IMPROVED CLASSES OF ESTIMATORS FOR POPULATION MEAN IN PRESENCE OF NON-RESPONSE
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1 Pak. J. Statist. 014 Vol. 30(1), IMPROVED CLASSES OF ESTIMATORS FOR POPULATION MEAN IN PRESENCE OF NON-RESPONSE Saba Riaz 1, Giancarlo Diana and Javid Shabbir 1 1 Department of Statistics, Quaid-i-Azam University, Islamabad, Pakistan. sabaqau@gmail.com; javidshabbir@gmail.com Department of Statistical Sciences, University of Padova, Italy. giancarlo.diana@unipd.it Corresponding author ABSTRACT Taking motivation from Rao (1991) and Diana et al. (011) biased regression estimators, we propose a general class of biased estimators for population mean under which considering four different situations and studying the occurrence of non-response in the study variable and (or not) in the auiliary variable with known or unknown mean of X. The bias and the mean square error of the estimators are obtained and compared with the linear regression estimators. Numerical results of the efficiency of the proposed estimators are displayed to confirm the superiority of the proposed class of estimators with the linear regression estimators. KEYWORDS Auiliary information, Non-response, Regression estimator, Efficiency comparison. 1. INTRODUCTION When sample surveys are conducted to estimate the unknown population parameter of interest the use of the auiliary information increases the efficiency of the estimators. Mostly the auiliary information is available from census or previous study. Ratio, product and regression estimators are used widely when auiliary information is utilized. Much work have been done over past few years for estimation of population mean of a variable of study when complete information is available about the study and auiliary variables [see Jain (1987), Naik and Gupta (1991), Diana and Perri (007) and Singh and Agnihotri (008)]. Nowadays, many authors are putting their contribution to the estimation of population parameter when complete information is not available or missing due to different reasons such as migration, refusal to respond, unavailability of sampling units when survey is conducted etc. To avoid this bias, its necessary to recontact those non-respondent groups through personal interview or using any other way to get complete information. It is also observed that dealing with non-response problem most of the work has been done for the regression, ratio and product estimators to estimate parameter of interest [see Hansen and Hurwitz (1946), Khare and Srivastava (1997), Okafor and Lee (000), Tabasum and Khan (004, 006), Singh and Kumar (008, 010)]. It is an important point of discussion from asymptotic results that, regression estimators are always better in the class of unbiased estimators as compared to 014 Pakistan Journal of Statistics 83
2 84 Improved classes of estimators for population mean. ratio and product estimators ecept when the study and the auiliary variables are approimately proportional to each other. Some other studies have been done for ratio and product estimators [see Upadhyaya and Singh (1999), Singh and Tailor (003) and Kadilar and Cingi (006)]. Most work has been done on using unbiased estimators and little attention is given on biased estimators [see Rao (1991), Gupta and Shabbir (008), Koyuncu and Kadilar (010) and Diana et al. (011)]. After getting a review of the previous studies, we are drawing our attention to general classes of estimators that can be more efficient than the usual regression estimators. In our present study, the purpose is to put some light on the issue of incomplete information when we are estimating the unknown population mean of the study variable with known or unknown mean of auiliary variable and for this we are taking motivation from Rao (1991) and Diana et al. (011) by using their idea of class of biased estimators, we are proposing four general classes of biased estimators in the presence of non-response.. NOTATIONS AND BACKGROUND Let us assume that U be a finite population consists of N distinct units. Let Y and X be the study and the auiliary variables having values y i and i, i=1,, N. Let X is correlated with Y and is used