A ratio chain-type exponential estimator for finite population mean using double sampling

Khan SpringerPlus (2016)5:86 DOI 10.1186/s40064-016-1717-4 Open Access RESEARCH A ratio chain‑type exponential estimator for finite population mean using double sampling Mursala Khan1,2* *Correspondence: 1 Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad 22060, Pakistan Full list of author information is available at the end of the article Abstract In this article, we have proposed a ratio chain-type exponential estimator for finite population mean of the study variable under double sampling scheme using auxiliary variables. The large sample properties of the suggested strategy are derived up to first order, of approximation, and its competence conditions are carried out under which the suggested estimator is performed better than the other existing estimators discussed in the literature. An empirical study shows that the suggested strategy is more efficient than the other relevant competing estimators under two phase sampling scheme. Keywords: Double sampling, Study variable, Bias, Auxiliary variable, Mean squared-error, Estimator, Efficiency Introduction and literature review To increase the precision of estimators for population mean of the study variable under double sampling design, a lot of works have been done in the field of sample survey and when the study variable is strongly connected with the auxiliary variables the precision of the estimators can be more and more. Using the knowledge of the auxiliary variables several authors have proposed different estimation technique for finite population mean of the study variable, Sukhatme (1962), have developed a general ratio-type estimator. Chand (1975), have suggested two chain ratio-type estimators to estimate the population mean using two auxiliary variables (Kiregyera 1980, 1984; Srivnstava et al. 1990; Bahl and Tuteja 1991; Srivastava 1970; Cochran 1977; Singh et al. 2006, 2007, 2011; Dash and Mishra 2011; Singh and Choudhury 2012; Khare et al. 2013; Khare and Rehman 2013) etc. Let us consider a finite population of size N of different units U = {U1 , U2 , U3 , . . . , UN }. Let y and x be the study and the auxiliary variable with corresponding values yi and xi respectively for i-th unit i = {1, 2, 3, . . . , N } is defined on a finite population U.    N    N Let Ȳ = 1 N i=1 yi and X̄ = 1 N i=1 xi be the corresponding population means of the study as well as auxiliary variable respectively. Also let 2    N  2    N  Sy2 = 1 N and Sx2 = 1 N be the corresponding i=1 xi − X̄ i=1 yi − Ȳ © 2016 Khan. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http:// creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Khan SpringerPlus (2016)5:86 Page 2 of 9 population variances of the study as well as auxiliary variable respectively and letCy and Cx be the coefficient of variation of the study as well as auxiliary variable respectively, and ρyx be the correlation coefficient between x and y. Let y and x be the study and the auxiliary variable with corresponding values yi and    n xi respectively for i-th unit i = {1, 2, 3, . . . , n} in the sample. Let ȳ = 1 n i=1 yi and    n x̄ = 1 n x be the corresponding unbiased sample means of the study as well as i=1 i auxiliary variable respectively. 2    n     n 2 and sx2 = 1 n − 1 Also let sy2 = 1 n − 1 i=1 yi − ȳ i=1 (xi − x̄) be the corresponding unbiased sample variances of the study as well as auxiliary variable respectively. Let Syx, Syz and Sxz be the co-variances between their respective subscripts respecs tively. Similarly byx = syx2 is the corresponding sample regression coefficient y on x based x S on a sample of size n. Also Cy = Ȳy , Cx = SX̄x and Cz = SZ̄z are the coefficients of variation of the study and auxiliary variables respectively. The usual unbiased estimator to estimate the population mean of the study variable is n ȳ0 = 1 yi n (1) i=1 The variance of the estimator ȳ up to first order of approximation is, given by   V ȳ0 = f1 Ȳ 2 Cy2 (2) The usual ratio and regression estimators in two phase sampling and their mean square error are, given as follows ȳ1 = ȳ ′ x̄ x̄ (3)   ȳ2 = ȳ + byx x̄′ − x̄ (4)      MSE ȳ1 = Ȳ 2 f1 Cy2 + f3 Cx2 − 2ρyx Cy Cx (5)       2 2 Var ȳ2 = Sy2 f1 1 − ρyx + f2 ρyx (6) The mean squared error and variance are given below       where f1 = n1 − N1 , f2 = n1′ − N1 and f3 = n1 − n1′ . Chand (1975), proposed the following chain ratio-type estimator in double sampling by incorporating the knowledge of two auxiliary variables, the suggested estimator is, given by ȳ3 = ȳ x̄′ Z̄ x̄ z̄ ′ The mean square error of the suggested estimator is, given as (7) Khan SpringerPlus (2016)5:86 Page 3 of 9  � � 2 2 f C + f − 2ρ C C C 1 3 yx y x y x � �   � � MSE ȳ3 = Ȳ 2   2 + f2 Cz − 2ρyz Cy Cz (8) Kiregyera (1984), suggested the following chain-type exponential estimators in two phase sampling, the suggested estimators are given as ȳ4 =   ȳ  ′ x̄ + bxz Z̄ − z̄ ′ x̄     ȳ5 = ȳ + byx x̄′ − x̄ − bxz Z̄ − z̄ ′ (9) (10) The mean square errors of the suggested estimators, up to first order of approximation are, given as follows   MSE ȳ4 = Ȳ 2    f1 Cy2 + f3 Cx Cx − 2ρyx Cy   +f2 ρxz Cx ρxz Cx − 2ρyz Cy    f ρ ρ ρ ρ − 2ρ   2 yx xz yx xz yz MSE ȳ5 = Ȳ 2 Cy2 2 + f1 − f3 ρyx (11) (12) Searls (1964), proposed an estimation procedure for population mean using known knowledge of the coefficient of variation of the auxiliary variable ȳ∗ = aȳ (13) var(ȳ∗ ) = (1 − B)f1 Ȳ 2 Cy2 (14)  −1 where a = 1 + f1 Ȳ 2 Cy2 and B = f1 Ȳ 2 Cy2 Khare and Rehman (2013), have proposed improved chain type estimators for population mean using auxiliary information, the suggested estimators are given by ȳ6 =   ȳ∗  ′ x̄ + b Z̄ − z̄ ′ x̄     ȳ7 = ȳ∗ + b1 x̄′ − x̄ − b2 Z̄ − z̄ ′ (15) (16) where b, b1 and b2 are constants. The mean square errors of the suggested estimators, are, given by  + b2 R2 f2 Cz2 − 2bR(1 − B)f2 ρyz Cy Cz � � � �   MSE ȳ6 = Ȳ 2  + f3 Cx Cx − 2(1 − B)ρyx Cy   + (1 − B)f1 Cy2 and (17) Khan SpringerPlus (2016)5:86 Page 4 of 9 � �  b1 f3 X̄Cx b1 X̄Cx − 2(1 − B)Ȳ ρyx Cy � � � �  MSE ȳ7 =  + b1 b2 Z̄f2 Cz b1 b2 Z̄Cz − 2Ȳ (1 − B)ρyz Cy   + Ȳ 2 (18) (1 − B)f1 Cy2  −k2 ± k22 −4k1 k3 (1−B)Cyz where the optimum values of b, b1 and b2 are bopt = RC 2 , b1opt = and 2k1 z M b2opt = b1opt .  Ȳ (1−B)Cyz , k1 = f3 b12 X̄ 2 Cx2, k2 = f3 (B − 1)Ȳ X̄b1 Cyx , R = Z̄ X̄ and Also M = Z̄C 2 z   k2 = M Z̄f2 Cz M Z̄Cz − Ȳ (1 − B)ρyz Cy . Singh et al. (2013), recommended a class of exponential chain ratio-product type estimator for estimating population (...truncated)