MINIMUM NORM CONSTRAINED ESTIMATORS IN RANDOMIZED-RESPONSE SURVEYS

Warner (1965) devised a randomized response (RR) sampling scheme, sampling from dichotomous populations. But the estimate for the binomial probability, though unbiased, is not the ML estimator, since it can be negative. For a multinomial problem, let p(ij) = P(response in category C(i) given that the respondent is truly in the category C(j)). Then the response probability vector eta(k) X 1 = Ppi(k X 1), where P = {p(ij)}kXk, The true probability vector, pi, is to be estimated. Bourke and Moran(ASA Proceedings on Survey Research Methods, 1986) applied EM algorithm to obtain the MLE for pi. In this paper we propose an alternative procedure called Minimum Norm Constrained Estimation (MINCE) procedure, which selects a matrix D as "close" to P-1 as possible and gives all estimates in [0, 1]. We study the MINCE in detail in an important special case when p(ij) = p for all i = j and p(ij) = (1 - p)/(k - 1) for all i not-equal j. The performance of MINCE is compared with that of MLE in terms of the MSE and GPN criteria. Asymptoticlly MINCE, MLE and the unbiased estimators are equivalent.