Research Article Estimation of Population Mean in Chain Ratio-Type Estimator under Systematic Sampling

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1 Probability and Statistics Volume 2015 Article ID pages Research Article Estimation of Population Mean in Chain Ratio-Type Estimator under Systematic Sampling Mursala Khan 1 and Rajesh Singh 2 1 Department of Mathematics COMSATS Institute of Information Technology Abbottabad Pakistan 2 Department of Statistics Banaras Hindu University Varanasi India Correspondence should be addressed to Mursala Khan; mursala.khan@yahoo.com Received 7 September 2015; Accepted 11 October 2015 Academic Editor: Shein-chung Chow Copyright 2015 M. Khan and R. Singh. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. A chain ratio-type estimator is proposed for the estimation of finite population mean under systematic sampling scheme using two auxiliary variables. The mean square error of the proposed estimator is derived up to the first order of approximation and is compared with other relevant existing estimators. To illustrate the performances of the different estimators in comparison with the usual simple estimator we have taken a real data set from the literature of survey sampling. 1. Introduction and Literature Review Incorporating the knowledge of the auxiliary variables is very important for the construction of efficient estimators for the estimation of population parameters and increasing the efficiency of the estimators in different sampling design. Using the knowledge of the auxiliary variables several authors have proposed different estimation technique for the finite population mean of the study variable; Cochran 1 Tripathy 2 Kadilar and Cingi 3 Singh et al. 5 Khan and Arunachalam 6 Lone and Tailor 7 Khan 8 Khan and Hussain9andKhanetal.10haveworkedontheestimation of population parameters using auxiliary information. In the present paper we will work on the estimation of population mean using the knowledge of the auxiliary variables under systematic sampling. Various statisticians have worked on the estimation of population mean in systematic sampling: Cochran 11 Hansen et al. 12 Robson 13 Swain 1 Singh 15 Shukla 16 Kushwaha and Singh 17 Banarasi et al. 18 R. Singh and H. P. Singh 19 Singh et al. 20 Singh and Solanki 21 Singh and Jatwa 22 Singh etal.232tailoretal.25vermaandsingh26and Verma et al. 27 and so forth. Consider a finite population U=U 1 U 2...U N } of size N units numbered from 1 to N in some order. A sample of size n units is taken at random from the first k units and every kth subsequent unit; then N=nkwhere n and k are positive integers; thus there will be k samples (clusters) each of size n andobservethestudyvariatey and auxiliary variate x for eachandeveryunitselectedinthesample.let(y ij x ij )fori= 12...kand j = 12...n: denote the value of jth unit in theith sample. Then the systematic sample means are defined as follows: y sy =t 0 =(1/n) n j=1 y ij and x sy =(1/n) n j=1 x ij are the unbiased estimators of the population means Y = (1/N) N j=1 y ij and X = (1/N) N j=1 x ij of y and x respectively. Let S 2 y = (1/(N 1)) k i=1 n j=1 (y ij Y) 2 S 2 x = (1/(N 1)) k i=1 n j=1 (x ij X) 2 ands 2 z =(1/(N 1)) k i=1 n j=1 (z ij Z) 2 be the population variances of the study variable and the auxiliary variables respectively with the corresponding population covariance s S yx = (1/(N 1)) k i=1 n j=1 (x ij X)(y ij Y) S yz = (1/(N 1)) k i=1 n j=1 (y ij Y)(z ij Z) and S xz = (1/(N 1)) k i=1 n j=1 (x ij X)(z ij Z) among the three variables y x andz respectively.alsoc 2 y C2 x andc2 z aretheknownpopulationcoefficientsofvariationofthestudy variable and the auxiliary variables respectively. To obtain the properties of the estimators up to first order of approximation we use the following errors terms:

