Simultaneous Estimation of Mean of Sensitive Variable and Sensitivity level by using Generalized Optional Scrambling

Randomized response technique introduces anonymity into subjects' responses hence encouraging more honest responses. In quantitative randomized response model, additive and multiplicative models have been developed to reduce bias. However, additive and multiplicative models may not be sufficient to reduce this bias so the generalized optional scrambling randomized response model proposed is able to reduce these problems. We also improved mean estimation utilizing information from a non-sensitive auxiliary variable by way of ratio and regression estimators in the proposed model.


Introduction
Randomized response technique (RRT), pioneered by (Warner, 1965) which helps interviewers extract reliable data corresponding to sensitive questions while maintaining respondent anonymity.The quantitative optional randomized response model was introduced by Gupta et al. (2002).In this model, the respondents decide themselves whether they want to tell the truth (or scramble their true response) depending upon whether the question being asked is perceived by them as non-sensitive (or sensitive).The sensitivity level of the question is the proportion of respondents who consider the question sensitive and is usually denoted by W. Gupta et al. (2002) and the Gupta and Shabbir (2004) models were based on multiplicative scrambling whereas the Gupta et al. (2010) model is based on additive scrambling which works better than the multiplicative scrambling, as shown in Gupta et al. (2012a).Mushtaq et al. (2016) proposed estimation of a population mean of a sensitive variable in stratified two-phase sampling.Mushtaq et al. (2017) presented a family of estimators of a sensitive variable using auxiliary information in stratified random sampling.
As we know that in a survey, different questions may have different sensitivity levels and it may be useful to quantify this sensitivity.In this paper we consider generalized optional scrambling that allows simultaneous estimation of mean of sensitive variable and the sensitivity level of a sensitive question.In this model we draw two subsamples and obtain two responses from each respondent using two different generalized scrambling variables.A theoretical comparison and simulation study is conducted to analyze the performance of the suggested estimators for proposed model.

Mean Estimator and Sensitivity Level
Let the population size N and sample size n and sample size split into two subsamples of sizes

Si
 respectively.In each subsample, we will observe X directly and the study variable Y will observe by using scramble response.In each subsample, respondents provide a scrambled response if they consider the question sensitive and a true response otherwise.Let W be the sensitivity level of the underlying sensitive question.So 1 k and 2 k are suitably chosen scalars.
According to the model, the reported response 1 Z and 2 Z are given by   with probability 1-W with probability W   with probability 1-W with probability W where The mean and variance of 1 Z and 2 Z are given by   and The proposed mean and sensitivity estimators are given by respectively The variances of ˆY  and Ŵ are given below: Theorem 2: Ŵ is unbiased estimator of W Proof: From (7) we have given as:

Sample Size Optimization for Model-II
We find optimum sub-sample sizes which help in minimizing variance which is helps to improve the efficiency of the model and estimation of the mean prevalence of the sensitive characteristic.Thus, taking both variances into account, one can try to find 1 n and 2 Var W    .We do this by taking partial derivatives with respect to 1 n and 2 n , respectively, setting the derivatives to zero, then solving for 1 n and 2 n to find specific optimal sample sizes.The optimal sample sizes subject to

Ratio Estimator following Proposed Model
We propose the following ratio estimator based on a two sample approach using two different generalized scrambling variables in optional randomized response technique for proposed model, is given as: Substituting for 1 2 1 ,, z z x and 2 x in (16) and we have the following: By solving (18), we have the following results: Let us define the following terms: Independent sample so we have the following: .
By squaring and solving (20), given as: And we noted that By using the above values in the (22), so we have:

Regression Estimator following Proposed Model
We propose the following regression estimator based on two sample approach using two different generalized scrambling variables in optional randomized response technique for proposed model given as:

Proof:
Now by expanding the (23), we have: By substituting the values of 1 Z  and 2 Z  in (24).And by solving we have: By solving (26), we have given as Where .

Simulation Study
In this simulation study is conducted to analyze the performance of the suggested estimators for proposed model.The comparison has been made by taking proposed mean estimator and Gupta et al. (2010).And the ratio and regression estimators compared with proposed mean estimator.For numerical comparison, we consider the following populations given as: In Tables 1 to 3, the empirical and theoretical MSE's of the estimators based on the firstorder approximation.And following expression is use to obtain percent relative efficiency (PRE) of different estimators with respect to ˆYG  : And the percent relative efficiency (PRE) for ratio and regression estimators for proposed model is given as: 1

stat.oper.res. Vol.XIII No.4 2017 pp856-866 863Table 1 : Empirical and Theoretical MSE, PRE for the Proposed Mean Estimator in proposed Model with respect to ˆYG  for Population 1 and 2
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