Efficient and Unbiased Estimation Procedure of Population Mean in Two-Phase Sampling

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1 Journal of Modern Applied Statistical Methods Volume 5 Issue Article --06 Efficient and Unbiased Estimation Procedure of Population Mean in Two-Phase Sampling Reba Maji Indian School of Mines, Dhanbad, India, rebamaji09@gmail.com Arnab Bandyopadhyay Asansol Engineering College, Asansol, India, arnabbandyopadhyay4@gmail.com G. N. Singh Indian School of Mines, Dhanbad, India, gnsingh_ism@yahoo.com Follow this and additional works at: Part of the Applied Statistics Commons, Social and Behavioral Sciences Commons, and the Statistical Theory Commons Recommended Citation Maji, Reba; Bandyopadhyay, Arnab; and Singh, G. N. (06) "Efficient and Unbiased Estimation Procedure of Population Mean in Two-Phase Sampling," Journal of Modern Applied Statistical Methods: Vol. 5 : Iss., Article. DOI: 0.7/jmasm/ Available at: This Regular Article is brought to you for free and open access by the Open Access Journals at DigitalCommons@WayneState. It has been accepted for inclusion in Journal of Modern Applied Statistical Methods by an authorized editor of DigitalCommons@WayneState.

2 Journal of Modern Applied Statistical Methods November 06, Vol. 5, No., doi: 0.7/jmasm/ Copyright 06 JMASM, Inc. ISSN Efficient and Unbiased Estimation Procedure of Population Mean in Two-Phase Sampling Reba Maji Indian School of Mines Dhanbad, India Arnab Bandyopadhyay Asansol Engineering College Asansol, Inida G. N. Singh Indian School of Mines Dhanbad, India In this paper, an unbiased regression-ratio type estimator has been developed for estimating the population mean using two auiliary variables in double sampling. Its properties are studied under two different cases. Empirical studies and graphical simulation have been done to demonstrate the efficiency of the proposed estimator over other estimators. Keywords: Double sampling, study variable, auiliary variable, chain-type, regression, bias, variance, efficiency Introduction The use of supplementary information on auiliary variable for estimating the finite population mean of the variable under study has played an eminent role in sampling theory and practices. Auiliary information may be truthfully utilized at the planning, design, and estimation stages to develop improved estimation procedures in sample surveys. Ratio, product, and regression methods of estimation are good eamples in this contet. Use of auiliary information at the estimation stage was introduced during the 90 s with a comprehensive theory provided by Cochran (940). Sometimes, information on auiliary variable may be readily available for all the units of a population; for eample, tonnage (or seat capacity) of each vehicle or ship is known in survey sampling of transportation, and number of beds available in different hospitals may be known well in advance in health care surveys. If such information is lacking, it is sometimes relatively cheap to take a large preliminary sample where an auiliary variable alone is measured. Such practice is applicable in two-phase (or double) sampling. Two-phase sampling happens to be a powerful and cost-effective (economical) technique to generate reliable estimates of the unknown population parameters of the auiliary variables in a first phase sample. Reba Maji is in the Department of Applied Mathematics. at: rebamaji09@gmail.com. 7

3 EFFICIENT AND UNBIASED ESTIMATION OF POPULATION MEAN In order to construct an efficient estimator of the population mean of the auiliary variable in a first-phase (preliminary) sample, Chand (975) introduced the technique of chaining another auiliary variable with the first auiliary variable by using the ratio estimator in the first phase sample. This estimator is known as the chain-type ratio estimator. This work was further etended by Kiregyera (980; 984), Sahoo and Sahoo (99), Tracy, Singh, and Singh (996), Singh and Espejo (007), Gupta and Shabbir (007), Dash and Mishra (0), Shukla, Pathak, and Thakur (0), and Choudhury and Singh (0), among others, who proposed various chain-type ratio and regression estimators. It may be noted that the most of these estimation procedures of the population mean in two-phase sampling are biased which becomes a serious drawback for their practical applications. Encouraged and fascinated with the work discussed earlier, we have proposed an unbiased regression-ratio type estimator of the population mean and studied its properties under two different structures of two-phase sampling. Performances of the proposed estimator have been eamined through empirical and graphical means of comparisons. Suitable recommendations to the survey statistician are made. Methodology Sample Structure and Some Eisting Estimation Procedures Let y k, k, and z k be the values of the study variable y, first auiliary variable, and second auiliary variable z, respectively, associated with the k th unit of the finite population U = (U, U, U,, U N ). The intent is to estimate the population mean Y of the study variable y in the presence of auiliary variables and z when the population mean X of is unknown but information on z is readily available for all the units of population. To estimate Y, a first-phase sample S' (S' U) of size n is drawn via a simple random sampling without replacement (SRSWOR) scheme from the entire population U and observed for the auiliary variable to furnish the estimate of X. Net, a second-phase sample S of size m (m n) is drawn by SRSWOR according to the following rules to observe the study variable y: Case I: Case II: Second-phase sample is drawn as a subsample of the first-phase sample Second-phase sample is drawn independently of the first-phase sample 7

