Kuk (1990) proposed the following randomized response method Ask the respondent to generate two independent binary variables X 1 and X 2 with P (X 1 1) 1 and P(X 2 1) 2 The probabilities 1 and 2 are known Now ask the respondent to tell you the value of X 1 if she is in the sensitive class, and X 2 ...

a P 1 P 1 sensitive p s P 1 not sensitive 1 p ...

Kuk (1990) proposed the following randomized response method. Ask the respondent to generate two independent binary variables

Question:

Kuk (1990) proposed the following randomized response method. Ask the respondent to generate two independent binary variables X₁ and X₂ with P (X₁ = 1) = θ₁ and

P(X₂ = 1) = θ₂. The probabilities θ₁ and θ₂ are known. Now ask the respondent to tell you the value of X₁ if she is in the sensitive class, and X₂ if she is not in the sensitive class. Suppose the true proportion of persons in the sensitive class is p_S.

a. What is the probability that the respondent reports 1?

b. Using your answer to (a), give an estimator Ṕ_S of p_S. What conditions must θ₁ and θ₂ satisfy?

c. What is V (Ṕ_S) if an SRS is taken?