An Improved Randomized Response Additive Model

In this paper a new randomized response model has been proposed. The theoretical properties of the proposed model have been studied. Comparisons of the proposed model with the existing additive models have been made. It is found that the proposed model is superior to the existing additive model under very realistic condition. Numerical illustrations are also given in support of the present study.


Introduction
have considered the usual additive model for gathering information on quantitative sensitive variables.Their model permits the interviewer to hide personal information using a scrambling variable to the response.Let Y be the true response variable for which we desire to estimate its mean Y  .Suppose S is a scrambling variable whose distribution is known, that is, its population mean  and variance 2  are known.For estimating Y  , a simple random sample of n respondents is selected with replacement from the population.Each respondent selected in the sample is requested to draw a value from the distribution of the scrambling variable, add it to the real response and report back to the interviewer.Thus, the observed i th scrambled response is given by Thus an unbiased estimator of the population mean Y  is given by ISBN-1391-4987 IASSL and the variance of the estimator Gjestvang and Singh (2009) have mentioned that "the practical application of an additive model is much easier than the multiplicative model, that is respondents may like to add two numbers rather than doing painstaking work of multiplying two numbers or dividing two numbers; thus the improvement of the additive model has its importance in the literature."This led Gjestvang and Singh (2009) to suggest an alternative additive randomized response model.The description of the additive model due to Gjestvang and Singh (2009) is given below.
Let  and  be two known positive real numbers [see Gjestvang and Singh (2006)].Consider a deck of cards in which P is the proportion of cards bearing the statement: "Multiply scrambling variable S with  and add to the real value of the sensitive variable Y" and (1-P) be the proportion of the cards bearing the statement: "Multiply scrambling variable S with  and subtract it from the real value of the sensitive variable Y".Let        P be known.Each respondent is asked to draw one card secretly and report the scrambled response accordingly.Mathematically, we have Then, Gjestvang and Singh (2009) proposed an unbiased estimator of the population mean It is not surprise to mention that the mean    and variance   2  of the scrambling variable S are known, but as far as our knowledge goes no one has used this information in defining the estimator of the population mean Y  of the true response variable Y.It is well known that the main advantage of standardization is the reduction in the size of the raw data so that the data can be handled easily.Here we have used the standardized scrambling variable to utilize the prior knowledge of mean () and standard deviation () of the scrambling variable.It is also well recognized that the use of prior knowledge of the parameters such as mean ( ) and standard deviation () either before the start of the survey or at the estimation stage to improve the precision of the estimate of the parameter [see, Thompson (1968), Mehta and Srinivasan (1971) and Hirano (1977)].Thus in our study we have used the standardized scrambling variable so that the proposed model perform better (or to increase the efficiency of the proposed resulting estimator) than some existing models.This fact has been shown both theoretically and empirically.For real situations where such models can be used, the reader is referred to Eichhorn

The Suggested Additive Model
The procedure is exactly the same as Gjestvang and Singh (2009) Then we suggest an unbiased estimator of the population mean and the variance of 6) and ( 9) we have The condition (11) is always true if  is always positive.

Relative Efficiency With Respect to Different Additive Model
The percent relative efficiencies of the proposed estimator are respectively given by the formula: We have computed the percent relative efficiencies    1 and 2. It should be mentioned here that the experience is must in real surveys while making a choice of randomization device to be used in practice.

Conclusion
This paper illustrates enrichment over the Gjestvang and Singh"s (2009) randomized response model.We have suggested the new additive randomized response model utilizing the prior knowledge of mean () and standard deviation () of scrambling variable S. The proposed model is found to be more efficient both theoretically as well as numerically than the additive randomized response model studied by Gjestvang and Singh (2009) and the additive model due to Himmelfarb and Edgell"s (1980).Thus the proposed randomized response procedure is therefore recommended for its use in practice as an alternative to Gjestvang and Singh"s (2009) model.
the only difference in the proposed procedure is to use standardized scrambled variable in place of the original scramble variable S. Replacing S by * S in (4), we have the observed ith scrambled response as

Table 1 :
The PRE (

Table 2 :
The PRE (