to estimate the unknown population mean Y. We take a sample of size n ( n< N ) by simple random sampling without replacement (SRSWOR) to collect information of Y and X. Now suppose that n 1 units out of n can supply information on Y and X and remaining n = n n1 units are taken as non-respondents. Using the technique proposed by Hansen and Hurwitz (1946), sub sample the n units and taking r = n / k, k >1 and assume that all r units show full response on second call. Note that r must be an integer and if it isn't so, it is necessary to round. The whole population can be said to be divided into two strata, one stratum is denoted by U 1 of N 1 units which would respond on the first call and the other stratum is denoted by U of N units which do not respond on the first call but will respond on the second call. We note that the strata sizes N 1 and N are not known in advance see Tripathi and Khare (1997). Hansen and Hurwitz (1946) proposed an estimator y for the population mean Y when non response occurs on Y. The same procedure can be adopted for estimation of X when there eists non response. When the population mean X of the auiliary variable X is unknown then we use two phase sampling scheme. At first phase, we select a large sample of size n ( n < N) by SRSWOR to estimate X and then at second phase we take a smaller sub-sample of size n from n ( n< n ) by SRSWOR sampling to estimate Y and X. For better understanding, let define a dummy variable = ( y, ) which will represent estimators as where = d d, 1 1 r
3 Saba, Diana and Shabbir 85 n n n 1 r i i i i i=1 i=1 i=1 i=1 n1 n 1 r d1 d n n n1 r n n ' =, =, =, =, =, =. Similarly we have V = DV 1 1 DV, where where N 1 N i i i=1 i=1 N1 N 1 1 N1 N N N V =, V =, D =, D =. The variance of is given by S S() S v Var = =, (1) S N N i V i V i =, =, N 1 N 1 i S() N( k1) =, =, =. n N n N nn In (1), =1 if non-response occurs in the dummy variable and 0 otherwise. One can define the covariance as where u Su usu() Svu Cov, = =, () N i V ui U i Vui U N Su = i i, Su() = N 1 N 1, = y, u =, In (), u =1 if non-response occurs in the dummy variable u and u =0, otherwise. Note that formulas (1) and (), according to the main part of the literature on this subject, are obtained under a deterministic scheme for the non-response and assuming a complete participation in second phase. Hansen and Hurwitz (1946) proposed the following estimator of population mean Y, y for the estimation y d1 y1 d yr. (3)
4 86 Improved classes of estimators for population mean. Khare and Srivastava (1997) defined the following regression estimator when X is known and there is incomplete information on Y and X reg(ks) y y = y ˆ X, (4) where ˆ s = y S y is an estimator of population regression coefficient = y y of y s S on, with s n 1 r i=1 i i i=1 i i y = y k y ny n 1 and s n 1 r i=1 i ki=1 i n = n 1 Okafor and Lee (000) proposed the following ratio and regression estimators when X is unknown using double sampling scheme when non-response occurs on Y and X both as, y ˆ r(ol) reg(ol) y y = ' and y = y '. (5) 3. SUGGESTED CLASSES OF ESTIMATORS We now introduce a general class of biased estimators representing four different situations. Taking motivation from Rao (1991), we study the occurrence of non response in the study and (or not) in the auiliary variable with known or unknown mean X in Situations 1 and. Furthermore, motivated by Diana et al. (011), we give a more general class in Situations 3 ( X known) and 4 ( X unknown). 3.1 Situation 1: We start with the most simple general class of biased regression estimator proposed by Rao (1991), assuming that X is known t = w y w X, (6) RA 0 1 where w 0 and w 1 are constants to be chosen properly. In the following, we assume that non-response occurs in the study variable Y and (or not) in the auiliary variable X then (6) becomes or where, S1 0 1 t = w y w X, (7) t = w Y w y Y w X, (8) S = if non-response occurs in X and = otherwise. We can write bias and mean square error (MSE) of the t S1 as t E t Y Y w Bias = = 1 (9) S1 S1 0.
