An Improved Class of Estimators Of Population Mean of Sensitive Variable Using Optional Randomized Response Technique

In this paper we have suggested a class of estimators of population mean of sensitive variable under optional randomized response technique as reported in Gupta et al (2014). We have obtained the mean squared error (MSE) of the suggested class of estimators up to the first order of approximation. The optimum conditions are obtained at which the (MSE) of the proposed class of estimators is minimum. An empirical study is carried out to show the performance of the suggested class of estimators over existing estimators .It is found that the performance of proposed class of estimators is better than the existing estimators including Grover and Kaur (2019)


INTRODUCTION
When dealing with the sensitive issues such as gambling, illegal income, alcoholism, sexual abuse, drug addiction, abortion, tax evasion and many others, people often do not respond truthfully or even refuse to answer. This can cause substantial bias in the estimation of population parameters. To eliminate or reduce the bias we have a technique known as randomized response (RRT) given by Warner (1965). There are also so much work has been done with situation when the response to a sensitive question is quantitative variable. For making the technique more efficient various methods have been suggested by many authors. These include Multiplicative Scrambling models by Eichorn and Hayre (1983). Gupta et al (2002) introduced an optional randomized response technique where the choice to provide a scrambled response or truthful response totally depends on the respondent weather respondent considered the question is sensitive or not. Gupta  In sampling practice direct techniques for collecting information about non sensitive characters make massive use of auxiliary variables to improve sampling design and to achieve higher precision in population parameters estimates furthermore development on RRM have been focused on the use of auxiliary variable. Yan (2005Yan ( -2006 used the auxiliary information directly at the estimation stage in simple random sampling to improve Warner (1965) estimator through the ratio method. Further many authors (Sousa et al. (2010) and Gupta et al. (2012)) have estimated the mean of the sensitive variable using auxiliary information. Perri (2009, 2011) also give best estimators using auxiliary information. When the study variable is sensitive many authors have done there tremendous work using auxiliary information, such as Bahl and Tuteja (1991), Grover and Kaur (2011), Singh and Solanki (2012), Singh and Vishwakarma (2007) and many more. Whereas when the study variable is sensitive, Sousa et al (2010), Gupta et al (2012), Koyuncu et al (2014), Kalucha et al (2015) and many more used auxiliary information in RRT under traditional additive model. But in both the condition auxiliary variable is non-sensitive. Koyuncu et al (2014) have studied exponential-type-estimator to get more efficient estimators. Grover  In this study, we have proposed a class of ratio cum exponential-type estimator of the mean of the sensitive variable using non sensitive auxiliary information. We also derived the Bias and MSE of the proposed class of estimators correct up to first order of approximation and compared it empirically with the Grover and Kaur (2019)'s estimator .

NOTATIONS AND EXISTING ESTIMATORS
Let a random sample of size n be drawn without replacement from a finite population ). ,..., , ( Let Y be the study variable, a sensitive variable which cannot be observed directly. Let X be a non-sensitive auxiliary variable that have a positive correlation with Y . Let W be the sensitivity level of the asked sensitive question. In the Optional RRT model respondent is asked to report a scrambled response for Y given by ST Y + =  but is asked to provide a true response for X . Where T is the Bernoulli random variable with parameter , . And , S be a scrambling variable, whose mean is supposed to be zero i.e.
and its variance 2 s  is supposed to be known quantity. It is assumed that the variables S and T are two mutually independent variables which are further independent of variables Y and X .
If we take 1 = W in the above model then it boils down to the additive RRT model, and scrambled response is then written as S Y + =  .
Now the population mean of variable  is given by Y is the population mean of variable. The population variance of variable  is given by . Let  C be the coefficient of variation of variable  . So

PROPOSED CLASS OF ESTIMATORS
We suggest the following class of estimators for population mean We assume that 1 1  e so that ( )  − + 1 1 e is expandable. Now expanding the right hand side of (2), multiplying out and neglecting terms of s e' having power greater than two, we have Taking expectation on both side of (3) we get the Bias of ( ) to the first degree of approximation as Squaring both side of (3) and neglecting terms of s e' having power greater than two we have Taking expectation of both sides of (5) we get the MSE of ( ) up to the first order of approximation as ( ) at (6) Putting (7) and (8) in eq (6) we get the resulting minimum MSE of ( ) Thus we arrived at the following theorem.  (7) and (8).
A large number of estimators can be generated from are shown in Table 1. S.no .
Choice of constants  Suggested by Gupta et al (2012) We note that the estimators listed in Table 1  will definitely have smaller (or equal to) minimum MSE than those estimators listed in Table 1 for some selected values of ( )  , .
To illustrate this we have carried out an empirical study in Section 4.

EMPIRICAL STUDY
To judge the merits of the suggested class of estimators over existing estimators, we have taken the same populations as considered by Grover