1. Introduction

It is common practice in sample survey we obtain the information about auxiliary variable(s) from a larger sample at first phase and relatively small sample from the second phase by using two-phase sampling procedure. Many authors worked on two-phase random sampling such as: Sukhatme (1962), Singh and Vishwakarma (2007), Sahoo et al. (2010) Noor-ul-amin and Hanif (2012), Sanaullah et al. (2014), etc.

In survey sampling information on sensitive variable would be collected by using randomized response technique introduced by Warner (1965), because direct reliable information on variable of interest is sometime may not possible. Several authors have worked on randomized response techniques on estimation of mean, including Eichhorn and Hayre (1983), Gupta et al. (2002), Chang et al. (2005), Huang (2008). Sousa et al. (2010) introduced ratio estimators by using non-sensitive auxiliary information. Gupta et al. (2014) presented ratio and regression estimator using optional scrambling. Mushtaq et al. (2017) presented a family of estimators of a sensitive variable using auxiliary information in stratified random sampling. Noor-ul-amin et al. (2018) proposed estimation of mean using generalized optional scrambled responses in the presence of non-sensitive auxiliary variable. Saleem I. et al (2019) presented estimation of mean of a sensitive quantitative variable in complex survey: improved estimator and scrambled randomized response model. Partha Parichha et al. (2020) discuss the development of estimation procedure of population mean in Two-Phase Stratified Sampling.

Encourage the above work, we have suggested a generalized class of estimators in two-phase sampling using randomized response technique. The main purpose is to suggest a strategy of two-phase (double) sampling in randomized response technique and proposed general family of estimators for estimating the finite population mean of a sensitive variable with non-sensitive auxiliary variable based on RRT in two- phase (double) random sampling.

2. Sampling Strategy

2.1. Notations & Scheme of Selection of Sample

We consider the finite population in which be the sensitive study variable, be non-sensitive auxiliary variable which is correlated with and be scrambling variable independent of and The reported response of the respondents is , and is the number of units in the first sample whereas n is the number of units in the second sample. Only in the second sample both study and auxiliary variables are observed, in the first sample only auxiliary variable is observed because study variable is expensive.

The two-phase sampling strategy is given below:

1. The first phase, a large sample of a fixed size is drawn from N to observe only or auxiliary variable.

2. The second phase sample, a sub-sample of fixed size is drawn from to observe and , so that .

Let define the following notations:

2.2. Discussion on Estimators in Double Sampling Strategy in RRT

Firstly, we introduce some existing estimators in double sampling using RRT.

The mean and variance of the usual mean estimator in RRT is given by

(2.1)

(2.2)

Where and .

A ratio estimator in two-phase sampling in RRT is given as:

(2.3)

(2.4)

And regression estimator in two-phase sampling given as

(2.5)

Where

(2.6)

3. Proposed Estimators

We propose the following a class of generalized estimators in two-phase sampling:

(3.1)

Where and are weights whose values are to be determined, and are the parameters of the auxiliary variables.

From for , we obtain the following estimators

From for we obtain the following estimators

Expanding (3.1), we have

(3.2)

Where

(3.3)

Using (3.3), the and of are given by

(3.4)

(3.5)

And optimum values of and , respectively, are found as,

Substituting these optimum values in (3.5), the minimum of is given by

(3.6)

By using (3.6), for different values of and or , we can get the minimum of .

4. Simulation Study & Efficiency Comparison of Proposed Estimators

We use the simulation studies for efficiency comparison by empirically and theoretically. Two populations for simulation studies of size 1000 each from bivariate normal populations for (Y, X), with different covariance matrices are used. The Scrambling variable and .

Population 1 Mean of given as

Population 2 Mean of given as

For all populations, we consider four sample sizes: and respectively.

The empirical and theoretical MSE’s for various sensitive mean estimators are given in Tables 1-2. We estimate the empirical MSE using 10000 samples of different sizes selected from each population.

5. Real Data Set Application of Two-Phase or Double Sampling in RRT

For this analysis, we consider the real population used in Gupta et al. (2012). Let be the monthly salaries amount in 2010, is the number of employees available from business data register and . For this population, we have:

, (in thousands of) and and respectively.

Numerical results of empirical and theoretical MSE based on population data is given in Table 3.

The following expression is used to obtain percent relative efficiency (PRE) of different estimators with respect to :

Where

In Table 3 we present the empirical and theoretical results of MSE estimates and PRE of the various estimators in the stratified sample.

6. Conclusions

In this study, we suggested the idea of two-phase sampling in randomized response technique. We consider a general class of estimators for mean of sensitive variable based on randomized response technique in two-phase sampling. In Tables 1-3, we present the results of the theoretical and empirical MSE and PRE of the estimators in two-phase sampling using randomized response technique. These results are computed with a simulation studies and using a real data set.

Appendix

Table 1. Empirical and Theoretical MSE, PRE for the estimators relative to RRT mean estimator for Population 1 in Two Phase Sampling

Table 2. Empirical and Theoretical MSE, PRE for the estimators relative to RRT mean estimator for Population 2 in Two Phase Sampling

Table 3. Empirical and Theoretical MSE, PRE for the estimators relative to RRT mean estimator for Real data in Two Phase Sampling