4 MAJI ET AL. The case where the second sample is drawn independent of the first was considered by Bose (94). In the sections below, we use the following notations: m, n, y m, z m, z n: Sample mean of the respective variables of the sample sizes shown in subscripts. X, Y, Z : Population mean of, y, and z, respectively. ρ y, ρ yz, ρ z : Correlation coefficient between the variables shown in subscripts. C, C y, C z : Coefficient of variance of, y, and z respectively. S yz : Population covariance between y and z. S : Population mean square of z. z s yz (m): Sample covariance between y and z based on the sample of size m. s m : Sample mean square of z based on the sample of size m. z β yz : Population regression coefficient between the variables y and z. b z (n), b yz (m), b y (m): Sample regression coefficient between the variables shown in subscripts and based on samples of the size indicated in braces. as To estimate the population mean Y, the classical ratio estimator is presented y r y m X () m If X is unknown, we estimate Y under the two-phase sampling set up as t y m n () m S. K. Srivastava (97) generalized the ratio method of estimation, and its structure in two-phase sampling is given as t n ym m () where α is a real scalar which can be suitably determined by minimizing the mean square error (MSE) of the estimator. 7

5 EFFICIENT AND UNBIASED ESTIMATION OF POPULATION MEAN The way in which the estimate of Y is improved using the auiliary information on can also be etended to improve the estimate of X in the firstphase sample if another auiliary variable, z, closely related to but remotely related to y is used. Thus, assuming that the population mean of the auiliary variable z is known, Chand (975) proposed a chain-type ratio estimator as t y m n Z m z (4) n Similarly, for negative correlation between the variables y and, the chain-type product estimator is defined as z (5) m n t4 ym n Z Kiregyera (984) suggested the chain linear regression estimator in double sampling as t y b m b n Z z 5 m y n z n m (6) Singh and Espejo (007) considered a ratio-product type estimator in double sampling as n t6 ym k k m m n (7) Proposed Estimator The suggested unbiased regression-ratio type estimator for estimating the population mean Y is * where ym ym byz mz zm chosen so that i * n R di ym i m T (8) and the d i (i =,, ) are real scalars suitably 74

6 MAJI ET AL. di (9) i Remark : The estimator T R is proposed under the following conditions:. The sum of the weights is one.. The weights of the linear form are chosen such that the approimate bias is zero.. The approimate variance attains minimum. Properties of the Estimator TR Note from (8) that the proposed estimator T R is biased for Y. Following Remark, it may be made unbiased for Y. The variance V(.) up to the first order of approimations are derived under large sample approimations using the following transformations:,,,, y Y e X e X e m m n z Z e s m S e s m S e m 4 yz yz 5 z z 6 where E(e i ) = 0 and e i < for all i =,, 6. Under the above transformations the estimator T R takes the following form: R i yz (0) i T d Y e Ze e e e e The bias and mean square error of the estimator was derived separately for the Cases I and II of the two-phase sampling structure. Case I: When the second phase sample is drawn as a subsample of the first phase sample. In this case we have the following epected values of the sample statistics: 75

7 EFFICIENT AND UNBIASED ESTIMATION OF POPULATION MEAN E e fmcy, E es fmc, E e fc, E e4 fmcz, E ee fm ycyc, E ee fyc yc, E ee 4 fmyzcycz, E ee fc, E ee4 fmzccz, 0 00 E ee4 f zccz, E e4e5 fm, E e4e6 f m ZS yz ZS z () where fm, f, m N n N N X y Y z Z p q r pqr i i i N i and p, q, r 0 are integers. Epanding the terms of (0) binomially and using the results from (), we have derived the epression of bias and mean square error of the estimator T R up to the first order of approimations as B TR ETR Y M f f PYC Y C C Z C C 00 0 m yz y y yz z z Sz S yz d f YC TR ETR Y Y f C Z f C YZ f C C P Y f C m y yz m z yz m yz y z PYf Z C C Y C C yz z z y y () () where P id f i, (4) i m n 76

8 MAJI ET AL. Minimization of mean square error in () with respect to P yields its optimum value as Cy P y yzz (5) C Substituting the optimum value of P in (), we obtain the minimum mean square error of T R as Further, from (4) and (5), T f S f S Min.M (6) R m y yz y y yz z C y idi y yzz (7) i C P which we will denote with R. From (9) and (7), it may be noted that the two equations in three unknowns are not sufficient to find the unique values of the d i (i =,, ). In order to get unique values of the d i, impose the linear constraint Thus, from (), where B T 0 (8) R Kd K Yf C d Kd M (9) K f 0 00 YC Y ycyc Z yzzccz, M fm yz Syz S z Equations (9), (7), and (9) can be written in matri form as 77