5 Saba, Diana and Shabbir 87 and or t E t Y E Y w w y Y w X MSE S1 = S1 = , t S1 Y w0 w0var y w1 Var w0w1 Cov y (10) MSE = 1,, (11) where y y() y Var y = S S = S, S S S Var = =, () Cov y, = S S = S. y y() y Note: In the above terms when =1 non-response occurs in the auiliary variable X and =0, otherwise. The MSE of proposed class will be minimum when w Y S Y S y 0 w 1 Y S SyS Sy Y S SyS Sy = and =. Remark: From the optimal values of positive quantity whereas the sign of We can write minimum MSE of t S1 as where 1 y y y w 0 and w 1, we can see that w 0 is always a w 1 depends from the correlation between Y and X. y S minmse ts1 =, S 1 1 Y y y y S =. S S Now consider a regression estimator of the population mean Y assuming that nonresponse is present in the study variable and (or not) in the auiliary variable reg1 1 (1) y = y w X, (13) where w 1 is a constant to be chosen such that the mean square error of the estimator y reg1 is minimum S w =. y 1 S
6 88 Improved classes of estimators for population mean. The minimum mean square error (MSE) of reg1 y y y reg1 is given by minmse y = S 1. (14) Substituting (14) in (1) the minimum MSE of t S1 can be rewritten as minmse t = minmse y reg1 S1 minmse yreg1 1 Y. (15) From (14) and (15), we can conclude that the estimator t S1 is always more efficient than the estimator 3. Situation : or y reg1. When X is unknown then proposed class becomes S 0 1 t = w y w, (16) ts = w0y w0 y Y w1 ' X X. The Bias and MSE of t S will be almost similar to t S1, the difference occurs only due to the sample selection scheme. In first situation, as X is known, we observe sample only in single phase while now we select samples in two phases. The first phase is used to estimate X and the second to estimate Y and X. We define Var ' = Cov, ' = S, Cov y, ' = S. The MSE of t S will be minimum y (17) where w Y S Y S y 0 w 1 Y S SyS Sy Y S SS Sy = and =, S S S S y y y S = and S =. minmse =, Y SyS Sy t S Y S SyS Sy (18)
7 Saba, Diana and Shabbir 89 S y Y Sy 1 S S y minmse ts =. S y Y Sy 1 S ys (19) Consider the regression estimator when X is unknown reg 1 y = y w ', (0) where w 1 is a constant to be chosen i.e. Sy w =. S 1 The minimum mean square error (MSE) of minmse y = 1 Sy reg Sy SyS y reg is By substituting (1) in (19) we can epress (19) as minmse t = 3.3 Situation 3: minmse y. (1) reg S minmse yreg 1 Diana et al. (011) have proposed the following class DP 0 1 Y. () t = w Y ˆ w u g( u), (3) where Y ˆ is an estimator of Y and u = X, w 0 and w 1 are constants and g is a generic function that satisfies the following mild conditions, g is continuous and bounded in a neighbourhood of zeros. g does not depend on n, N and 1,, N. g is a three times differentiable function with continuous and bounded derivatives. Taking motivation from the Diana et al. (011), we consider a more general class of estimators assuming X known and there is non-response on the study variable Y and (or not) in the auiliary variable X,
8 90 Improved classes of estimators for population mean. ˆ t = w Y w u g u, (4) S3 0 1 t = w Y w v w u g u, (5) S where u = X and v Y ˆ Y =. If we assume that the complete information is available from the sampling units then the proposed class is eactly the Diana et al. (011)'s class. But here we are assuming that there is non-response problem in the study variable and (may or may not be) in the auiliary variable so we can write Y ˆ = y, keeping all the terms related to y and same which are described in Situation 1. Epanding resulting epression will be gu in Taylor's series up to including terms which are p o u, the 1 ts3 w0y g(0) g(0) u g(0) u w0v g(0) g(0) u w1u g(0) g(0) u, where g (0) is a constant term, g (0) is first order partial derivative in zero and g (0) is second order partial derivative in zero. For simplicity we can write g(0) = a, g (0) = b and 1 (0) = g c, ts3 w0y a bu cu w0v a bu w1 u a bu. (6) (7) The bias (B) and mean square error (MSE) of t S3 up to first order of approimation are and B ts3 Y aw0 1 w0yc w1b S w0bs y (8) MSE ts3 a w0y w0 S y w1 S w0w1 S y y y 1 aw0y b w1s w0s Y cw0s 1 Y b Yw 0S b w1s w0s Y cw0s. (9) Now we are minimizing MSE t S3 to achieve the optimum values of the constants w 0 and w 1
