An Improved Exponential Type Estimator Of Population Mean of Sensitive Variable Using Optional Randomized Response Technique

In this paper, we improve the efficiency of Koyuncu et al (2014)’s estimator of population mean of sensitive variable by replacing Traditional Randomized response technique with Optional Randomized response technique as suggested by Gupta et al (2014). The mean square error of proposed estimator is obtained, up to first order of approximation, and is compared with mean square error of various existing estimators theoretically as well as numerically.


Introduction
We know that auxiliary information plays an important role to improve the efficiency of an estimator of parameter of interest when the study variable is sensitive or non-sensitive in nature. Bahl and Tuteja (1991), Grover Tarray and Singh (2017) have suggested optional RRT with new additive model. The importance of Optional RRT model lies in the fact that a question may not be sensitive for the entire population. One person consider a particular question as sensitive question and other may consider it non sensitive question. In an Optional RRT Model, scrambled answer is given by the respondent only if he/she consider the question is sensitive otherwise true answer is given by the respondent. Gupta et al (2014) suggested an efficient estimator of population mean of sensitive variable by replacing traditional RRT model used in Sousa et al (2010) and Gupta et al (2012) with Optional RRT model. In this article, we use Optional RRT model to improve the efficiency of an exponential type estimator suggested by Koyuncu et al (2014). Our proposed estimator is also more efficient than the estimators suggested by Gupta et al (2014). In this article, we will deal with the quantitative study variable, whereas some authors in the literature like Singh and Tarray (2014), Tarray et al (2015), etc studied optional randomized response model for qualitative study variable. To support theoretical results obtained, a numerical illustration is considered finally.

Notations and existing estimators
Consider a population = ( 1 , 2 , … , ) of size N from which a sample of size is drawn using simple random sampling without replacement. Let be the study variable which is sensitive in nature. Let be a non-sensitive auxiliary variable which is positively correlated with the study variable . Let be the sensitivity level of the asked sensitive question. The respondent gives the correct response for the auxiliary variable but has optional randomized response for variable . In this Optional RRT model, respondent gives the response as Ƶ = + for the study variable , where T is a Bernoulli random variable with parameter , so that 0 ≤ ≤ 1 and is a scrambling variable whose mean is assumed to be zero i.e. ( ) = ̅ = 0 and its variance 2 is assumed to be known quantity. It is assumed that the variables and are two mutually independent variables which are further independent of variables and .

Remark 2.1:
If we take = 1 in the above model then it reduces to the traditional additive RRT model, and scrambled response is then written as = + . , where ̅ is the population mean of auxiliary variable . Let be the coefficient of variation of variable .

Remark 2.2:
The estimate of sensitivity level in the above Optional RRT model may be obtained by using the same approach of Gupta et al (2014). According to them, the estimated value of , where ̂( ) is the estimate of variance of y. They further found that , when is assumed to follow Poisson distribution and , when it is assumed that = .
Assume that ̅ is known. Now we consider estimator suggested by Koyuncu et al (2014) and various other estimators under Traditional RRT Model: = + and also various estimators suggested by Gupta et al (2014) under Optional RRT Model: Ƶ = + . These estimators and their mean square errors, up to first order of approximation, are given in the following

Proposed exponential estimator and its properties
If ̅ is known then we propose the following estimator of ̅ by replacing scrambled variable = + in Koyuncu et al (2014) with the scrambled variable Ƶ = + : (1) where 1 and 2 are suitable chosen constants.
The Bias and Mean square error, up to first order of approximation, are respectively given by When we minimise (̂Ƶ) w.r.t. 1 and 2 , then optimum values of 1 and 2 are obtained as follows The minimum mean square error of ̂Ƶ corresponding to these optimum values of 1 and 2 is given by .
Here we have obtained the minimum mean square errors of the proposed estimators ̂Ƶ and ̂Ƶ in terms of (̂Ƶ) because this makes possible to perform easily the comparative study of mean square error of proposed estimator with that of the existing estimators.

Comparison of the proposed estimator with the existing estimators
From the above results, we note the following observations: (A) Our proposed estimator ̂Ƶ is more efficient than the various existing estimators discussed in this article. (B) The estimator ̂Ƶ is more efficient than estimators ̂Ƶ,̂Ƶ and ̂Ƶ under the condition 2 < 1 and also more efficient than ̂,̂,̂ and ̂ under the conditions < 1 and 2 < 1. (C) It is interesting to note that the condition 2 < 1 is very likely to hold true and also in the present paper, the condition 0 ≤ ≤ 1 is always true. (D) The proposed estimator ̂Ƶ in the Section 3 is more efficient than the other proposed estimator ̂Ƶ if the condition (7) hold.

Remarks 4.1:
As we know that, bias has negligible impact on the accuracy of an estimator when the bias is less than one tenth of the standard deviation of estimator. We can be certain that the proportion Bias ⁄ St. Dev will not surpass 0.1 if the sample size is sufficiently large. So in the above comparison, we have considered only mean square errors of various estimators and not taking their biases [see pages 14-15 of Cochran (1977)].

Numerical illustration
We compare the efficiencies of various estimators numerically by using the some empirical populations. We obtain the percent relative efficiencies (PRE) of various estimators, with respect to ̂ by using the formula      From the Table 5.1 to Table 5.6, we observe the following facts: (i) The percent relative efficiencies of the all estimators with optional RRT decrease as the value of increases.
(ii) The proposed estimator ̂Ƶ is always more efficient than the various existing estimators considered in this paper.
(iii) It is important to note that various estimators with optional RRT model are always more efficient than the corresponding estimator with traditional RRT model.