite population. J Indian Soc Agric Stat 41: Sukhatme BV 196 Some ratio type estimators in two-phase sampling. J Am Stat Assoc 57:68 63