9 Saba, Diana and Shabbir 91 and or Y S a b S acs w0 =, 4 a a Y S SyS S y Y ac 3b S Y a YSy a cys S y b Y S S ys S y ab YSS y b Y S w1 =, 4 a a Y S S ys S y Y ac b S Y a S S S a S b S S S c Y S 4 4 y y y y ab cy S b Y S minmse ts3, (30) 4 a a Y S S ys S y Y S ac 3b 1 1 y 1 y 3 Y a S a S b S Y c S 4 y y y y 4 Y b S ac b min MSE ts 3. a a Y S Y S ac b (31) For simplicity we can use (14) for minimum MSE of t S3 as 1 min 1 min reg1 3 Y a MSE y a S b MSE y Y c S 4 min reg reg 4 Y b S ac b min MSE ts 3. a a Y MSE y Y S ac b (3) 3.4 Situation 4: Following the estimator t S3, now assuming that X is unknown and using two-phase sampling scheme to estimate X at first phase and then Y and X at second phase t = w Y w v w u g u, (33) S and v y Y where u = ' =. The Bias and MSE of t S4 can be written as Bias t Y aw 1 w Yc w b S w bs, (34) S y
10 9 Improved classes of estimators for population mean. and MSE S S ts4 a Y w0 w0 Sy w1 w0w1 y y y ayw 0 b w1s w0s Y cw0s 1 Y b Yw0S b w1s w0s Y cw0s 1, the difference between t S4 and t S3 will be the same like the difference which is in t S and t S1 that is the presence of terms such as S y and S. It may be interesting to note that the results of Situations 1 and 3 can be obtained from those of and 4, respectively by placing =0. Now we can obtain the minimum MSE of t S4 when (35) and Y S a acs b S w0 = a a Y S b Y acy 4 4 S ys Sy 3 S S 3 3 Y a YSy a cyssy b Y S S ys Sy ab YSSy b Y S w1 =. 4 4 a a Y S SyS Sy 3b Y S acy S minmse t S4 Y a S a b S c Y 4 4 ys Sy S ys Sy S ab cy S b Y S. (36) 4 4 a a Y S S ys Sy 3b Y S acy S 4 S S y y Y a S y 1 a S 1 b S y SyS SyS 4 c Y S ac b Y S minmse ts4. S y a a Y S y 1 ac 3b Y S S ys Using (1) to epress minimum MSE of t S4 as (37)
11 Saba, Diana and Shabbir 93 reg reg reg Y a minmse y a S b minmse y 4 4 c Y S ac b Y S minmse ts4. a a Y minmse y ac 3b Y S It can be seen in Situation 3 (see 3.3) that at first glance the results of the proposed class look similar to Diana et al. (011). But all their work is done when the complete information is available from the sampling units while the purpose of the proposed class is to put some contribution when some of sampling units do not give response and we face problem of incomplete information. 3.5 Choice of Function g: The proposed classes t S3 and t S4 depend on the choice of function g. Many possible choices can be considered from theoretical and practical point of view. Before moving our attention to the selection of function g, first we evaluate the efficiency of the considered estimators on the basis of their minimum MSE. It is well known fact that linear regression estimators are always more efficient than Hansen and Hurwitz (1946) estimator and for this reason we make efficiency comparison of the proposed classes with the linear regression estimators y reg1 and yregj tsi minmse minmse 0, y reg as where j = (1, ) and i = (3,4). After some computations we obtain a minmse yregj Y S b ac minmse regj 3 a a y Y Y S ac b where i indicates both cases 0, ts3, t S4 and j for reg1, reg y y. This epression will be certainly true if the condition ac 3b 0 is satisfied. Then we can say that the proposed classes t S3 and t S4 are more efficient than the linear regression estimators y reg1 and For simplification we can also write as y reg, respectively. (38) 3b c. (39) a There are many possible choices for the selection of function g proposed by different authors but we are considering only two possible choices. The first choice may be an eponential function which fulfil the Condition (39) and the second one is a different non linear function which may or may not fulfil this condition.
12 94 Improved classes of estimators for population mean. Consider an eponential function as a possible choice for g assuming that X is known. See Bahl and Tuteja (1991) and Singh et al. (010) for details of eponential function. t = w y w u g u, (40) where u X u =, u X u = ep, g u when we epand a= g (0) =1, 1 b= g(0) =, 1 3 c= g (0) =. 8 gu by Taylor's theorem, we get 3b It can be seen that c = is satisfied and we can use eponential function as g for a Situation 3. The same results can be obtained if X is unknown. Consider a ratio function as a choice for g proposed by Ray and Singh (1981) and Kadilar and Cingi (004) X g u X u and, t = w y w u g u. (41) 0 1 In this case we have a= g (0) =1, b= g(0) =, 1 ( 1) c= g (0) =, where is a constant. If we consider =0 then t will be the same as t S1. Now if we take =1, as in Ray and Singh (1981) then
