Some Optimality Issues in Estimating Two-Stage Optional Randomized Response Models

A Three-Stage Optional Randomized Response Model

“…In Gupta et al (2010;2011), it was observed that introduction of T into the Gupta et al (2006) model doesn't always decrease the variance of mean estimator. In order to reduce this variance, T should be chosen carefully.…”

“…The following theorem compares the variances of the two two-stage models under the assumption that the scrambling variables of both the models have the same mean and the same variance. In Gupta et al (2011), it was observed that for questions with small sensitivity level, any value of T would make the Gupta et al (2010) model perform better than the Gupta et al (2006) model, but for questions with larger sensitivity level, a larger value of T was needed, which is not recommended as it affects respondent cooperation. So instead of using a large value of T, one should rather opt for the new two-stage model.…”

“…Thus, for more sensitive questions, introducing F helps. In view of the observations in Gupta et al (2010;2011) and Table 1, an appropriate Downloaded by [Florida Atlantic University] at 18:32 20 November 2014 Table 1 V(μ X ) F (in bold) and V(μ X ) for various values of W and F, with n = 1,000, n 1 = n 2 = 500, X ∼ Poisson(4), S 1 ∼ Poisson(2) and S 2 ∼ Poisson(5) Gupta et al (2010) model. The following theorem compares the variances of the two two-stage models under the assumption that the scrambling variables of both the models have the same mean and the same variance.…”

“…It was shown in that paper that a two-stage optional RRT model will always perform better than the corresponding one-stage optional RRT model (where T = 0) except that a larger value of T may be needed if the question is highly sensitive. Gupta et al (2011) presented a method for selecting an optimum value of T.…”

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In Gupta et al (2010;2011), it was observed that introduction of a truth element in an optional randomized response model can improve the efficiency of the mean estimator. However, a large value of the truth parameter (T) may be needed if the underlying question is highly sensitive.

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