2 2 Probability and Statistics e 0 =(y sys Y)/Y e 1 =(x sys X)/Xande 2 =(z sys Z)/Z such that E(e i )=0fori=01and2. The first order of approximation of the above errors terms is given by The classical ratio and product estimators for finite population mean suggested by Swain 1 and Shukla 16 are given by E(e 2 0 )=θρ y C2 y E(e 2 1 )=θρ x C2 x E(e 2 2 )=θρ z C2 z E(e 0 e 1 )=θkcx ρ 2 y ρ x E(e 0 e 2 )=θk Cz ρ 2 y ρ z (1) t 1 = y sy ( X x sy ) t 2 = y sy exp ( z sy Z ). The mean square errors of the estimators to the first order of approximation are given as follows: () MSE (t 1 )=λy 2 ρ y C2 y +ρ x C2 x (1 2k ρ ) (5) E(e 1 e 2 )=θk C 2 z ρ x ρ z MSE (t 2 ) =λy 2 ρ y C2 y +ρ z C2 z (1+2k ρ2 ). (6) where λ=( N 1 nn ) ρ yx = S yx S y S x ρ yz = S yz S y S z The usual regression estimator using single auxiliary variable and its variance is given as follows: t 3 = y sy +b(x x sy ) (7) MSE (t 3 )=λρ y S2 y 1 ρ2 yx. (8) Utilizing the known knowledge of the auxiliary variable Singh et al. 20 suggested the following ratio and product type exponential estimators: ρ xz = S xz S x S z k= ρ yxc y C x k = ρ yzc y C z ρ y =1+(n 1) ρ y} (2) t = y sy exp ( X x sy X+x sy ) t 5 = y sy exp ( x sy X x sy + X ). The mean square errors of the estimators up to first order of approximation are given by (9) ρ x =1+(n 1) ρ x} ρ z =1+(n 1) ρ z} ρ = ρ y ρx ρ 2 = ρ y ρz ρ 1 = ρ x ρz where ρ y ρ x andρ z are the intraclass correlation among the pair of units for the variables y xandzrespectively. The variance of the usual unbiased estimator for population mean is Var (t 0 )=λy 2 ρ y C2 y. (3) MSE (t )=λy 2 ρ y C2 y + ρ x C2 x MSE (t 5 )=λy 2 ρ y C2 y + ρ x C2 x (1 k ρ ) (10) (1 + k ρ ). (11) After that Tailor et al. 25 define the following ratio-cumproduct estimator for the population mean Y: t 6 = y sy ( X )( z sy ). (12) x sy Z The mean square error of the estimator t 6 up to first order of approximation is given by MSE (t 6 )=λy 2 ρ y C2 y +ρ x C2 x (1 2k ρ ) +ρ z C2 z (1 2k ρ1 )+2k Cz ρ 2 y ρ z. (13)

3 Probability and Statistics 3 2. Proposed Estimator In this section we have proposed the following regression in ratio-cum-product type estimator for the unknown population mean under systematic sampling: X t m = y sy ( X+b yx (x sy X) ) ( Z+b yz (z sy Z) )δ2 Z δ 1 (1) where δ 1 and δ 2 are the unknown constants whose values are to be found for the minimum mean square error. The mean square error (MSE) of the estimator up to first order of approximation is MSE (t m )=λy 2 ρ y C2 y +δ2 1 β2 yx ρ x C2 x +δ2 2 β2 yz ρ z C2 z (ii) By (16) and (5) MSE(t m ) MSE(t 1 ) if ρ y k 2 (Cx 2 k2 Cz 2) } } +ρ x C2 x (1 2k ρ ) 0. (iii) By (16) and (6) MSE(t m ) MSE(t 2 ) if ρ y k 2 (Cx 2 k2 Cz 2) } } +ρ z C2 z (1 + 2k ρ2 ) 0. (iv) By (16) and (8) MSE(t m ) MSE(t 3 ) if (18) (19) 2δ 1 β yx kc 2 x ρ y ρ x +2δ 2β yz k C 2 z ρ y ρ z 2δ 1 δ 2 β yx β yz k C 2 z ρ x ρ z. (15) k 2 x C2 z C2 y ρ2 yx + (k k C 2 z kc2 x )2 (Cx 2 k2 Cz 2) 0. (20) On differentiating (15) with respect to δ 1 and δ 2 weobtain theminimummeansquarederroroftheestimatort m which is given by MSE (t m ) =λy 2 ρ y C 2 y k2 x C2 z (k k C 2 z kc2 x )2 (Cx 2 k2 Cz 2) (16) where the optimum values are δ 1 = ρ y (kc2 x k k C 2 z )/ β yx ρx (C2 x k2 C 2 z ) and δ 2 =( ρy /β yz ρz )k (kc 2 x k k C 2 z )/(C2 x k2 C 2 z ) k }. 3. Comparison In this section we have compared the MSE of the proposed estimator with the MSEs of simple estimator Swain 1 estimator Shukla 16 estimator Singh et al. 20 estimators and Tailor et al. 25 estimator and found some theoretical conditions under which the proposed estimator will always perform better: (i) By (16) and (3) MSE(t m ) MSE(t 0 ) if k 2 x C2 z + (k k C 2 z kc2 x )2 (Cx 2 k2 Cz 2) 0. (17) (v) By (16) and (10) MSE(t m ) MSE(t ) if ρ y k 2 (Cx 2 k2 Cz 2) } } + ρ x C2 x (1 k ρ ) 0. (vi) By (16) and (11) MSE(t m ) MSE(t 5 ) if ρ x C2 x +ρ y (1 + k ρ ) k 2 (Cx 2 k2 Cz 2) } 0. } (vii) By (16) and (13) MSE(t m ) MSE(t 6 ) if ρ x C2 x (1 2k ρ )+ρ z C2 z (1 2k ρ +2k C 2 z ρ y ρ z +ρ y 1 ) k 2 (Cx 2 k2 Cz 2) } 0. } (21) (22) (23)