13 Saba, Diana and Shabbir 95 a= g (0) =1, b= g(0) = 1, 1 c= g (0) = 1, 1 which do not fulfil the required Condition (39). Also we can see that if = then it fulfils the condition. In general, we can say that the Condition (39) is satisfied for 1 0<. It can also be observed that the classes proposed by Ray and Singh (1981) and Kadilar and Cingi (004) may be efficient but not optimum, as we know that optimum class always fulfil the condition of efficiency. From a practical point of view we should eclude all those choices of selection of function g which could not fulfil the Condition (39). 4. NUMERICAL STUDY For the numerical study, we are taking population from Khare and Kumar (011) of 96 villages of rural areas under polish station, Singur, District Hooghly from District Census Handbook (1981), published by government of India. The data on the number of cultivators ( y ), as a study character and the population of villages, as an auiliary character ( ) have been taken. The values of the parameters of the population are given as follows: X , Y 185., S , S , S , y y y The non response rate in the population is considered to be 5 percent. S = , S = 97.8, S = , = () y() y() y() The results are shown in Table 1 and Table. The comparison is performed in terms of Percent Relative Efficiency (PRE) as PRE t Si regj min MSE yregj(.) PRE tsi (.) Si min MSE tsi(.) min MSE y 100, 100 min MSE t, where j = (1, ), t Si and t S i(.) are presenting Situations i ( i = 1,,4) and (.) mark is used to represent X having full response. In Table 1 the following estimators reg1(.), S1(.), S3(.) y t t are used when it is assumed that complete information is available for the auiliary variable X, then the minimum MSE of t S1(.) and t S3(.) can be written as
14 96 Improved classes of estimators for population mean. y reg1(.) minmse minmse t =, S1(.) minmse yreg1(.) 1 Y reg1(.) minmse reg1(.) 4 Y c S Y b S ac b { minmse reg1(.) 3 Y a minmse y a S b y 4 minmse ts3(.), a a Y y Y S ac b S y reg1(.) Sy S minmse y =. In Table the following estimators yreg(.), ts (.), t S4(.) represent non-response is present only in Y then the minimum MSE of t S(.) and t S4(.) can be written same like the minimum MSE of t S1(.) and t S3(.), the only difference is to replace ( ) with as reg() Sy S y ( ') S minmse y =. For the possible choice of function g, two cases t 1 and t are studied and the results are displayed for different values of sample size n and n. Table 1: PRE of the estimators with respect to yreg1(.), yreg1 for different values of k. k n case Situation t t 1 t t 1 t S1(.) t S t S3(.) t S t S3(.) t S t S1(.) t S t S3(.) t S t S3(.) t S3
15 Saba, Diana and Shabbir 97 We are taking a =1, b =0, and c =0 for Situations 1 and. The PREs of the both estimators are same in case t 1 and t. To avoid repetition of same results we have shown 1 those only one time in the tables. For t S3 and t S4, we are taking a =1, b = and 3 c = in case t 1 and, a =1, b = 1 and c =1 in case t. The PREs of 8 ts1(.), ts1, ts3(.), ts3 are shown in Table 1 for n only as we are working in single phase sampling to estimate Y. The results of ts(.), ts, ts4(.), ts4 for n and n are displayed in Table. In Table 1 it is observed that the PREs of ts3(.), ts3 are higher in case t 1 than for the case of t because the Condition (39) is fulfilled for the selected function g in t 1 but could not proved in the case t when =1. The same behaviour is observed for ts4(.), t S4 in Table. In numerical comparison it is observed that all the considered estimators tsi(.), tsi, i 1,,4 are more efficient than the linear regression estimators yreg1(.), yreg1, yreg(.), yreg. Also t S4 achieve higher efficiency than the other proposed estimators. It is also observed that the PREs of the class of estimators are higher when non-response eists in the study variable only and it decrease a little when non-response occurs in both the study and the auiliary variables. It can be shown that as we increase the inverse sampling rate k, the PREs of the proposed estimators also increase.