9 EFFICIENT AND UNBIASED ESTIMATION OF POPULATION MEAN d d R K K Yf C K d M (0) Solving (0), the unique values of the d i are d M R K fyc fyc d RK M fyc d fyc M RK R fyc () From (), substituting the values of d, d, and d into (8) yields the unbiased optimum regression-ratio type estimator as * n TR M R K fyc ym fyc m * n RK M y m fyc m * n M RK R f YC ym fyc m () whose variance up to the first degree of approimations is given by T f S f S V () R m y yz y y yz z Case II: When the second-phase sample is drawn independently of the firstphase sample. In this case, the epected values of the sample statistics are: 78

10 MAJI ET AL. 4 4 E e fmcy, E e fmc, E e fc, E e fmcz, E e e fm ycyc, E e e fmyzcyc, z 0 00 E ee4 fmzccz, E e4e5 fm, E e4e6 fm, ZS yz ZSz Eee Eee Eee4 0 (4) Proceeding as in Case I, the unbiased optimum regression-ratio type estimator is obtained as * n TR f fm G WG Aym f fm YC m * n WG A f G ym f fm YC m * n fm G WG A ym f fm YC m (5) with variance up to the first order of approimations as fm V f f TR fmsy yz Sy y yz z m (6) where W f YC f YC f Y C C f Z C C f m m y y m yz z z m y G y yzz f f m C A fm yz S 0 00 yz Sz Remark : The unique value of the scalars d i depend on unknown population parameters such as β z, β yz, μ 0, μ 00, C, C y, C z, X, Y, ρ y, and ρ z. Thus, to make the estimator practicable, these unknown population parameters may be estimated with C 79

11 EFFICIENT AND UNBIASED ESTIMATION OF POPULATION MEAN their respective sample estimates or from past data or guessed from eperience gathered over time. Such problems are also considered by Reddy (978), Tracy et al. (996), and Singh and Espejo (007). Results Efficiency Comparison To eamine the performance of our proposed estimator, we have considered some contemporary estimators of population mean which are discussed in a previous section. The mean square errors/minimum mean square errors of the estimators t i (i =,,, 6) are given below for both cases of two-phase sampling structure considered in this paper: Case I: Case II: M t Y fmcy fc f ycyc Min.M t5 S f f M M t 6 t S y fm f y M t Y fmcy fc fc z fycyc fyzcyc z M t4 Y fmcy fc fc z fycyc fyzcyc z 6 y m y y z y z yz S y fm f y M t Y fmcy fmc fc fm ycyc Min.M t fms y y M t Y fmcy fmc fc z fmycyc fzcc z M t4 Y fmcy fmc fc fc z fmycyc fzcc z M M t5 S y fm f y f y z t f S m y y 80

12 MAJI ET AL. where f fm f m. The superiority of the suggested estimator has been demonstrated over the estimators t i (i =,,, 6) through numerical illustrations and graphical interpretation. Numerical Illustrations Five natural population data sets were selected to illustrate the efficiency of the proposed estimator. The source of the populations, the nature of the variables y,, z and the values of the various parameters are as follows: Population I: (Murthy, 967) y: Area under wheat in 964. : Area under wheat in 96. z: Cultivated area in 96. Population II: (Sukhatme & Sukhatme, 970) y: Area (acres) under wheat in 97. : Area (acres) under wheat in 96. z: Total cultivated area (acres) in 9. Population III: (S. K. Srivastava, 97) y: yield per plant. : Height of the plant. z: Base diameter. Population IV: (Anderson, 958) y: Head length of second son. : Head length of first son. z: Head breadth of second son. Population V: (R. S. Srivastava, Srivastava, & Khare, 989) y: measurement of weight of children. : Mid-arm circumference of children. z: Skull circumference of children. 8

13 EFFICIENT AND UNBIASED ESTIMATION OF POPULATION MEAN Table. Parametric values of different populations Population N n m Y r ρ y ρ yz ρ z C y C C z I II III IV V Table. PREs of different estimators (Case I) Population y m t t t t 4 t 5 t 6 T R I 00 * * * II * III * IV * V * Table. PREs of different estimators (Case II) Population y m t t t t 4 t 5 t 6 T R I 00 * * * II 00 * * III 00 * * IV 00 * * V 00 * * The values of various parameters obtained from the above populations are presented in Table. To have a tangible idea about the performance of the proposed estimator T R, the percent relative efficiencies (PREs) of T R and other estimators were computed with respect to the sample mean estimator y m, and the results are demonstrated in Tables -. The PRE of an estimator T with respect to a sample mean estimator y is defined as V y PRE 00 (7) M T where M(T) denotes the MSE/Minimum MSE of an estimator T. 8

