Two-Phase Stratified Sampling Estimator for Population Mean in the Presence of Nonresponse Using One Auxiliary Variable

Received: November 13, 2019 Revised: January 27, 2020 Accepted: February 03, 2020 Published Online: March 30, 2020 In this article, a new estimator for the mean of population in stratified two-phase sampling in the presence of nonresponse using one auxiliary variable is been suggested. The Mean Squared Error (MSE) and the bias of the suggested estimator have been given using large sample approximation. The empirical study shows that the MSE of the suggested estimator is better than existing estimators in terms of efficiency. The optimal values of first and second phase sample have been determined.


Introduction
Double Sampling for stratification is one of those sample survey designs that uses auxiliary information in the process of estimation. It was introduced by Neyman (1938). The efficacy of the procedure of estimation using auxiliary information relies on technique whereby the estimator is been suggested. Research works abound whereby the accuracy of the estimators increases by using auxiliary information. Shabbir and Gupta (2005) and Kadilar and Cingi (2003) extended Singh (1999) estimators in evaluating the mean of the population that has been stratified. Although stratified double sampling is useful, the nonresponse problem is intrinsic in every survey which may lead to incorrect evaluation of the parameters. Hansen and Hurwitz (1946) suggested a method of sub-sampling of non-respondents so as to modify the non-response in their mail surveys. Khoshnevisan et al. (2007), Chaudhary et al. (2009), Chaudhary and Singh (2013) and Chaudhary and Kumar (2015) have all suggested different estimators in two-phase stratified sampling under nonresponse.
In this article, an efficient ratio-product estimator in stratified two-phase sampling under nonresponse using one auxiliary variable is been suggested. The characteristics of the estimator suggested have been given.

Sampling Design
Consider a population of size N divided into k strata. Let the size of the i th stratum be N i (i = 1,2,….k) such that ∑ = i k i N N. A large first sample of size n i ' is drawn from N i units by (SRSWOR) scheme for the i th stratum and auxiliary variable x i ' is observed to estimate the population mean X , which is unknown. Secondly, a smaller second phase sample of size n i is drawn from n i ' unit by SRSWOR. Let Y be the study variable with population mean Y pY i k i i = ∑ and assume that at the second phase, non-response is examined on the study variable while the auxiliary variable does not undergo nonresponse. Also, assume that at the second phase, there are n i1 respondent units and n i 2 non-respondent units in n i units. Using the method of subsampling the nonrespondent introduced by Hansen and Hurwitz (1946) , L i ≥1 being a sub-sample is drawn from the sample of n i2 non-respondents units and the needed data is obtained from them all.

Some Existing Estimators
Some existing estimators for the mean of the population in stratified two-phase sampling with one auxiliary variable under non-response shall be presented in this section.

Khare and Srivastava (1993) Estimator
Khare and Srivastava suggested a ratio estimator of the form With mean square error, With mean square error, Chaudhary et. al. (2009) suggested a family of combinedtype estimators given as

3.4.Chaudhary et al. (2009) Estimator
; and X i are mean based on n i units and mean based on N i individually in each stratum for auxiliary variable. The mean squared error (MSE) was given as Chaudhary et al. (2012) gave a combined type estimator given as

Chaudhary And Kumar (2015) Estimator
A combined type estimator was proposed by Chaudhary and Kumar (2015) as follows: With mean square error given as

Proposed Estimator and its Properties
The proposed estimator for population mean in twophase stratified in the presence of nonresponse using single auxiliary variable is given as,

The Mean Squared Error (MSE) and the Bias of the Estimator Proposed
To derive the expression for the mean squared error and the bias of the estimator proposed for the mean of population in stratified two-phase sampling under nonresponse, the following symbolization shall be used. This method is adopted from Chaudhary and Kumar (2015).
are the relative error terms and are defined as Such that the following expectations are applied E ( ) ( ) ( ) e e e 0 1 2 Using the large sample approximation used by Chaudhary and Kumar (2015) the estimator proposed has its bias stated as Hence, the Mean Squared Error (MSE) of the estimator proposed is stated as  Sci. Vol. 8, No.2 (2020) p.52

The Survey Cost and Obtaining Optimal Values of n i , n i ' and L i for T ae
Let c i ' represent the cost in each unit related to the sample size of the first phase n i ' also let c i0 represent the cost of each unit of the first attempt on the study variable having the sample size of second phase, n i . Consider c i1 and c i 2 to represent the cost in each unit of computing the n i1 units that responded and h i 2 units that did not respond. Hence the overall cost for each stratum is stated as The expected cost for each stratum is given by Where W i1 is the response rate in the i th stratum, W i 2 represents the rate of non-response rate in the strata and L i is the inverse sampling rate. Hence the overall cost over the entire strata is given as Consider the Lagrange function Where λ is the multiplier of the Lagrange function To derive the optimum values of n n i i , ' , and L i , φ is differentiated with regard to n n i i , ' , and L i individually and the derivatives equated to zero. Hence, for stratum i, we get Where Q a R S aR S S abR S X iY X X putting the value of λ in (23) into (21), we get  Sci. Vol. 8, No.2 (2020) p.53 On substituting the value of L i opt ( ) from (57) into (54), we can express n i as To obtain the value of λ with regard to the total cost C 0 , the value of n n i i , ' , and L i opt ( ) are put into (17)