16 98 Improved classes of estimators for population mean. Table : PRE of the estimators with respect to yreg(.), yreg for different values of k. n n case Situation t 1 t t 1 t t 1 t t 1 t k t S(.) t S t S4(.) t S t S4(.) t S t S(.) t S t S4(.) t S t S4(.) t S t S(.) t S t S4(.) t S t S4(.) t S t S(.) t S t S4(.) t S t S4(.) t S
17 Saba, Diana and Shabbir CONCLUSIONS In this paper taking motivation from Rao (1991)and Diana et al. (011), we propose four general classes of estimators for the estimation of population mean Y when auiliary information is available along with considering the problem of non response on the study and (or not) on the auiliary variable. We determine the minimum mean square error of the proposed estimators. The results obtained from four classes hold asymptotically when the parameters ( w 0, w 1 ) are known or estimated. To investigate the efficiency of the proposed classes we take linear regression estimators rather than Hansen and Hurwitz (1946) estimator which is always less efficient than linear regression estimators. The classes t S3 and t S4 depend on the choice for the function g. We consider only two possible choices that fulfil Condition (39) and eclude all those which do not hold this condition. The eponential function seems to be good to reach the optimal condition and the second may not always fulfil this condition. We can conclude that t S3 and t S4 is optimum class of estimators which fulfil the sufficient condition. Numerical results of the efficiency comparison of the proposed estimators with the linear regression estimators presented in Table 1 and Table confirm the superiority of the proposed class of estimators and at the same time show the goodness of linear regression estimators, even in presence of non-responses. The purpose of this study is to throw some light on the issue of incomplete information when we estimate the unknown population parameter of interest with known or unknown mean of the auiliary variable. At first glance our results look similar to Rao (1991) and Diana et al. (011) but the difference lies in the fact that we consider the case of non-response using the procedure of Hansen and Hurwitz (1946). ACKNOWLEDGEMENTS The authors wish to thank anonymous referees for their careful reading and constructive suggestions which led to improvement over an earlier version of the paper. Thanks to the Department of Statistical Sciences, University of Padova, Italy and the Higher Education Commission (HEC), Islamabad, Pakistan for their logistic and financial support for this research. REFERENCES 1. Bahl, S. and Tuteja, R.K. (1991). Ratio and product type eponential estimator. Information and Optimization Sciences, 1, Diana, G., and Perri, P.F. (007). Estimation of finite population mean using multiauiliary information. Metron, LXV, Diana, G., Giordan, M. and Perri, P.F. (011). An improved class of estimators for the population mean. Statistical Methods and Applications, 0, Gupta, S. and Shabbir, J. (008). On improvement in estimating the population mean in simple random sampling. J. App. Statist., 35, Hansen, M.H. and Hurwitz, W.N. (1946). The problems of non-response in sample surveys. J. Amer. Statist. Assoc., 41,
18 100 Improved classes of estimators for population mean. 6. Jain, R.K. (1987). Properties of estimators in simple random sampling using auiliary variable. Metron, XLV, Kadilar, C. and Cingi, H. (004). Ratio estimators in simple random sampling. App. Math. and Computa., 151, Kadilar, C. and Cingi, H. (006). Improvement in estimating the population mean in simple random sampling. Applied Mathematics Letters, 19, Khare, B.B. and Kumar, S. (011). Estimating of population mean using known coefficient of variation of the study character in the presence of non-response. Commun. in Statist.-Theo. and Meth., 40, Khare, B.B. and Srivastava, S. (1997). Transformed ratio type estimators for the population mean in the presence of non-response. Commun. in Statist.-Theo. and Meth., 6, Koyuncu, N. and Kadilar, C. (010). On improvement in estimating population mean in stratified random sampling. J. App. Statist., 37, Naik, V.D. and Gupta, P.C. (1991). A general class of estimators for estimating population mean using auiliary information. Metrika, 38, Okafor, F.C. and Lee, H. (000). Double sampling for ratio and regression estimation with sub-sampling the non-respondents. Survey Methodology, 6(), Rao, T.J. (1991). On certain methods of improving ratio and regression estimators. Commun. in Statist.-Theo. and Meth., 0, Ray, S.K. and Singh, R.K. (1981). Difference-cum-ration type estimators. J. Ind. Statist. Assoc., 19, Singh, H.P. and Agnihotri, N. (008). A general procedure of estimating population mean using auiliary information in sample surveys. Statistics in Transition, 9, Singh, H.P. and Kumar, S. (008). A regression approach to the estimation of the finite population mean in the presence of non-response. Aust. and N.Z. J. Statist., 50(4), Singh, H.P. and Kumar, S. (010). Improves estimation of population mean under two phase sampling with subsampling the non-respondents. J. Statist. Plann. and Infer., 140, Singh, H.P. and Tailor, R. (003). Use of known correlation coefficients in estimating the finite population mean. Statistics in Transition, 6, Singh, H.P., Kumar, S. and Kozak, M. (010). Improved estimation of finite population mean using sub-sampling to deal with non-response in two phase sampling scheme. Commun. in Statist.-Theo. and Meth., 39, Tabasum, R. and Khan, I.A. (004). Double sampling for ratio estimation with nonresponse. J. Ind. Social and Agri. Statist., 58, Tabasum, R. and Khan, I.A. (006). Double sampling for ratio estimator for the population mean in presence of non-response. Assam Statistical Review, 0, Tripathi, T.P. and Khare, B.B. (1997). Estimation of mean vector in presence of nonresponse. Commun. in Statist.-Theo. and Meth., 6, Upadhyaya, L.N. and Singh, H.P. (1999). Use of transformed auiliary variable in estimating the finite population mean. Biometrical Journal, 45,
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