14 MAJI ET AL. Graphical Interpretation The performance of the proposed estimator is illustrated by means of pictorial representation for different choices of correlations. This could not only improve the readability of the results but also allows the comparison of a much denser grid of different correlation values. For N = 00, n = 50, m = 0, and different values of ρ y, ρ yz, ρ z, the PREs of the proposed estimator T R with respect to y m are computed and presented in Figures -. Note that the X-ais, Y-ais, and Z-ais are denoting ρ y, ρ yz, and PRE, respectively, and that ρ z is assumed to be 0.5. Figure. PRE of TR (Case I) 8

15 EFFICIENT AND UNBIASED ESTIMATION OF POPULATION MEAN Figure. PRE of TR (Case II) Conclusion From Table and Table, it may be observed that, under different structures of two-phase sampling set up, the suggested estimator T R is superior to the eisting ones. It can also be noted that, for high positive values of correlation coefficients, the estimator T R yields impressive gains in efficiencies over the conventional estimators of population mean. From Figures and, it is observed that, for fied values of ρ z, the PRE of the proposed estimator is increasing with increasing values of ρ y and ρ yz. This phenomenon indicates that suggested estimator could perform satisfactorily if highly positive correlated auiliary variables are available. Therefore, the proposed estimator T R is more justified in comparison with the previous work of similar nature. Hence, it may be recommended to the survey practitioners for their use in real life problems. 84

16 MAJI ET AL. References Anderson, T. W. (958). An introduction to multivariate statistical analysis. New York: John Wiley & Sons. Bose, C. (94). Note on the sampling error in the method of double sampling. Sankhyā: The Indian Journal of Statistics, 6(), 9-0. Chand, L. (975). Some ratio type estimators based on two or more auiliary variables (Unpublished doctoral thesis). Iowa State University, Ames, IA. Choudhury, S., & Singh, B. K. (0). A class of chain ratio-product type estimators with two auiliary variables under double sampling scheme. Journal of the Korean Statistical Society, 4(), doi: 0.06/j.jkss Cochran, W. G. (940). The estimation of the yields of cereal eperiments by sampling for ratio gain to total produce. The Journal of Agricultural Science, 0(0), doi: 0.07/S Dash, P. R., & Misra, G. (0). An improved class of estimators in twophase sampling using two auiliary variables. Communication in Statistics Theory and Methods, 40(4), doi: 0.080/ Gupta, S., & Shabbir, J. (007). On the use of transformed auiliary variables in estimating population mean by using two auiliary variables. Journal of Statistical Planning and Inference, 7(5), doi: 0.06/j.jspi Kiregyera, B. (980). A chain ratio-type estimator in finite population double sampling using two auiliary variables. Metrika, 7(), 7-. doi: 0.007/BF Kiregyera, B. (984). Regression type estimators using two auiliary variables and the model of double sampling from finite populations. Metrika, (), 5-6. doi: 0.007/BF0950 Murthy, M. N. (967). Sampling theory and methods. Calcutta, India: Statistical Publishing Society. Reddy, V. N. (978). A study on the use of prior knowledge on certain population parameters in estimation. Sankhyā: The Indian Journal of Statistics, 40(), 9-7. Sahoo, J., & Sahoo, L. N. (99). A class of estimators in two-phase sampling using two auiliary variables. Journal of the Indian Statistical Association, 7,

17 EFFICIENT AND UNBIASED ESTIMATION OF POPULATION MEAN Shukla, D., Pathak, S., & Thakur, N. S. (0). Estimation of population mean using two auiliary sources in sample surveys. Statistics in Transition new series, (), -6. Singh, H. P., & Espejo, M. R. (007). Double sampling ratio-product estimator of a finite population mean in sampling surveys. Journal of Applied Statistics, 4(), doi: 0.080/ Srivastava, R. S., Srivastava, S. P., & Khare, B. B. (989). Chain ratio type estimator for ratio of two population means using auiliary character. Communications in Statistics Theory and Methods, 8(0), doi: 0.080/ Srivastava, S. K. (97). Generalized estimator for the mean of a finite population using multi auiliary information. Journal of the American Statistical Association, 66(4), doi: 0.080/ Sukhatme, P. V., & Sukhatme, B. V. (970). Sampling theory of surveys with application. Ames, IA: Iowa State University Press. Tracy, D. S., Singh, H. P., & Singh, R. (996). An alternative to the ratiocum-product estimator in sample surveys. Journal of Statistical Planning and Inference, 5(), doi: 0.06/ (95)