Randomized Response Technique in General Sampling Design
We start with a survey population of a known number of N people labeled on identification by i = 1, …, N, the universe being denoted by U = (1, …, i, …, N). From this U, a sample s of labeled persons is supposed to be selected with a probability p(s) according to a probability design p. For this p, the inclusion probability of an i is πi s i p s i U= ∑ ∈∋ ( ), and that of a distinct pair of individuals i, j (i ≠ j) is πij s i j p s= ∑ ∋ , ( ). We restrict to designs for which πi i U> ∀ ∈0 and πij i j U i j> ∀ ∈ ≠0 , , . Letting y denote a real-valued variable with values yi for i ∈ U, in sample surveys, the major problem is to estimate the total Y ylN i= ∑ and the mean Y Y N= / on surveying a sample s of individuals bearing the values yi for i in s. In this book we are mostly concerned with the situation when
yi = 1 if i bears a stigmatizing feature, say A = 0 if i bears the complementary feature, Ac.