4 Probability and Statistics Table 1: The percent relative efficiency of different estimators with respect to t 0. Population Estimator MSE(t α ) PRE(t α t 0 ) t t t t t t t t m Numerical Comparison For comparing the theoretical efficiency conditions of the different estimators numerically we have used the following real data set. Population 1 (source: Tailor et al. 25). Consider N = 15 n=3 X =.7 Y = 80 Z = 8.0 C y = 0.56 C x = 0.28 C z = 0.3 S 2 y = 2000 S 2 x = S 2 z = S yx = S yz = S xz = ρ yx = ρ yz = ρ xz = ρ y = ρ x = ρ z = (2) For the percent relative efficiencies (PREs) of the estimator we use the following formula and the results are shown in Table 1: PRE(t α t 0 )=MSE(t 0 )/MSE(t α ) 100forα = and m. 5. Conclusion A chain ratio-type estimator is proposed under double sampling scheme using two auxiliary variables and the properties of the proposed estimator are derived up to first order of approximations. Both theoretically and empirically it has been shown that the recommended estimator performed better than the other competing estimators in terms of higher percent relative efficiency. Hence looking on the dominance nature of the proposed estimator may be suggested for its practical applications. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgment The authors are thankful to the anonymous learned referees for their valuable suggestions regarding the improvement of the paper. References 1 W. G. Cochran Sampling TechniquesJohnWiley&SonsNew York NY USA T. P. Tripathy A general class of estimators of population ratio Sankhya Series Cvol.2pp C. Kadilar and H. Cingi Ratio estimators in simple random sampling Applied Mathematics and Computation vol.151no. 3pp C. Kadilar and H. Cingi Improvement in estimating the population mean in simple random sampling Applied Mathematics Lettersvol.19no.1pp R. Singh P. Chauhan N. Sawan and F. Smarandache Improved exponential estimator for population variance using two auxiliary variables Italian Pure and Applied Mathematicsvol.28pp M. Khan and A. Arunachalam Estimation of population variance using the knowledge of kurtosis of an auxiliary variable under simple random sampling International Journal ofappliedscienceandmathematicsvol.1no.2pp H.A.LoneandR.Tailor Dualtoseparateproducttype exponential estimator in sample surveys Statistics Applications & Probability Lettersvol.2no.2pp M. Khan Improvement in estimating the finite population mean under maximum and minimum values in double sampling scheme Statistics Applications & Probability Lettersvol.2no.2pp M.KhanandS.Hussain Animprovedclassofratio-type estimators for finite population mean under maximum and minimum values Science Internationalvol.27no.2pp M. Khan S. Ullah A. Y. Al-Hossain and N. Bashir Improved ratio-type estimators using maximum and minimum values under simple random sampling scheme Hacettepe Mathematics and Statisticsvol.no.pp

5 Probability and Statistics 5 11 W. G. Cochran The estimation of the yields of cereal experiments by sampling for the ratio of grain to total produce The JournalofAgriculturalScience vol. 30 no. 2 pp M.H.HansenW.N.HurwitzandM.Gurney Problemsand methods of the sample survey of business The the American Statistical Association vol.1no.23pp D. S. Robson Application of multivariate polykays to the theory of unbiased ratio-type estimation TheJournalofthe American Statistical Associationvol.59pp A. K. P. C. Swain The use of systematic sampling ratio estimate Indian Statistical Associationvol.2pp M. P. Singh Ratio cum product method of estimation Metrikavol.12no.1pp N. D. Shukla Systematic sampling and product method of estimation in Proceeding of All India Seminar on Demography and Statistics BHU Varanasi India K. S. Kushwaha and H. P. Singh Class of almost unbiased ratio and product estimators in systematic sampling the Indian Society of Agricultural Statistics vol.1no.2pp BanarasiS.N.S.KushwahaandK.S.Kushwaha Aclassof ratio product and difference (RPD) estimators in systematic sampling Microelectronics Reliability vol.33no.pp R. Singh and H. P. Singh Almost unbiased ratio and product type estimators in systematic sampling Questiiovol.22no.3 pp H. P. Singh R. Tailor and N. K. Jatwa Modified ratio and product estimators for population mean in systematic sampling JournalofModernAppliedStatisticalMethodsvol.10no.2pp H. P. Singh and R. S. Solanki An efficient class of estimators for the population mean using auxiliary information in systematic sampling Statistical Theory and Practice vol.6no. 2 pp H. P. Singh and N. K. Jatwa A class of exponential type estimators in systematic sampling Economic Quality Control vol.27no.2pp R. Singh S. Malik and V. K. Singh An improved estimator in systematic sampling Scientific Research vol.56pp R. Singh S. Malik M. K. Chaudhary H. Verma and A. A. Adewara A general family of ratio-type estimators in systematic sampling Reliability and Statistical Studiesvol.5no. 1pp T. Tailor N. K. Jatwa and H. P. Singh A ratio-cum-product estimator of finite population mean in systematic sampling Statistics in Transition vol. 1 no. 3 pp H. K. Verma and R. Singh A family of efficient estimator in circular systematic sampling JournalofAdvancedComputing vol. 3 no. 2 pp H. K. Verma R. D. Singh and R. Singh Some improved estimators in systematic sampling under non-response National Academy Science Lettersvol.37no.1pp

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