Reply to the Correction by Grover and Kaur: A New Randomized Response Model

原文摘要

In Gjestvang and Singh (2006), the properties of the estimator π^New are investigated when v = 1 and w = 0, which are obtained by taking p=α1/(β1+α1) and T=α2/(β2+α2)⁠. In this case, the correction that was made by Grover and Kaur (2020) is proper, i.e. any choice of α1, β1⁠, α2 and β2 will make v = 1 and w = 0. Note that the meaning of ‘specifically’ is ‘done in a definite and exact way’. Gjestvang and Singh (2006) have pointed out that ‘specifically’ we have the following three cases. Now model (7) results in a binary outcome of either a or b, similarly to Warner (1965), but nevertheless this model is not same model as the Warner (1965) model. The response ‘a’ may come from both types of respondent—those belonging to the sensitive group and those belonging to the non-sensitive group, and the response ‘b’ may also come from both types of respondent, so privacy is maintained. In other words, given p0∈[0,1]⁠, which represents the device parameter of the Warner model, if v = p0 and w = 1 − p0, estimator (2) has variance equivalent to that of the Warner estimator for innumerable choices of α2 and β2⁠. Moreover estimator (2) coincides with the Warner estimator when a = 1 and b = 0 (i.e., when α2=β1=0⁠, α1θ1=β2θ2=1⁠, in such a way that p = p0 and T = 1 − p0). When v = 1−(1 − p0)(1 − T0) and w = (1 − p0)(1 − T0), i.e. if p = {1−(1 − p0)(1 − T0)− (1−α1θ1)}/{θ1(β1+α1)} and T={(1−p0)(1−T0)+α2θ2}/{θ2(β2+α2)}⁠, the variance of estimator (2) coincides with the variance of the Mangat–Singh estimator (with device parameters p0 and T0) for particular values of p0 and T0, whereas estimator (2) coincides with the Mangat–Singh estimator for a = 1 and b = 0 (i.e., when α2=β1=0⁠, α1θ1=β2θ2=1⁠, in such a way that p = 1−(1 − p0)(1 − T0) and T = (1 − p0)(1 − T0)). When v = 1 and w = 1 − p0, i.e. if p=α1/(β1+α1) and T=(1−p0+α2θ2)/{θ2(β2+α2)}⁠, estimator (2) coincides with the Mangat (1994) estimator and has the same variance as the Mangat estimator (with device parameter p0) and it coincides with the Mangat estimator for a = 1 and b = 0 (i.e. when α2=β1=0 and α1θ1=β2θ2=1⁠, in such a way that p = 1 and T = 1 − p0). It is worth pointing out that the processes of two (or more) randomized response models may not be the same, but when the variances of the estimators from two (or more) models are found to be the same then those models are considered as the same models. Gjestvang and Singh (2006) gave three ‘if’ conditions to be satisfied by their randomization device to make three different models as special cases. It is common sense that the parameters α1,α2,β1,β2,θ1 and θ2 are selected parameters and should be adjusted so that p is a practical choice. It seems that Grover and Kaur (2020) failed to differentiate between the randomized response model (1) and estimator (3), which is only one special case of Gjestvang and Singh’s (2006) model (1), but they made a nice and proper observation for the case when v = 1 and w = 0. It is hoped that this reply to the correction by Grover and Kaur (2020) will lead to a better understanding of the Gjestvang and Singh (2006) model in future studies. Finally, we conclude that the correction that was proposed by Grover and Kaur (2020) is proper for v = 1 and w = 0. After introducing the new estimator (2), the corrections that were proposed by Grover and Kaur (2020) are no longer valid owing to cases I–III. The author is grateful to the Joint Editor and a learned referee for very valuable comments on the original version of this reply. The author is also grateful to Professor Stephen A. Sedory, Department of Mathematics, Texas A&M University, for editing the original version of this reply.

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